Electrosensory Dynamics:

Dendrites and Delays

by

Brent Doiron

A thesis presented to the University of Ottawa

in fulfilment of the thesis requirement for the degree of

Doctor of philosophy in Physics

Ottawa, OntarioDecember 21, 2004

c© Brent Doiron 2004

Prologue

“...it is impossible for us to think of anything, which we have notantecedently felt, either by our external or internal senses.”David Hume

“Let your senses guide you”Slogan for Baileys Fine Liquors.

Mature sciences develop a formal or mathematical description of their theories. Many sub-disciplines of

physics and chemistry have progressed as such and now have intertwined communities of experimentalists

and theorists. In contrast, the biological sciences have had a long evolution of relatively informal inquiry

where taxonomy and characterization were the prime tasks. The estrangement of biology and mathematics

is most likely due to the fact that biological systems are highly complex and composed of many nonlinear

and interacting agents. At the time of early biological studies, physics and mathematics traditions were

ill-equipped to be of use in solving these problems. However, over the last fifty years explosive advances in

nonequilibrium statistical mechanics, nonlinear systems theory, and computational science have occurred.

This has recently allowed physicists and mathematicians to explore biological problems. As a result theorists

are starting to be accepted in biological and medical communities and theory is even beginning to be a

component in the education of younger biological scientists. A formalization of biology is up and running,

and it will change and enrich both biology and mathematics during its course.

i

Summary

Every sensory system is composed of complex cells that connect to one another in complicated ways. This

‘system complexity’ that spans multiple spatial scales in the brain is undoubtably involved in sensory

perception. This thesis explores both single neuron and network dynamics of the electrosensory system used

by wave-type weakly electric fish. Isolated electrosensory pyramidal neurons show a dendritic dependent

burst patterning of their action potentials. In the first three chapters we present and explore both high and

low-dimensional mathematical neuron models of these cells. Using dynamical systems theory we show how

specific bifurcations in the neural dynamics impart both a threshold and slow passage effects to the burst

firing. This gives a wide continuum of time-scales to the neural behaviour. These predictions are verified in

experiment. Further, we show how stochastically driven real cell, and associated models, process in parallel

high and low frequency stimuli respectively with isolated spikes and bursts.

The final three chapters explore how networks of pyramidal neurons can be induced to oscillate syn-

chronously via a delayed closed loop architecture in the peripheral electrosensory system. A dynamical

systems analysis shows that the oscillatory dynamics are born via a delay-induced bifurcation in the net-

work firing. We present experimental and modelling results showing that the oscillatory dynamics support

a distinct neural response to communication-like stimuli in comparison to prey-like stimuli. By using novel

experimental and theoretical techniques we further dissect the oscillation mechanism and uncover a new

form of oscillatory control. We show that the degree of spatial correlation in a stochastic stimulus sets the

intensity of the network oscillations. The combination of these results show how neural organizations in

the electrosensory brain, at both single cell and network levels, are appropriate for coding stimuli that the

animal routinely observes in its natural environm ent.

ii

Sommaire

Tout systeme sensoriel est compose de cellules complexes interconnectees de maniere compliquee. Cette

complexite systemique, etalee sur plusieurs echelles spatiales du cerveau, est sans doute impliquee dans la

perception sensorielle. Cette these explore la dynamique de neurones isoles et en reseau dans le systeme

electrosensoriel du poisson electrique a faible champ de type onde. Les cellules pyramidales de ce systeme

manifeste des bouffees de potentiels d’action, dont le patron depend des dendrites. Dans les trois premiers

chapitres, nous presentons et analysons deux modeles mathematiques de ces cellules, l’un de dimension

elevee, l’autre de dimension reduite. Nous demontrons comment une bifurcation de type point de selle-

noeud determine la presence d’un seuil et des effets de passage lent pour le patron de bouffees. Il en resulte

un grand eventail d’echelles de temps pour l’activite neuronale. Ces predictions sont par la suite confirmees

experimentalement. De plus, nous montrons comment des cellules reelles sujettes a un stimulus stochastique,

ainsi que les modeles correspondant a ces cellules, traitent parallelement les composantes a haute et a basse

frequence du stimulus a l’aide, respectivement, de decharges isolees et de bouffees.

Les trois derniers chapitres explorent l’activite oscillante de cellules pyramidales par l’entremise d’un

retroaction avec delai dans la peripherie sensorielle. Une analyse theorique de cette architecture demontre

qu’une bifurcation de Hopf dans l’activite neuronale est a l’origine de cette oscillation. Nous presentons des

resultats experimentaux et computationnels qui associent cette oscillation a des stimuli relies a la commu-

nication entre poissons, et non a des proies. Une combinaison de techniques experimentales et theoriques

originales permet une dissection du mecanisme causant l’oscillation, et met a jour une nouvelle forme de

controle oscillatoire. Le degre de correlation spatiale d’un stimulus stochastique determine l’intensite des

oscillations, ce qui appuie la notion d’une specificite de ces oscillations a la communication. L’ensemble des

resultats montre comment l’organisation neuronale, tant au niveau cellulaire qu’au niveau reseau neuronal,

dans la partie sensorielle du cerveau est adaptee a l’encodage de stimuli qu’un animal rencontre regulierement

dans son environnement.

iii

Acknowledgements

Andre and Len let me make the mistakes that helped me learn, but steered me away from the ones that

would make me fail. Ray and Joe gave me endless data sets that filled my mind with problems. Carlo and

Benjamin helped me solve them. Jay let me be a slob and made me laugh as much as think. Everybody

needs a friend who is a nemesis; mine was Moe. John, Bill, Kristina, Eric, Jan, Connie, Dave, and Martin

tolerated me. Actually, Bill didn’t. My parents and brother still don’t know what I do, but they supported

it all along. The two women of the thesis were Anne and Lori; one listened and engaged/corrected, the

other made sure that the spiral into geekdom was slow. Their support was a scholarship I could not have

gone without. NSERC and OGSST fed me and bought video games and beer. The University of Ottawa

gave me a chair and a table, and for some of the stay even a window. However, the most important person

was the guy who invented spellcheck, without him there would be even more typos in this thesis.

iv

Thesis format

This is a paper-format thesis. Four of the six chapters are direct reprints of refereed journal articles on

which I was the first author. The remaining two chapters are unpublished manuscripts. Both of these latter

chapters contain material that is either published or submitted to refereed journals. I have written them as

new manuscripts, which will be exclusive to this thesis, in order to highlight the contributions I have made

yet without over-reporting the work that my co-authors have done.

This thesis is a combination of computational, theoretical, and experimental work. All computational

work and modelling that is presented was done by myself. Unless otherwise stated below, all data analysis

was also done by myself. I was the primary writer of each chapter with corrections offered by various

co-authors. I treat the specifics of my contributions to each chapter separately:

I. This chapter is a reprint of the article:

B. Doiron, A. Longtin, R. W. Turner and L. Maler.Model of gamma frequency burst discharge generated by conditional backpropagation.Journal of Neurophysiology. 86: 1523-1545, 2001.

This paper introduces a large scale compartmental model of a basilar electrosensory pyramidal neuron.

This model is used to explore the novel burst mechanism employed by these cells. The co-authors

contributions were supervisory in nature.

II. This chapter is a reprint of the article:

B. Doiron, C. R. Laing, A. Longtin. and L. Maler.Ghostbursting: A novel neuronal burst mechanism.Journal of Computational Neuroscience. 12: 5-25, 2002.

This paper reduces the large compartmental model introduced in chapter I to a much lower dimensional

model. This model is then used to characterize the burst mechanism using nonlinear dynamics and

bifurcation theory. Carlo Laing was involved in many motivating discussions as well as in numerically

computing the Lyapunov exponents shown in Fig. 5C as well as the bifurcation curve in Fig 6. The

other co-authors contributions were supervisory in nature.

III. This chapter is an unpublished manuscript. It is a compendium of the work presented in the following

articles:

B. Doiron, L. Noonan, N. Lemon and R. W. TurnerPersistent Na+ current modifies burst discharge by regulating conditional backpropagation of dendritic

v

vi

spikes.Journal of Neurophysiology. 89: 338-354, 2003.

C. R. Laing, B. Doiron, A. Longtin, L. Noonan, R. W. Turner and L. MalerType I Burst Excitability.Journal of Computational Neuroscience. 14: 329-342, 2003.

A. M. M. Oswald, M. J. Chacron, B. Doiron, J. Bastian and L. MalerParallel processing of sensory stimuli by bursts and isolated spikes.Journal of Neuroscience. 24: 4351-4362, 2004.

A. M. M. Oswald, B. Doiron and L. MalerDendrite dependent interval coding Submitted to the Journal of Neuroscience.

All the data that is presented was recorded either by Liza Noonan or Anne-Marie Oswald. The data

measures for the burst threshold component (Fig. 2) were developed by myself. The signal processing

measures for bursts and isolated spikes (Fig. 7) were developed in conjunction with Anne-Marie

Oswald and Maurice Chacron.

IV. This chapter is an unpublished manuscript. A subsection of the paper is a direct excerpt from a

submitted manuscript:

N. Masuda, B. Doiron, A. Longtin, and K. AiharaCoding of oscillatory signals by globally coupled networks of spiking neurons.Submitted to Neural Computation.

N. Masuda’s component of the submitted manuscript was omitted. The entire set of results presented

were done by myself.

V. This chapter is a direct reprint of the article:

B. Doiron, M. J. Chacron, L. Maler, A. Longtin and J.BastianInhibitory feedback required for network oscillatory responses to communication but not prey stimuli.Nature. 421: 539-543, 2003.

All data were collected by Joseph Bastian. Maurice Chacron performed the information-theoretic

analysis that is presented in the supplementary materials section.

VI. B. Doiron, B. Lindner, A. Longtin, L. Maler and J. BastianOscillatory Activity in Electrosensory Neurons increases with the spatial correlation of the stochasticinput stimulus.Physical Review Letters. 93: 048101, 2004.

The set of experiments and modelling was conceived and implemented by myself and Joe Bastian.

The theoretical component was primarily done by Benjamin Lindner; however, Fig. 3 was conceived

of and computed by myself.

Contents

Prologue i

Summary ii

Sommaire iii

Acknowledgements iv

Thesis format v

Contents vii

0 INTRODUCTION 1The basics of neuroscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4The neuroethology of weakly electric fish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Electrosensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6The advantages of the electrosense in sensory studies . . . . . . . . . . . . . . . . . . . . . . . 9

Single neuron dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10The Hodgkin-Huxley model of the action potential . . . . . . . . . . . . . . . . . . . . . . . . 11Dendrites and the cable equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Simple spiking models and stochastic neuroscience . . . . . . . . . . . . . . . . . . . . . . . . . . . 16The integrate-and-fire model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Stochastic differential systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Stochastic neuroscience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Saddle-Node bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1 ARTICLE I 27Model of gamma frequency burst discharge generated by conditional backpropagation.Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Cell morphology and discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Channel distribution and kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

vii

Contents viii

IAptKv3.3: high voltage activated K+ channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Action potential discharge in ELL pyramidal cells . . . . . . . . . . . . . . . . . . . . . . . . . 31INa,s, IDr,s, and IAptKv3.3: somatic spike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32INa,d, IDr,d, and IAptKv3.3: dendritic spike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33soma-dendritic interactions underlying the DAP . . . . . . . . . . . . . . . . . . . . . . . . . . 34spike backpropagation and refractory period . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35INap: determining RMP, nonlinear EPSP boosting, and latency to first spike shifts. . . . . . . 35IA: latency to first spike from hyperpolarized potentials . . . . . . . . . . . . . . . . . . . . . 37IKA, IKB : somatic K+ currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39ELL pyramidal cell bursting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Standard burst mechanism: slow activating K+ current . . . . . . . . . . . . . . . . . . . . . 40ELL burst mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Slow inactivation of dendritic Na+ channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Disection of burst mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47ELL burst mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47AptKv3.3 as a potential candidate for IDr,d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Relation to in vivo bursting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Application of ELL burst model to mammalian chattering cells . . . . . . . . . . . . . . . . . 48References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2 ARTICLE II 51Ghostbursting: A novel neuronal burst mechanism.Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

ELL pyramidal cell bursting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Two-compartment model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Model performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Bifurcation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57The burst mechanism: reconstructing the burst attractor . . . . . . . . . . . . . . . . . . . . 60The interburst interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64The burst interval: intermittency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Gallery of bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Ghostbursting: a novel burst mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Predictions for bursting in the ELL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Contents ix

3 ARTICLE III 73Thresholds to Sensory Coding with GhostburstingAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74Burst threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Burst threshold in experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Ghosting and ISI return maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Parallel processing with bursts and isolated spikes . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Ghostbusting the ghostburster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Spike train processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Thresholds in bursting systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Sensory processing by bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4 ARTICLE IV 83Oscillations and Synchrony with Delays in Neural NetworksAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Global network oscillatory activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Feedback strength - g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85Excitability - µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Hopf bifurcation in network activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86From integral to differential systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Hopf bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5 ARTICLE V 92Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli.Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Physiology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96Network simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97Supplementary material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Intrinsic burst mechanisms and network oscillations . . . . . . . . . . . . . . . . . . . . . . . 98Information transfer and network oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Contents x

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6 ARTICLE VI 102Oscillatory Activity in Electrosensory Neurons increases with the spatial correlation of the stochastic

input stimulus.Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7 CONCLUSION 107Ghostbursting: Large, Medium, and Small Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Hopf Bifurcations and Stimulus Induced Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 110Oscillatory Networks of Ghostbursters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

INTRODUCTION

1

INTRODUCTION 2

Living beings sense their world. The act of sensing is to gain information; to categorize and interpret

inputs. We typically associate ourselves with four senses: visual, auditory, somatosensory, and olfactory.

The information collected by these separate modalities constructs the perception of our environment.

It is universally accepted that the brain is the organ responsible for perception. A relevant question is

then: what are the mechanics of sensory perception employed by the brain? Even though my thesis falls

far short of answering this question, the question stands as a prime motivation of sensory science.

The subject of this thesis is sensory dynamics. Typically once a subject is introduced the next task

for the author is to woo the reader to the importance of the subject. If I were studying a disease I would

present various statistics on the widespread nature of the disease. Rather, if I were focusing on laser

physics I would enumerate the various industrial uses of lasers. These statements would be aimed at

the layperson with the intent of highlighting any experiences that the reader may have with the subject.

In many sensory studies I have read the following paraphrased introduction: ‘As you read this sentence

photoreceptors in your retina are responding to the exact visual scene that this sentence produces’. The

point of this sentence, I assume, is to demonstrate that sensory systems are somewhat special in that it is

simple to link it to some experience of a reader. This is because sensory processing is a major component

in almost all experiences of all readers. Sensory processing and the brain dynamics that underlie it link

us to our environments. It is thus paramount to unfold their inner workings. This unfolding could allow

for significant control over sensory processing e.g. to cure sensory malfunction or even push sensory

systems beyond their current limits.

In all of brain research, sensory and beyond, one pervasive thread is common: the brain operates

on many spatial scales. Ion channels measuring mere nanometers influence the behaviour of a patch of

cellular membrane. Dendrites that stretch for hundreds of micrometers can influence single cell activity.

Small local networks that cover millimeter distances exhibit interesting dynamics. Finally, large networks

of neurons stretching for centimetres, such as the cerebellum, the cortex, and the hippocampus, interact

with one another and are the substrate of cognition. There are open questions at all spatial scales, and

perhaps one of the deepest questions in neurosciences is how to link behaviour across the different scales.

This thesis will address sensory dynamics at two spatial scales: the single neuron and the large network.

A general theory of sensory science is elusive and perhaps not even feasible. The approach often

taken is to choose a specific sensory system in a specific organism, identify its strengths and weaknesses,

and then study the strengths. For example, the auditory system of barn owls has offered tremendous

insights into the neural computation of sound localization [5]. In stark contrast, the visual system of

barn owls has offered no real insight to the computation of colour fields. This is because barn owls

1

INTRODUCTION 3

primarily use their ears and not their eyes to detect prey. This science of studying the neural aspects

of an appropriate computation or behaviour, in a specific organism, is called neuroethology. The general

hope of neuroethology is that the results obtained are just one instance of a more general principle that

may be applicable across many organisms, including humans. In this thesis I will use the electrosensory

system of weakly electric fish to explore two central questions in sensory science:

I. Single neurons often exhibit complex spatiotemporal activity. What are the cellular mechanisms

responsible for a specific response? How does this activity code stimulus? Chapter 1 of this the-

sis introduces a large multi-compartmental model of an electrosensory pyramidal neuron. This

model reproduces the complex ‘bursting’ neural firing pattern that is observed in in vitro experi-

ments. Chapter 2 reduces the large model to a low-dimensional dynamical system that captures

the essence of pyramidal cell bursting. A dynamical systems analysis predicts a threshold between

tonic and bursting behaviour along with certain slow passage effects imparted by this threshold.

Chapter 3 first analyzes in vitro recordings of electrosensory pyramidal neurons so as to verify the

predictions of Chapter 2. Chapter 3 next outlines a coding scheme whereby bursts and isolated

spikes respectively code for the low and high frequency components of a dynamic input stimulus.

II. Sensory processing is often done by networks of interconnecting neurons. What are the relations

between specific stimuli and specific network architectures? Chapter 4 studies the oscillatory be-

haviour of networks of leaky integrate-and-fire neurons coupled via delayed inhibitory feedback.

Chapter 5 uses such a network to model a particular electrosensory network. Experimental record-

ings and modelling show how this electrosensory network gives an oscillatory dynamic only in

response to communication-like stimuli. Finally, Chapter 6 uses novel experimental and theo-

retical paradigms to study the effects of spatial correlations in stimuli on electrosensory network

dynamics.

I begin with an introductory section that briefly reviews several of the basic disciplines that will be

of use in this thesis. Section 0.1 introduces the fundamentals of brain circuitry and neural interaction.

Section 0.2 reviews the electrosensory system of weakly electric fish. This will be the model system

of study throughout the thesis. The Hodgkin-Huxley model of action potential production and the

dendritic cable equation are presented in section 0.3. These results give the basics of single neuron

activity. Section 0.4 discuses a reduced model of neural spiking: the integrate and fire neuron. This

model is used to introduce the fundamentals of stochastic neuroscience. Finally, section 0.5 is a brief

break from neuroscience to review nonlinear effects known as the saddle-node and Hopf bifurcations.

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INTRODUCTION 4

0.1 The Basics of Neuroscience

Neurons are cells in the brain. It is widely believed that neurons are the principal brain cells that

self-organize to bring about cognitive function. Specific groups of neurons are responsible for memory

and learning, perception, and the regulation of bodily function [21]. This section will give the briefest

of introductions to neurons and how they communicate with one another.

The human brain has about 1012 neurons, with varying shapes and sizes, and most importantly

different behaviours. The brain is a classic example of cellular diversity. This ‘diversity property’ is by

no means exclusive to humans. Rat, owl, and fish brains all have distinct groups of neurons, some of

them similar to the distinct groups that make up the primate brain. In fact, even invertebrates like crabs

and leeches have distinctly different neurons, even though they are not centrally localized and often only

thousands in total. In spite of this diversity, there are some near universal properties of neurons. These

shared aspects are what links the neuroscience of rat, owl, and fish to primates and back to crabs and

leeches. These properties are specifically that neurons are designed to communicate via a neural code

[33, 6]. Indeed, if neuroscience has a paradigm then it is the following: neurons use a communication

system based on a neural code to self-organize into wondrous cognitive machines.

Most neurons have three main parts: an axon, a cell body called the soma, and dendrites [30, 21, 32].

Axons are thin cables that stretch from the soma sometimes for long distances in the brain. An axon

can come quite close to another neuron, usually near the dendrites of that neuron. Dendrites are also

thin cable-like branches that protrude from the soma, often in wild and complex patterns. The axon

will connect, or in neural terms, it will synapse onto the dendrite. For this reason the cell that the

axon comes from is called the pre-synaptic cell while the cell that the axon synapses onto is called the

post-synaptic cell. These are the basics of the circuitry in the brain. Next to discuss is how this circuitry

is used for a communication system.

The cellular membrane of a neuron has the property of excitability [30, 21, 20]: if it gets enough

inputs of a specific type then it will give a stereotyped response. The excitable nature of neurons is

due in part from that fact that they are electrochemical cells; their stereotyped response is a brief (1-2

ms) voltage spike in their membrane potential called an action potential. The site of action potential

initiation is often the soma or at least near it. Once initiated an action potential propagates along the

axon. When it reaches a synapse it will commence a process that perturbes the cellular membrane of the

post-synaptic cell, this perturbation is called a synaptic potential. The dendrites transfer the synaptic

potential to the soma of the post-synaptic cell, and there it will influence the generation of an action

potential. Synapses have a polarity, i.e. they can inhibit or excite the post-synaptic membrane to action

3

INTRODUCTION 5

Figure 1: Neurons and neural communication. A pre-synaptic neuron (right) synapses onto a postsynaptic neuron (left). Their dendrites, somas, and an axon are are labelled. The synaptic connectionis expanded in the insert. Examples of the time series for an action potential and a synaptic potentialare also shown. Anne-Marie Oswald generously created the figure.

potential production. This process of action potential initiation → propagation → synaptic potential →

action potential initiation/inhibition is the core of the communication system in the brain. A schematic

of the circuitry and action/synaptic potentials are given in Fig. 1.

The language of the brain, the so called ‘neural code’, should be treated next. How do action

potentials and synapse code for sensory input? That question is daunting and remains for the large part

unanswered, see [33, 6]. The best I can hope for at the end of this thesis is that the appreciation of the

question is somewhat deeper.

4

INTRODUCTION 6

0.2 The Neuroethology of Weakly Electric Fish

There are numerous aquatic animals that have developed specialized electric organs. Most famous are

eels that use massive electric discharges to stun predators or prey. In the early 1950s Hans Lissmann

reported that certain African fish discharge a weak electric field [26]. Lissmann with Machin [27, 28] later

showed that these animals use their electric fields as a sensory modality to locate objects and navigate

their surroundings. These animals were naturally labelled weakly electric fish. Since Lissmann’s studies

many distinct weakly electric fish species have been documented in both Africa and South America [15].

Further, the electrosensory system has been shown to be essential for these nocturnal animals and is used

not only for location but as an interesting communication system as well [15, 45]. This section outlines

the basics of electrosensory processing as well as detailing the advantages of this unique preparation for

neuroethology.

0.2.1 Electrosensing

The electric organ discharge (EOD) is quite different between the African and South American species

of electric fish. The Mormyriformes of Africa have a pulsatile EOD with a brief 1 ms discharge time

that is separated by long periods of silence. In contrast, the EOD discharge of the South American

Gymnotiformes fish is a continuous quasi-sinsuiodial wave. This thesis will deal exclusively with the

electrosensory system of the Gymnotiform fish Apteronotous leptorynchus, more commonly known as

the ‘brown ghost knife’ fish. I will thus briefly outline the electrosensory system of the wave-type

Gymnotiform fish.

Figure 2A is a schematic of the electric field lines that surround a wave-type weakly electric fish. The

high frequency EOD (500-1100 Hz) of the fish are shown in Fig. 2B. When an object with a conductivity

different from that of the surrounding water is near the fish the field lines are distorted. The specific

distortion of the field lines depends on the object’s electrical properties (insulator vs. conductor), the

size and shape of the object, and the distance between the animal and the object. In all cases the

distortion produces an effective ‘electric image’ of the object on the surface of the fish’s skin [1]. A

dynamic stimulus produces amplitude modulations of the EOD so that the electric image is coded in

time by the envelope of the high frequency EOD carrier signal; this is shown in Fig. 2B. The skin of

the animal has arrays of electroreceptor cells that are sensitive to an electric potential drop across the

skin. Field distortions are recorded by the activity of these receptors which are then transmitted to the

brains of these fish.

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INTRODUCTION 7

Figure 2: Gymnotiform electrosensory processing. A Schematic of electric field lines produced from theelectric organ. A conducting object (gray) distorts the field lines and produces an ‘electric image’ onthe skin. B The quasi-sinusoidal EOD (top) which acts as a carrier wave for amplitude modulationsproduced by objects or conspecifics (bottom). The figure is modified from [23].

Two major functions of the electrosensory system are electrolocation and electrocommunication (see

[15, 45] for reviews). Electrolocation is the use of this unique sensory modality to locate objects such as

prey and predators, as well as to code background scenes. These inputs all give unique electric images

and significant neural processing is devoted to coding these inputs [2]. Electrocommunication occurs

when a fish’s own electric field acts as a signal to a conspecific [17]. Its simplest form is when two fish

with different EOD frequencies are near one another. The resulting electric image for both animals is

a beating of their EODs with a beat frequency that is the difference of their own EOD frequencies.

The fish detect this beat as an environmental signal, and behave accordingly. A more active form

of communication is when a fish actively changes his EOD frequency to cause a rapid amplitude and

frequency modulations of a neighbouring fish’s electric field. This act is called ‘chirping’ in analogy to

the acoustic communication of birds.

The electrosensory receptors project directly and exclusively to a laminar brain region known as

the electrosensory lateral line lobe (ELL) [29, 3]. The ELL consists of four segments: the lateral (LS),

centrolateral (CLS), centromedial (CMS), and medial (M) segments. The medial segment receives input

from ampullary electroreceptors; I will not consider this segment in my thesis. The three remaining

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INTRODUCTION 8

Figure 3: Reduced schematic of the ELL and descending feedback pathways. ELL basilar (BP) andnon-basilar (NBP) pyramidal cells are shown. The ELL granule cell (GC) is shown; these come in twovarieties but this is ignored here. ELL pyramidal cells output to the nucleus praminentialis dorsalis (Pd);this is shown here as an arrow from the NBP cell to Pd, the arrow from the BP cell is omitted to notoverload the figure. The Pd houses three distinct cell types: stellate (ST), bipolar (BP), and multipolar(MP) cells. The Pd projects back to the ELL along the stratum fibrosum (StF) with BP cells providingdirect inhibition and ST cells excitation to both BP and NBP cells. The MP cells of the Pd also projectto another higher brain structure: the eminentia granularis pars posterior (EGP). Granule cells (G) inthe EGP project back to the ELL along the parallel fibre (PF) pathway which also has synapses of bothpolarities (inhibition is through the VML interneuron). The figure is a reduction of a schematic givenin [3].

7

INTRODUCTION 9

segments all receive ‘tuberous’ electroreceptor input that monitors EOD amplitude information; these

segments will be the focus of my thesis, however segmental differences will not be discussed.

The major output cells of the ELL are its pyramidal cells, see Fig. 3. They appear in two distinct

types: basilar (E-cells) and non-basilar (I-cells). E-cells receive direct excitatory input from the elec-

troreceptor afferents which synapse onto their basilar bushes. I-cells, which lack basilar bushes, do not

receive direct input from the afferents but only from granule interneurons. When afferents fire they

excite the granule cells which inhibit the I-cells. When afferents are silent the inhibition from granule

cells is removed and the pyramidal cells can fire; this process is sometimes called ‘disinhibition’. This

circuitry leads to E and I-cells firing out of phase with one another. The combined activity of both E

and I-cells codes all electrosensory information for the higher neural processing centres of these fish.

In Gymnotiform fish pyramidal cells do not have any direct local connections between one another

[29]. Rather they output to higher brain centres, most notably the nucleus praminentialis dorsalis (Pd)

[3]. Pd neurons project directly back to the ELL with both excitatory and inhibitory projections; the pro-

jection pathway is called the direct feedback pathway. In addition, Pd neurons also project to cells in the

eminentia granularis pars posterior (EGP) which in turn project back to ELL with excitatory/inhibitory

paths. This pathway is called the indirect pathway. The direct and indirect pathways synapse onto

pyramidal cells in different locations on the dendritic tree; the direct approximately 200 µm from the

cell’s soma whereas the indirect 600 µm from the soma. Figure 3 omits pathways from Pd and EGP to

even higher brain structures that feedback back to Pd and/or EGP. Similar forms of recurrent neural

pathways are common in most brain structures [30, 21], however not often as segregated as they are in

electric fish. These pathways give the higher brain the ability to modulate the processing of lower brain

areas, such as the ELL. This will be a significant theme in the later chapters of the thesis.

0.2.2 Advantages of the electrosense in sensory studies

The study of electrosensory processing has several advantages over more traditional sensory systems. In

this section we give motivation that highlights electric fish for their neuroethogical worth, their complex

single cell dynamics, as well their characterized feedback pathways.

Electrosensory stimuli are simply electric field distortions. An entire electric scene is a dynamic

three-dimensional vector field that obeys Maxwell’s laws. This may seem complex, however, in com-

parison to the colour [11] and orientation fields [38] of the visual system, or the somewhat innumerable

chemical stimuli of the olfactory system [25], the relative simplicity is attractive. Further, creating and

manipulating electric scenes in the laboratory is easy by using electric dipoles or even passing metal or

8

INTRODUCTION 10

plastic objects near the fish [2]. In addition, communication calls can be mimicked in the laboratory

which elicit a response from paralysed fish. This gives a useful behavioural paradigm for neuroethological

studies [17]. Finally, electric fish can exclusively use their electric sense to perceive their world [13]. This

is not always the case for other organisms where integration of separate sensory information is required

for perception.

Neuronal burst discharge is the phenomena whereby action potentials are grouped in time to form

clusters that are separated by silence. Burst response is commonly observed in thalamic relay neurons and

cortical neurons [46] involved in visual processing. The role of bursting in neural coding is still unknown,

though hotly debated (see [23] for a recent review). ELL pyramidal cells have bursting dynamics that

are dependent on a somatic-dendritic interplay, hinting at a complex spatiotemporal phenomenon inside

a single neuron [44]. The advantage of the ELL is that it is only one synapse away from the sensory

periphery. Thus the synaptic inputs that drive ELL pyramidal cells are closely related to the actual

electric scenes that drive electroreceptors. This facilitates interpretations of how a complex bursting

dynamic codes input stimuli.

Finally, the direct and indirect feedback pathways offer a simple neural system from which to study

the influence of top-down processing in the brain. The segregated axon pathways allows experimentalists

to open one feedback loop and study its effects on pyramidal cell behaviour, without affecting the other

pathway. One often overlooked advantage of electrosensory processing is that ELL pyramidal cells do

not locally connect to one another [29]; they can only influence each other’s activity via feedback. Some

of the feedback projections are spatially diffuse and a single axon may connect to many pyramidal cells

[3]. Thus a single pyramidal cell influences a large number of its neighbouring cells with an equal weight.

This sets ELL pyramidal cell networks as an excellent candidate for a mean field theory of sensory

processing. This is a simplification that can be profitable for any theoretical studies, and the final two

chapters of this thesis make use of it.

0.3 Single Neuron Dynamics

Without controversy it can be said that a major currency or unit of information in the brain is the action

potential. It is then not surprising that a vast majority of neuroscience research focuses on the mechanics

of action potential generation. This section will first introduce the most widely accepted cellular model

of neuronal action potential production: the Hodgkin-Huxley model.

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INTRODUCTION 11

0.3.1 The Hodgkin Huxley Model of the Action Potential

Neurons, like all cells, have membranes (3 to 4 nm thick) that separate the inside of the cell from the

extra-cellular medium. Both the intra and extra-cellular regions contain various charged ions: Na+, K+,

Ca2+, and Cl−. For the most part the cellular membrane is a near perfect insulator and acts like a

capacitor in its charged environment, its capacitance given by Cm. However, unlike other cells, there

are ion selective channels (holes) in the membrane that allow specific ions to diffuse through them. This

imparts conductive properties to the membrane. With charge carriers and a conductive medium defined,

neurons then have an electrical potential across their membrane, labelled Vm, and a total current flowing

through their membrane, labelled Im.

Im is determined by the combined activity of all the ions passing through the channels in the mem-

brane [16]. Channels can be grouped into subtypes by considering the ion that they select for, their

physical structure, as well as their operating dynamics. Over a macroscopic patch of membrane (hun-

dreds of µm2) there may be a large ensemble (thousands) of channels of a specific type. It is then

appropriate to consider the average activity of all the channels in a specific group when determining

their contribution to the membrane current. This is commonly called determining the channel’s ensem-

ble dynamics. We decompose Im by considering the influence of each channel ensemble separately.

Ions are driven through channels by both electric and diffusive forces. Each channel ensemble has

an equilibrium point where the diffusive flow is counterbalanced by the electrical current flow. Let the

membrane potential at the equilibrium point for channel i be Ei, Ei is often called the Nernst reversal

potential of the channel. For Vm 6= Ei then there is a net current flow and the i channels influence the

electric properties of the membrane. The magnitude of the influence scales with the total conductance

or permeability of the channel ensemble i, labelled gi. If we treat the membrane as a simple RC circuit,

then we can describe its ‘neuroelectronics’ with the following:

CmdVm

dt= Im (1)

Im =∑

i

gi(Ei − Vm). (2)

The above sum ranges over all distinct channel ensemble subtypes in a given patch of cellular membrane.

See [20, 16, 22] for a further introduction into membrane biophysics.

In their seminal paper, Hodgkin and Huxley [18] gave a detailed understanding of the generation of

an action potential in the giant axon of the squid. Their model includes only three channel subtypes:

Na+, K+, and a leak channel L (L is a combination of nonspecific K+ and Cl−). The total membrane

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INTRODUCTION 12

current from each of these channel subtypes is given by:

Im = gNam3h(ENa − Vm) + gKn4(EK − Vm) + gL(EL − Vm). (3)

The reversal potentials for a typical neuron are ENa = 40mV , EK = −85mV , and EL = −70mV [22, 20].

Notice that the channel conductances gNa and gK are decomposed as gNam3h and gKn4 respectively.

The variables m, h, and n model a selective permeability of the membrane to both Na+ and K+ ions.

Their dynamics are what gives action potentials their precise shape; this will be the focus of the next

few paragraphs.

Channels have microscopic ‘gates’ [16]. These gates may be open or closed, and a channel can only

pass ions when its gate is open. To model the opening and closing of a subset of the channel gates in

an ensemble, Hodgkin and Huxley assigned the ensemble activation and inactivation gating variables.

These variables collectively determine the average dynamics of the channel gates. Let Na+ channel

activation be given by m ∈ (0, 1) and its inactivation by h ∈ (0, 1). Similarity n ∈ (0, 1) is the activation

of the K+; the channel ensemble does not classically have an inactivation variable. A gate opens as it is

activated (m → 1, n → 1) and once activated can close either through deactivation (m → 0, n → 0), or

in the case of Na+, through inactivation (h → 0). The products m3h ≤ 1 and n4 ≤ 1 then represent the

normalized fraction of open channels in either the Na+ or the K+ ensembles respectively. The terms

gNa and gK are the maximal channel conductances that would occur if all channels in an ensemble were

simultaneously open (m = 1,h = 1, and n = 1).

The dynamics of m, h, and n are given by:

dm

dt=

m∞(Vm)−m

τm(Vm)(4)

dh

dt=

h∞(Vm)− h

τh(Vm)(5)

dn

dt=

n∞(Vm)− n

τn(Vm). (6)

These equations are linear in themselves and show that at constant Vm, m, h, and n relax exponentially

to steady state values m∞(Vm), h∞(Vm), and n∞(Vm) respectively (the ∞ symbol is used for t → ∞).

They each relax with their own time-scale τm(Vm), τh(Vm), or τn(Vm). There is an equivalent description

of Hodgkin-Huxley gating dynamics that model open and closed gating states [18, 22, 20]. However, I

choose to motivate gating dynamics as presented above to better agree with later thesis chapters.

Action potential production is inherently a nonlinear phenomena; in the Hodgkin-Huxley model that

nonlinearity appears in the dependence of the steady state values and time-scales of ionic gates on Vm.

By developing an experimental protocol whereby the membrane potential of a neuron is fixed, or as

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INTRODUCTION 13

Figure 4: Typical m∞(Vm), h∞(Vm), and n∞(Vm) curves for a Hodgkin-Huxley type model

is commonly termed ‘clamped’, Hodgkin and Huxley were able to isolate the functional forms of these

dependencies. Their fitted forms for the steady state conductances were sigmoidal:

m∞(Vm) =1

1 + exp ((V1/2 − Vm)/k). (7)

V1/2 and k are the respective half activation and slopes of the activation curve. Similar curves are given

for h∞(Vm) and n∞(Vm). I do not report the functional forms for the gating time-constants τ(Vm);

rather I take then to be independent of Vm. This is an often used simplification of classic Hodgkin-

Huxley dynamics [22] and it is used throughout the thesis.

Figure 4 shows standard steady state curves for m, h, and n. Na+ and K+ activate with depolar-

izations; i.e. both m and n tend to 1 as Vm increases. Na+ inactivates with depolarizations, i.e. h tends

to zero as Vm increases. With standard values of gNa, gK , and gL the equilibrium or resting state of the

membrane potential is typically -70 mV. Upon consideration of the reversal potentials for the various ion

channels this means that, in equilibrium conditions, there is an excess of K+ and a lack of Na+ inside

the cell as compared the extracellular medium. During an action potential this situation changes.

A key requirement for proper action potential generation is that the gating time-scales are such

that τm < τh and τn. This ensures that given a depolarization in Vm the Na+ m gate is the first to

respond. The opening of the Na+ channel allows Na+ to rush into the cell. This raises Vm further and

begins a positive feedback interaction between Vm and m that forces Vm to explode to quite depolarized

potentials. This is the upstroke of an action potential. Once Vm is depolarized, the n gate opens and

K+ rushes out of the cell. This drops, or hyperpolarizes, Vm rapidly and forms the downstroke of the

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INTRODUCTION 14

Figure 5: Vm (top) and m, h, and n (bottom) during an action potential

action potential. In addition, Na+ also inactivates with h dropping from 1 during the spike. This has

some effect on spike repolarization, but h must also return back to high values before another spike can

be initiated. This produces a refractory time for the spiking dynamics, meaning that directly after a

spike another spike may not be initiated until a minimum time has elapsed. Figure 5 shows both the

membrane potential and the associated gating variables during an action potential.

The specifics of an action potential, its height and width, and the sensitivity of action potential

production to inputs, can differ from neuron to neuron. These differences can be modelled by fitting the

specific model channel parameters or even adding more channel types to the membrane equation. As it

currently stands the conductance based Hodgkin-Huxley model and its variants are the most realistic

models of action potential production and has received widespread use over the past fifty years, often

with little to no modification.

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INTRODUCTION 15

0.3.2 Dendrites and the Cable Equation

The Hodgkin-Huxley model describes only the time dependent electrical properties of a neuron. The

model neglects the spatial flow of current in a neuron. However, it has been known for roughly a

century that many neurons possess a wickedly complex spatial morphology [32]. Dendrites of various

shapes, widths, and lengths often contribute a large portion of the total cell surface area. Furthermore,

some dendrites have recently been shown to have active Na+,K+, and Ca2+ channels that impart an

excitability to the dendritic membrane [34, 42]. This section will expand the Hodgkin-Huxley model to

a spatiotemporal description so as to model dendritic dynamics as well as somatic dynamics (note that

the original Hodgkin and Huxley study [18] included analysis of the spatial effects of propagation of an

action potential down an axon).

The narrow width of a dendrite permits a one-dimensional description of current flow along the

somatic-dendritic axis. We thus extend our membrane description to be Vm(x, t), where x is the longi-

tudinal distance from the soma to a point on the dendrite. Current flows within a neuron via a diffusive

process. It is relatively straightforward to derive the governing diffusion equation for electrical flow along

the dendrite [19, 22, 6]:

τm∂Vm

∂t= λ2 ∂2Vm

∂x2+ Im. (8)

Equation (8) is a classic cable equation since it models flow of electricity along a cable. τm = cm/gL

is the membrane time constant while the space (or diffusion) constant is given as λ =√

agax

2gL. In these

definitions gL is the leak conductance across the membrane, gax is the axial conductance along the

membrane, and a is the cross-sectional area of a dendrite. The derivation of Eq. (8) assumes that a is

independent of x. A specific solution of Eq. (8) requires the specification of the appropriate boundary

conditions. It is typical to assume that no current flows out of the tip of a dendrite. Mathematically

this amounts to setting ∂Vm∂x |L = 0 where x = L is the tip of a dendrite.

The term Im(x, t) is a reaction term that models all of the ionic dynamics that occur across the

membrane at the point x along the dendrite. Classically this is treated linearly in Vm(x, t) and the

solutions exponentially decay (in space) from a source input depolarization. However, it is now well

established that dendrites can have nonlinear ionic dynamics along the dendrite [42]. These dynamics

can often support action potential generation i.e. Im(x, t) would have a form similar to the Hodgkin-

Huxley model given in Eq. (3). However, the ensuing dynamics can be much richer than those from a

point neuron model. Spatially extended excitable systems are able to exhibit wave propagation or even

wave destruction if certain areas of the dendrite are refractory while other parts are not.

If Im(x, t) is nonlinear in Vm(x, t) then analytic solutions for Eq. (8) are difficult, if not impossible,

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INTRODUCTION 16

to find. This problem is compounded with an intricate dendritic morphology, often making boundary

conditions complicated. This morphology implies that a cable equation must be solved in each branch

of the dendritic tree, with proper continuity relations at the branch points. It is then appropriate to

use computational methods such as finite element numerical schemes to solve Eq. (8). Computational

neurosciectists often call such a computational model a ‘compartmental model’ [22]. Compartmental

models dissect a dendritic cable into a set of N isopotential cable segments {Vm,i(t)} where the length

of each segment is ∆x << λ. This allows for an approximation of the spatial derivative transforming

the partial differential system in Eq. (8) into a set of N ordinary differential equations:

τmdVm,i

dt= λ2 Vm,i+1 − 2Vm,i + Vm,i−1

∆x2i = 1, . . . , N (9)

Matrix inversion schemes required to numerically iterate the system (9) can be made efficient due

to the sparse connectivity typical for tree-like structures such as dendrites [22]. There exists sev-

eral software packages that perform the integration of compartmental models, for example NEURON

(www.neuron.yale.edu) or Genesis (www.genesis-sim.org/GENESIS/). These packages often use semi-

implicit integration schemes such as Crank-Nicholson for the integration of (9). These schemes offer

better convergence than standard explicit Euler schemes.

Compartmental models of spatially extended neurons have been developed for a wide variety of

neurons: cortical pyramidal neurons, hippocampal neurons, and Purkinje cells in the cerebellum to name

but a few (see Chapter 1 for references). The first chapter of this thesis introduces a compartmental

model developed for ELL pyramidal neurons.

0.4 Simple Spiking Models and Stochastic Neuroscience

Conductance-based models can be quite realistic and reproduce a variety of complex single neuron

behaviour. However, there are a few drawbacks when using these models to describe membrane dynamics.

First, these models are often computationally expensive and large networks of conductance-based neurons

can take a long time to simulate. Second, neurons are stochastic creatures and models of them should

often include noisy forcing terms. The high dimensionality of conductance-based models often disallows

any theoretical treatment when stochastic forcing is present. In this section I discuss a simplified ‘toy’

model for spiking neurons called the integrate-and-fire (IF) neuron [24, 43, 22]. I use this model to

motivate stochastic neuron dynamics and introduce some measures that are appropriate for noisy neural

data.

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INTRODUCTION 17

0.4.1 Integrate-and-Fire Models

IF neurons model cellular dynamics with an isopotential compartment having a capacitance Cm and a

resistance Rm = 1/g. The membrane dynamics are given by the linear equation:

CmdVm

dt+ gVm = I(t). (10)

I(t) is the sum of any applied or synaptic currents. The dynamics of spike generation are where IF

neurons differ from Hodgkin-Huxley neurons. Rather than use a set of gating variables for Na+ and K+

currents to produce the dynamics of action potentials, a simple threshold crossing rule is used. When

Vm equals a threshold value Vth it is assumed that a spike occurs and the membrane is reset to Vr < Vth.

This resetting models the hyperpolarization of the membrane after the upstroke of an action potential.

It is also typical to force the membrane to be Vr for a period of time tr, which is independent of ongoing

inputs. This thereby models a refractory period. The simplicity of IF neurons is at the expense of

realism; spiking dynamics are forced rather than emergent from the model dynamics. However, that

simplicity is often an advantage when faced with either hopes of analytic solution to the membrane

equation, or large scale neural networks.

0.4.2 Stochastic Differential Systems

Despite the common analogy that the brain is a computer, an organic brain does not have the clockwork

design of modern day silicon. The brain is noisy, and thus often unpredictable. Stochastic behaviour is

found at all spatial scales in the brain. Single ion channels have a stochastic flickering opening/closing

dynamic [16] and large scale brain scans also have a distinctly random component. With the IF neuron

now defined I will use it to introduce the concept of the stochastic dynamical system and measures of

random neural activity. I briefly mention that all measures of the stochastic behaviour of a generic

function f(t) are rooted in the time average, defined as:

〈f(t)〉 = limT→∞

1T

∫ T

0f(t)dt. (11)

Neurons often receive synaptic inputs from thousands of neighbouring neurons. Their summed effect

often looks unpredictable or random. Stein [39] introduced the approximation of reducing many infini-

tesimal depolarizing and hyperpolarizing inputs that arrive at high frequency to a single Gaussian white

noise process [10]. Mathematically this amounts to setting the input I(t) to be:

I(t) = I0 +√

2Dξ(t); 〈ξ(t)〉 = 0, 〈ξ(t)ξ(t′)〉 = δ(t− t′). (12)

16

INTRODUCTION 18

I0 is the time-independent component of the input and the intensity of the stochastic input ξ(t) is given

by D. The correlation function of ξ(t) is a Dirac delta function; this means that the noise forcing is only

correlated with itself for an infinitesimal instant and then is uncorrelated for all other times, past and

future. This is a working definition of white noise.

Equations (10) and (12) are stochastic and it is thus necessary to solve them with a stochastic calculus

[10, 35]. Combining the two equations we have:

dVm(t) = (I0 − Vm(t))dt +√

2DdW (t), (13)

In going from Eq. (10) to Eq. (13) I have introduced a nondimensional time gt/C → t. W (t) is a

Wiener process, more commonly known as Brownian motion [10, 7]. We associate dWdt with ξ(t) since

Gaussian white noise is conceived as the derivative of a Wiener process. Note that it is not proper to

write dWdt since a Brownian path is nowhere differentiable. This formal non-differentiability of W (t) sets

ξ(t) as having an infinite variance. A Wiener process has the important property that for 0 < s < t

the statistics of W (t)−W (s) are√

t− sN(0, 1) where N(0, 1) is a normally distributed random variable

with zero mean and unit variance. This scaling of variance with time is a characteristic of a diffusion

process, of which the Wiener process is a canonical example.

I will always treat Eq. (13) in the Ito sense [10, 35]. Noise whose variance is dependent on V (t)

(multiplicative noise) will not be used in this thesis so a discussion of Ito versus Stratonovich stochastic

integration is not required [10, 35]. The numerical integration of Eq. (13) is done using and Euler-

Maruyama (EM) approximation scheme [10, 35]. EM stochastic integration, like all numerical schemes,

discretizes continuous time into a set of instances 0 < ∆t < 2∆t < . . . < (N − 1)∆t < N∆t = T . A

discretized Wiener process follows the rule

W (t + ∆t) = W (t) + ∆W (t) t = M∆t, M < N. (14)

Here the ∆W are independent random variables identically distributed according to√

∆tN(0, 1). The

full numerical approximation for Eq. (13) is then written as:

Vm(t + ∆t) = Vm(t) + ∆t(I0 − Vm(t)) +√

2D∆tN(0, 1) t = M∆t, M < N. (15)

Note that D = 0 we have the standard explicit Euler scheme. Unlike deterministic integration schemes

more sophisticated numerical schemes for stochastic integration do not offer large gains in convergence.

As a result EM integration is simple and widely used. Figures 6A shows a realization of Eq. (15)

complete with spiking resets due to threshold crossings.

17

INTRODUCTION 19

Figure 6: Stochastic LIF model. I show Vm(t), complete with spiking resets due to threshold crossings,for a short segment of time (A). The spikes (vertical lines) were manually added to easily identify spiketimes; the mean firing r was 7.5 Hz. The ISI histogram (B), the spike train autocorrelation (C), andthe spike train power spectrum (D) are all shown. All statistics were computed from a simulation thatproduced in excess of 40 000 spikes; simulation parameters were I0 = 0.8, D = 0.01, and we assumed amembrane time constant of 10 ms.

18

INTRODUCTION 20

0.4.3 Stochastic Neuroscience

The relevant output of an IF neuron is typically a sequence of threshold crossing times {ti} (t = 1 . . . N).

Since the membrane is stochastic, {ti} is a random point process. I now introduce some standard

statistical measures I will use to investigate the temporal patterning of {ti} [8]. In what follows I assume

that the stochastic spike train processes are stationary in the weak sense (their first few statistical

moments are time-independent). From {ti} it is easy to define the spike train x(t) =∑

i δ(t − ti) and

the interspike interval (ISI) sequence Ti = ti − ti−1 (t0 = 0). Figure 6B shows a typical ISI distribution

computed from simulations of Eq. (15). The statistical structure of both the spike train and the ISI

sequence are indicators of neural behaviour. From the ISI sequence I define the firing rate r of a spike

train as

r =1〈T 〉

. (16)

The autocorrelation function of the spike train A(τ) is defined as

A(τ) =〈x(t)x(t + τ)〉〈x(t)2〉 − 〈x(t)〉2

. (17)

A(τ) is the mean corrected probability of observing a spike τ time units after (or −τ time units before)

another spike. ‘Mean corrected’ implies that I have replaced x(t) with x(t)−〈x(t)〉 so that when A(τ) = 0

it should be interpreted as having the equivalent event probability of a simple Poisson process with the

same firing rate. Structure in A(τ) is indicative of specific features of the spike generating mechanism,

or of neural network interactions. Figure 6C shows A(τ) computed from simulations of Eq. (15). Note

the negative component for small τ , this is because of the refractory dynamics in Eq. (15).

Neurons often produce an oscillatory spike train pattern [36]. This can be seen from A(τ), however

it is often more useful to use a spectral measure. The Weiner-Khichin theorem [10] relates the power

spectrum of a stochastic process to its autocorrelation function via a Fourier transform:

S(ω) =∫ ∞

−∞eiωτA(τ)dτ. (18)

Figure 6D shows S computed from simulations of Eq. (15). There is a slight peak at 20 Hz, this is

because the most common ISI in the spike train is 50 ms (see Fig. 6B). The spike train then naturally

oscillates weakly with that period.

The statistical behaviour of the spike train as observed from the ISI histogram, A(τ), and S(ω) can

change significantly when new mechansisms are introduced into the spiking dynamics described by Eq.

(13). I will use these measures as tools to uncover the underlying dynamics in stochastic electrosensory

spiking.

19

INTRODUCTION 21

0.5 Bifurcations

Neuron models are systems of nonlinear differential equations. Nonlinearities often make differential

systems intractable, with no hope of obtaining closed form solutions. Fortunately, Henri Poincare began

devising geometric theories for nonlinear dynamical systems that offer a qualitative understanding of

their varied behaviours [31]. The application of these theories to problems has been fruitful in a variety

of physical and life sciences, including neuroscience.

One distinction between linear and nonlinear differential systems is that nonlinear systems can exhibit

qualitatively distinct behaviours depending on system parameters. A transition between one behaviour

and another as parameters are varied is called a bifurcation [14, 41]. I next will mathematically state

what is meant by ‘transition in behaviour’.

Consider the general n dimensional nonlinear system

dx

dt= fµ(x); x ∈ Rn, µ ∈ R. (19)

fµ : R → R is a smooth vector-valued function; often fµ is said to be the vectorfield for Eq. (19). µ is

a parameter that describes the vector field. Equilibrium solutions of (19) at some x = x∗ must satisfy

fµ(x∗) = 0. To describe the local dynamics of the vector field f around an equilibrium solution one

linearizes the vector field around the point x∗. Let the Jacobian derivative of fµ(x) with respect to x

and evaluated at x∗ be Dxfµ|x∗ . The eigenvalue spectrum of Dxfµ|x∗ determines the stability of x∗. If

for all eigenvalues λ we have that Re(λ) < 0, then x∗ is stable and is an attractor in the vector field.

If any or all of the eigenvalues have Re(λ) > 0 then x∗ is a saddle or repeller in the vector field. A

bifurcation occurs when there exists an eigenvalue or set of eigenvalues that are completely imaginary,

i.e. Re(λ) = 0. This typically implies that an equilibrium point is transitioning from stable to unstable

(or vice-versa), and the vector field is undergoing a qualitative change. At these points in parameter

space a full linearization around x∗ is insufficient to determine the fate of the solution. In what follows

we assume that this condition occurs when µ = µ0. I will further motivate bifurcations by reviewing two

of the simplest bifurcations: the saddle-node bifurcation and the Hopf bifurcation [14, 47, 41].

0.5.1 Saddle-node bifurcation

Consider the case when one and only one of the eigenvalues is zero when µ = µ0. Center manifold theory

shows that there exists a one-dimensional manifold that is tangential to the zero (or center) eigendirection

in phase space space [14, 47]. The motion along the center manifold is dictated by a nonlinear flow and

20

INTRODUCTION 22

the other n− 1 eigendirections will have a linearization near x∗. It is sufficient to study only the motion

along the center manifold; due to their linear dynamics, all the other eigendirections will have only simple

exponential expansion or contraction near x∗. Further, motion on the center manifold can be reduced to

its simplest form, called its normal form, by a process called ‘normal form reduction’ [47]. The normal

form of any dynamical system with a saddle-node bifurcation is given as:

dy

dt= µ + y2. (20)

Equation (20) admits two equilibrium solutions y∗ = ±√|µ| when µ < 0. Linearization around

y∗ = −√|µ| shows that it is stable to perturbation while y∗ =

√|µ| is not. However, when µ > 0

the roots of the right hand side of Eq. (20) are both complex and no equilibrium solutions exist. The

transition between these two cases, when µ = 0, is when the system (20) is at a saddle-node bifurcation

(in this case µ0 = 0). What happens is that as µ → 0 from negative values the stable and unstable

solutions approach one another and annialate each other at µ = 0. This is schematically shown in Fig.

7.

For µ > 0, there are no equilibrium solutions and we have that limt→∞ x(t) = ∞. However, due to

the quadratic vector field the passage of x(t) to ∞ happens in finite time. Consider the passage time,

T , for a particle placed in the vector field at x = −∞ to pass through x = 0 and go to x = ∞. For the

simple normal form of equation (20) this time can be explicitly computed:

T =∫ ∞

−∞

dy

µ + y2=

π√

µ(21)

Thus as µ → 0 we have that the passage time diverges, and specifically it diverges as an inverse square

root in the bifurcation parameter µ. This gives a large continuum of time-scales for the flow of trajectories

in the vector field governed by Eq. (20). The reduction of velocity near y = 0 for small, but positive,

µ is often called sensing the ghost of the saddle-node bifurcation [41]. This is because it can be thought

that the disappeared stable node is still ‘haunting’ the vector field and causing a slow passage through

the origin. This effect plays a crucial role in Chapters 2 and 3 of this thesis.

0.5.2 Hopf bifurcation

The saddle-node bifurcation happens when one real eigenvalue passes through zero. However, the eigen-

values of a fixed point can either be real or complex. A Hopf bifurcation (or Andronov-Hopf as it is

sometimes called) occurs when two complex eigenvalues (conjugates of one another) pass through the

imaginary axis, i.e have Re(λ) = 0. Since there must be two eigenvalues for this to occur, it follows that

21

INTRODUCTION 23

Figure 7: Phase space description of saddle-node and Hopf bifurcations.

Hopf bifurcations can only occur in systems with n ≥ 2. Center manifold and normal form techniques

can be applied to a system with a Hopf bifurcation; the associated normal form for a Hopf bifurcation

in polar coordinates is:

dr

dt= µr − r3 (22)

dθ

dt= ω + br2. (23)

Equations (22)-(23) have a fixed point at r = 0 (recall that when r = 0, θ is undefined in polar

systems). For µ < 0, r = 0 is a stable spiral with all vector field lines flowing into the origin. However,

when µ > 0 we have that r = 0 is an unstable spiral and there now exists a stable limit cycle at r =√

µ.

A limit cycle is a closed orbit in phase space; a stable limit cycle is an attracting orbit while an

unstable limit cycle is a repelling one. The Hopf bifurcation transition is schematically shown in Fig. 7.

If we replace Eq. (22) with drdt = µr + r3 then an unstable limit cycle collapses onto a stable fixed point

at r = 0 when µ goes through zero from negative values. When a stable limit cycle is created, the Hopf

bifurcation is called supercritical (shown in Fig. 7), while it is subcritical when an unstable limit cycle is

lost (not shown).

Unlike the saddle-node, Hopf bifurcations do not induce any slow passage effects in phase space. The

frequency of the limit cycle that emerges is roughly ω near the bifurcation. Note that the radius of the

limit cycle is however variable and is proportional to√

µ. The Hopf bifurcation and the limit cycle that

it produces will be a central theme in Chapter 4 and will influence the interpretation of Chapters 5 and

6.

22

INTRODUCTION 24

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ARTICLE I

B. Doiron, A. Longtin, R. W. Turner and L. Maler.Model of gamma frequency burst discharge generated by conditional backpropaga-tion.Journal of Neurophysiology. 86: 1523-1545, 2001.

27

ARTICLE I 28

Model of Gamma Frequency Burst Discharge Generated byConditional Backpropagation

BRENT DOIRON,1 ANDRE LONGTIN,1 RAY W. TURNER,2 AND LEONARD MALER3

1Physics Department, University of Ottawa, Ottawa, Ontario K1N 6N5;2Department of Cell Biology and Anatomy,Neuroscience Research Group, University of Calgary, Calgary, Alberta T2N 4N1; and3Department of Cellular andMolecular Medicine, University of Ottawa, Ottawa, Ontario K1H 8M5, Canada

Received 18 January 2001; accepted in final form 10 May 2001

Doiron, Brent, Andre Longtin, Ray W. Turner, and LeonardMaler. Model of gamma frequency burst discharge generated byconditional backpropagation.J Neurophysiol86: 1523–1545, 2001.Pyramidal cells of the electrosensory lateral line lobe (ELL) of theweakly electric fishApteronotus leptorhynchushave been shown toproduce oscillatory burst discharge in theg-frequency range (20–80Hz) in response to constant depolarizing stimuli. Previous in vitrostudies have shown that these bursts arise through a recurring spikebackpropagation from soma to apical dendrites that is conditional onthe frequency of action potential discharge (“conditional backpropa-gation”). Spike bursts are characterized by a progressive decrease ininter-spike intervals (ISIs), and an increase of dendritic spike durationand the amplitude of a somatic depolarizing afterpotential (DAP). Thebursts are terminated when a high-frequency somatic spike doubletexceeds the dendritic spike refractory period, preventing spike back-propagation. We present a detailed multi-compartmental model of anELL basilar pyramidal cell to simulate somatic and dendritic spikedischarge and test the conditions necessary to produce a burst output.The model ionic channels are described by modified Hodgkin-Huxleyequations and distributed over both soma and dendrites under theconstraint of available immunocytochemical and electrophysiologicaldata. The currents modeled are somatic and dendritic sodium andpotassium involved in action potential generation, somatic and prox-imal apical dendritic persistent sodium, and KV3.3 and fast transientA-like potassium channels distributed over the entire model cell. Thecore model produces realistic somatic and dendritic spikes, differen-tial spike refractory periods, and a somatic DAP. However, the coremodel does not produce oscillatory spike bursts with constant depo-larizing stimuli. We find that a cumulative inactivation of potassiumchannels underlying dendritic spike repolarization is a necessary con-dition for the model to produce a sustainedg-frequency burst patternmatching experimental results. This cumulative inactivation accountsfor a frequency-dependent broadening of dendritic spikes and resultsin a conditional failure of backpropagation when the intraburst ISIexceeds dendritic spike refractory period, terminating the burst. Thesefindings implicate ion channels involved in repolarizing dendriticspikes as being central to the process of conditional backpropagationand oscillatory burst discharge in this principal sensory output neuronof the ELL.

I N T R O D U C T I O N

The temporal discharge pattern of central neurons is animportant element of signal processing and information trans-

fer. Cortical neurons have traditionally been grouped into threebroad classes according to their discharge patterns in responseto depolarizing current injection: regular spiking, fast spiking,and intrinsic bursting (Connors and Gutnick 1990; Connors etal. 1982; McCormick et al. 1985). Both regular and fast spikingcells respond to depolarizing current with a repetitive dischargeof action potentials but differ in that regular spiking neuronsshow significant frequency adaptation in their firing patterncompared with the consistent discharge frequency of fast spik-ing cells. However, the discharge pattern of intrinsic burstingneurons is quite distinct in generating a phasic burst followedby a tonic discharge of action potentials. Several studies havefocused on distinguishing morphological and electrophysiolog-ical characteristics of neurons exhibiting these three patterns ofspike output (Franceschetti et al. 1995; Jensen et al. 1994;Mason and Larkman 1990; Nun˜ez et al. 1993; Schwindt et al.1997; Williams and Stuart 1999). A fourth discharge patternconsisting of rhythmic spike bursts in theg-frequency range(20–80 Hz) has now been identified in cortical as well assub-cortical and medullary neurons (Brumburg et al. 2000;Gray and McCormick 1996; Lemon and Turner 2000; Lo et al.1998; Pare´ et al. 1995; Steriade et al. 1998; Turner et al. 1994).This pattern differs from intrinsic bursting cells by exhibitinga continuous and nonadapting series of spike bursts duringcurrent injection (Gray and McCormick 1996; Steriade et al.1998; Turner et al. 1994).

Gamma frequency discharge is thought to be important toseveral aspects of signal processing and neuronal synchroni-zation (Gray and McCormick 1996; Gray and Singer 1989;Lisman 1997; Ribary et al. 1991), yet comparatively fewstudies have examined the mechanisms underlying burst outputat such a high-frequency. Gamma frequency bursting in hip-pocampus is known to involve extensive interneuronal synapticcircuitry (Buzsaki and Chrobak 1995; Stanford et al. 1998;Traub et al. 1998). Other studies have revealed that backpropa-gating dendritic spikes contribute to burst discharge by gener-ating a depolarizing afterpotential (DAP) at the soma (Mainenand Sejnowski 1996; Turner et al. 1994; Wang 1999; Williamsand Stuart 1999). The amplitude of the DAP can be augmentedby a persistent sodium current (INaP) (Brumburg et al. 2000;Franceschetti et al. 1995; Wang 1999) or dendritic Ca21 cur-

Address for reprint requests: B. Doiron, Physics Dept., University of Ot-tawa, 150 Louis Pasteur, Ottawa, Ontario K1N 6N5, Canada (E-mail:bdoiron@science.uottawa.ca).

The costs of publication of this article were defrayed in part by the paymentof page charges. The article must therefore be hereby marked ‘‘advertisement’’in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

15230022-3077/01 $5.00 Copyright © 2001 The American Physiological Societywww.jn.org

ARTICLE I 29

rent (Magee and Carruth 1999; Williams and Stuart 1999).Alternatively, the amplitude of the DAP can be influenced bydendritic morphology because the dendrite-to-soma currentflow increases with the relative dendritic to somatic surfacearea and decreases with axial resistance (Mainen and Sej-nowski 1996; Quadroni and Knofnel 1994). Lemon and Turner(2000) recently described a novel mechanism of “conditionalspike backpropagation” that modulates DAP amplitude andproduces ag-frequency oscillatory burst discharge in pyrami-dal neurons of the electrosensory system.

Electrosensory lateral line lobe (ELL) pyramidal cells areprincipal output cells in the medulla that respond to AM ofelectric fields detected by peripheral electroreceptors (Bastian1981; Shumway 1989). Several studies have described theproperties of burst discharge in ELL pyramidal cells (Bastianand Nguyenkim 2001; Gabbiani and Metzner 1999; Gabbianiet al. 1996; Lemon and Turner 2000; Metzner et al. 1998;Rashid et al. 2001; Turner and Maler 1999; Turner et al. 1994,1996). Signal detection analysis has shown that ELL pyramidalcells generate burst discharge in relation to specific signalfeatures, such as up or down strokes in the external electricfield (Gabbiani and Metzner 1999; Gabbiani et al. 1996;Metzner et al. 1998). Further, significant progress has beenmade in identifying how conditional backpropagation gener-ates an oscillatory pattern of spike bursts in ELL pyramidalcells in vitro. Pyramidal cell spike bursts are initiated when aNa1 spike backpropagating over the initial 200mm of apicaldendrites generates a somatic DAP (Turner et al. 1994). Afrequency-dependent broadening of dendritic spikes potenti-ates the DAP until a high-frequency spike doublet is triggeredat the soma (Lemon and Turner 2000). The short inter-spikeinterval (ISI) of the doublet falls within the dendritic refractoryperiod and blocks spike backpropagation, removing the den-dritic depolarization that drives the burst. Repetition of thisconditional process of backpropagation groups repetitive spikedischarges into bursts in theg-frequency range. A key issuethat remains in understanding the mechanism of ELL burstdischarge is the identity of factor(s) underlying the frequency-dependent broadening of dendritic spikes that drives burstdischarge.

Our present knowledge of spike discharge in ELL pyramidalcells and the simple mechanism underlying conditional back-propagation provides an excellent opportunity to model a formof g-frequency burst discharge and test hypotheses about burstgeneration. This study presents a detailed compartmentalmodel of an ELL pyramidal cell that is based on extensiveelectrophysiological and morphological data. We establish thedistribution and complement of ion channels that are necessaryto fit various aspects of Na1 spike discharge and spike back-propagation to physiological data. However, the resultingmodel neuron fails to reproduce the change in dendritic spikerepolarization and somatic afterpotentials required to induceg-frequency bursting. Hence we test potential ionic mecha-nisms that could underlie burst discharge. Our results show thatcumulative inactivation of a dendritic K1 current is necessaryand sufficient to generate a burst discharge that is remarkablysimilar to that found in ELL pyramidal cells in vitro. Some ofthis work has been previously reported in abstract form (Doi-ron et al. 2000).

M E T H O D S

The compartmental model we use in this investigation builds on theearlier one introduced in Doiron et al. (2001). Simulations are per-formed with the software package NEURON (Hines and Carnevale1997), which uses a central difference algorithm (Crank-Nicholson) tointegrate forward in time. The time step for all simulations is 0.025ms, well below the time scale of the synaptic and ionic responsespresent in the ELL (Berman and Maler 1999; Berman et al. 1997).

Cell morphology and discretization

Pyramidal cell somata are located within a pyramidal cell bodylayer, a distinct lamina in the ELL. Basal and apical dendrites emanatefrom the ventral and dorsal aspects of the cell soma, respectively; thebasal dendrites receive electrosensory afferent input while the apicaldendrites receive feedback input (Berman and Maler 1999). There aretwo classes of pyramidal cells, basilar and nonbasilar (lacking a basaldendrite), both of which generateg-frequency oscillatory spike burststhat depend on conditional backpropagation into the apical dendrites(Lemon and Turner 2000; Turner et al. 1994). The present model isfocused on the activity associated with basilar pyramidal cells to allowfuture analysis of the effect of electrosensory afferent input on burstdischarge.

Figure 1 shows our two-dimensional spatial compartmentalizationof a basilar pyramidal neuron based on detailed spatial measurementsof confocal images of a Lucifer yellow-filled neuron (Berman et al.1997). The model contains 153 compartments with lengths rangingfrom 0.8 to 669.2mm and diameters spanning from 0.5 to 11.6mm.

FIG. 1. Basilar pyramidal cell morphology and ionic channel distributionincorporated into the compartmental model. Typical basilar pyramidal cellshave somata of 10–25mm in diameter; a basilar dendritic trunk of 5–12mmin diameter extending 200–400mm distance before branching to form a distalbasilar bush (Maler 1979). A thick proximal apical dendritic shaft of;10 mmin diameter extends dorsally;200mm prior to branching in a molecular layer;the apical dendrites continue to branch over a distance of#800 mm. Theparticular pyramidal cell used for the model was reconstructed from a Lucifer-yellow filled cell (Berman et al. 1997). The distribution of ionic channels in thecore model is indicated in 3 boxes according to their location on the soma,proximal apical dendrite, or all dendrites, respectively. An enlargement of theproximal apical dendrite is shown on theleft to detail the placement of Na1

and K1 channels in separate proximal dendritic compartments over the regionfor backpropagation of dendritic Na1 spikes.

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The total model cell surface area is 65,706mm2, comparable to otherpyramidal cell models (Koch et al. 1995). Longer compartments arefurther subdivided to guarantee that no isopotential segment is oflength .25 mm, ensuring sufficient computational precision. Thisresults in a total spatial discretization of the full model cell into 312isopotential regions. For simplicity, the initial segment and soma arerepresented as a single compartment.

The proximal apical dendrite is modeled by ten connected com-partments (total length, 200mm) whose diameter decreases withdistance from the somatic compartment (initial diameter of 11.6 mm,final diameter of 6mm; see Fig. 1,inset). The dendrite then bifurcatesto give rise to daughter branches of 6mm diameter, which furtherbifurcate with an associated step decrease in diameter, extending atotal distance of;800 mm in an overlying molecular layer. Theproximal basilar dendrite is modeled as a single compartment of 7.4mm diameter and 194mm in length. The distal extent of the dendriteseparates into many compartments that form a bush-like pattern (seeFig. 1). The lengths of dendrites within the basilar bush range from 3to 56mm with diameters as thick as 4mm to as thin as 0.5mm. As theaxial resistivity (Ra) and membrane capacitance per unit area (Cm) arenot precisely known for ELL pyramidal cells,Ra is set to 250VcmandCm to 0.75mF/cm2, both realistic values for vertebrate neurons(Mainen and Sejnowski 1998). The model cell temperature is set to28°C, similar to the natural habitat of the fish.

Model equations

Each ionic current,Ix, is modeled as a modified Hodgkin/Huxleychannel (Hodgkin and Huxley 1952) governed by

I x 5 gmax,x z mxi hx

j z ~Vm 2 Erev,x! (1.1)

dmx

dt5

m`,x~Vm! 2 mx

tm,x

(1.2)

m`,x~Vm! 5 ~1 1 e2~Vm2V1/2,m,x!/km,x!21 (1.3)

hx is given by a similar description (equations omitted). Heregmax,x

is the maximal channel conductance of a generic channel x, whilemx

and hx are the activation and inactivation state variables, raised topowersi and j, respectively.m`,x is the steady state activation curve,andtm,x the activation time constant, both governing the evolution ofthe variablemx. The voltage dependence ofm`,x is given by a sigmoidrelationship (Eq. 1.3) determined by the half voltageV1/2,m,x and slopeparameterkm,x. The channel reversal potential isErev,x. To facilitateparameter fitting, our model follows Koch et al. (1995) in assumingtm,x is voltage independent.

Figure 1 introduces the specific ionic currents and indicates theirdistribution over the cell. Table 1 gives the parameters stated inEq.1.1–1.3for each current. In this study, “proximal dendrite” refers tothe initial 200mm of the apical dendrite, which is in most pyramidalcells a single nonbranching shaft over this distance. A detailed anal-ysis of basilar dendritic electrophysiology is not available, hence fewionic channels are localized to the basilar dendritic region. The ionicchannels (Ix) influence the potential of each compartment according to

Cm

dVm

dt5 O

x

Ix 1 Iapp (2)

where the sum is over all the ionic currentsIx present in a givencompartment as indicated in the distribution of Fig. 1.Iapp representsan externally applied current that is always constant in time andinjected into the somatic compartment for this study.

R E S U L T S

For each ionic channel incorporated into the model, we firstjustify the chosen distribution over the cell (Fig. 1) and identifythe influence of each conductance on cell membrane potential.Finally, the parameters are fit so that model performancematches known experimental recordings. In the case of insuf-ficient experimental evidence from pyramidal cells, valueswere taken from experimental or computational results re-ported for other cells.

Channel distribution and kinetics

PASSIVE CURRENTILEAK. I leak represents the classic leak chan-nel with voltage-independent conductance and is present in allcompartments. The channel density,gmax,leakis chosen to es-tablish the model input resistance as 76.1 MV and the passivemembrane time constant as 26.8 ms, similar to values mea-sured in intracellular recordings of pyramidal cells in vitro(Lemon and Turner 2000). The leak reversal potential,Erev,leak,is set to270 mV (Koch et al. 1995; Mainen et al. 1995; Rappet al. 1996). Due to the high-density of K1 currents (seefollowing text) the final resting membrane potential (RMP) ofthe core model is273.3 mV, a value within the range recordedin pyramidal cells in vitro (Berman and Maler 1998a; Turner etal. 1994).

TABLE 1. Model parameters

Channel gmax, S/cm2 i / j Erev, mV t, ms V1/2, mV k, mV

INa,s 1.8 2/1 40 0.2/0.6 240/245 3/235

IDr 0.7 2/0 288.5 0.39 240 3IKA 0.27 1/0 288.5 1 230 4IKB 0.015 1/0 288.5 2000 230 1IKV3.3 6.5: soma. 3/1 288.5 0.8/1.5 0/23 19/240

1: proximal apical dendrite.0.5: basal and distal apical dendrite.

INaP 0.0035: soma. 3 40 0.3 258.5 60.001: proximal apical dendrite. 3 40 0.3 258.5 6

INa,d 0.6 2/1 40 0.5/1 240/252 5/25IA 0.0012 1/1 288.5 10/100 275/285 4/22IDr,d 0.2 2/0 288.5 0.9 240 5ILeak 2 1025 N/A 270 N/A N/A N/A

Ionic channel parameters. All parameters correspond to the notation given inEqs. 1.1–1.3.If the channel is present, and of varying density, in multiplecompartments (see Fig. 1), the density for the specific compartments is indicated in column 2. If the channel is modeled with both activation and inactivationassociated entries will be double; the left is associated with activation parameters and the left with inactivation parameters.

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NA1 AND K1 CURRENTS. Previous work has established thedistribution pattern of immunolabel for both Na1 channels andan apteronotid homologue of the Kv3.3 K1 channel,AptKv3.3,over the dendritic-somatic axis of pyramidal cells (Rashid et al.2001; Turner et al. 1994). Electrophysiological studies havefurther mapped the site for Na1 spike initiation and conductionover the soma-dendritic axis (Lemon and Turner 2000; Turneret al. 1994) and identified the kinetic properties ofAptKv3.3K1 channels in both native pyramidal cells and when ex-pressed in a heterologous expression system (Rashid et al.2001). Although other K1 channels are incorporated into themodel, our existing knowledge of the distribution and proper-ties of Na1 andAptKv3.3 K1 channels are used when possibleto constrain our parameters and improve the representation ofaction potential waveforms. We begin by matching the distri-bution and kinetic properties ofAptKv3.3 channels to experi-mental data.

IAptKv3.3: high-voltage-activated K1 channel

AptKv3.3 K1 channels are distributed over ELL pyramidal cellsomata as well as apical and basal dendrites (Rashid et al. 2001).A dendritic distribution ofAptKv3.3 channels is unique in that allprevious studies on Kv3 channels have shown a distribution thatis restricted to the soma, axon, and presynaptic terminals (Perneyand Kaczmarek 1997; Sekirnjak et al. 1997; Weiser et al. 1995).Rashid et al. (2001) also showed thatAptKv3.3 K1 channelscontribute to spike repolarization in both somatic and apicaldendritic membranes. This role is particularly relevant in dendriticregions where a reduction inAptKv3.3 current enhances thesomatic DAP and lowers burst threshold.

Figure 2 shows the fit ofAptKv3.3 current in the model towhole cell recordings ofAptKv3.3 K1 current when expressedin human embryonic kidney (HEK) cells.AptKv3.3 channelsproduce an outwardly rectifying current that exhibits a high-threshold voltage for initial activation (more than220 mV;Fig. 2, A andB). Over a 100-ms time frame, step commandsproduce little inactivation ofAptKv3.3 current in the whole cellrecording configuration (Fig. 2A), although inactivation can berecorded over longer time frames (Rashid et al. 2001). Asindicated in Table 1, both activation (mAptKv3.3) and inactiva-tion (hAptKv3.3) curves are used to describeIAptKv3.3. The V1/2for mAptKv3.3 is set to 0 mV so as to produce the high-thresholdvoltage necessary for activation as shown in Fig. 2B. A shallowslope of activation,km,AptKv3.3, is required to produce a gradualrise of IAptKv3.3 with voltage.

As reported for Kv3 channels in some expression systems,AptKv3.3 current exhibits an early peak on initial activationfollowed by a relaxation to a lower amplitude current for theduration of a 100-ms step command (Fig. 2A). The origin ofthis early peak and relaxation is currently unknown, but it hasbeen proposed to reflect a transient accumulation of extracel-lular K1 when Kv3 channels are expressed at high-density inexpression systems (Rudy et al. 1999). Since this has not beenfirmly established, we incorporate inactivation kinetics(V1/2,h,AptKv3.3, kh,AptKv3.3, th,AptKv3.3) that allow the model toreproduce the early transient peak of current (Fig. 2A). Anadditional characteristic of Kv3 channels is a fast rate of bothactivation and deactivation (Fig. 2C). The activation time con-stant, tm,Kv3.3, is chosen to produce rates of activation anddeactivation that most closely match the experimental data

(Fig. 2C). BecauseAptKv3.3 channels are found with highprevalence in the soma and dendrites, they are incorporatedwith the above kinetics over the entire axis of the model cell(Fig. 1). The conductance level is adjusted accordingly to thefit to experimental data (see following text).

Action potential discharge in ELL pyramidal cells

We first treat somatic and dendritic spike discharge sepa-rately to determine the necessary descriptions of the model

FIG. 2. AptKv3.3 K1 channel kinetics.A: whole cell K1 currents recordedin tsA 201 HEK cells transiently transfected withAptKv3.3 K1 channels (left).Shown are currents activated by step commands in 10-mV increments from aninitial prepotential of290 mV. ModelAptKv3.3 currents (right) in response toan equivalent voltage-clamp protocol as the experimental recordings inA,restricted to the somatic compartment for space-clamp considerations. Simu-latedAptKv3.3 whole cell currents shown here and inC have calibration barsnormalized to the current evoked at 30 mV. We normalize the simulationresults because we do not wish to quantitatively compare the simulation to theHEK cell experiments.B: normalized current-voltage plots for both experi-mental and modelAptKv3.3 currents in response to the voltage-clamp protocolshown inA. A high initial voltage for activation ofAptKv3.3 current is evidentin that whole cell current becomes significant only for voltage steps above220mV. C: time expanded plots of the beginning and end ofAptKv3.3 whole cellcurrents in response to the voltage steps shown inA. This shows both the fastactivation and deactivation ofAptKv3.3 current. See Rashid et al. (2001) for adescription of experimental methods.

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Na1 and K1 channels required to produce action potentialwaveforms that match in vitro recordings. Next we focus onthe soma-dendritic interaction that gives rise to the DAP thatdrives burst discharge.

INa,s, IDr,s, and IAptKv3.3: somatic spike

Figure 3A illustrates an action potential recorded in the somaof pyramidal cells in an in vitro slice preparation (see alsoTable 2 for quantitative comparison of model and experimentalsomatic action potential features). As we are interested in firstmodeling the somatic spike in the absence of backpropagatingspikes, the recording in Fig. 3A was obtained after TTX had

been locally applied to the entire dendritic tree. This effectivelyblocks all spike backpropagation and selectively removes theDAP at the soma (Lemon and Turner 2000). The somatic spikein pyramidal cells is of large amplitude (typically reaching20-mV peak voltage) and very narrow half-width (width athalf-maximal amplitude) of;0.45 ms (Table 2) (Berman andMaler 1998a; Turner et al. 1994). In the absence of a DAP, afast afterhyperpolarization (fAHP) that follows the somaticspike is readily apparent (Fig. 3A). A slow Ca21-sensitive AHP(sAHP) follows the fAHP but does not contribute directly tothe soma-dendritic interaction that underlies burst discharge(Lemon and Turner 2000).

ELL pyramidal cell somatic spikes are generated by TTX-sensitive Na1 channels (Mathieson and Maler 1988; Turner etal. 1994), which are modeled here as a fast activating andinactivating current,INa,s. Rashid et al. (2001) established thatKv3 channels contribute to repolarizing the somatic spike inpyramidal cells. However, local blockade of somatic Kv3channels using focal ejections of TEA in vitro only blocks afraction of the total repolarization, indicating the additionalinvolvement of other voltage-dependent K1 channels. Pyrami-dal cells are known to express large conductance (BK) Ca21-activated K1 channels in both somatic and dendritic mem-

FIG. 3. Fit of somatic and dendritic spikes to experimental data.A: in vitro recording of an electrosensory lateral line lobe (ELL)pyramidal cell somatic action potential under conditions of complete block of all spike backpropagation by dendritic TTXapplication (Lemon and Turner 2000). This serves to selectively remove the depolarizing afterpotential (DAP) from the somaticrecording, unmasking a large fast afterhyperpolarization (fAHP).B and C: superimposed simulations comparing the effects ofvarious combinations of K1 currents on the rate of somatic spike repolarization. All simulations have KA and KB channels presentat the soma and Na,d channels have been removed to block backpropagation. All action potentials are produced via a square wavestep depolarization forIapp at the soma (amplitude of 0.3 nA, 0.5 ms in duration).B: insertion of IDr,s at the soma results in asubstantial increase in the rate of spike repolarization with an associated decrease in spike height (trace 1 vs. trace 2).C: a seriesof superimposed simulations to illustrate the effects of combiningIDr,s andIAptKv3.3 on spike repolarization. The results show thatIAptKv3.3 alone affects primarily the initial falling phase of the spike but only a fraction of the total repolarization (trace 1 vs. 2).The best fit is obtained by combiningIDr,s andIAptKv3.3 (trace 4). Trace 3 indicates that subsequent removal ofIAptKv3.3 at the somahas a moderate effect on spike repolarization but little effect on spike amplitude.D: in vitro recording of an ELL pyramidal celldendritic spike (;150mm). Note the slower rate of rise, lower amplitude, and slower rate of repolarization as compared with thesomatic spike inA. E andF: superimposed simulations comparing the effects of various combinations of K1 currents on the rateof dendritic spike repolarization.E: with no dendritic K1 channels (trace 1), the dendritic spike has a long duration with therepolarizing phase interrupted by 2 small positive-going potentials reflecting the passive conduction of spikes generated at the soma(*). Trace 2 shows thatIDr,d can account for a large extent of the repolarizing phase of dendritic spikes.F: a comparison of the rolefor IAptKv3.3 andIDr,d in combination on dendritic spike repolarization.IAptKv3.3 can account for a proportion of spike repolarization(trace 1 vs. 2), but the best fit is obtained when bothIDr,d and IAptKv3.3 are present (trace 4). Removal ofIAptKv3.3 reveals a finalrole in adjusting both spike amplitude and repolarization (trace 3). Experimental methods for the recordings inA andD are as inLemon and Turner (2000). Calibration bars inB andE also apply toC andF, respectively.

TABLE 2. Somatic action potential; model and experiment

Action Potential Features Pyramidal Cell Soma Model Result

Half-width, ms 0.4546 0.159 0.585Rise time, ms 0.2996 0.278 0.125Decay time, ms 0.5156 0.3892 0.800Amplitude, mV 80.96 10.3 74.3

Comparison of model and experimental somatic spike features. All experi-mental results are taken from in vitro analysis presented in Berman and Maler(1998a). Values are means6 SD.

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branes (E. Morales and R. W. Turner, unpublished observa-tions), but they do not appear to contribute to spike repolar-ization (Noonan et al. 2000; Rashid et al. 2001). A largefraction of the somatic spike repolarization is likely mediatedby a small conductance K1 channel (,10 pS) found with highprevalence in patch recordings (Morales and Turner, unpub-lished observations). Although the kinetic properties of thischannel have not been established, whole cell currents inpyramidal cell somata indicate that it will have propertiesconsistent with a fast activating and deactivating delayed rec-tifier-like (Dr) current, referred to here asIDr,s. We thereforemodel somatic spike depolarization and repolarization using acombination ofINa,s, IDr,s, andIAptKv3.3.

INa,s and IDr,s are confined to the somatic compartment,which includes both the initial segment and somatic mem-brane, and represents the site for Na1 spike initiation in the cell(Turner et al. 1994). The activation (bothINa,s and IDr,s) andinactivation (only INa,s) steady-state conductance curve halfvoltages,V1/2, and slopes,k, are similar to values used inprevious compartmental models (Koch et al. 1995; Mainen etal. 1995; Rapp et al. 1996). Spike threshold occurs atVthres.260 mV, a value close to that reported for pyramidal cell spikethreshold in vitro (Berman and Maler 1998a).

Previous compartmental models of mammalian neurons typ-ically incorporate action potential half-widths of 1–2 ms (deSchutter and Bower 1994; Koch et al. 1995; Mainen et al.1995; Rapp et al. 1996). Given the comparatively narrowhalf-width of ELL pyramidal cell spikes (Table 2), we chooserelatively shortINa,s and IDr,s time constants of activation andinactivation,tm and th (Table 1), andgmax,Na,s is set to 1.8S/cm2. To compensate the Na1 conductance and achieveproper repolarization,gmax,Dr,s is set to a value of 0.7 S/cm2.We note that the model density of Na, s and Dr, s channels arecomparable to that used in the spike initiation zone of othercompartmental models (de Schutter and Bower 1994; Koch etal. 1995; Mainen et al. 1995; Rapp et al. 1996). Figure 3Bshows somatic action potentials withINa,s alone (trace 1) andboth INa,s and IDr,s (trace 2) inserted in the model. The reduc-tion in spike half-width and amplitude whenIDr,s is presentallows the model somatic spike to better approximate experi-mental recordings (compare Fig. 3,A andB).

In considering the role ofAptKv3.3 K1 current, we find thatAptKv3.3 channels alone could account for only a fraction ofthe total somatic spike repolarization. Figure 3C compares thesomatic action potential with noIDr,s or IAptKv3.3-mediatedrepolarization (trace 1) to one with onlyIAptKv3.3 current (trace2). A reduction of spike width byIAptKv3.3 during the initialfalling phase of the action potential with a minimal reductionin peak voltage matches the results of earlier modeling studiesthat considered the role of Kv3.1 K1 channels in spike repo-larization (Perney and Kaczmarek 1997; Wang et al. 1998). Asshown in Fig. 3C (trace 4), spike repolarization is most closelymodeled whenIDr,s and IAptKv3.3 are both present at the soma,with AptKv3.3 conductance set to 6.5 S/cm2 (Rashid et al.2001). RemovingAptKv3.3 conductance slightly reduces therate of spike repolarization without significantly affecting ac-tion potential amplitude. Such an effect on spike repolarizationby AptKv3.3 is also consistent with pharmacological tests invitro (Rashid et al. 2001).

INa,d, IDr,d, and IAptKv3.3: dendritic spike

Many central neurons are known to be capable of activelybackpropagating Na1 spikes from the soma and over themajority of the dendritic tree (Stuart et al. 1997b; Turner et al.1991, 1994; Williams and Stuart 1999; see Ha¨usser et al. 2000for review). Patch-clamp recordings in hippocampal pyramidalcells further indicate that Na1 channels are distributed with arelatively uniform density over the entire dendritic tree (Mageeand Johnston 1995); a distribution that has been used in othermodeling studies (Mainen et al. 1995; Rapp et al. 1996). Wehave previously determined that Na1 channel immunolabel inELL pyramidal cells is uniformly distributed in the cell bodyregion but exhibits a distinct punctate distribution over theproximal 200mm of apical dendrites (Turner et al. 1994). Thisdistribution correlated with the distance over which TTX-sensitive spike discharge was recorded, suggesting that theimmunolabel correspond to Na1 channels involved in spikegeneration. Electrophysiological recordings further establishedthat Na1 spikes are initiated in the somatic region but back-propagate over only;200 mm of the dendritic tree that canextend as far as 800mm (Maler 1979; Turner et al. 1994). Inthis respect, spike backpropagation in ELL pyramidal celldendrites falls between the active conduction of Na1 spikesseen over the entire axis of cortical neuron dendrites and thepassive decay of Na1 spikes seen over the proximal dendritesof cerebellar Purkinje cells (Ha¨usser et al. 2000; Hoffman et al.1997; Stuart and Ha¨usser 1994; Stuart et al. 1997a).

Figure 3D shows an ELL pyramidal cell dendritic spikerecorded;150 mm from the soma. The dendritic spike half-width is substantially larger than that of the somatic spike, withthe total duration approaching as much as 12 ms in recordings;200 mm from the soma. Recordings in the slice preparationfurther indicate that even proximal dendritic spikes (50mm)exhibit a sharp decrease in the rate of rise and rate of repolar-ization with respect to somatic spikes. This difference in rate ofrepolarization leads to a substantial delay in the peak latency ofdendritic versus somatic spikes with a rapid increase in thisdifference beyond;100 mm (Turner et al. 1994). Modelingthis rapid change in spike characteristics in proximal dendritesrequires a specific set of parameters for both dendritic Na1 andK1 conductances.

Given the immunocytochemical data (Turner et al. 1994),the distribution of dendritic Na1 channels is confined to fivepunctate zones of 5mm in length (200mm2 area) along theproximal apical dendrite that are assigned a high Na1 channeldensity (Fig. 1). The distance between active dendritic zonesalso increases with distance from the soma. Specifically, activeINa,d zones 1–3 are separated by 15mm of passive dendrite,while zones 3–5 are separated by 60mm of passive dendrite.Lacking a cytochemical localization ofIDr,d K1 channels indendrites, we co-localized these channels to the five activedendritic zones. As stated in the preceding text,AptKv3.3channels are incorporated over the entire soma-dendritic axis,although with varying levels of conductance for somatic, prox-imal apical dendritic, and distal apical and basal dendriticcompartments.

We first attempted to equate the kinetic parameters of Na, dand Dr, d channels to those of somatic Na, s and Dr, s,respectively, but with reduced channel densities to produce alower amplitude dendritic spike, as done in previous studies

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(Mainen et al. 1995; Traub et al. 1994). However, this led todendritic spikes with too short of a half-width and delay timeto peak incompatible with intracellular dendritic recordings. Afirst attempt at correcting this discrepancy was raising themodel axial resistance, thereby reducing the passive cablepropagation of the spike along the dendrite. However, to obtainmodel agreement with experimental data,Ra had to be setoutside acceptable norms by a factor of 10 (Mainen and Sej-nowski 1998). This approach was problematic and hence aban-doned.

Recently differences in ion channel kinetics have been ob-served for dendritic Na1 and K1 channels as compared withthose at the soma, indicating precedence for applying differ-ential kinetic properties to somatic versus dendritic ion chan-nels (Colbert et al. 1997; Hoffman et al. 1997; Jung et al.1997). Since a similar level of analysis is not yet available forELL pyramidal cells, we explored a broad range of kineticparameters describing bothINa,d and IDr,d. More successfulmodeling of experimental results is achieved by increasing thesteady-state conductance curve slope,k, and time constant,t,for both the activation and inactivation ofINa,d and IDr,d ascompared with their somatic counterparts. Simulations usingonly INa,d and IDr,d then begin to reproduce the delay indendritic spike peak as well as the increase in spike half-width,the longer rate of rise, and the slower rate of repolarizationwhen compared with the somatic spike (Fig. 3B). By compar-ison, modeling the dendritic spike with onlyINa,dandIAptKv3.3,does not achieve a sufficient rate of spike repolarization (Fig.3F, trace 1 vs. 2). By combiningIDr,d with IAptKv3.3 set to adensity of 1 S/cm2 in the proximal apical dendrite and a lowerdensity of 0.5 S/cm2 in other dendritic compartments a good fitto experimental data is obtained (cf. Fig. 3,D andF, trace 4).

Note that incorporation ofAptKv3.3 current also reduces den-dritic spike amplitude (Fig. 3F; trace 4). This result is consis-tent with experimental data indicating a slight increase indendritic spike amplitude following local ejections of 1 mMTEA to block dendriticAptKv3 channels (Rashid et al. 2001).

Soma-dendritic interactions underlying the DAP

Figure 4 illustrates the effects of combining active somaticand dendritic compartments on spike waveforms. Previousintracellular recordings have indicated substantial differencesin the duration and peak response of somatic versus dendriticspike waveforms (Fig. 4A). Both experimental recordings inFig. 4A were obtained in intact slices with full backpropagationof spikes into dendrites. Under these conditions, the somaticspike is followed by a clear DAP that superimposes on thesomatic fAHP (Fig. 4A). The DAP is due to the dendritic fastsodium currents,INa,d, which boost the backpropagating actionpotential to elevated voltages in the dendrite, and promotesreturn electrotonic current flow during the longer durationdendritic spike (Turner et al. 1994). When the model includesactive spike discharge in both somatic and dendritic compart-ments, it successfully reproduces a DAP at the soma that isoffset by the fAHP (Fig. 4B). If the gmax of all INa,d currents isset to zero, in effect removing the active zones from the apicaldendrite, the influence of the DAP at the soma is lost, uncov-ering a clear fAHP at the soma. Only a low-amplitude passivereflection of the somatic spike occurs in the apical dendrites(Fig. 4C), as recorded in vitro when TTX is focally applied tothe dendrites to block active spike backpropagation (Turner etal. 1994).

FIG. 4. Generation of a DAP through spike backpropaga-tion. A: superimposed single spike discharge recorded in sepa-rate intracellular somatic and dendritic impalements (;200mmfrom the soma), aligned temporally at the onset of spike dis-charge (Lemon and Turner 2000). The somatic DAP generatedby the backpropagating spike is enlarged within theinset.Thescale bars shown inA also apply toB andC. B: superimposedsomatic and dendritic spike simulations (measured 200mmfrom the soma) evoked by depolarizing somatic current injec-tion. The somatic DAP is enlarged within the insert for com-parison withA. Simulations inB andC also includeIKA andIKB

at the soma that underlies a late slow AHP (sAHP; not shown).C: superimposed model somatic and dendritic spike responsesto somatic current injection with allINa,d removed from thedendritic compartments. The somatic DAP is removed, uncov-ering the prominent fAHP. Note that only a small passiveresponse is reflected into apical dendrites on the discharge ofsomatic spikes in the absence ofINa,d. D: Plot of the peakvoltage as a function of distance from the soma (0mm) afterdischarging a somatic spike, with and without active dendriticINa,d zones. The location of active sites in the proximal den-dritic region are indicatedabovethe abscissa. The slight devi-ation from expected passive decay in the proximal dendriticregion is due to a reduction in the diameter of the modelproximal dendritic shaft.

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Spike backpropagation and refractory period

Figure 4D plots how the fitted kinetics ofINa,d and IDr,daffect the backpropagation of action potentials initiated at thesoma. Shown is the peak voltage of the response measured ata select number of compartments for both active propagation(gmax,Na,d 5 0.6 S/cm2) and passive electronic conduction(gmax,Na,d 5 0 S/cm2) in the dendritic compartments. In theproximal apical dendrite (,200mm), the five activeINa,d sitesincorporated into the model boost the peak of the backpropa-gating action potential over that measured during passive con-duction. In the mid-distal dendrite (.200 mm), the peak volt-age decays exponentially in both cases because no activeinward currents are believed to boost the voltage beyond thisdistance (Turner et al. 1994). This decrease in spike amplitudenear 200mm also qualitatively mimics the properties of back-propagating spikes as measured through laminar profile fieldpotential analysis (Turner et al. 1994).

To test the accuracy of fit of channel parameters forINa,s,INa,d, IDr,s, andIDr,d, we measure the refractory period of bothsomatic and dendritic spikes (Fig. 5,A andB). The simulationprotocol is equivalent to the condition-test (C-T) interval ex-periment used by Lemon and Turner (2000). In the model, thisconsists of first injecting a brief somatic current pulse sufficientto induce a single somatic spike that backpropagates into thedendrites. A second identical current pulse is then applied at avariable time interval following the first pulse to identifyrelative and absolute refractory periods. At sufficiently longtest intervals (Fig. 5,A andB, 8 ms) the second pulse evokesfull somatic and dendritic spikes, both identical to the condi-tion responses. At the soma, a reduction in the C-T interval to;4 ms results in a select blockade of the DAP without signif-icant effect on the somatic spike (Fig. 5A). Somatic spikeamplitude remains essentially stable for C-T intervals above2.5 ms, although C-T intervals below this slightly reduce spikeamplitude, given thatINa,s has not completely recovered frominactivation. An absolute somatic refractory period is evidentat a C-T interval of 1.5 ms (Fig. 5A). The effects of a similarseries of C-T intervals monitored 200mm from the somareveals a relative refractory period for dendritic spikes between4 and 6 ms, as reflected by a gradual decline in spike ampli-tude. This reduction in amplitude ends at C-T intervals be-tween 2.0 and 3.0 ms (Fig. 5B), with subsequent failure of thesmall potential evoked at C-T intervals of;1.5 ms.

Each of these properties qualitatively match experimentalresults in pyramidal cells of an absolute somatic refractoryperiod at;2 ms, a relative dendritic refractory period between5 and 7 ms and an absolute dendritic refractory period between3 and 4 ms (Lemon and Turner 2000). Furthermore a selectiveblock of the somatic DAP over the same range of C-T intervalsthat reduce dendritic spike amplitude is also a characteristicobserved in intact pyramidal cells (Turner et al. 1994).

INaP—determining RMP, nonlinear EPSP boosting, andlatency to first spike shifts

A TTX-sensitive and persistent Na1 current (INaP) (Frenchet al. 1990) has been recorded at both the somatic and dendriticlevel of ELL pyramidal cells (Berman et al. 2001; Turner et al.1994). Blocking this current with focal ejections of TTX invitro results in a clear hyperpolarizing shift in cell resting

membrane potential (RMP) and an increase in the latency todischarge a spike at the soma (Turner et al. 1994). There is alsoa prominent voltage-dependent late component to the dendriticand somatic excitatory postsynaptic potential (EPSP) evokedby stimulation of descending tractus stratum fibrosum (tSF)

FIG. 5. Refractory period of model somatic and dendritic spikes. Spikedischarge was evoked with a short-duration depolarization at the soma and acondition-test (C-T) pulse pair presented at varying intervals to determine ifthe model reproduced the observed differential refractory period betweensomatic and dendritic regions of pyramidal cells.A: several superimposed C-Tstimulus pairs at the soma. The C-T interval is indicated below the traces(arrows). The absolute refractory period of the model somatic spike occurs at1.5 ms. Note that the somatic DAP is also selectively blocked at a C-T intervalof 4 ms (slanted arrows) with no significant change in the somatic spike.B:superimposed C-T stimuli monitored at 200mm in the dendrite. The dendriticspike exhibits a relative refractory period beginning at;6 ms as shown by adecline in spike amplitude at progressively shorter C-T intervals. An absoluterefractory period at 3 ms is indicated by the final presence of only a passivelyreflected response associated with the somatic spike. Each of these propertiesaccurately reflect the conditions found in ELL pyramidal cells in vitro.

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inputs that terminate in the proximal dendritic region (Bermanand Maler 1998c; Berman et al. 1997). Recent work indicatesthat focal ejection of TTX at the soma selectively blocks thislate component of the tSF-evoked EPSP (Berman et al. 2001).Each of these results identifies an important contribution byINaP in determining the resting membrane potential (RMP) ofthe cell, in boosting the tSF synaptic depolarization, and insetting the latency to spike discharge. We determine the modelNaP channel parameters through a detailed match to theseconstraints.

NaP channels are modeled in both the soma and proximalapical dendrite as suggested by experimental and modelingstudies of ELL pyramidal cells (Fig. 1) (Berman and Maler1998c; Turner et al. 1994). In the absence of definitive knowl-edge of the distribution of dendritic NaP channels, a uniformdistribution of NaP channels over the entire proximal apicaldendrite is chosen. Local TTX applications in vitro producequantitatively similar shifts in RMP at both the somatic anddendritic level of pyramidal cells (Turner et al. 1994). Wehence choose a relative factor of 3.5 for the ratio of totalsomatic/dendritic NaP channel density to account for thegreater proximal apical dendritic surface area. This producesapproximately equal netINaP current from both soma anddendrite in response to similar depolarizations.

To set the NaP channel activation parameters,V1/2 andk, weconsider its effects on the model cell RMP. To do so, we applya small constant depolarizing somatic current injection to themodel (Iapp , 100 pA), which is insufficient to cause spiking.After a transient period of depolarization, the model membranevoltage settles to a steady-state value, which we label as the“effective” RMP. The shape of the increase in RMP asIappincreases will be determined by the specific NaP distribution.Figure 6A plots this final equilibrium somatic voltage (t .200ms) as a function ofIappwith NaP present in various compart-ments (see figure legend for description). For small currents(Iapp, 50 pA), the rise in RMP is linear (Ohm’s law) becauseat these potentials (Vm , 270 mV) NaP is not significantlyactivated. However, for larger applied currents (Iapp . 60 pA)yielding higher RMP values (Vm . 270 mV), a significantnonlinear RMP increase occurs when NaP is present (trace 1,Fig. 6A). This effect requires a steep subthreshold voltagedependency of activation that forces the half activation ofINaP,V1/2,m,NaP, to be set to258.5 mV and the activation slopeparameter,km,NaP, to be low, 6 mV. This agrees with the

FIG. 6. The effects of somatic or dendriticINaP on resting membranepotential (RMP), excitatory postsynaptic potential (EPSP) amplitude, andspike latency.A: RMP in the model cell plotted as a function of a series ofsubthreshold depolarizing current injections (Iapp) 200 ms after the onset of thedepolarization, at which point dVsoma/dt ' 0. Several superimposed plots areshown labeled according to the model conditions: control (all parameters are asgiven in Table 1), somaticgNaP5 0, dendriticgNaP5 0, and both somatic anddendritic gNaP 5 0 (see figure legend). Note the linear displacement ofpotential with noINaP present (trace 4), but a nonlinear increase in membranevoltage with current injection whenINaP is present in both somatic andproximal dendritic compartments (trace 1).B: model somatic responses to anevoked EPSP in the proximal dendritic region (200mm from soma; alphafunction,t 5 1.5 ms,gmax 5 16 nA) for the NaP conditions as labeled inA.Inset: pyramidal cell somatic EPSPs evoked by tractus stratum fibrosum (tSF)inputs to the proximal dendrites before and after focal somatic TTX ejection toblock INaP(Berman et al. 2001).C: latency to 1st spike is plotted as a functionof suprathreshold depolarizations for the NaP conditions shown inA. Trace 1best approximates previous experimental latency measurements from ELLpyramidal cells (Berman and Maler 1998a).

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experimental effects of TTX application on pyramidal cellRMP in vitro (Berman and Maler 1998c; Turner et al. 1994)and with experimentally determinedV1/2 (256.92 mV) andk(9.09) voltages for NaP in rat and cat thalamocortical neurons(Parri and Crunelli 1998). The quantitative agreement withexperiment is achieved by adjustinggmax,NaP. However, sincef-I characteristics (see Fig. 8) are compromised with a signif-icant modification ofgmax,NaP, the fit is balanced to produce arealistic RMP, and properf-I relationship for the model cell.

As illustrated in Fig. 6A, restricting the NaP distribution toeither somatic or dendritic membranes (traces 2 and 3) revealsthe nonlinear effect of NaP density in determining the RMP(for Iapp. 80 pA the difference between traces 1 and 4 is largerthan the sum of the differences between traces 2 and 3 with 4).The nonlinear effect will be treated when we analyze the roleof NaP in EPSP boosting (see following text). The near exactquantitative agreement of traces 2 and 3 in Fig. 6A indicate thatthe effect of NaP on the RMP is not site-specific between somaand proximal dendrite. This is expected since the net NaPcurrents of both somatic and dendritic membranes are set to beequal (see preceding text). The parameters that model thesteady-state characterization of the NaP,V1/2,m,NaP, andkm,NaP,are now set; however, the dynamics of theINaP current, deter-mined bytm,NaP, remains to be addressed.

INaP currents have been shown to boost the amplitude ofsubthreshold EPSPs in cortical pyramidal cells (Andreasen andLambert 1999; Lipowsky et al. 1996; Schwindt and Crill 1995;Stafstrom et al. 1985; Stuart and Sakmann 1995); there isrecent evidence in ELL pyramidal cells for a similar effect(Berman and Maler 1998c; Berman et al. 2001). To set the timeconstant of NaP activation,tm,NaP, we consider the EPSPboosting properties of the modelINaP. The inset in Fig. 6Bshows somatic recordings of a distally evoked EPSP undercontrol and somatic TTX conditions. The boost provided byTTX-sensitive currents begins with the fast rising phase of theEPSP, suggesting that NaP channels activate quickly (Fig. 6B;compare theinsetcontrol trace to the TTX trace). To match thedata, the modeltm,NaPis set to a small value (t 5 0.3 ms). Thisfast activation coincides with the treatment ofINaP in otherionic models (Lipowsky et al. 1996; Wang 1999). Figure 6Bshows the model somatic voltage response due to dendriticEPSP activation under the same various NaP distributionsconsidered in Fig. 6A. The EPSP boost observed with alldistributions shows that the subthreshold boost of the EPSP byINaP is substantial and that indeed the boost begins with therising edge of the potential, thereby matching experimentaldata.

The nonlinear nature ofINaP boosting of EPSPs is apparentin Fig. 6B when we compare EPSP amplitudes when NaP ispresent in both the somatic and proximal apical dendriticcompartments (trace 1) to NaP distributions in only in thesomatic (trace 2) or dendritic (trace 3) compartments. Figure6B shows approximately a 3-mV boost of the model EPSPamplitude when comparing the control case to complete NaPremoval (trace 1 as compared with 4). However, with onlysomatic or dendritic NaP distribution (trace 2 or 3), a boost of,1 mV at the peak of the EPSP is observed. A linear increasein the degree of EPSP enhancement would require that theeffects of traces 2 and 3 summate algebraically to produce trace1. The nonlinear boosting is a result of the steep sigmoidalvoltage activation of NaP (plot not shown).

It has been experimentally shown in thalamocortical neuronsthat INaP activation reduces the latency to spike in response todepolarizing current (Parri and Crunelli 1998). We hence fur-ther test the fit of NaP parameters by considering the modelcell’s latency to discharge a spike on membrane depolarization.Figure 6C plots the model latency to first spike as a function ofapplied somatic suprathreshold depolarizing current under allfour NaP conditions described in Fig. 6 (A andB). All tracesshow that just above the rheobase current (minimum currentrequired to induce spiking) the latency to first spike is long, yetas the input current is increased, the latency decays to shortervalues. Comparing the various NaP conditions reveals thatincreasingINaPreduces the rheobase current of the cell, with afull 0.71-nA shift occurring between the condition in which allsomatic and dendritic NaP removed (trace 4) as compared withthe condition in whichINaPis distributed over the both somaticand dendritic membranes (trace 1). As a result of the rheobaseshift, Fig. 6C shows the latency to first spike atIapp5 0.81 nAwith all NaP removed to be 62.4 ms as compared with thecontrol latency of 4.3 ms at the same current.

Figure 6C also indicates that the nonlinear effects ofINaPalready seen on the RMP and EPSP amplitude are even moredramatic on both the rheobase current and latency shift. Partialremoval of NaP (traces 2 and 3) only shifts the rheobase andthe latency to first spike by slight amounts from the controlcase (trace 1) when compared with the dramatic shifts observedwith a block of both somatic and dendriticINaP (trace 4). Thequantitative agreement between latency shifts observed wheneither somatic or dendritic NaP are removed shows that thereis no spatial differences of NaP effects on rheobase current orlatency as expected by the spatially balanced net ionic current.

Finally, we compare the model spike latency characteristicswith experimental measurements from ELL pyramidal celllatency to spike (Berman and Maler 1998b). In ELL pyramidalcells, the transition from long to short latencies to spike doesindeed occur over a small depolarizing current interval (;0.1nA), as in the model control case (trace 1, Fig. 6C). The modelresults show a power law decay (spike latency;1/=Iapp2Irheobase). However, experimental work indicates astep decline from long latencies (.20 ms) to a constant latency(;4 ms) for large depolarization (Berman and Maler 1998b).This could not be fit with a power law as in the model results.The model latency decay occurred with allINaPparameter setsexplored; however, the given fitted parameters (Table 1) pro-duced the best approximation to experimental latency shifts.The discrepancy between the experimental and model latencydecay is presently unexplained.

IA—latency to first spike from hyperpolarized potentials

Previous studies of ELL pyramidal cell electrophysiology(Berman and Maler 1998b; Mathieson and Maler 1988) havesuggested the existence of a transient outward current similarto IA (Connor and Stevens 1971). Both studies observed anincrease in spike latency if a depolarization was preceded bymembrane hyperpolarization, an effect previously attributed tothe activity of anIA current (Connor and Stevens 1971; Mc-Cormick 1991). At this time, patch-clamp recordings have notconclusively shown the possible contribution of anIA currentat the somatic or dendritic level of pyramidal cells (Turner,unpublished observations). Our attempts to simulate the exper-

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iments performed in Mathieson and Maler (1988) now showthat to reproduce the observed effects of membrane potentialon the latency to spike discharge, a low density ofIA-like K1

current must be introduced into the model cell.Because patch-clamp recordings in pyramidal cells have not

yet isolated transiently activating and inactivating channelsconsistent with a traditional A-type K1 current, the choice ofchannel distribution over the model cell must be postulated. Aprevious compartmental model has shown that altering thedistribution of IA channels over a cell gives only small devia-tions in observedIA character, with the exception of cells withlarge axial resistance (;2,000Vcm) (Sanchez et al. 1998). Ourmodel neuron axial resistance (250Vcm) is much lower, andhence any chosen channel distribution should not significantlyaffect cell output. Considering this result, we distributeIAuniformly over the whole cell (Fig. 1).

The half voltages for the steady-state conductance curves forIA currents are set toV1/2,m,A 5 275 mV andV1/2,h,A 5 285mV and the curve slope factors tokm,A 5 4 mV andkh,A 5 22mV. Other models ofIA (Huguenard and McCormick 1992;Johnston and Miao-Sin Wu 1997; Sanchez et al. 1998) showIAwindow currents at more depolarized levels ranging over alarger voltage interval. However, in the absence of definitiveIAcharacterization in ELL pyramidal cells, we choose to fit theactivation/inactivation parameters to ensure thatIA does notaffect the model RMP yet that inactivation could be removed

with moderate hyperpolarization (in correspondence with theresults of Mathieson and Maler 1988).tm,A is set to 10 ms tohave sufficiently rapid A-type K1 channel activation (Johnstonand Miao-Sin Wu 1997). However, the time constant ofIA

inactivation, th,A, is chosen to be 100 ms to produce anappropriate latency to first spike and a subsequent transientincrease in spike frequency over the first 200 ms (Doiron et al.2001).

Figure 7A shows the model cell response withoutIA present(gmax,IA 5 0) to a depolarizing somatic current injection of 0.6nA. The model cell is at a resting potential of273 mV prior tothe stimulus. A repetitive firing pattern results with a latency tofirst spike of 6.05 ms (stimulus onset is att 5 0) and ISI of 9.5ms. Figure 7B repeats this simulation but with a 50-ms pre-stimulus hyperpolarization of211 mV induced via20.2-nAcurrent injection. Again, repetitive firing occurs, yet with aslightly longer latency to first spike of 12.1 ms followed by arepeating ISI of;9.5 ms. These results qualitatively matchthose obtained by Mathieson and Maler (1988) when 1 mM4-aminopyridine (4-AP, a knownIA channel blocker) was bathapplied to ELL pyramidal cells in vitro. However, under con-trol conditions, a substantially longer latency to first spike wasobserved in ELL recordings when a prestimulus hyperpolar-ization preceded depolarization (Mathieson and Maler 1988).Furthermore when depolarizing current is applied from a hy-perpolarized level, the first ISI is long and then subsequent ISIs

FIG. 7. The effects ofIA on latency to spike dis-charge. All panels show the model response to a partic-ular applied current protocol as indicated below eachsimulation. Each panel marks theIA channel conduc-tance used in the simulation and the value of the mem-brane potential at the time of depolarization.A andB:core model response in the absence ofIA to equivalentdepolarizations from a resting potential of273 mV (A)or following a prestimulus hyperpolarization to284mV (B). C and D: the model response to a similarcurrent injection protocol whenIA is inserted into themodel. At 273 mV resting potential (C) IA is inacti-vated, leading to an equivalent response to that shown inA. Note that depolarizing the cell following a prestimu-lus hyperpolarization (D) results in a substantiallylonger latency to spike whenIA is present (cf.B andD).The set of calibration bars shown inC applies to allpanels.

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slowly shorten (an increase in cell firing rate) in the first 200ms of the stimulus, presumably due toIA inactivation.

Figure 7,C andD, illustrates simulation protocols identicalto those shown inA andB, respectively, yet withIA channelsdistributed across the model cell.gmax,IA is fit to match theexperimental findings of Matheison and Maler (1988) with afinal assigned value ofgmax,IA 5 1.2 mS/cm2. At a restingpotential of 273 mV, IA is substantially inactivated so thatdepolarizing the somatic compartment via a 0.6-nA currentinjection results in a latency to first spike and subsequent ISIshift that is nearly identical to the results obtained whenIA isnot incorporated into the model (cf. Fig. 7,A andC). However,with a prestimulus hyperpolarization, the same net currentinjection produces a latency to first spike of 78.5 ms, anincrease of 66.3 ms as compared with spike discharge withouta prestimulus hyperpolarization (Fig. 7D). In addition, in thepresence ofIA the ISI begins at 14.4 ms and reduces to 10 msafter 200 ms of depolarization, successfully reproducing thegradual increase in spike frequency observed in pyramidal cells(not shown). Smaller depolarizations (Iapp, 0.6 nA) followingequivalent prestimulus hyperpolarizations (Iapp 5 20.2 nA)induce larger differences in latencies to first spike for caseswithout IA. Similarly, larger depolarizations (Iapp . 0.6 nA)give smaller latency shifts. These results are also qualitativelysimilar to the ELL experimental results of Matheison andMaler (1988).

Sincegmax,x represents the maximal channel conductance ofcurrentIx in the compartment of interest, it provides an indirectvalue for the expected channel density of the channel in thatcompartment. It is therefore interesting to note that comparingthe model currentsIAptKv3.3 to IA shows thatgmax,AptKv3.3 isthree orders of magnitude larger thangmax,A when spike dis-charge properties are properly reproduced. If this is an accuraterepresentation of the situation in pyramidal cells, the high-density ofAptKv3.3 could have masked evidence of anIA inpatch-clamp recordings.

IKA, IKB—somatic K1 currents

Somatic K1 currents in ELL pyramidal cells have onlyrecently been subjected to voltage-clamp analysis and cannotbe fitted as stringently as currents discussed in previous sec-tions. However, without a proper treatment of somatic K1,both the modelf-I relationship and spike frequency adaptationdisagree with experimental results. In this section, we intro-duce two somatic K1 currents,IKA andIKB, to improve modelperformance in these areas.

Whole cell recordings in ELL pyramidal cell somata indicatea prominent expression of Ca21-dependent large conductance(BK) K1 channels (Morales and Turner, unpublished observa-tions). The effect of these channels is modeled by a voltage-dependent current,IKA, because of the uncertainties on thelocation and magnitude of Ca21 influx in pyramidal cells. Thedynamics is similar to that ofIAptKv3.3, given the fast rate of BKactivation and deactivation in ELL pyramidal cells (Moralesand Turner, unpublished observations). Specifically, a fast ac-tivation time constant (smalltm,KA) is chosen forIKA dynam-ics, and the current is set to be high threshold in its initialactivation (smallkm,KA and depolarizedV1/2,m,KA, see Table 1).

Whole cell recordings reveal that iberiotoxin-sensitive BKK1 channels contribute;75% of the somatic K1 current

(Morales and Turner, unpublished observations). Consequentlythe gmax of IKA is adjusted to first establish the BK simulatedcurrent to a level approximately triple that of somaticIAptKV3.3

at voltages.0 mV (results not shown). Because of its fastactivation and the lack of inactivation,IKA overlaps the sAHPfollowing a somatic spike and hence significantly affects themodel f-I relationship. Thus in addition, the channel density,gmax,KA, is set so thatf-I characteristics of the model moreclosely approximate ELL pyramidal cells (see Fig. 8). Thef-Icurves show a rheobase current of 0.1 nA with a saturating risein spike frequency as current increases. This is equivalent tomeasuredf-I curves from ELL pyramidal cells in control situ-ations (Berman and Maler 1998b; Lemon and Turner 2000).

Spike frequency adaptation during long current pulses(Mathesion and Maler 1988) has been reported in some ELLpyramidal cells, although it is typically not prominent. Theoriginal version of our model incorporated a slow, noninacti-vating K1 current termedIKB to provide correct frequencyadaptation under constant depolarizing current (Doiron et al.2001). The kinetic parameters of this current are not modifiedfrom the original model, where comparisons between experi-mental and model frequency adaptation are presented.

In summary, the core pyramidal cell model includes 10 ioniccurrents, each described by separate Hodgkin/Huxley dynamics.Model AptKv3.3 currents match HEK cell experiments that iso-late and describe properties of this current. The model cell givesthe correct shape and refractory properties for both somatic anddendritic spikes and the generation of a DAP at the soma by thebackpropagating spike. The model somatic and dendritic NaPchannels produce the correct shifts in RMP, boosting of EPSPs,and reduction in spike latencies from rest, as observed in exper-imental recordings in vitro. A-type K1 channels are included so asto give correct latency to spike from hyperpolarized potentials.Finally, the model shows correct passive andf-I characteristics asdetermined from intracellular recordings.

FIG. 8. Spike frequency-current (f-I) characteristic of the model. The fre-quency of spike discharge of the model for a given bias currentIappapplied tothe soma is measured as the mean discharge rate over 500 ms, after a 100-mstransient period.Iapp is incremented in steps of 0.02 nA.

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ELL pyramidal cell bursting

Figure 9A shows the model’s repetitive discharge properties(nonburst activity) in response to depolarizing somatic currentinjection. However, the lack of a transition tog-burst dischargesuggests that there are missing elements in the model. Weshould note that electrophysiological work to date has reportedno evidence for a low-threshold “T”-type Ca21 current orIh inELL pyramidal cells, two currents known to underlie burstdischarge in many cell types (Huguenard 1996; Pape 1996).Rather, burst discharge in ELL pyramidal cells comes aboutthrough a progressive shift in the interaction between soma anddendrite during repetitive discharge (Lemon and Turner 2000).In the following sections, we alter the kinetic properties ofspecific ion channels to test their ability to generate burstdischarge in the core model. We first introduce a well-studiedbursting mechanism involving slow activating K1 channels(Chay and Keizer 1983) in the hopes of achieving bursting.

Burst discharge in the model does occur but fails to qualita-tively reproduce several burst properties in somatic and den-dritic recordings of ELL pyramidal neurons. We then insertvarious potential burst mechanisms into the ionic description ofthe model. It will be shown that a slow, cumulative, inactiva-tion of a dendritic K1 current coupled with the longer refrac-tory period of the dendritic spike as compared to the somaticspike is required to reproduce ELL pyramidal cellg-bursting.

Standard burst mechanism: slow activating K1 current

Reduced dynamical burst models show a characteristic abil-ity to switch between stable oscillatory discharge (burst) andquiescent rest voltage (inter-burst) (see Izhikevich 2000; Rin-zel 1987; Wang and Rinzel 1995; for a review of dynamicalbursting models). What is required to switch between thesestates is a slow dynamical variable. In most previous compart-mental models of intrinsic bursting, this slow variable is a

FIG. 9. The influence of a slowly activating K1 current(IKA) does not reproduce burst discharge observed in ELLpyramidal cells. All simulations have a tonic depolarizingcurrent of 0.6 nA applied to the somatic compartment.A:repetitive firing of the core model.B: when the activationtime constant ofIKA, tKA, is set to 50 ms, a very-low-frequency (;6 Hz) burst pattern results.C: when the acti-vation time constant ofIKA is set to 10 ms, a higherfrequency (;30 Hz) burst pattern results.D: an expandedview of a single burst from the somatic spike train shownin C. Although a higher frequency of spike bursts is gen-erated withtKA 5 10 ms, the existence of a repeating seriesof spike doublets, a progressively increasing inter-doubletinterval, and a large terminating DAP all disagree withexperimentally observed ELL bursts.E: the dendritic re-sponse (200mm) during the single somatic spike burstshown inD. Multiple dendritic spike failures (arrows) andthe lack of a slow summating depolarization during thespike train disagree with experimental observations of py-ramidal cells in vitro.

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Ca21 and/or voltage-dependent K1 channel,K(Ca) (Chay andKeizer 1983; Mainen and Sejnowski 1996; Pinsky and Rinzel1994; Rhodes and Gray 1994; Traub et al. 1994; Wang 1999;to name but a few). The mechanism involves increases inintracellular Ca21 due to repetitive firing. This causes a slowactivation ofK(Ca), thereby hyperpolarizing the cell. The hy-perpolarization byK(Ca) first acts as a voltage shunt, increasingthe ISIs at the tail of a burst. WhenK(Ca) is sufficientlyactivated, spike discharge stops completely, and the burst ter-minates. At hyperpolarized levels,K(Ca) deactivates, and spik-ing (bursting) may once again occur, given sufficient depolar-ization. This gives a characteristic burst pattern, with anincreasing ISI during the length of the burst. This pattern hasbeen observed in both experiment (Gray and McCormick1996) and models that useK(Ca) (Wang 1999).

Burst discharge in ELL pyramidal cells is insensitive tothe Ca21 channel blocker Cd21 (Rashid et al. 2001), andhenceK(Ca) is presumably not implicated in the burst mech-anism. Yet a slow, increasing K1 current that terminates theburst is still a potential mechanism. Rather than hypothesizea new ionic K1 channel, we modify the existing somaticIKAto produce the desired effects. Specifically, Fig. 9 shows themodel somatic voltage under constant somatic current in-jection after the activation time constanttm,KA is increasedfrom 1 ms (Fig. 9A) to 50 ms (Fig. 9B). Bursting occurs, yetwith a burst frequency of;5 Hz, whereas pyramidal neu-rons exhibit typical burst frequencies in theg range (20 – 80Hz) (Lemon and Turner 2000). To rectify the discrepancy,tm,KA is reduced to 10 ms with the corresponding spike trainshown in Fig. 9C. The burst frequency now approachesexperimental values, but the spike pattern within a burstdisagrees with experimental data. We elaborate on the dis-crepancies in the following text.

Figure 9 (D and E) illustrates the somatic and dendritic(200 mm) responses during a single burst from the trainshown in Fig. 9C. During a burst, the model somatic voltageshows the existence ofmultiple spike doublets (2 spikes at.200-Hz frequency), while the dendritic voltage showsmultiple spike failures (Fig. 9,D andE). Both somatic anddendritic bursts show no growing depolarization during theburst as evident from the lack of both an increasing DAP atthe soma and spike summation in the dendrite. Dendriticspike broadening is not observed during the length of aburst, and the somatic spike train is followed by a large DAPthat fails to elicit a spike due to a large KA conductance atthe end of the burst. Finally, Fig. 9D shows anincreasinginter-doublet interval during a burst. This result is similar toother bursting models incorporating a slow activating K1

current (Rhodes and Gray 1994; Wang 1999) but does notmatch the properties of burst discharge in ELL pyramidalcells (Lemon and Turner 2000). Therefore these discrepan-cies indicate that a slow activating K1 current is inadequateto model the burst discharge that incorporates conditionalbackpropagation in ELL pyramidal neurons.

ELL burst mechanism

To produce an output that more closely simulates pyramidalcell burst discharge, we concentrate on methods in which adecreasing ISI pattern during a burst can be realized in themodel. The duration of an ISI is determined by the amount of

somatic depolarization that produces the next spike in a spiketrain. Hence changes in ISIs must be linked to changes in theamount of depolarization during a spike train. In the presentstudy, somatic depolarization is determined from two sources:a constant applied currentIapp and a dendritic componentactivated by spike backpropagation that generates the DAP.BecauseIapp is constant, the ISI decrease observed during aburst should be correlated with increases in the DAP as shownby Lemon and Turner (2000). Four possible alternatives forDAP increase present themselves. First, cumulative inactiva-tion of dendritic fast Na1 current could broaden dendriticspikes indirectly by diminishing the activation of dendriticrepolarizing currents (IDr,d andIAptKv3.3). The broadening of thedendritic spike could lead to DAP growth at the soma. Second,a slowly activating inward current could increase the dendriticdepolarization during repetitive discharge and hence augmentthe somatic DAP. Third, a cumulative inactivation of a den-dritic K1 current involved in dendritic spike repolarizationwould increase dendritic spike duration during a burst andincrease the DAP. The fourth potential mechanism is a cumu-lative decrease in K1 currents underlying AHPs at the somathat would allow the DAP to become progressively moreeffective in depolarizing somatic membrane. However, thismechanism has been ruled out because somatic AHPs areentirely stable in amplitude at the frequencies of spike dis-charge encountered during burst discharge (Lemon and Turner2000). We consider each of the remaining alternatives in thefollowing text.

Slow inactivation of dendritic Na1 channels

Patch-clamp recordings in hippocampal pyramidal cellshave shown that the amplitude of dendritic spikes are reducedsubstantially during repetitive discharge (Colbert et al. 1997;Golding and Spruston 1998; Jung et al. 1997; Mickus et al.1999). The reduction in dendritic spike amplitude saturatesduring long stimulus trains, giving a final steady-state potentialheight. Jung et al. (1997) and Colbert et al. (1997) haveidentified the cause to be a slow inactivation of dendritic Na1

channels. This process may also contribute to a decrease indendritic spike amplitude in ELL pyramidal cells observedduring antidromic spike trains (Lemon and Turner 2000). Asstated in the preceding text, a slow inactivation of dendriticNa1 channels could contribute to burst discharge by reducingthe activation of K1 currents that repolarize the dendritic spike(IDr,d and IAptKv3.3 in the model).

Mickus et al. (1999) presented a detailed kinetic model ofINa,d inactivation that considers two separate inactivation statesvariables: fast and slow. Slow inactivation is the state variablethat cumulatively grows from spike to spike, producing spikeattenuation during repetitive discharge. To incorporate thisconcept into the model Hodgkin-Huxley framework, we pro-pose to modify the existingINa,d description through the addi-tion of a second inactivation state variablepNa,d designed torepresent a slow inactivation according to

INa,d5 gmax,Na,dz mNa,d2 hNa,dpNa,d z ~Vm 2 Vrev,Na,d! (4)

where mNa,d and hNa,d are the original fast activation andinactivation state variables introduced inMETHODS. mNa,d, hNa,d,andpNa,d are each described byEqs. 1.2and1.3, with param-eters specific to the channel (see Table 1 formNa,d andhNa,d,

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and the following text forpNa,dparameters). To our knowledge,no previous studies have characterized dendritic Na1 slowinactivation kinetics sufficiently so as to fully justify our choiceof the model parameters. We thus set theV1/2,pNa,dto 265 mVandkpNa,dto 5 mV thereby ensuring that spike backpropagationis required to significantly inactivate Na, d, similar to previousexperimental observations (Colbert et al. 1997; Jung et al.1997). However, to properly settp,Na,d, which determines boththe time scale of onset and recovery from inactivation, furtherassumptions must be made.

Colbert et al. (1997) and Jung et al. (1997) both reported thatfull recovery of Na1 channel inactivation occurred in the orderof seconds. This time scale is to long reproduce ELL pyramidalcell spike bursts because significantINa,d recovery needs tooccur within the inter-burst interval, which is 10–15 ms(Lemon and Turner 2000). However, simply settingtp,Na,dsothat recovery from Na1 channel inactivation occurs on a 15-mstime scale also results in significant recovery from inactivationduring repetitive spike discharge, removing the decrease inspike amplitude during the burst. Recovery from slow inacti-vation of dendritic Na1 channels after a stimulus train has beenshown to be accelerated through membrane hyperpolarization(Mickus et al. 1999). This is ideal for allowing only significantinactivation recovery of dendritic Na1 in the inter-burst inter-val, which occurs in proximal apical dendrites at potentials upto 210 mV lower than intra-burst subthreshold potentials(Lemon and Turner 2000). We therefore extend the descriptionof tp,Na,dto include voltage dependence as originally modeledby Hodgkin and Huxley (1952). Rather than using thea andbrate formalism, we simply assumed a sigmoidal relation be-tweentp,Na,d andVm

tp,Na,d~Vm! 5tmax

~1 1 e2~Vm2V1/2,t!/kt!(5)

wheretmax is the maximum time constant ofpNa,d, while V1/2,tandkt describe the voltage dependence in analogous fashion tothe steady-state conductance relations given byEq. 1.3.We setV1/2,t to 260 mV, kt to 23 mV, andtmax to 20 ms to separatethe pNa,d inactivation into two distinct time scales; slow atmembrane potentials typical for intra-burst depolarizations andfast for the hyperpolarized potentials associated with the burstAHPs.

Figure 10A shows the response of the model dendritic com-partment (200mm) to two 50-ms step somatic depolarizationsof 1 nA, separated by 15 ms to simulate the occurrence of aburst AHP. Spike amplitude during both trains of backpropa-gating model spikes show attenuation similar in both magni-tude and time scale to that observed in recordings of burstingELL pyramidal cell proximal apical dendrites (Lemon andTurner 2000). The model dendritic spikes do indeed show afull recovery from slow inactivation during a 15-ms pausebetween depolarizations (Fig. 10A). This indicates that theinclusion of voltage dependence oftp,Na,dproduces a recoveryfrom Na1 channel inactivation that would be necessary tosustain bursting. Nevertheless Fig. 10B shows that under con-stant somatic depolarization, only rapid dendritic spike atten-uation, which saturates at a fixed amplitude, and not burstdischarge is observed. This pattern is identical to the attenua-tion and saturation of dendritic spike amplitude observed in ratCA1 pyramidal cells during antidromic repetitive stimulation(Colbert et al. 1997; Jung et al. 1997). It is also important to

note that significant broadening of model dendritic spikes doesnot occur under these conditions, and an ISI of 8 ms issustained during repetitive discharge (Fig. 10B). Further ad-justment of modelINa,d parameters does not qualitativelychange these results. These simulations indicate that a cumu-lative inactivation of Na, d and the associated decrease in spikeamplitude is not sufficient to produce burst discharge in thecore model. Although this mechanism may contribute to the

FIG. 10. Slow inactivation ofINa,d is not sufficient to generate conditionalbackpropagation and burst discharge.A: model dendritic voltage response (200mm from soma) during repetitive spike discharge at;130 Hz with slowinactivation inserted intoINa,d kinetics. Two depolarizing pulses (Iapp) of40-ms duration are applied, separated by a return to rest for 15 ms to simulatethe time frame of a typical burst and burst AHP in ELL pyramidal cells in vitro.The amplitude of the dendritic spike decreases during repetitive discharge witha similar magnitude and time scale as found in ELL pyramidal cell dendrites(cf. Fig. 12C). The 15-ms pause inIapp is sufficient to allow significantrecovery fromINa,d slow inactivation, as indicated by the identical dendriticresponse to both pulses. This behavior requiredtp,Na,dto be voltage dependent,as described byEq. 5. B: the response of the model to a sustainedIapp of thesame magnitude as inA but without the 15-ms pause between pulses. Note thatthe steady-state discharge pattern is tonic, indicating that slow inactivation ofINa,d is not sufficient to cause the intermittent failure of backpropagation thatcharacterizes burst discharge in ELL pyramidal cells.

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processes underlying dendritic spike failure in the intact cell,we remove slow inactivation of Na, d from subsequent simu-lations to simplify the analysis.

Slow activation of INaP

Recent experimental studies have suggested thatINaP con-tributes substantially to the depolarization that drives burstdischarge in several mammalian neurons (Azouz et al. 1996;Brumburg et al. 2000; Franceschetti et al. 1995; Magee andCarruth 1999). In ELL pyramidal cells, we find thatINaPkinetics that lead to appropriate shifts in RMP and EPSPamplitude are not able to promote burst discharge. To identifythe potential forINaP to contribute to burst discharge inApter-onotuspyramidal cells, a slow activating component to thetypical fast activatingINaP is hypothesized. If such a compo-nent was added to the model description, then NaP couldcumulatively grow during a burst, thereby broadening dendriticspikes. This would produce a slow growth in DAP amplitude,a reduction in spike latency to discharge, and a shortening inISI through a burst. We modify the existing description ofINaPin both somatic and dendritic compartments by creating asecond activation variable,qNaP, in the NaP current equation

INaP5 gmax,NaPz mNaP3 qNaP z ~V 2 Vrev,Na! (6)

wheremNaP is the original activation variable, andqNaP is alsodescribed byEqs. 1.2and 1.3. We force themNaP and qNaPvoltage dependencies to be equal by settingV1/2,q,NaP andkq,NaP to be identical to those ofmNaP (see Table 1). We settq,NaP to 10 ms, which is of the same order of magnitude asburst oscillation periods observed in the ELL (Lemon andTurner); recall thattm,NaP is an order of magnitude smaller,being set to 0.3 ms. With the addition of a second statevariable, the overall conductance will decrease; hencegmax,NaPis increased by a factor of 5 in both the soma and dendrites tocompensate. Figure 11A shows the time course of the modelsomatic INaP during a repetitive spike train with Fig. 11Bshowing the associated somatic voltage. A cumulative increasein INaPduring repetitive spike discharge (Fig. 11A) produces aslow depolarizing envelope at the soma that reduces the ISI andleads to a transition from tonic to spike doublet discharge (Fig.11B). The doublet firing is due to NaP promoting high-fre-quency spike discharge, which results in the failure of back-propagation of the second spike of the doublet, as described inFig. 5 and Lemon and Turner (2000). The ability forINaP topromote a switch to burst discharge that incorporates condi-tional backpropagation is encouraging. However, we find thatthis mechanism is unable to generate bursts composed of fouror more spikes, as is typically the case in pyramidal cells invitro. In fact, all reasonable modifications ofqNaP parametersare unable to generate bursts longer than recurring spike dou-blets. To increase the possibility of producing realistic burstdischarge,tq,NaP was made voltage dependent (not shown),with similar dependency as that described byEq. 5.This causesfast deactivation ofqNaP during the hyperpolarization associ-ated with inter-burst periods, and slow activation ofqNaPduring the intra-burst depolarization. Once again, the simula-tions only generated doublet firing, as shown for the voltage-independenttNaP case.

As reported previously, these results indicate thatINaP iscapable of contributing to burst discharge in theg-frequency

range (Wang 1999). However, in ELL pyramidal cells, thiscurrent is unable to promote burst discharge without the addi-tion of a slow activation rate intoINaP channel kinetics. Thefailure to produce multiple spike bursts even with the slowactivation time constant may be a result of the steeply nonlin-ear positive feedback incorporated into NaP (Fig. 6). Thereforewe remove the slow activation of NaP introduced in thissection from all subsequent analyses.

Cumulative inactivation of dendritic K1 channels

Previous studies have shown that a cumulative inactivationof K1 channels is essential for producing a frequency-depen-dent spike broadening (Aldrich et al. 1979; Ma and Koester1995; Shao et al. 1999). Direct experimental evidence for theexistence of a cumulatively inactivating K1 channel withinELL pyramidal cell dendrites has yet to be established. None-theless the assumption is indirectly supported by the dendriticspike broadening observed during bursting (Lemon and Turner2000) (Fig. 13). Instead of inserting a separate K1 channel intothe model, we choose to modify the dendritic K1 channel, Dr,d, so that it inactivates substantially over the length of a burstyet remains constant during an individual action potential.

The core model Dr, d kinetics include only fast activation,

FIG. 11. Slow accumulation ofINaP is insufficient to produce model burstdischarge comparable to in vitro experiment.A: gNaP response to constantdepolarization (0.6 nA) that evokes repetitive spike discharge at a frequency of;150 Hz. The conductance rapidly tracks somatic spiking through fast acti-vation and deactivation, controlled bymNaP. The slow activation, mediated byqNaP, is the depolarizing envelope allowing cumulative growth ofgNaP fromspike to spike during repetitive discharge.B: somatic voltage response duringthe same simulation described inA. Spike discharge frequency increases untilthe appearance of a spike doublet (onset of spike doublets marked by *).

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represented by the state variablemDr,d. To incorporate cumu-lative inactivation within the model, we introduce a secondstate variablehDr,d, representing inactivation, inserted into thecurrent equation for Dr, d

IDr,d 5 gmax,Dr,dz mDr,dhDr,d z ~Vm 2 Vrev,Dr,d! (7)

where hDr,d is described byEqs. 1.1–1.3,with parametersth,Dr,d, V1/2,h,Dr,d, andkh,Dr,d. We set the inactivation to be lowthreshold (V1/2,h,Dr,d5 265 mV), steeply nonlinear (kh,Dr,d 526), and following a time course so that its inactivation issubstantial over the length of a burst, yet negligible during anindividual spike (th,Dr,d 5 5 ms). Modification of these param-eters will be explored later.

Figure 12 (A andB) illustrates the effects of incorporatingcumulative inactivation of Dr, d into the core model accordingto Eq. 7.The qualitative change from nonbursting to burstingbehavior is evident. Direct comparisons to experimental dataare presented in the following text. Figure 12C plots the statevariablehDr,d from the same simulation as in Fig. 12B. Theslow increase of inactivation (hDr,d decreases) occurs over thelength of each individual burst. At burst termination, inactiva-tion is removed quickly (hDr,d increases) due to the largehyperpolarization associated with a burst AHP (bAHP), andslow inactivation begins again with the next burst. Because ofthe addition of this second state variable,hDr,d, which oscillatesbetween values of 0.3 and 0.1 during bursting,gmax,Dr,d isincreased from 0.2 to 0.65 S/cm2 to compensate. This ensuresthat the currentIDr,d remains comparable between nonburstingand bursting models.

Figure 13 compares in detail the characteristics of burstdischarge in ELL pyramidal cells to burst simulations when acumulative inactivation ofIDr,d is incorporated into the model.Shown are somatic and dendritic recordings of a single spikeburst with the equivalent simulation results at the same scalebelow. Comparing the somatic recording and simulation (Fig.13, A andB) reveals that the model successfully reproduces aprogressive increase in the DAP, a reduction in ISI during theburst, and a high-frequency spike doublet that is followed by alarge-amplitude burst AHP; all characteristic and key elementsin the process of conditional spike backpropagation (Lemonand Turner 2000). Note that potentiation of the DAP in themodel occurs at the same initial intra-burst ISI as pyramidalcells, with a matching decrease in ISI over the course of asix-spike burst. The final burst frequency in both experimentand simulation is 25 Hz (not shown), within the expected rangeof oscillatory burst frequency near burst threshold (Lemon andTurner 2000). At the dendritic level (Fig. 13,C and D), themodel now reproduces a progressive frequency-dependentspike broadening that underlies a temporal summation of den-dritic spikes (Fig. 13,C andD, insets). This temporal summa-tion leads to the further development of a depolarizing enve-lope that contributes to potentiation of the DAP at the soma.The dendritic simulation also correctly replicates the condi-tional failure of spike backpropagation when a spike doublet isgenerated at the soma as indicated by a passively reflectedpartial spike response at the end of the simulated burst (Fig.13D). Finally, the hyperpolarization produced by the bAHP issufficient to promote recovery fromIDr,d inactivation (Fig.12C), essentially resetting the duration of dendritic spikes toallow the cell to repeat this process and generate the next burst.

One discrepancy between the dendritic simulation and ex-

perimental recordings is the lack of dendritic spike attenuationin the model output (Fig. 13D). Although the ionic mechanismunderlying this process in pyramidal cells is unknown, it mayinvolve a slow inactivation of dendritic Na1 channels as mod-eled in Fig. 10. However, the results of Fig. 13 clearly indicatethat it is not essential to reproducing the major features of burstdischarge. Hence, for simplicity, it is not incorporated into themodel at this time.

It should be noted that the voltage and time dependencies ofIDr,d inactivation,hDr,d, are chosen so as to best reproduce theexperimental data. In fact, several characteristics of burst out-put depend strongly on the kinetic properties ofIDr,d inactiva-tion. Specifically, reduction oftDr,d 2 ms removes all burstoutput from the simulations, indicating thatIDr,d inactivationmust be cumulative to achieve burst discharge. SettingtDr,d tovalues above 2 ms does not prevent bursting but rather modi-fies burst frequency, withtDr,d .10 ms producing oscillatoryburst discharge outside theg-frequency range (,20 Hz). Re-

FIG. 12. Cumulative inactivation of the dendritic K1 currentIDr,d is suffi-cient to promote burst discharge. All simulations have a depolarizing currentof 0.6 nA applied to the somatic compartment.A: the core model generatesonly repetitive somatic spike discharge in response to depolarization.B:cumulative inactivation ofIDr,d promotes a gradual decrease in the ISI, anincrease in DAP amplitude, and spike doublet discharge and subsequent bAHPassociated with conditional spike backpropagation.C: the dynamics of thegating variablehDr,d as the burst shown inB evolves. The cumulative inacti-vation occurs ashDr,d decreases slowly during a burst. Removal of inactivationoccurs rapidly with the bAHP that follows each spike doublet (8).

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alistic modifications ofkh,Dr,d do not qualitatively change burstoutput in the model. However, increases or decreases ofV1/2,h,Dr,doutside the voltage range attained by dendritic spikes(below 280 or above220 mV) blocks burst discharge. Thisagain emphasizes the importance of the properties of back-propagating dendritic spikes in activating and inactivatingIDr,dchannels to bring about burst discharge.

Dissection of the burst mechanism

We investigated the role of several potential ionic mecha-nisms to simulate ELL pyramidal cellg-frequency burst dis-charge. We could not match experimental results forg-fre-quency burst output by introducing a slow activation ofsomatic K1 conductance, a slow inactivation of Na, d chan-nels, or a slow activation of NaP, indicating that these condi-tions are not by themselves sufficient components of the burstmechanism. In contrast, introducing a cumulative inactivationof dendritic K1 current was very successful in producing arealistic burst output, suggesting that this mechanism is anessential factor in pyramidal cellg bursting. We now dissectthe complete burst mechanism into the main ionic currents thatunderlie its evolution and termination.

We first elicit a single burst by somatic depolarization,shown in Fig. 14A complete with spike doublet and bAHP; allsubsequent analysis will pertain to this burst. Figure 14B plotsthe time series of the channel conductancegDr,d from the apicaldendritic compartment 200mm from the soma (last activezone) over the duration of the burst. Recall thatgDr,d 5gmax,Dr,d z mDr,dhDr,d, wheremDr,d andhDr,d are the respectiveactivation and cumulative inactivation parameters of the den-

dritic K1 channel. Figure 14B shows the conductance of Dr, dtracking each spike in the burst as it activates and deactivatesquickly (the activation time constanttm,Dr,d 5 0.9 ms). Acumulative inactivation is clear from the attenuation ofgDr,dthat occurs during the burst. This attenuation leads to dendriticspike broadening and a temporal summation of dendritic spikesthat produces a slow depolarizing envelope (Fig. 13). Thecombination of an increase in dendritic spike duration and aslow depolarizing envelope allows the DAP at the soma toincrease from spike to spike during the burst. The increase inthe DAP further depolarizes the soma to reduce the ISI as theburst evolves. Thus cumulative inactivation of a dendritic K1

channels is sufficient in itself to account for each of the keyproperties of spike discharge that characterize burst generationin pyramidal cells (Lemon and Turner 2000). Another factorthat merits further consideration in future electrophysiologicalstudies is the kinetic properties ofINaPthat might contribute tothe dendritic depolarization observed during repetitive dis-charge.

The termination of the burst is quite separate from themechanism driving spike discharge. Figure 14C plotsgNa,d, thedendritic Na1 conductance 200mm from the soma during thelength of the burst. Recall that the conductance is given bygNa,d 5 gmax,Na,dandmNa,dhNa,d. HeregNa,d decreases slightlywith each spike in the burst but exhibits a pronounced decreaseon the generation of a spike doublet at the end of the spike train(Fig. 14C). This corresponds to a failure of spike backpropa-gation when the high frequency of the spike doublet exceedsthe dendritic refractory period as reflected by the significantdrop ingNa,don the final spike. In contrast, doublet frequenciesof firing can easily be sustained by the somatic conductance

FIG. 13. A comparison between simulatedburst discharge and experimentally recordedbursts in ELL pyramidal cells.A and B: so-matic and dendritic burst trains recorded fromELL pyramidal cells [see Lemon and Turner(2000) for experimental protocol]. In pyrami-dal cell somatic recordings (A), a potentiationof the DAP and a burst AHP (bAHP) at theend of a burst are prominent (arrows). Den-dritic recordings (B) are associated with afrequency-dependent broadening of the den-dritic spike and final failure (arrow) on gen-eration of a spike doublet at the soma.Inset: amagnified and superimposed view of the baseof spikes marked by the plus and the asteriskin the dendritic recording to reveal a differ-ence of 0.26 ms (D) in dendritic spike dura-tions. C andD: model somatic and dendriticburst response withIapp5 0.6 nA. The modelsomatic train (C) indicates the same pattern ofDAP potentiation at the soma, dendritic spikesummation and broadening, spike doublet andbAHP as found in ELL pyramidal cell record-ings (A). The model duration of the indicateddendritic spikes (as inB) are again shown inexpanded form in theinset.

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FIG. 14. Analysis of the burst mechanism in pyramidal cells.A: the potential at the soma during a burst simulation, includingtermination by a spike doublet and bAHP. This burst is used in the analysis of the other panels. The time calibration shown alsoapplies toB andC. B: a plot ofgDr,d as the burst shown inA evolves indicates a cumulative inactivation that steadily reduces peakconductance during repetitive discharge.C: a plot ofgNa,d as the burst shown inA evolves. The conductance peak shows a slightdecrease until the spike doublet at the end of the burst, at which point the ISI is inside the refractory period ofINa,d and hence thechannel cannot respond. Dendritic backpropagation then fails, preventing the current flow that produces a DAP at the soma.D: plotsof the peak conductance ofgDr,d (B) andgNa,d (C) as a function of spike number for the burst shown inA. The slow drop ingDr,d

due to cumulative inactivation is shown as well as the complete failure ofgNa,d at the occurrence of the spike doublet when spikefrequency exceeds the dendritic refractory period.E: a schematic diagram of the soma-dendritic interactions that underliesconditional backpropagation in the simulation. The burst begins with initiation of a spike at the soma which backpropagates over;200mm of the apical dendrite through the activation ofINa,d (upward arrow). The long duration of the dendritic spike comparedwith somatic spike results in return current flow to generate a DAP at the soma (downward arrow). The DAP depolarizes the soma,thereby contributing to the next spike. Cumulative inactivation ofIDr,d during repetitive discharge increases dendritic spike widthand thus the depolarization that determines DAP amplitude and duration (shown by an increasing width of the downward arrows).The increase in the somatic DAP reduces subsequent ISIs in the burst until triggering a final spike doublet. At spike doubletfrequencies dendritic membrane is refractory and backpropagation fails, removing the DAP at the soma (small arrow blocked byan X). The sudden loss of dendritic depolarization uncovers a large somatic bAHP, which hyperpolarizes the soma and terminatesthe burst.

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gNa,s (not shown), allowing two full somatic spikes to begenerated (Fig. 14A). When the dendritic spike fails to activate,the DAP is selectively removed at the somatic level, uncover-ing a large bAHP that signifies the end of a spike burst (Fig.14A). To test the significance of the dendritic refractory periodfor burst generation, we performed additional simulationswhereINa,d was replaced byINa,s, thereby establishing equiv-alent dendritic and somatic refractory periods. Spike bursts donot occur in these simulations with only repetitive dischargeobserved for all levels of depolarizing somatic current injec-tion. Hence the longer refractory period of the dendrite ascompared with the soma is necessary for burst termination asproposed by Lemon and Turner (2000).

The ionic basis of the bAHP has not been fully determined,but the initial early phase of this response is insensitive to bothTEA and Cd21, eliminating many candidate K1 currents inpyramidal cells that could actively contribute to the bAHP(Noonan et al. 2000). However, we should note that the largehyperpolarization of the model bAHP is also due to slightsummation of the somaticIKA (tKA 5 1 ms) that occurs onlyat doublet frequencies (data not shown). Although this effectenhances burst termination, elimination ofIKA does not preventbursting in the model.

Figure 14D presents the cumulative inactivation ofgDr,d andthe sudden failure ofgNa,das a function of spike number of theburst presented in Fig. 14A. This superimposes the results ofFig. 14,B andC, and considers the evolution and terminationof the burst as events driven by action potentials. Figure 14Eschematically summarizes the burst mechanism by consideringthe end effect ofIDr,d and INa,d in shaping the soma-dendriticinteraction. Their action results in a cumulative DAP growthand eventually a sharp DAP failure that is the manifestation ofthe burst mechanism presented in Fig. 14,B–D, at the level ofmembrane voltage.

The currentsIA and IKB were included for all burstingsimulations. However, the presence of these channels is notrequired for model bursting (data not shown). This is clearbecauseIA requires hyperpolarization to remove inactivationand therefore was inactivated at the depolarized membranepotentials required for bursting.IKB has a time course in theorder seconds and can be approximated as static on the shorttime scale of bursting.

D I S C U S S I O N

Importance of a realistic model of ELL pyramidal cells

Recently the ELL has been the focus of numerous investi-gations into the role of bursting in sensory information pro-cessing (Gabbiani and Metzner 1999; Gabbiani et al. 1996;Metzner et al. 1998). Feedforward information transfer is cur-rently under study in the ELL through modeling of P-afferentdynamics and coding (Chacron et al. 2000; Kreiman et al.2000; Nelson et al. 1997; Ratnam and Nelson 2000; Wessel etal. 1996). Studies of synaptic feedback to the ELL involve bothexperimental and computational work (Berman and Maler1998a–c; 1999; Berman et al. 1997; Doiron et al. 2001; Nelson1994). In fact recent experimental work has suggested thatfeedback activity modulates bursting behavior in ELL pyrami-dal cells (Bastian and Nguyenkim 2001). The subtle nature ofthese issues points to the usefulness of a detailed and realistic

compartmental model for creating and testing hypotheses con-cerning burst mechanisms and their regulation by feedforwardand feedback synaptic input.

ELL burst mechanism

Our results suggest the following necessary and sufficientconditions for ELL pyramidal cell bursting. First, there must bea dendritic Na1 current to support spike backpropagation intothe proximal apical dendrites, yielding a DAP at the soma.Second, there must be a (proposed) cumulative inactivation ofa K1 conductance involved in dendritic spike repolarization.This inactivation effectively results in a dendritic spike broad-ening during repetitive discharge that is known to potentiateDAP amplitude at the soma. Finally there must be a longerspike refractory period in the dendrites compared with thesoma, causing backpropagation to be conditional on the so-matic spike discharge frequency, which terminates a burst atsufficiently high discharge rates.

AptKv3.3 as a possible candidate for Dr, d

In the present study, we assigned kinetic properties to the Dr,d channel that allow it to be activated by dendritic spikes andto exhibit cumulative inactivation during repetitive spike dis-charge by virtue of a relatively lowV1/2 for inactivation. It isimportant to note that the Dr, d channel is a hypotheticalchannel subtype inserted in the model to allow for propersimulation of the dendritic spike response. The question re-mains as to which, if any, dendritic channel in intact ELLpyramidal cells matches the description of Dr, d. One potentialcurrent isAptKv3.3, which is located with high prevalenceover the entire axis of pyramidal cells (Rashid et al. 2001).Indeed, pharmacological blockade of dendriticAptKv3 chan-nels has been shown to decrease dendritic spike repolarizationand lower the threshold for burst discharge (Rashid et al.2001). The possibility therefore exists that dendriticAptKv3.3K1 channels may serve a similar capacity as the Dr, d channelsin the current model.

We modeledIAptKv3.3 according to the kinetic propertiesinherent to whole cell currents whenAptKv3.3 channels aretransiently expressed in HEK cells (Rashid et al. 2001). Thesecurrents share several properties with Dr, d channels, includingfast activation and deactivation kinetics. The major differenceis the V1/2 of activation: V1/2,m,Dr,d 5 240 mV andV1/2,m,AptKv3.3 5 0 mV. We found that burst discharge can stillbe produced in the model ifV1/2,m,Dr,dis raised no higher than220 mV with corresponding adjustments to channel density tooffset the smaller degree of current activation. However, withV1/2,m,Dr,d . 220 mV, the model could not produce burstingwith realistic values ofgmax,Dr,d. Similarly, AptKv3.3 as mod-eled in the cell could not produce bursting with aV1/2 of 0 mV,even if a cumulative inactivation similar to Dr, d was intro-duced (results not shown). We have recently begun to charac-terize an additional slow inactivation process inAptKv3.3channels that has not been incorporated into the present model.Recent work further indicates that theV1/2 for AptKv3.3 slowinactivation can exhibit a leftward (negative) shift in the out-side-out as compared with whole cell recording configuration(Morales and Turner, unpublished observations). This suggeststhat AptKv3.3 kinetics are subject to second-messenger regu-

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lation as previously shown for other Kv3 channel subtypes(Atzori et al. 2000; Covarrubias et al. 1994; Moreno et al.1995; Velasco et al. 1998). The potential therefore exists for aselective modulation of dendriticAptKv3.3 kinetics that wouldallow this channel to exhibit a cumulative inactivation duringrepetitive spike discharge. Further experimental work will beneeded to determine the exact voltage-dependence and regula-tion of somatic versus dendriticAptKv3.3 channels to test thishypothesis.

Relation to in vivo bursting

A recent quantitative analysis of burst discharge in ELLpyramidal cells has shown that the ISIs during spontaneousbursts recorded in vivo remain relatively constant, and oftenlack a terminating spike doublet; results that differ from theburst mechanism routinely recorded in vitro (Bastian andNguyenkim 2001). At present the discrepancy between thebursts that are driven by current-evoked depolarizations invitro and baseline discharge recorded in vivo is not clear. Thereare many differences between the state of pyramidal cells invivo versus in vitro: the resting membrane potential of pyra-midal cells in vivo is closer to spike threshold (J. Bastian,personal communication), the input resistance of pyramidalcells is likely to be far lower in vivo because of synapticbombardment (Bernander et al. 1991; Pare´ et al. 1998; Bastian,personal communication), stimulation in vitro is via constantcurrent injection, whereas in vivo the primary stimulus is astochastic synaptic input to dendrites, and in vivo studiescontain network effects which in vitro studies necessarily lack.Further exploration with the model ionic channel parameters,to modify these features, as well introducing simulated synap-tic input, may bridge the gap between in vivo and in vitroresults. However, more detailed experimental analysis is re-quired before any modeling attempt is to be made along theselines.

Application of ELL burst model to mammalianchattering cells

Sustainedg-frequency bursting, or chattering behavior, hasbeen observed in mammalian cortical neurons (Brumburg et al.2000; Gray and McCormick 1996; Llina´s et al. 1991) andcorticothalamic neurons (Steriade et al. 1998). Wang (1999)produced a “chattering” behavior in a two-compartment neuronmodel that incorporates a similar “ping-pong” reciprocal inter-action between the cell soma and dendrites we have describedin ELL pyramidal cells. The evolution and termination of theburst in Wang’s model relies on a cumulative activation ofK(Ca). This produces a characteristic increase of intra-burst ISIsand a prominent DAP in the inter-burst interval (Wang 1999).We have shown (Fig. 9) that this mechanism is not able toproduce realistic ELL pyramidal cell bursts because it fails toreproduce a decrease in intra-burst ISIs, the slow somaticdepolarization that increases the DAP, and the lack of a DAPat burst termination as observed with pyramidal cells in vitro(Lemon and Turner 2000). Our proposed mechanism success-fully reproduces all the preceding criteria. Brumburg et al.(2000) report the cumulative reduction of spike fAHPs as aburst evolves in supragranular cortical neurons. This result alsocannot be reproduced by cumulative activation ofK(Ca) in our

model, as used in Wang (1999) or in other detailed IB neuroncompartmental models (Mainen and Sejnowski 1996; Pinskyand Rinzel 1994; Rhodes and Gray 1994; Traub et al. 1994).Brumberg et al. (2000) hypothesize that the mechanism under-lying g-burst discharge in mammalian visual cortex could beeither a slow increase of a Na1 current or a slow decrease ofa K1 current. Our modeling of ELL pyramidal cells revealsthat cumulative inactivation of a dendritic K1 current can playa key role in generating burst discharge, a result that may havewide applicability to cells discharging in theg-frequencyrange.

The NEURON codes for our ELL pyramidal cell model arefreely available at http://www.science.uottawa.ca/phy/grad/doiron/

The authors thank N. Berman, E. Morales, N. Lemon, and L. Noonan for thegenerous use of data. Many useful discussions with M. Chacron, C. Laing, andJ. Lewis were greatly appreciated during the writing of the manuscript. R. W.Turner is an Alberta Heritage Foundation for Medical Research SeniorScholar.

This research was supported by operating grants from National Science andEngineering Research Council (B. Doiron and A. Longtin) and CanadianInstitute for Health Research (L. Maler and R. W. Turner).

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ARTICLE II

B. Doiron, C. R. Laing, A. Longtin. and L. Maler.Ghostbursting: A novel neuronal burst mechanism.Journal of Computational Neuroscience. 12: 5-25, 2002.

51

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Journal of Computational Neuroscience 12, 5–25, 2002c© 2002 Kluwer Academic Publishers. Manufactured in The Netherlands.

Ghostbursting: A Novel Neuronal Burst Mechanism

BRENT DOIRON, CARLO LAING AND ANDRE LONGTINPhysics Department, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, Canada K1N 6N5

bdoiron@physics.uottawa.ca

LEONARD MALERDepartment of Cellular and Molecular Medicine, University of Ottawa, 451 Smyth Road, Ottawa,

Canada K1H 8M5

Received June 8, 2001; Revised October 17, 2001; Accepted November 1, 2001

Action Editor: John Rinzel

Abstract. Pyramidal cells in the electrosensory lateral line lobe (ELL) of weakly electric fish have been observedto produce high-frequency burst discharge with constant depolarizing current (Turner et al., 1994). We present a two-compartment model of an ELL pyramidal cell that produces burst discharges similar to those seen in experiments.The burst mechanism involves a slowly changing interaction between the somatic and dendritic action potentials.Burst termination occurs when the trajectory of the system is reinjected in phase space near the “ghost” of a saddle-node bifurcation of fixed points. The burst trajectory reinjection is studied using quasi-static bifurcation theory,that shows a period doubling transition in the fast subsystem as the cause of burst termination. As the applieddepolarization is increased, the model exhibits first resting, then tonic firing, and finally chaotic bursting behavior,in contrast with many other burst models. The transition between tonic firing and burst firing is due to a saddle-nodebifurcation of limit cycles. Analysis of this bifurcation shows that the route to chaos in these neurons is type Iintermittency, and we present experimental analysis of ELL pyramidal cell burst trains that support this modelprediction. By varying parameters in a way that changes the positions of both saddle-node bifurcations in parameterspace, we produce a wide gallery of burst patterns, which span a significant range of burst time scales.

Keywords: bursting, electric fish, compartmental model, backpropagation, pyramidal cell

1. Introduction

Burst discharge of action potentials is a distinct andcomplex class of neuron behavior (Connors et al., 1982;McCormick et al., 1985; Connors and Gutnick, 1990).Burst responses show a large range of time scales andtemporal patterns of activity. Many electrophysiologi-cal studies of cortical neurons have identified cells thatintrinsically burst at low frequencies (<20 Hz) (Blandand Colom, 1993; Steriade et al., 1993; Franceschettiet al., 1995). However, recent work in numerous sys-tems has now identified the existence of “chattering”

cells that show burst patterns in the high-frequencyγ range (>20 Hz) (Turner et al., 1994; Pare et al.,1998; Gray and McCormick, 1996; Steriade et al.,1998; Lemon and Turner, 2000; Brumburg et al., 2000).Also, the specific interspike interval (ISI) pattern withinthe active phase of bursting varies considerably acrossburst cell types. Certain bursting cells show a lengthen-ing of ISIs as a burst evolves (e.g., pancreatic-β cells,Sherman et al., 1990), others a parabolic trend in theISI pattern (e.g., Aplysia R15 neuron, Adams, 1985),and yet others show no change in the ISI during a burst(e.g., thalamic reticular neuron, Pinault and Deschenes,

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6 Doiron et al.

1992). This diversity of specific time scales and ISIpatterns suggests that numerous distinct burst mecha-nisms exist. Knowledge of the burst mechanisms allowsone to predict how the burst output may be modifiedor halted completely in response to stimuli. This mayhave consequences for the information content of thecell’s output (Lisman, 1997).

Pyramidal cells in the electrosensory lateral linelobe (ELL) of the weakly electric fish Apteronotus lep-torhynchus have been shown to produce both tonic fir-ing and γ frequency sustained burst patterns of actionpotential discharge (Turner et al., 1994; Turner andMaler, 1999; Lemon and Turner, 2000). These sec-ondary sensory neurons are responsible for transmit-ting information from populations of electroreceptorafferents that connect to their basal bushes (see Bermanand Maler, 1999, and references therein). In vivorecordings from ELL pyramidal cells have indicatedthat their bursts are correlated with certain relevantstimulus features, suggesting the possible importanceof ELL bursts for feature detection (Gabianni et al.,1996; Metzner et al., 1998; Gabianni and Metzner,1999). Thus, both the proximity of ELL pyramidal cellsto the sensory periphery and the known relevance oftheir bursts to signal detection suggest that studies ofELL bursting may provide novel results concerning therole of burst output in sensory processing.

Previous in vitro and in vivo experiments have fo-cused both on specifying the mechanism for burst dis-charge of ELL pyramidal cells and showing methodsfor the modulation of burst output (Turner et al., 1994,1996; Turner and Maler, 1999; Lemon and Turner,2000; Bastian and Nguyenkim, 2001; Rashid et al.,2001). Lemon and Turner (2000) have shown that afrequency dependent or “conditional” action potentialbackpropagation along the proximal apical dendrite un-derlies both the evolution and termination of ELL burstoutput. Recently, through the construction and analy-sis of a detailed multicompartmental model of an ELLpyramidal cell, we have reproduced burst dischargessimilar to those seen in experiment. This model allowedus to make strong predictions about the characteristicsof the various ionic channels that could underlie theburst mechanism (Doiron et al., 2001b). However, adeeper understanding of the dynamics of the ELL burstmechanism could not be achieved due to the high di-mensionality (312 compartments and 10 ionic currents)of the model system.

The analysis of bursting neurons using dynami-cal systems and bifurcation theory is well established

(Rinzel, 1987; Rinzel and Ermentrout, 1989; Wangand Rinzel, 1995; Bertram et al., 1995; Hoppensteadtand Izhikevich, 1997; Izhikevich, 2000; Golubitskyet al., 2001). These studies have reduced complexneural behavior to flows of low-dimensional nonlin-ear dynamical systems. In the same spirit, we presenthere a two-compartment reduction of our detailed ionicmodel of an ELL pyramidal cell (Doiron et al., 2001a,2001b). The reduced model, referred to as the ghost-burster (this term is explained in the text), producesburst discharges similar to both the full model andin vitro recordings of bursting ELL pyramidal cells.This analysis supports our previous predictions onthe sufficient ionic and morphological requirements ofthe ELL pyramidal cell burst mechanism. In additionto this, the low dimension of this model allows fora detailed dynamical systems treatment of the burstmechanism.

When applied depolarization is treated as a bifur-cation parameter, the model cell shows three distinctdynamical behaviors: resting with low-intensity depo-larizing current, tonic firing at intermediate levels, andchaotic burst discharge at high levels of depolarization.This is contrary to other burst mechanisms that showburst discharge for low levels of depolarization and thentransition to tonic firing as applied current is increased(Hayashi and Ishizuka, 1992; Gray and McCormick,1996; Steriade et al., 1998; Wang, 1999). Both of the bi-furcations separating the three dynamical behaviors ofthe ghostburster are shown to be saddle-node bifurca-tions of either fixed points (quiescent to tonic firing) orlimit cycles (tonic firing to bursting). Treating our burstmodel as a fast-slow burster (Rinzel, 1987; Rinzel andErmentrout, 1989; Wang and Rinzel, 1995; Izhikevich,2000) and using quasi-static bifurcation analysis, weshow that the burst termination is linked to a transi-tion from period-one to period-two firing in the fastsubsystem, causing the burst trajectory to be reinjectednear the “ghost” of the saddle-node bifurcation of fixedpoints. The time spent near the saddle-node determinesthe interburst interval length.

This concept of burst discharge is quite differentfrom the two-bifurcation analysis used to understandmost other burst models (Rinzel, 1987; Rinzel andErmentrout, 1989; Wang and Rinzel, 1995; de Vries,1998; Izhikevich, 2000; Golubitsky et al., 2001). Fur-ther analysis predicts that the route to chaos in transi-tioning from tonic to chaotic burst firing is through typeI intermittency (Pomeau and Manneville, 1980). Com-parisons of both model and experimental ELL burst

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Ghostbursting: A Novel Neuronal Burst Mechanism 7

recordings support this prediction. Furthermore, bychanging the relative position of the two-saddle-nodebifurcations in a two-parameter bifurcation set, thetime scales of both the burst and interburst period canbe chosen independently, allowing for wide variationsin possible burst outputs.

2. Methods

2.1. ELL Pyramidal Cell Bursting

Figure 1A shows in vitro recordings from the somaof a bursting ELL pyramidal cell with a constant de-polarizing input. The bursts comprise a sequence ofaction potentials, which appear on top of a slow depo-larization of the subthreshold membrane potential. Thedepolarization causes the interspike intervals (ISIs) todecrease as the burst evolves. The ISI decrease culmi-nates in a high-frequency spike doublet that triggers arelatively large after-hyperpolarization (AHP) labeleda burst-AHP (bAHP). The bAHP causes a long ISI thatseparates the train of action potentials into bursts, twoof which are shown in Fig. 1A. The full characteriza-tion of the burst sequence has been presented in Lemonand Turner (2000).

Immunohistochemical studies of the apical dendritesof ELL pyramidal cells have indicated a patched dis-tribution of sodium channels along the first ∼200 µmof the apical dendrite (Turner et al., 1994). Figure 1Billustrates schematically such a Na+ channel distribu-tion over the dendrite. The active dendritic Na+ allowsfor action potential backpropagation along the apicaldendrite, producing a dendritic spike response (Fig. 1B;Turner et al., 1994). Na+ or Ca2+ mediated action po-tential backpropagation has been observed in severalother central neurons (Turner et al., 1994; Stuart andSakmann, 1994; for a review of active dendrites, seeStuart et al., 1997) and has been modeled in many stud-ies (Traub et al., 1994; Mainen et al., 1995; Vetter et al.,2001; Doiron et al., 2001b). Action potential backprop-agation produces a somatic depolarizing after-potential(DAP) after the somatic spike, as shown in Fig. 1B.The DAP is the result of a dendritic reflection of thesomatic action potential. This requires both a long den-dritic action potential half-width as compared to thatof a somatic action potential and a large somatic hy-perpolarization succeeding an action potential. Thesetwo features allow for passive electrotonic current flowfrom the dendrite to the soma subsequent to the somaticspike, yielding a DAP.

Figure 1. ELL burst discharge and dendritic backpropagation. A:In vitro recording of burst discharge from the soma of an ELL pyra-midal cell with constant applied depolarizing current. Two bursts ofaction potentials are shown, each exhibiting a growing depolarizationas the burst evolves, causing the ISI to decrease; the burst ends witha high-frequency doublet ISI. The doublet triggers a sharp removalof the depolarization, uncovering a prominent AHP, labeled a burst-AHP. B: Active Na+ conductances are distributed along the somaand proximal apical dendrite of ELL pyramidal cells (left). Na+ re-gions are indicated with vertical bars to the left of the schematic. Notethat the distribution of dendritic Na+ is punctuate, giving regions ofhigh Na+ concentration (often referred to as “hot spots”) separatedby regions of passive dendrite. The active dendritic regions allow forbackpropagation of a somatic action potential through a dendriticaction potential response, as seen from ELL recordings from boththe soma and proximal (∼150 µm) dendrite (right). Somatic actionpotential rectification by K+ currents and the broader action potentialin the dendrite allow for electrotonic conduction of the dendritic ac-tion potential to the soma, resulting in a DAP at the soma (inset). Wethank R.W. Turner for generously providing his data for the figure.

Recent work has shown the necessity of spike back-propagation in ELL pyramidal cells for burst dis-charge (Turner et al., 1994; Turner and Maler, 1999;Lemon and Turner, 2000). These studies blocked spike

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8 Doiron et al.

backpropagation by locally applying tetrodoxin (TTX,a Na+ channel blocker) to apical dendrites of ELL pyra-midal cells, after which all bursting ceased and onlytonic firing persisted. Our previous modeling study(Doiron et al., 2001b) reproduced this result, sincewhen active Na+ conductances were removed fromall dendritic compartments, similar results were ob-tained. However, in that study we modeled the prox-imal apical dendrite with 10 compartments, five ofwhich contained active spiking Na+ channels. Thelarge number of variables in such a model is incom-patible with the objectives of the present study. In lightof this and following previous modeling studies in-volving action-potential backpropagation (Pinsky andRinzel, 1994; Bressloff, 1995; Mainen and Sejnowski,1996; Lansky and Rodriguez, 1999; Wang, 1999; Boothand Bose, 2001), we investigate a two-compartmentmodel of an ELL pyramidal cell, where one compart-ment represents the somatic region, and the secondthe entire proximal apical dendrite. Note that a two-compartment treatment of dendritic action potentialbackpropagation is a simplification of the cable equa-tion (Keener and Sneyd, 1998). However, in considera-tion of the goals of the present study, which require onlyDAP production, the two-compartment assumption issufficient.

2.2. Two-Compartment Model

A schematic of our two-compartment model of an ELLpyramidal cell is shown in Fig. 2, together with the ac-tive inward and outward currents that determine thecompartment membrane potentials. Both the soma anddendrite contain fast inward Na+ currents, INa,s andINa,d , and outward delayed rectifying (Dr) K+ currents,respectively IDr,s and IDr,d . These currents are neces-sary to reproduce somatic action potentials and properspike backpropagation that yields somatic DAPs. In ad-dition, both the soma and dendrite contain passive leakcurrents Ileak. The membrane potentials Vs (somatic)and Vd (dendritic) are determined through a modi-fied Hodgkin/Huxley (1952) treatment of each com-partment. The coupling between the compartments isassumed to be through simple electrotonic diffusiongiving currents from soma to dendrite Is/d , or vice-versa Id/s . In total, the dynamical system comprisessix nonlinear differential equations, Eqs. (1) through(6); henceforth, we refer to Eqs. (1) through (6) as theghostburster model, and the justification for the nameis presented in the Results section.

Figure 2. Schematic of two-compartment model representation ofan ELL pyramidal cell. The ionic currents that influence both the so-matic and dendritic compartment potentials are indicated. Arrowsthat point into the compartment represent inward Na+ currents,whereas arrows pointing outward represent K+ currents (the specificcurrents are introduced in the text). The compartments are joinedthrough an axial resistance, 1/gc , allowing current to be passed be-tween the somatic and dendritic compartments.

Soma:

dVs

dt= IS + gNa,s · m2

∞,s(Vs) · (1 − ns) · (VNa − Vs)

+ gDr,s · n2s · (VK − Vs) + gc

κ· (Vd − Vs)

+ gleak · (Vl − Vs) (1)dns

dt= n∞,s(Vs) − ns

τn,s(2)

Dendrite:

dVd

dt= gNa,d · m2

∞,d(Vd) · hd · (VNa − Vd)

+ gDr,d · n2d · pd · (VK − Vd)

+ gc

(1 − κ)· (Vs − Vd) + gleak · (Vl − Vd) (3)

dhd

dt= h∞,d(Vd) − hd

τh,d(4)

dnd

dt= n∞,d(Vd) − nd

τn,d(5)

dpd

dt= p∞,d(Vd) − pd

τp,d(6)

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Ghostbursting: A Novel Neuronal Burst Mechanism 9

Table 1. Model parameter values. The values correspond to theparameters introduced in Eqs. (1) through (6). Each ionic current(INa,s ; IDr,s ; INa,d ; IDr,d ) is modeled by a maximal conductancegmax (in units of mS/cm2), sigmoidal activation and possibly in-activation, infinite conductance curves involving both V1/2 and kparameters m∞,s(Vs) = 1

1+e−(Vs −V1/2)/k , and a channel time con-stant τ (in units of ms). Double entries x /y correspond to channelswith both activation (x) and inactivation (y), respectively. If theactivation time constant value is N/A, then the channel activationtracks the membrane potential instantaneously. Other parametersvalues are gc = 1, κ = 0.4, VNa = 40 mV, VK = −88.5 mV,Vleak = −70 mV, gleak = 0.18, and Cm = 1 µF/cm2. These valuescompare in magnitude to those of other two-compartment models(Pinsky and Rinzel, 1994; Mainen and Sejnowski, 1996).

Current gmax V1/2 K τ

INa,s(m∞,s(Vs)) 55 −40 3 N/A

IDr,s(ns(Vs)) 20 −40 3 0.39

INa,d (m∞,d (Vd )/hd (Vd )) 5 −40/−52 5/−5 N/A/1

IDr,d (nd (Vd )/pd (Vd )) 15 −40/−65 5/−6 0.9/5

Table 1 lists the values of all channel parametersused in the simulations. The soma is modeled with twovariables (see Eqs. (1) and (2)). The reduction from theclassic four dimensional Hodgkin-Huxley model is ac-complished by slaving INa,s activation, m∞,s , to Vs (i.e.,the Na+ activation ms tracks Vs instantaneously), andmodeling its inactivation, hs , through IDr,s activation,ns (we set hs ≡ 1 − ns). This second approximation isa result of observing in our large compartmental model(Doiron et al., 2001b) that hs + ns ≈ 1 during spikingbehavior. Both of these approximations have been usedin various other models of spiking neurons (Keener andSneyd, 1998). The dendrite is modeled with four vari-ables (see Eqs. (3) through (6)). Similar to the treatmentof INa,s , we slave INa,d activation m∞,d to Vd but modelits inactivation with a separate dynamical variable hd .Lemon and Turner (2000) have shown that the refrac-tory period of dendritic action potentials is larger thanthat of somatic in ELL pyramidal neurons. This resulthas previously been shown to be necessary for bursttermination (Doiron et al., 2001b). To model the dif-ferential somatic/dendritic refractory period we havechosen τh,d to be longer than τn,s (similar to our largecompartmental model, Doiron et al., 2001b). This re-sult has not been directly verified through immunohis-tochemical experiments of ELL pyramidal cells Na+

channels; thus, at present, this remains an assumptionin our model.

The crucial element for the success of our model inreproducing bursts is the treatment of IDr,d . Dendritic

recordings from bursting ELL pyramidal cells showa slow, frequency-dependent broadening of the actionpotential width as a burst evolves (Lemon and Turner,2000). Such a cumulative increase in action potentialwidth has been observed in other experimental prepa-rations and has been linked to a slow inactivation ofrectifier-like K+ channels (Aldrich et al., 1979; Ma andKoester, 1996; Shao et al., 1999). In light of this, ourprevious study (Doiron et al., 2001b) modeled the den-dritic K+ responsible for spike rectification with bothactivation and inactivation variables. When the timeconstant governing the inactivation was relatively long(5 ms) compared with the time constants of the spik-ing currents (∼1 ms), the model produced a burst dis-charge comparable to ELL pyramidal cell burst record-ings. Doiron et al. (2001b) also considered other po-tential burst mechanisms, including slow activation ofpersistent sodium; however, only slow inactivation ofdendritic K+ produced burst results comparable to ex-periment. At this time there is no direct evidence fora cumulative inactivation of dendritic K+ channels inELL pyramidal cells, and these results remain a modelassumption. However, preliminary work suggests thatthe shaw-like AptKv3.3 channels may express such aslow inactivation (R.W. Turner personal communica-tion); these channels have been shown to be highly ex-pressed in the apical dendrites of ELL pyramidal cells(Rashid et al., 2001). In the present work, our dynam-ical system also models dendritic K+ current IDr,d , ashaving both activation nd and slow inactivation pd vari-ables (see Eqs. (3), (5), and (6)). Slow inactivation ofK+ channels, although not a mechanism in contempo-rary burst models, was proposed by Carpenter (1979) inthe early stages of mathematical treatment of burstingin excitable cells. We do not implement a similar slowinactivation of somatic Dr,s since somatic spikes ob-served in bursting ELL pyramidal cells do not exhibitbroadening as the burst evolves (Lemon and Turner,2000).

The somatic-dendritic interaction is modeled as sim-ple electrotonic diffusion with coupling coefficient gc

and is scaled by the ratio of somatic-to-total surfacearea κ . This form of coupling has been used in pre-vious two-compartment neural models (Mainen andSejnowski, 1996; Wang, 1999; Kepecs and Wang,2000; Booth and Bose, 2001). IS represents either anapplied or synaptic current flowing into the somaticcompartment. In the present study IS is always constantin time and is used as a bifurcation parameter. Physi-ological justification for the parameter values given in

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10 Doiron et al.

Table 1 is presented in detail in Doiron et al. (2001b).Equations (1) through (6) are integrated by a fourth-order Runge-Kutta algorithm with a fixed time step of�t = 5 × 10−6 s.

3. Results

3.1. Model Performance

Figure 3A and B shows simulation time series of Vs

and pd , respectively, for the ghostburster with constantdepolarization of IS = 9. We see a repetitive burst trainsimilar to that shown in Fig. 1A. Figure 4 compares thetime series of Vs and Vd for the ghostburster (bottomrow) during a single burst to both a somatic and den-dritic burst from ELL pyramidal cell recordings (toprow) and the large compartmental model presented inDoiron et al. (2001b) (middle row). All burst sequencesare produced with constant somatic depolarization. Thesomatic bursts all show the same characteristic growthin depolarization (DAP growth) and consequent de-creases in ISI leading to the high-frequency doublet.The dendritic bursts all show that a dendritic spike fail-ure is associated with both doublet spiking and bursttermination. The somatic AHPs in the simulation of theghostburster do not show a gradual depolarization dur-

Figure 3. Model bursting. A: Time series of the somatic poten-tial Vs during burst output. B: Dendritic IDr,d inactivation variablepd during the same burst simulation as in A. Note the cumulative(slow) inactivation as the burst evolves and the rapid recovery frominactivation during the interburst period.

ing the burst, as do both the AHPs in the ELL pyrami-dal cell recordings and the large compartmental modelsimulations. This is a minor discrepancy, which is notrelevant for the understanding of the burst mechanism.

The mechanism involved in the burst sequencesshown in Figs. 3 and 4 has been explained in de-tail (although not from a dynamical systems point ofview) in past experimental and computational studies(Lemon and Turner, 2000; Doiron et al., 2001b). Wegive a short overview of this explanation. Action po-tential backpropagation is the process of a somatic ac-tion potential actively propagating along the dendritedue to activation of dendritic Na+ channels. Rapid hy-perpolarization of the somatic membrane, mediated bysomatic potassium activation ns , allows electrotonicdiffusion of the dendritic action potential, creating aDAP in the somatic compartment. However, with repet-itive spiking the dendritic action potentials, shown byVd , broaden in width and show a baseline summation(Fig. 4). This is due to the slow inactivation of IDr,d ,mediated by pd , as shown in Fig. 3B. This further driveselectrotonic diffusion of the dendritic action potentialback to the soma; consequently, the DAP at the somagrows, producing an increased somatic depolarizationas the burst evolves. This results in decreasing somaticISIs, as experimentally observed during ELL burst out-put. This positive feedback loop between the soma anddendrite finally produces a high-frequency spike dou-blet (Fig. 4).

Doublet ISIs are within the refractory period of den-dritic spikes but not that of somatic spikes (Lemon andTurner, 2000). This causes the backpropagation of thesecond somatic spike in the doublet to fail, due to lackof recovery of INa,d from its inactivation, as shown inthe dendritic recordings (Fig. 4). This backpropaga-tion failure removes any DAP at the soma, uncoveringa large bAHP, and thus terminates the burst. This cre-ates a long ISI, the interburst period, which allows pd

and hd to recover, in preparation for the next burst (seeFig. 3B).

3.2. Bifurcation Analysis

In the following sections we use dynamical systemstheory to explore various aspects of the ghostbursterequations (Eqs. (1) through (6)). An introduction tosome of the concepts we use can be found, for example,in Strogatz (1994). An alternative explanation of theburst mechanism, given in physiological terms, waspresented in Doiron et al. (2001b).

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Ghostbursting: A Novel Neuronal Burst Mechanism 11

Figure 4. Model performance. A single burst is obtained from ELL pyramidal cell recordings (top row; data donated by R.W. Turner),full multicompartmental model simulations (middle row; simulation presented in Doiron et al., 2001b), and reduced two-compartment modelsimulations (bottom row; Eqs. (1) through (6)). All bursts are produced by applying constant depolarization to the soma (0.3 nA top; 0.6nA middle; Is = 9, bottom). The columns show both somatic and dendritic responses for each row. The reduced-model somatic spike trainreproduces both the in vitro data and full-model simulation spike trains by showing the growth of DAPs and reduction in ISI as the burst evolves.All somatic bursts are terminated with a large bAHP, which is connected to the dendritic spike failure.

Figure 5A gives the bifurcation diagram of hd ascomputed from the ghostburster with IS treated as thebifurcation parameter. We chose IS since this is both anexperimentally and physiologically relevant parameter

to vary. Three distinct dynamical behaviors are ob-served. For IS < IS1 two fixed points exist—one sta-ble, representing the resting state, and one unstablesaddle. When IS = IS1, the stable and unstable fixed

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12 Doiron et al.

Figure 5. A: Bifurcation diagram of the ghostburster equations (Eqs. (1) through (6)) as a function of the bifurcation parameter IS . We choosehd as the representative dynamic variable and plot hd on the vertical axis. For IS < IS1 a stable fixed point (solid line) and a saddle (dashed line)coexist. A saddle-node bifurcation of fixed points (SNFP) occurs at IS = IS1. For IS1 < IS < IS2 stable (filled circles) and unstable (open circles)limit cycles coexist, the maximum and minimum of which are plotted. A saddle-node bifurcation of limit cycles (SNLC) occurs at IS = IS2. ForIS > IS2 a chaotic attractor exists; we show this by plotting the maximum and minimum of hd for all ISIs that occur in a 1 s simulation for fixedIS . A reverse-period doubling cascade out of chaos is observed for large IS . The software package AUTO (Doedel, 1981) was used to constructthe leftmost part of the diagram. B: Instantaneous frequency (1/ISI) is plotted for IS simulations of the ghostburster model for each incrementin IS . The transitions from rest to tonic firing and tonic firing to chaotic bursting are clear. C: The maximum Lyapunov exponent λ as a functionof IS .

points coalesce in a saddle-node bifurcation of fixedpoints on an invariant circle, after which a stable limitcycle exists. This is characteristic of Class I spike ex-citability (Ermentrout, 1996), of which the canonical

model is the well-studied θ neuron (Hoppensteadt andIzhikevich, 1997). For IS1 < IS < IS2 the stable limitcycle coexists with an unstable limit cycle. Both limitcycles coalesce at IS = IS2 in a saddle-node bifurcation

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Ghostbursting: A Novel Neuronal Burst Mechanism 13

of limit cycles. For IS > IS2 the model dynamics, lack-ing any stable periodic limit cycle, evolve on a chaoticattractor giving bursting solutions as shown in Figs. 3and 4 (lower panel). As IS increases further a perioddoubling cascade out of chaos is observed, and a period-two solution exists for high IS . The importance of bothof the saddle-node bifurcations will be explored in latersections.

Figure 5B shows the observed spike discharge fre-quencies f (≡1/ISI) from the ghostburster as IS is var-ied over the same range as in Fig. 5A. The rest stateIS < IS1 admits no firing, indicated by setting f = 0.For IS1 < IS < IS2 the stable-limit cycle attractor pro-duces repetitive spike discharge giving a single nonzerof value for each value of IS . f becomes arbitrarily smallas IS approaches IS1 from above due to the infinite-period bifurcation at IS1. However, for IS > IS2 theattractor produces a varied ISI pattern, as shown inFigs. 3 and 4. This involves a range of observed f val-ues for a given fixed IS , ranging from ∼100 Hz in theinterburst interval to almost 700 Hz at the doublet fir-ing. The burst regime, IS > IS2 does admit windowsof periodic behavior. A particularly large window ofIS ∈ (13.13, 13.73) shows a stable period six solutionthat undergoes a period doubling cascade into chaos asIS is decreased. Finally, the period doubling cascadeout of chaos for IS � IS2 is evident.

Figure 5C shows the most positive Lyapunov expo-nent λ of the ghostburster as a function of IS . We seethat λ < 0 for IS < IS1 because the only attractor isa stable fixed point. For IS1 < IS < IS2, λ = 0 becausethe attractor is a stable limit cycle. Of particular inter-est is that λ is positive for a range of IS greater thanIS2, indicating that the bursting is chaotic. The win-dows of periodic behavior within the chaotic burstingare indicated by λ being zero (e.g., the large windowfor IS ∈ (13.13, 13.73)). For IS > 17.65, λ = 0 becausethe ghostburster undergoes a period doubling cascadeout of chaos, resulting in a stable period two solution.

Figure 6 is a two parameter bifurcation set showingcurves for both the saddle-node bifurcation of fixedpoints (SNFP) and of limit cycles (SNLC). The pa-rameters are the applied current IS , already studied inFig. 5, and gDr,d , which controls the influence of theslow dynamical variable pd (see Eq. (3)). It is natural tochoose gDr,d as the second bifurcation parameter sincethe burst mechanism involves dendritic backpropaga-tion, which IDr,d regulates, and gDr,d can be experi-mentally adjusted by focal application of K+ channelblockers to the apical dendrites of ELL pyramidal cells

Figure 6. Two-parameter bifurcation set. Both the saddle-node bi-furcations of fixed points (SNFP) and limit cycles (SNLC) bifurca-tions were tracked, using AUTO (Doedel, 1981) in the (IS , gDr,d )subspace of parameter space. The curves partition the space intoquiescence, tonic firing, and chaotic bursting regimes.

(Rashid et al., 2001). A vertical line in Fig. 6 corre-sponds to a bifurcation diagram similar to that pre-sented in Fig. 5A. The diagram in Fig. 5A correspondsto the rightmost value of gDr,d in Fig. 6 (gDr,d = 15).The intersection of the curves SNFP and SNLC withany vertical line gives the values IS1 and IS2 for that par-ticular value of gDr,d . Thus, the curves SNFP and SNLCpartition parameter space into regions corresponding toquiescence, tonic firing, and chaotic bursting solutionsof the ghostburster equations, as indicated in Fig. 6.The curves intersect at a codimension-two bifurcationpoint corresponding to simultaneous fixed-point andlimit-cycle saddle-node bifurcations. The curve to theleft of the intersection point corresponds to the codi-mension one SNFP curve; there is no stable period-onelimit cycle corresponding to tonic firing in this region.Figure 6 demonstrates that it is possible to make IS1 andIS2 arbitrarily close, by choosing gDr,d appropriately.This property is of use later in the study.

3.3. The Burst Mechanism: Reconstructingthe Burst Attractor

The dynamical system described by the ghostbursterequations possesses two separate time scales. The timeconstants governing the active ionic channels ns , hd ,and nd , are all ∼1 ms, and the half width of the spike

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14 Doiron et al.

response of the membrane potentials Vs and Vd are∼0.5 ms and 1.1 ms, respectively. However, the timescale of pd is characterized by τp,d , which is a factor offive times larger than any of the other time scales. Pre-vious studies of other burst models have profited from asimilar coexistence of at least two time scales of activityduring bursting (Rinzel, 1987; Rinzel and Ermentrout,1989; Wang and Rinzel, 1995; Bertram et al., 1995; deVries, 1998; Izhikevich, 2000; Golubitsky et al., 2001).This allowed for a separation of the full dynamical sys-tem into two smaller subsystems, one fast and one slow.We also treat our burst model as a fast-slow burster. Thenatural variable separation is to group Vs , ns , Vd , hd ,and nd into a fast subsystem, denoted by the vector x,

while the slow subsystem consists solely of pd . Thisgives the simplified notation of our model,

dx

dt= f (x, pd) (7)

dpd

dt= pd,∞(x) − pd

τp,d, (8)

where f (x, pd) represents the right-hand side ofEqs. (1) through (5) and Eq. (8) is simply Eq. (6)restated.

Since pd changes on a slower time scale than x ,we approximate pd as constant and use pd as a bi-furcation parameter of the fast subsystem (quasi-static

Figure 7. A: Quasistatic bifurcation diagram. pd is fixed as a bifurcation parameter, while Vd is chosen as a representative variable from thefast-subsystem x . The maxima in the dendritic voltage ( dVd

dt = 0 and d2Vddt2

< 0) are plotted for each value of pd . At pd = pd1, the maxima of Vd

switch to two values, corresponding to the values taken during each ISI of a period-two solution. B: Time series of the dendritic voltage, Vd (t),while pd = 0.13 > pd1. The fast subsystem follows a period-one solution. C: Time series of the dendritic voltage, Vd (t), while pd = 0.08 < pd1.The fast subsystem follows a period-two solution. A constant value of IS = 9 > IS1 is chosen for all simulations in A, B, and C .

approximation; see, e.g., Hoppensteadt and Izhikevich,1997). We note that with pd constant the fast subsystem(7) cannot produce bursting comparable to that seenfrom ELL pyramidal cells. Bursting requires the slowvariable to modulate DAP growth dynamically (Doironet al., 2001b). Treating pd as a bifurcation parameterwill show how changes in pd produce the character-istics of ELL bursting through the bifurcation struc-ture of the fast subsystem. Since pd directly affects thefast subsystem only through the dynamics of Vd (seeEq. (3)), we choose Vd as a representative variable ofthe fast subsystem x .

Figure 7A shows the local maxima of Vd on a peri-odic orbit as a function of pd , while the fast subsystemis driven with IS = 9 > IS2. At a critical value of pd , la-beled pd1, the fast subsystem goes through a transitionfrom a period-one to a period-two limit cycle. Thisis shown by only one maximum in Vd for pd > pd1,whereas there are two maxima for pd < pd1. Figure 7Bshows a time series of Vd (t) following the period-onelimit cycle when pd = 0.13 > pd1, while Fig. 7C showsthe period-two limit cycle when pd = 0.08 < pd1. Thesecond dendritic action potential in the period-two or-bit (Fig. 7C) is of reduced amplitude; this correspondsto the dendritic failure observed in the full dynamicalsystem (Eqs. (7) and (8)) when pd is low (see right col-umn of Fig. 4). The bifurcation diagram in Fig. 7A maybe thought of as a “burst shell” in a projection of phase

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Ghostbursting: A Novel Neuronal Burst Mechanism 15

Figure 8. pd (t) and pd computed from integration of the ghost-burster equations with IS = 9 > IS2. Four bursts are shown with thecorresponding time-stamped spikes given above for reference. Aslow burst oscillation in pd (t) is observed. It is evident that the dis-crete function pd (solid circles) tracks the burst oscillation in pd (t).pd shows a monotonic decrease throughout the burst until the in-terburst interval, at which point pd is reinjected to a higher value.The horizontal lines are the values pd1, corresponding to the perioddoubling transition, and pd2, corresponding to the crossing of thenullcline curve with the 〈Vd 〉 curve. The pd (t) reinjection occursafter pd (t) < pd2 as explained in the text and in Fig. 9A. pd has beentranslated downward to lie on top of the pd (t) time series. This isrequired because Eq. (10) uses a unweighted average of Vd , given inEq. (9). This produces a pd series that occurs at higher values thanpd (t) because Eqs. (9) and (10) ignore the low-pass characteristicsof Eq. (6). However, only the shape of pd is of interest, and this isnot affected by the downward translation.

space. The full burst dynamics will evolve on the burstshell as pd is modulated slowly by the fast subsystem.We therefore next address the dynamics of pd (t) duringthe burst trajectory in the fast subsystem.

On inspection of Fig. 3B it is clear that there exist twooscillations in pd (t)—one fast oscillation occurring onthe time scale of spikes and the other on a much longertime scale tracking the bursts. Figure 8 shows pd (t)during a burst solution of the full dynamical system. Itis clear that the fast spike oscillations in pd (t) are drivenby the instantaneous value of Vd (t). This is due to τp

being small enough to allow pd (t) to be affected bythe spiking in the fast subsystem. In addition, there isa general decrease in pd (t) as the burst evolves anda sharp increase in pd (t) after the doublet ISI. The

increase reinjects pd (t) to a higher value allowing theburst oscillation to begin again. The period of a burstoscillation encompasses several spikes and thus cannotbe analyzed in terms of the instantaneous dynamics ofthe fast subsystem.

Due to the separation of time scales and the fact thatdpddt depends only on Vd (Eq. (6)), we expect that the

burst oscillation depends on the average of Vd betweenconsecutive spikes, defined as

〈Vd〉 = 1

ti+1 − ti

∫ ti+1

ti

Vd(t) dt, (9)

where ti is the time of the i th spike. We construct adiscrete function pd

pd = pd,∞(〈Vd〉), (10)

where pd,∞(·) is the infinite conductance curve as inEq. (6). Figure 8 shows a sequence of pd values con-structed by using 〈Vd〉 from the burst solution of thefull dynamical system. This sequence is plotted (solidcircles) on top of the full pd (t) dynamics during theburst train. It is evident that the time sequence of pd

is of the same shape as the burst oscillation in pd (t).This is evidence that the slow burst oscillation can beanalyzed by considering 〈Vd〉 .

We now complete the burst shell by adding to Fig. 7Athe nullcline for pd (from Eq. (6)) as well as 〈Vd〉 com-puted for the stable periodic solutions of the fast sub-system. This is shown in Fig. 9A. Note that as pd de-creases through pd1, 〈Vd〉 decreases by ∼10 mV. Thisis due to the dendritic spike failure and subsequentlong ISI occurring when pd < pd1, both contributingto lower Vd on average (see Fig. 7C). The pd nullclineand 〈Vd〉 curves cross at pd = pd2 < pd1. Since we haveshown that the burst oscillation is sensitive to 〈Vd〉, thecrossing corresponds to 〈 dpd

dt 〉 changing from negativeto positive (see Fig. 9D).

A saddle-node bifurcation of fixed points occurs atpd = p∗

d for some p∗d > pd1 (data not shown). This bi-

furcation is similar to the saddle-node bifurcation offixed points in Fig. 5A, where IS is the bifurcation pa-rameter. This is expected, since pd is the coefficientto a hyperpolarizing ionic current (see Eq. (3)); hencean increase in pd is equivalent to a decrease in depo-larizing IS . Because of the saddle-node bifurcation atpd = p∗

d , the period of the period-one limit cycle scalesas 1√

|pd−p∗d | for pd near p∗

d (Guckenheimer and Holmes,

1983).

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16 Doiron et al.

Figure 9. A: The bifurcation diagram of Fig. 6A is replotted along with the pd nullcline pd,∞(Vd ) (dashed line labeled N ). Note that the pd

nullcline is inverted so as to give Vd,∞(pd ) = V1/2,p − kp ln( 11−pd

). We plot the average of Vd over a whole period of Vd , 〈Vd 〉 (solid line),at a fixed pd . Note the sharp decline in 〈Vd 〉 for pd below pd1. B: The diagram in A is replotted with the labels removed. A single directedburst trajectory projected in the (Vd , pd ) plane obtained by integrating the full dynamical system (Eqs. (1) through (6)) is plotted on top of theburst shell. C: All observed discharge frequencies of the fast subsystem are plotted as a function of pd . At pd = pd1 a stable period-one firingpattern of ∼200 Hz changes to a period-two solution with one ISI being ∼(700 Hz)−1 and the other ∼(100 Hz)−1. The inverse of the ISIs ofthe single burst shown in Fig. 9B are plotted as well. The ISIs are numbered from 1 (the first ISI) through to 5 (doublet ISI) and 6 (interburstinterval). D: The average of the derivative of pd , 〈 dpd

dt 〉, is plotted for each ISI in the single burst shown in Fig. 8B. Only the long interburst ISIhas 〈 dpd

dt 〉 > 0; all other ISIs have 〈 dpddt 〉 < 0. A constant value of IS = 9 > IS1 is chosen for all simulations in A, B, C , and D.

With the burst shell now fully constructed (Fig. 9A),we place the full burst dynamics (Eqs. (7) and (8)) ontothe shell. This is shown in Fig. 9B. The directed tra-jectory is the full six-dimensional burst trajectory pro-jected into the Vd − pd subspace. As the burst evolves,pd (t) decreases from spike to spike in the burst. Thiscauses the frequency of spike discharge to increasedue to the gradual shift away from the saddle-nodebifurcation of fixed points at pd = p∗

d . However, oncepd(t) < pd1, the spike dynamics shift from period-onespiking to period-two spiking. This first produces ahigh-frequency spike doublet, which is then followedby a dendritic potential of reduced amplitude, caus-ing 〈Vd〉 to decrease. When pd(t) < pd2, 〈 dpd

dt 〉 > 0(see Fig. 9D), and pd (t) increases and is reinjected toa higher value. The reinjection toward the “ghost” of

the saddle-node bifurcation of fixed points at pd = p∗d

causes the ISI (the interburst interval) to be long, sincethe velocity through phase space is lower in this region.

Figure 9C shows the burst trajectory in the fre-quency domain. The period doubling is evident atpd = pd1 since two distinct frequencies are observedfor pd < pd1, corresponding to a period-two solution ofthe fast subsystem, whereas for pd > pd1 only a period-one solution is found. As pd is reduced in the period-one regime (pd > pd1), the frequency of the limit cycleincreases, due to the reduced effect of the hyperpolariz-ing current IDr,d . We superimpose the ISIs of the bursttrajectory shown in Fig. 9B on the frequency bifurca-tion diagram in Fig. 9C. The sequence begins with along ISI (numbered 1) with subsequent ISIs decreasing,culminating with the short doublet ISI (numbered 5).

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Ghostbursting: A Novel Neuronal Burst Mechanism 17

The reinjection of pd near p∗d occurs during the next

ISI (numbered 6). The reinjection causes this next ISIto be long; it separates the action potentials into bursts.Figure 9D shows the average of the derivative of pd ,〈 dpd

dt 〉 = 1ti+1−ti

∫ ti+1

ti(

dpddt ) dt, during each ISI in the burst

shown in Fig. 9B and C. Notice that 〈 dpddt 〉 is nega-

tive and decreases as the burst evolves. This is becausethe ISI length reduces as the burst evolves, allowing theburst trajectory to spend less time in the region wheredpddt > 0. However, a large fraction of the burst trajectory

during the interburst ISI (6) occurs in the region wheredpddt > 0. Hence, the average 〈 dpd

dt 〉 is greater than zerofor the interburst interval, producing the reinjection ofpd (t) to higher values.

Izhikevich (2000) has labeled the burst mechanismsaccording to the bifurcations in the fast subsystem thatoccur in the transition from quiescence to limit cycleand vice versa. Even though there is never a true “qui-escent” period during the burst-phase trajectory, theinterburst interval for our model is determined by theapproach to an infinite period bifurcation. This phe-nomenon is often labeled as sensing the “ghost” of abifurcation (Strogatz, 1994), and we naturally label theburst mechanism as ghostbursting.

The ghostburster system exhibits bursting, for somerange of IS , only for 2 < τp < 110 ms, with all otherparameters as given in Table 1. The lower bound of τp

is due to the fact that the inactivation of IDr,d must becumulative for there to be a reduction of the ISIs asthe burst evolves. This requires a τp larger than thatof the ionic channels responsible for spike production(<1 ms). The upper bound on τp is also expected sincesignificant removal of pd inactivation during the inter-burst interval is necessary for another burst to occur.Too large a value of τp will not allow sufficient recoveryof IDr,d from inactivation, and therefore bursting willnot occur.

3.4. The Interburst Interval

By varying IS it is possible to set the interburst inter-val, TIB, to be different lengths. This is because afterthe dendrite has failed (removing the DAP at the soma),the time required to produce an action potential in thesomatic compartment (which is TIB) is dictated almostsolely by IS . The spike excitability of the somatic com-partment is Type I (Ermentrout, 1996), as evident fromthe saddle-node bifurcation of fixed points at IS = IS1.As a consequence TIB is determined from the well-known scaling law associated with saddle-node bifur-

Figure 10. Interburst interval. 〈TIB〉 is plotted as a function ofIS − IS1. The averaging was performed on 100 bursts produced bythe ghostburster equations at a specific IS . gDr,d was set to 12.14.〈TIB〉 shows a similar functional form to that described by Eq. (11).The dips in 〈TIB〉 are discussed in the text.

cations on a circle (Guckenheimer and Holmes, 1983),

TIB ∼ 1√IS − IS1

. (11)

Figure 10 shows the average interburst interval, 〈TIB〉,as a function of IS − IS1 for the ghostburster withgDr,d = 12.14. This value of gDr,d sets IS1 and IS2 closeto one another (see Fig. 6), allowing the system to burstwith values of IS close to IS1. It is necessary to forman average due to the chaotic nature of burst solutions.Nevertheless, 〈TIB〉 increases as IS approaches IS1, assuggested by Eq. (11). A linear regression fit of 1/〈TB〉2

against IS − IS1 gives a correlation coefficient of 0.845,further verifying that Eq. (11) holds. Figure 10 alsoshows downward dips in 〈TIB〉 that occur more fre-quently as IS − IS1 goes to zero. Time series of burstswith IS corresponding to the dips in 〈TIB〉 show scat-tered bursts with short interburst intervals that deviatefrom Eq. (11), amongst bursts with longer interburstintervals, which fit the trend described by Eq. (11).These scattered small values of TIB reduce 〈TIB〉 forthese particular values of IS . These dips contribute tothe deviation of the linear correlation coefficient citedabove from 1. We do not study the dips further since thebehavior has yet to be observed experimentally. How-ever, experimental measurements of ELL pyramidalcell-burst period do indeed show a lengthening of theperiod as the applied current is reduced (R.W. Turner,personal communication). This corresponds to the gen-eral trend shown in Fig. 10. Equation (11) and Fig. 10

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18 Doiron et al.

show that by choosing the model parameters properlyit is possible to regulate the effect of the ghost of thesaddle-node bifurcation of fixed points on the burstsolutions. We will show later how this property yieldsgreat diversity of time scales of possible burst solutionsof the ghostburster model.

3.5. The Burst Interval: Intermittency

Regions of chaotic and periodic behavior exist inmany burst models (Chay and Rinzel, 1985; Terman,1991, 1992; Hayashi and Ishizuka, 1992; Wang, 1993;Komendantov and Kononenko, 1996). The results ofFig. 5 show that periodic spiking and chaotic burstingare also two distinct dynamical behaviors of the ghost-burster. Moreover, the bifurcation parameter we haveused to move between both dynamical regimes is theapplied current IS , which mimics an average synapticinput to the cell. This indicates that changing the mag-nitude of input to the cell may cause a transition fromperiodic spiking to chaotic bursting. In ELL pyramidalcells a transition from tonic firing to highly variablebursting has been observed as applied depolarizing cur-rent is increased (Lemon and Turner, 2000; Bastian andNguyenkim, 2001; Doiron and Turner, unpublished re-sults). It remains to be shown that the experimentallyobserved bursting is indeed chaotic; preliminary re-sults suggest that such an analysis is difficult due tononstationarity in the data (Doiron and Turner, unpub-lished observations). Nonetheless, understanding thetransitions or routes to chaos in the model separatingtonic and chaotic burst regimes is necessary not onlyfor a complete description of the dynamics of the modelbut also for characterizing the input-output relation ofbursting ELL pyramidal cells.

Figure 5A shows that the transition from periodicspiking to chaotic bursting occurs at IS = IS2 when astable limit cycle collides with an unstable limit cyclein a saddle-node bifurcation of limit cycles. Since weare analyzing spiking behavior on both sides of the bi-furcation, it is natural to consider the ISI return map forIS near IS2. We choose IS slightly larger than IS2 andplot in Fig. 11A the ISI return map for a single burstsequence from the ghostburster (for IS1 < IS < IS2 thereturn map is a single point). We have labeled the re-gions of interest in the figure and explain each regionin order: (1) The burst begins here. (2) The ISI se-quence approaches the diagonal. This produces a clus-tering of points corresponding to the pseudo-periodicbehavior observed in the center of the burst. We refer

Figure 11. Burst intermittency. A: The ISI return map for a single-burst sequence with IS = 6.587 and gDr,d = 13 is shown (for theseparameters IS1 = 5.736 and IS2 = 6.5775). The diagonal is plottedas well (dashed line). The labels (1) through (5) are explained in thetext. B: The ISI return map for a single-burst sequence with IS = 9and gDr,d = 15 as in Fig. 3. C: The ISI return map for a single-burst recording from an ELL pyramidal cell (data courtesy of R.W.Turner). Compare with the model burst sequence in B.

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Ghostbursting: A Novel Neuronal Burst Mechanism 19

to this region of the map as a trapping region. (3) TheISI sequence leaves the trapping region with a down-ward trend. (4) The interburst interval involves a sharptransition from small ISI to large ISI. (5) The ISI se-quence returns to the trapping region and another burstbegins.

The above description indicates that the route tochaos is Type I intermittency (Manneville and Pomeau,1980; Guckenheimer and Holmes, 1983). Intermit-tency involves seemingly periodic behavior separatedby brief excursions in phase space. The clustering ofpoints in the ISI return map in the trapping region ofFig. 11A (labeled 2) is a manifestation of this appar-ent periodic firing. A trapping region is a characteristicfeature of Type I intermittency and corresponds to asaddle-node bifurcation of fixed points in the returnmap (which is the saddle-node bifurcation of limit cy-cles in the continuous system), occurring specificallyat IS = IS2 for the ghostburster equations. The escapeand return to the trapping region (regions 3, 4, 5 inFig. 11A) are the brief excursions. These events corre-spond to the period doubling transition and the cross ofthe 〈Vd〉 curve and pd nullcline, in the fast subsystem,as explained in Fig. 9. Figure 11B shows the ISI returnmap for a model burst of seven spikes and Fig. 11Cthe same map for a seven-spike burst recording froman ELL pyramidal cell. Both maps show the qualita-tive structure similar to in Fig. 11A, including a clearescape from and reinjection into a trapping region nearthe diagonal. Wang (1993) has also observed Type Iintermittency in the Hindmarsh-Rose model.

Since intermittent behavior is connected to a saddle-node bifurcation, the time spent in the trapping regionTB, corresponding to the burst period (the duration ofthe spikes in the cluster making up the burst), has awell-defined scaling law:

TB ∼ 1√IS − IS2

. (12)

Similar to Fig. 10 we consider the average of the burstperiod 〈TB〉 because of the chaotic nature of the burst-ing. Figure 12 shows that 〈TB〉 asymptotes to infinityas IS approaches IS2. Linear regression fits to 1/〈TB〉2

against IS − IS2 give a correlation coefficient of 0.886.These results validate Eq. (12) for the ghostburster burstsequences. Again the deviation in the correlation coef-ficient from 1 is caused by slight dips in 〈TB〉, similarto the dips observed in 〈TIB〉 (Fig. 10). By choosingthe quantity IS − IS2 we can obtain bursts with spikenumbers comparable to experiment.

Figure 12. Burst interval 〈TB〉 plotted as a function of IS − IS2. Theaveraging was performed on 100 bursts produced by the ghostbursterequations at a specific IS . gDr,d was set to 12.14. 〈TB〉 shows a similarfunctional form to that described by Eq. (12).

3.6. Gallery of Bursts

Equations (11) and (12) give the inverse square-rootscaling relations of TB and TIB, respectively. These re-sults showed that TB is determined by IS − IS2 and TIB

by IS − IS1. Using this fact and the ability to vary thedifference between IS2 and IS1 (see Fig. 6) we can pro-duce a wide array of burst patterns with differing timescales.

Figure 13A reproduces the (IS , gDr,d ) bifurcationset shown in Fig. 6. The letters B through F mark(IS , gDr,d ) parameters used to produce the spike trainsshown in the associated panels B through F of Fig. 13.Figure 13B uses (IS , gDr,d ) values such that the ghost-burster is in the tonic firing regime. The burst trainsshown in Figs. 3 and 4 correspond to (IS , gDr,d ) valuesin the burst regime of Fig. 6, which are not close toeither of the SNFP or SNLC curves. An example of aburst train with such a parameter choice is shown inFig. 13C. However, if we approach the SNLC curvebut remain distant from the SNFP curve, we can in-crease TB by one order of magnitude yet keep TIB thesame. The burst train in Fig. 13D shows an example ofthis. If we choose IS and gDr,d to be close to both theSNFP and SNLC curves, we can now increase TIB aswell (Fig. 13E). The interburst period TIB has now alsoincreased dramatically from that shown in Figs. 13Cand D.

Finally, for IS and gDr,d values to the left of thecodimension two bifurcation point, burst sequencesshow only a period-two solution (Fig. 13F). The burst

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20 Doiron et al.

Figure 13. Burst gallery. A: Reproduction of the two-parameter bifurcation set shown in Fig. 6. The letters B through F marked inside thefigure correspond to the (IS , gDr,d ) parameter values used to produce panels B through F, respectively. Examples of the interburst period TIB andburst period TB for each burst train are indicated (except for the tonic solution shown in B). The exact IS and gDr,d values used to produce eachspike train are as follows: B: IS = 6.5, gDr,d = 14. C: IS = 7.7, gDr,d = 13. D: IS = 7.6, gDr,d = 14. E: IS = 5.748, gDr,d = 12.14. F: IS = 5.75,gDr,d = 11. The vertical mV scale bar in C applies to all panels; however, each panel has its own horizontal time-scale bar.

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Ghostbursting: A Novel Neuronal Burst Mechanism 21

sequences are no longer chaotic. This is to be expectedsince there no longer is a saddle-node bifurcation oflimit cycles, which gave rise to the intermittency routeto chaos in the ghostburster equations (Fig. 11). Ap-proaching the SNFP curve allows for a large TIB, butthe fact that only bursts of two spikes can appear forcesTB to be small. The restriction of bursts to only doubletswhen gDr,d is small occurs because gDr,d is the coeffi-cient to the hyperpolarizing K+ current in the dendrite(Eq. (3)) and as such controls the effect of the DAP atthe soma. For gDr,d to the left of the codimension, two-bifurcation point, the first somatic spike in the doubletproduces a DAP of sufficient strength to cause the sec-ond somatic spike, which is within the refractory periodof the dendrite. Thus dendritic failure occurs after thefirst reflection, and the burst contains only two spikes.

4. Discussion

4.1. Ghostbursting: A Novel Burst Mechanism

We have introduced a two-compartment modelof bursting ELL pyramidal cells—the ghostburster.The model is a significant reduction of a largemulticompartmental ionic model of these cells(Doiron et al., 2001b). The large model was moti-vated by the conditional backpropagation burst mech-anism that has been experimentally characterized inELL pyramidal cells (Lemon and Turner, 2000). Theresults of Lemon and Turner (2000) and Doiron et al.(2001b) suggest that the ionic requirements necessaryand sufficient to support bursting as observed in theELL are (1) action-potential backpropagation along theapical dendrite sufficient to produce somatic DAPs, (2)the refractory period of dendritic action potentials thatis longer than that of the somatic potentials, and (3)slow inactivation of a dendritic K+ channel involved inrepolarization. The fact that the ghostburster was de-signed to contain only these three requirements and yetsucceeds in producing burst discharge comparable toexperiment suggests that these three requirements cap-ture the essential basis of the burst mechanism used inELL pyramidal cells.

The simplicity of the ghostburster, as compared tothe large compartmental model, has allowed us tounderstand, from a dynamical systems perspective,the mechanism involved in this type of bursting. Theghostburster was analyzed using a separation of thefull dynamical system into fast and slow subspaces

(Eqs. (7) and (8)), similar to the analysis of many otherburst models (Rinzel, 1987; Rinzel and Ermentrout,1989; Wang and Rinzel, 1995; Bertram et al., 1995;Hoppensteadt and Izhikevich, 1997; de Vries, 1998;Izhikevich, 2000; Golubitsky et al., 2001). Treating theslow dynamical variable pd as a bifurcation parameterwith respect to the fast subsystem allowed us to con-struct a burst shell on which the full burst dynamicsevolve. The shell shows that a transition from a period-one limit cycle to a period-two limit cycle occurs inthe dynamics of the fast subsystem as pd is reduced.The period-two limit cycle causes a sharp reduction in〈Vd〉 since the second spike of the limit cycle is of re-duced amplitude, due to dendritic refractoriness. Thereduction in 〈Vd〉 causes the 〈Vd〉(pd) curve to crossthe pd nullcline, and pd (t)grows during the second ISIof the period-two orbit. The growth in pd (t) reinjectspd (t) near a saddle-node bifurcation of fixed pointsoccurring at high pd . This passage near the ghost ofthe saddle-node bifurcation causes the ISI to be long,separating the action potentials into bursts.

Recently, Izhikevich (2000) has approached the clas-sification of bursters from a combinatorial point ofview. This has been successful in producing a largenumber of new fast-slow bursting mechanisms. One ofthese burst mechanisms has been recently observed in abiophysically plausible model of bursting corticotrophcells of the pituitary (Shorten and Wall, 2000). In con-trast, Golubitsky et al. (2001) (extending the work ofBertram et al., 1995, and de Vries, 1998) have classifiedbursters in terms of the unfoldings of high codimensionbifurcations. Both these methods have used the im-plicit assumption that burst initiation and terminationinvolve bifurcations from quiescence (or subthresholdoscillation) to limit cycle and vice-versa. However, ourburst mechanism does not appear in any of the aboveclassifications. This is because the trajectories in thefast subsystem of the ghostburster are always followinga limit cycle and are never in true quiescence, corre-sponding to a stable fixed point. The period of the limitcycle changes dynamically because the slow subsys-tem is oscillating, forcing the fast system to sometimespass near the ghost of an infinite period bifurcation.Furthermore, in the ghostburster, burst termination isconnected with a bifurcation from a period-one to aperiod-two limit cycle in the fast subsystem. This is anovel concept, since burst termination in all other burstmodels is connected with a transition from a period-onelimit cycle to a stable fixed point in the fast subsystem(Izhikevich, 2000; Golubitsky et al., 2001). Thus, while

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22 Doiron et al.

classifying burst phenomena through the bifurcationsfrom quiescence to a period-one limit cycle and vice-versa in the fast subsystem of a dynamical burstingmodel has had much success, our work requires an ex-tension of the classification of bursting to include analternative definition of quiescence and a burst attrac-tor that is composed of only period-one and period-twolimit cycles with no stable fixed points.

Rinzel (1987) shows that burst mechanisms with aone-dimensional slow subsystem require bistability inthe fast subsystem to exhibit bursting. The slow sub-system of ghostbuster equations is one dimensional, yetFig. 9 shows that the fast subsystem x is not bistable.This would seem to be a contradiction; however, recallthat as τp approaches values that are similar to otherbursting mechanisms, bursting is not observed. Thusour results do not contradict Rinzel’s previous studyand yet do support a separate mechanism entirely. Thisillustrates a key distinction between the ghostbursterand conventional bursting systems; the timescale of theslow variable has an upper bound in the ghostburster.The fast and slow timescales are sufficiently separateto allow us to successfully study the burst mechanismusing a quasistatic approximation. Thus ghostburst-ing, while distinct, does share similarities with conven-tional burst mechanisms. Note that mechanisms similarto ghostbursting, which involve a slow-passage phe-nomena (requiring saddle-node or homoclinic bifurca-tions), may exist, placing the ghostburster as only onein a family of new burst mechanisms.

The ghostburster model exhibits a threshold betweentonic firing and bursting behavior. Both Terman (1991,1992) and Wang (1993) have also identified thresholdsbetween these behaviors in the Hindmarsh-Rose modeland a modified version of the Morris-Lecar equations,respectively. Both of these models exhibited a homo-clinic orbit in the fast subsystem as the spiking phaseof a burst terminated. As a result, the bifurcations fromcontinuous spiking to bursting in the full dynamicswere complicated. Wang observed a crisis bifurcationat the transition (Grebogi et al., 1993), whereas Termanshowed that a series of bifurcations occurs during thetransition, which could be shown to exhibit dynamicssimilar to the Smale horseshoe map (Guckenheimerand Holmes, 1983). The saddle-node bifurcation oflimit cycles that separates the two regimes in theghostburster model is a great deal simpler than ei-ther of these bifurcations. However, Wang has shownthat an intermittent route to chaos is also observedin the Hindmarsh-Rose model as continuous spiking

transitions into bursting, much like the ghostburstersystem.

The fact that the transition from tonic firing to burst-ing in the ghostburster system occurs as depolariza-tion is increased is in contrast to both experimentaland modeling results of other bursting cells (Terman,1992; Hayashi and Ishizuka, 1992; Wang, 1993; Grayand McCormick, 1996; Steriade et al., 1998; Wang,1999). However, since many experimental and model-ing results, separate from ELL, show burst thresholdbehavior, the concept of burst excitability may havebroader implications. To expand, the saddle-node bi-furcation of limit cycles marking burst threshold canbe compared to the saddle-node bifurcation of fixedpoints, which is connected to the spike excitability ofType I membranes (Ermentrout, 1996; Hoppensteadtand Izhikevich, 1997). The functional implication of aburst threshold have yet to be fully understood; how-ever, recent work suggests that it may have importantimplications for both the signaling of inputs (Eguiaet al., 2000) and dividing cell response into stimulusestimation (tonic firing) and signal detection (bursting)(Sherman, 2001).

4.2. Predictions for Bursting in the ELL

An integral part of the burst mechanism in ELL pyra-midal cells is the interaction between the soma and den-drite through action potential backpropagation. Onepotential function of backpropagation is thought tobe retrograde signaling to dendritic synapses (Häusseret al., 2000). Further, a recent experimental study hasshown that the coincidence of action potential back-propagation and EPSPs produces a significant ampli-fication in membrane potential depolarization (Stuartand Häusser, 2001). These results may have conse-quences for both synaptic plasticity and dendritic com-putation. Our results (and those of others; see Häusseret al., 2000, for a review) imply that backpropagationcan also determine action potential patterning.

As mentioned above, the ghostburster exhibits athreshold separating tonic firing and bursting as de-polarization is increased. Similar behavior has beenobserved in both in vitro and in vivo experimentalrecordings of ELL pyramidal cells (Lemon and Turner,2000; Bastian and Nguyenkim, 2001) and in our fullcompartmental model simulations (data not shown).A reduction of burst threshold was observed in ELLpyramidal cells when TEA (K+ channel blocker) wasfocally applied to the proximal apical dendrite (Noonan

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Ghostbursting: A Novel Neuronal Burst Mechanism 23

et al., 2000; Rashid et al., 2001). Our work is consistentwith this observation, since dendritic TEA applicationis equivalent to a reduction in gDr,d conductance in ourmodel. Figure 6 shows that as gDr,d is reduced, burstthreshold is lowered.

Bursts, as opposed to individual spikes, have beensuggested to be a fundamental unit of information(Lisman, 1997). In fact, Gabbiani et al. (1996) havecorrelated bursts from ELL pyramidal cells with fea-tures in the stimulus driving the cell. Considering theseresults, it is possible that the time scale of burst-ing, T (=TB + TIB), could be tuned to sensory input;hence, the ability of a bursting cell to alter T may im-prove its coding efficiency. A natural method to al-ter T would be to change the time constants, τ , thatdetermine the slow process of the burst mechanism(Giannakopoulous et al., 2000). Nevertheless, toachieve an order of magnitude change in T requires apotentially large change in τ . Recently, Booth and Bose(2001) have shown, in a two-compartmental model of abursting CA3 pyramidal cell, that the precise timing ofinhibitory synaptic potentials can change the burst pe-riod T . Their results have potential implications for therate and temporal coding of hippocampal place cells.However, the ghostburster shows that both TB and TIB

can be changed by an order of magnitude but with onlysmall changes in either depolarizing input and/or den-dritic K+ conductances (see Fig. 13). Small changesin IS are conceivable through realistic modulations offeedforward and feedback input that occur during elec-trolocation and electrocommunication in weakly elec-tric fish (Heiligenburg, 1991). Changes in gDr,d canfurther occur through the phosphorylation of dendriticK+ channels, such as AptKv3.3, which has been shownto be abundant over the whole dendritic tree of ELLpyramidal cells (Rashid et al., 2001). Hence, the ghost-bursting mechanism may offer ELL pyramidal cells aviable method by which to optimize sensory codingwith regulated burst output. Further studies quantify-ing the information-theoretic relevance of bursting arerequired to confirm these speculations.

We conclude our study with a concrete prediction.Figures 10, 12, and 13 show that the full-burst period Tof ELL pyramidal cells can be significantly decreasedas either depolarizing current (IS) is increased or den-dritic K+ conductance (gDr,d ) is decreased by a smallamount. This prediction can be easily verified by ex-perimentally measuring T in bursting ELL pyramidalcells for (1) step changes in IS and (2) before and af-ter TEA application to the apical dendrites, which will

change gDr,d . Modification of other ionic currents (per-sistent sodium and somatic K+, in particular) may alsobe used to create similar bifurcation sets as in Fig. 13.

Acknowledgments

We would like to thank our colleague Ray W. Turnerfor the generous use of his data and fruitful discus-sions. Valuable insight on the analysis of our modelwas provided by John Lewis, Kashayar Pakdaman,Eugene Izhikevich, Gerda DeVries, Maurice Chacron,and Egon Spengler. This research was supported by op-erating grants from NSERC (B.D., A.L.), the OPREA(C.L.), and CIHR (L.M.).

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Ghostbursting: A Novel Neuronal Burst Mechanism 25

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ARTICLE III

B. Doiron, A.-M. Oswald, L. Noonan and L. Maler.Thresholds to Sensory Coding with Ghostbursting.Unpublished review article of the following journal articles:

B. Doiron, L. Noonan, N. Lemon and R. W. Turner.Persistent Na+ current modifies burst discharge by regulating conditional backpropagation of den-dritic spikes.Journal of Neurophysiology. 89: 338-354, 2003.

C. R. Laing, B. Doiron, A. Longtin, L. Noonan, R. W. Turner and L. Maler.Type I Burst Excitability.Journal of Computational Neuroscience. 14: 329-342, 2003.

A. M. M. Oswald, M. J. Chacron, B. Doiron, J. Bastian and L. Maler.Parallel processing of sensory stimuli by bursts and isolated spikes.Journal of Neuroscience. 24: 4351-4362, 2004.

A. M. M. Oswald, B. Doiron and L. Maler.Dendrite dependent interval coding.Submitted to the Journal of Neuroscience.

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Thresholds to Sensory Coding with Ghostbursting

Brent Doiron1,2, Anne-Marie Oswald2, Liza Noonan3, and Leonard Maler21 Physics Department, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, Canada, K1N 6N5

2 Department of Cellular and Molecular Medicine, University ofOttawa, 451 Smyth Rd., Ottawa, Ontario, Canada, K1H 8M5

3 Department of Cell Biology and Anatomy, Neuroscience ResearchGroup, University of Calgary, Calgary, Alberta, Canada, T2N 4N1.

This chapter is a compendium of several results surrounding the Ghostbursting mechanism [6]used by sensory pyramidal neurons in weakly electric fish [24, 41]. Firstly, a threshold between tonicfiring and bursting behaviour is experimentally shown to occur in in vitro recordings of pyramidalcells, as predicted in [6]. Secondly, this threshold imparts slow passage effects to the burst dynamicsas required by the ghostburst mechanism [6]. Thirdly, the response of pyramidal cells to broad-band dynamics stimuli is investigated. A reduced mathematical model of stochastic pyramidal cellbehaviour is developed. This model is shown to reproduce all first and second order statistics ofpyramidal cell spike and inter-spike interval trains. We use this model to reproduce the pyramidalcell encoding of a time varying stimulus. It is shown that bursts and isolated spikes code in parallelfor low frequency and high frequency stimuli components respectively.

INTRODUCTION

Neurons are nonlinear systems. Their trademarknonlinearity is an excitable threshold separating twotypes of dynamic behaviour: resting and spiking. Apervasive paradigm in neuroscience is that spike eventsare responsible for coding and transmitting informationwithin the brain [33]. However, a third form of neuralbehaviour is gaining attention: bursting. Technically,bursting refers to spiking mechanisms that produce aclustered pattern of spikes. However, there is growingevidence that the role of bursts in coding and informa-tion transmission is distinct from that of isolated spikes[14, 18–20, 26, 32, 36]. Many neurons have the capabil-ity to respond in either a burst or tonic (regular) modedepending on the applied stimulus [36]. This is sugges-tive of a burst threshold in the system dynamics whichseparates both activity patterns.

Weakly electric fish are nocturnal fresh water ani-mals that thrive in South America and Africa. Over60 million years they have developed a unique sensorymodality - an electrosense. These fish have an electricorgan that generates a weak and quasi-sinusoidal elec-tric field. Prey, background scenes, and even communi-cation calls from con-specifics modulate the field ampli-tude. The electrosensory system detects and processesthese modulations to produce a neural representationof their environment [16, 43]. The use of naturalis-tic dynamic inputs, well-charted neural feedback path-ways, and detailed single neuron characterization haverecently made weakly electric fish a popular avenue inthe study neurocomputational dynamics and process-ing [20, 42].

ELL pyramidal neurons are the principal output cellswithin the ELL and their activity is devoted to theprocessing of a variety of sensory inputs. These cellsexhibit a form of burst discharge termed “Ghostburst-

ing” [6, 7, 24, 41]. The ELL and ghostbursting offerdistinct advantages for the study of bursting and sen-sory coding. ELL pyramidal cells are only one synapsefrom the outside world - an outside world that is sim-ply describable in terms of electric field modulationsrather than complex visual or olfactory scenes. Feed-back from higher brain centers is segregated and isknown to contextually modulate pyramidal cell spik-ing behaviour [5, 9]. Finally, the burst discharge isdependent on active dendritic processes [41] which of-fers several methods for modulation of burst behaviour[8, 28].

In this paper we review a host of experimental andcomputational results on ELL pyramidal cell burst dis-charge. Our main results are divided into two sections.The first section details the existence and slow passageeffects of a burst threshold that is dependent on a sta-tic depolarizing input. These results have been compo-nents of two previous publications [8, 21]. The secondsection presents the response of ELL pyramidal cells toa broadband stochastic stimulus. A reduced model ofan ELL pyramidal neuron is presented. This model re-produces the first and second order statistics of both thespike and inter-spike interval train. We use this modelto motivate a parallel processing of broadband input bypyramidal neurons: bursts code for the low frequencycomponents, while isolated spikes simultaneously codefor the high frequency components. These results havebeen presented in other publications [29, 30].

BURST THRESHOLD

In Doiron et al. [6] the authors studied the burstmechanism used by pyramidal cells in weakly electricfish. They presented a two-compartment approxima-tion of a large compartmental model [7] that repro-

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duced the burst discharge recorded in slice experiments[24, 41]. Despite the severe simplification of spatial den-dritic processes the reduced model, entitled the Ghost-burster, qualitatively reproduced the main features ofbursting observed in the large model [7]. The advan-tage of performing this simplification is that it alloweda detailed dynamical systems treatment of the burstmechanism.

An interesting scenario unfolded when the intensityof a static depolarizing input was used as a bifurcationparameter. For weak depolarizing inputs, resting be-haviour (no spiking) occurred. As the input increased,there was a transition from resting to tonic spike dis-charge at intermediate intensities, and finally a chaoticburst output occurred at high input intensities. Math-ematically these two transitions were identified to berespectively a saddle-node bifurcation of fixed pointson an invariant circle (rest → tonic) and a saddle-nodebifurcation of limit cycles (tonic → burst). It is thesecond bifurcation, which we label the burst threshold,that will be addressed in this section.

Burst Threshold in Experiment

In Doiron et al. [8] it was shown that pyramidalcells give a non-stationary response over time whena long (4 s) constant depolarizing input is presented.The spike train slowly shifted in time from producinglong inter-spike intervals (ISIs), to short ISIs, and theneventually a burst response. In [8] experimental andcomputational evidence was given for a slow activationof a persistent sodium current producing an intrinsicramped depolarization. This slow effect was responsiblefor the non-stationary spiking behaviour. Nonetheless,whatever the ionic mechanism is, this non-stationarityprevents the use of classic statistical analysis on ELLpyramidal cell spike trains [13]. To quantitatively studya threshold between spiking and bursting behaviour itwas thus necessary to develop measures dependent overshort time scales.

A large burst after-hyperpolarization (AHP) consis-tently follows a spike doublet [24]; this is the most re-liable indicator of the occurrence of burst discharge.We thus used the difference between consecutive AHPamplitudes to detect spike bursts. Specifically, we cal-culated the local difference between consecutive AHPsi − 1 and i as σi = (AHPi − AHPi−1)2. AHP am-plitudes were computed as the minimum potential be-tween two consecutive action potentials and the dif-ference was squared so that the measure was positivedefinite. Given a spike train of N spikes, this gavea sequence of (N − 1) σ values. The σ measured be-tween the AHP within a spike doublet and the followingburst AHP was much larger than the σ measured be-tween other AHPs; see Fig. 1 for example trains. The

FIG. 1: Burst statistics. A σ values (bottom) as computedfrom a tonic spike train (top). B σ values (bottom) ascomputed from a bursting spike train (top). Note the largeincrease in σ that occurs at the termination of a burst.

computation of a given σ was insensitive to the non-stationarity in the membrane since typical consecutiveAHPs are separated in time on the order of millisec-onds, while the ramped depolarization led to membranechanges on the order of seconds. For more details onthis measure see [8].

We compute the average of σ, labelled Σ =1

N−1

∑i σi, over long static depolarizations (4 s). Since

bursts promote large positive values in σ then Σ willbe significantly positive for a bursting train whereasit will be near zero for a tonic train. A higher burstfrequency, meaning more burst AHPs per unit time,will also tend to increase Σ. Unfortunately, the full σtrain is also non-stationary since burst frequency of-ten increases over time [8], hence Σ is not statisticallywell-defined. However, since the Σ measure is relativelystraightforward to interpret (Σ ≈ 0 means tonic whileΣ > 0 means bursting) we can use it to quantify burstdischarge without fear of confusion.

Figure 2 shows Σ computed from the spike responseof both the large compartmental model (with the non-stationary INaP current modelled as in [8]) and a realpyramidal cell to a range of static depolarization cur-rents (I). At a critical value I = IC , Σ becomes sig-nificantly positive; this occurred in both the large scalemodel [7] and several pyramidal cell recordings (n=8out of 13). Thus, there exists a burst threshold thatcan be crossed as the intensity of a depolarizing inputis increased. This qualitative change which occurred atI = IC is reminiscent of the saddle-node bifurcation oflimit cycles that also occurred for a critical value of I[6]. Saddle-node bifurcations also impart slow passageeffects to the dynamics of a system near such a bifur-cation. This will be the focus of the next subsection.

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FIG. 2: Burst threshold in the large model [8] and exper-imental data. Σ values computed from the spike train re-sponse to a static 4s depolarization of amplitude I for boththe model A and a typical pyramidal cell B (8/13 cellsshowed this behaviour). Note the critical value of I, la-belled IC , at which Σ becomes significantly positive.

Ghosting and ISI return maps

A convenient method to illustrate burst dynamics isto construct ISI return maps (ISIi+1 vs ISIi). All thepanels of Fig. 3 show the somatic voltage from eitherthe large model [7] or from experiments during a singleevoked burst. Below the voltage trace is the associatedISI return map, where the diagonal ISIi+1 = ISIi par-titions the map into burst and inter-burst regions. Asa single burst evolves its ISI sequence traces a curvein the return map. The sequence begins at a high valuein the burst region, and then moves along the diagonalas the ISIs within the burst decrease (1). The dou-blet ISI invokes a sharp descent in the sequence (2),which is followed by an injection to a higher value in theinter-burst region corresponding to the burst AHP du-ration (3). Finally, the sequence is reinjected back intothe burst region and a new burst begins (4). Viewingthe burst trajectory with ISI return maps illustratesthe trapping region and its ‘ghosting’ effects on systemdynamics.

Dynamical systems near saddle-node bifurcationshave a region in phase space where trajectories havelow velocities; this is termed a trapping region [38]. Thetrapping region is the ‘haunting’ effect of the ghost ofthe stable attractor that was there before the saddle-node bifurcation. Near the saddle-node bifurcation, thewidth of the trapping region is inversely proportionalto the square of the time that it takes to exit the region.

Unfortunately, a direct experimental measure of theaverage burst time as a function of depolarizaing cur-rent is problematic due to the non-stationary behaviourof pyramidal cells [8]. Luckily, the trapping region as-sociated with the saddle-node bifurcation of limit cy-cles (SNLC) in the Ghostburster system is reflected

FIG. 3: Ghosts in ISI return maps. Time series (top) andISI return maps (bottom) of long bursts in both the largecompartmental model [8] (A) and experiment (B; n=8 outof 13). This is to be compared to those of short bursts forthe model (C) and experiment (D ; n=8/13). The stagesof bursting (1-4) labelled on all panels are explained in thetext. Note the variability of the spike height in the exper-imental data; this is due to the finite sampling used in theexperiments.

by the ISI sequence clustering about the diagonalISIi+1 = ISIi (region (1)). The sequence must passthrough the trapping region before it can finally es-cape and produce the somatic doublet (2), terminatingthe burst. The time required to traverse the trappingregion (the sum of all the ISIs within the region) is ap-proximately the duration of the burst. Hence, control

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of the average burst period is equivalent to determin-ing the factors that control the width of the trappingregion in the ISI return map.

Near the SNLC the trapping region is thin and pas-sage through it is slow. The farther away parametersare from the SNLC, the wider the trapping region is andthe faster the passage becomes. Figure 3 shows bothlong and short period bursts in the model and data; thelong bursts have parameter values near the SNLC whilethe short burst are farther from the SNLC. ISI cluster-ing near the diagonal is much more significant for thelong bursts (see Fig. 3 A and B) as compared to thatof short bursts (see Fig. 3 C and D). This behaviourwas observed in a number of cells (n=8 out of 13).

The combination of the results shown in Figs. 2and 3 show that a threshold between tonic and burst-ing behaviour exists in real ELL pyramidal cells andthat it imparts slow passage effects to system dynam-ics. These results are consistent with the dynamicalsystems analysis presented in [6] that showed that asaddle-node bifurcation of limit cycles separated tonicand bursting behaviour in a simplified model of ELLpyramidal cells.

PARALLEL PROCESSING WITH BURSTS ANDISOLATED SPIKES

The original experimental characterization of ELLpyramidal cell burst discharge was in response to staticdepolarizing inputs [24, 41, 42]. There have been suc-cessful theoretical efforts to characterise the response ofthe ghostburster system to periodic inputs as well as toinstantaneous step depolarizations [21, 23]. However,weakly electric fish thrive in a sensory environment withvery rich temporal structure. Prey inputs and naturalscenes, such as the electric image of the fish’s own tail,are typically composed of only low frequency compo-nents (0-10 Hz). This is in contrast to intra-specificcommunication calls that are broadband in their spec-tral content (0-200 Hz). An often used simplificationof natural scenes is to model its temporal structure asGaussian. It should be mentioned that this simplifi-cation is not always appropriate, most notably in theauditory system [35]. Nevertheless, this strategy hasbeen successfully employed on numerous occasions inin vivo electric fish studies [4, 5, 14]. A study of howthe in vitro ghostburster system processes stochasticinputs has only recently been undertaken [29, 30]. Thissection will briefly describe the main findings of theselast studies.

Ghostbusting the Ghostburster

In Oswald et al. [30] broadband Gaussian stimuli (0-60 Hz) was applied to ELL pyramidal cells in a slicepreparation. A welcome result of the high frequencydepolarizations and hyperolarizations induced by thestimulus is a removal of the non-stationarity in cell be-haviour that occurs for static inputs [8]. This allowsfor a standard statistical analysis of spike train data.The spike train was quite distinct from those observedin response to static depolarizations. Notably, the ran-dom stimuli elicited a mixture of both isolated spikedischarge and bursting, with bursts often only consist-ing of two spikes and without the characteristic largeAHP at burst termination. Figure 4A shows an ex-ample spike train from a real in vitro pyramidal celldriven with a stochastic input. The differences betweenautonomous and driven burst discharges is presumablythat the high frequency components, specifically rapidhyperpolarizations in the stimulus, prematurely termi-nate burst discharges before a full burst is completed.This assumption can be used to greatly simplify themathematical modelling of the burst mechanism.

The ghostburst mechanism described in [6, 7] in-volves three main components: 1) a dendritic depen-dent DAP at the soma, 2) a potentiation of the DAPover the course of a burst event, and 3) a longer refrac-tory period of the dendritic action potential as com-pared to the somatic action potential. The relativelyslow time-scale of the dendritic potassium inactivationtypically allows for significant DAP potentiation onlyover bursts containing many spikes. Somewhat less ap-parent is that large dendritic refractory periods alsorequire bursts composed of many spikes. This was ex-perimentally and computationally demonstrated in [28]by forcing dendrites to discharge at small inter-spikeperiods (6 ms) and observing that dendritic spike fail-ure occurred only after multiple (approximately 5) den-dritic spikes.

The lack of many spike bursts in response to broad-band Gaussian stimuli permits an approximation ofELL burst dynamics that contains only DAPs andignores burst mechanism components 2) and 3). Moti-vated by past leaky integrate-and-fire (LIF) modellingof ghostbursting [22, 28] the following description wasthen postulated [29]:

dV

dt= −V + γx(t− τd) + I(t) (1)

dx

dt= y (2)

dy

dt= −α2x− 2αy + α2

∑i

δ(t− ti) (3)

The dynamics of Eqs. (1)-(3) are quite simple in

ARTICLE III 785

FIG. 4: Stochastic bursting in ELL pyramidal cells. Ain vitro recordings of an ELL pyramidal cell membranepotential when driven by Gaussian current (0-60 Hz) ap-plied via a sharp electrode located in the soma. The stim-ulus waveform (Istim) is shown below. B Simulations ofEqs. (1)-(3) when Istim is Gaussian (0-60 Hz). The mem-brane potential V (top), dendritic DAP current x (mid-dle), and Istim (bottom) are shown. The time-scales forpanels A and B are identical. Model parameters wereγ = 1.2, τd = 0.2, τr = 0.2, α = 12.5, 〈Istim〉 = 0.88, andp〈I2

stim〉 − 〈Istim〉2 = 0.185. The spikes in B were addedmanually. The experimental methodology is as described in[30]

comparison to that of the full conductance based ghost-burst system [6] or even that of the reduced LIF model[22, 28]. Equation (1) is a standard LIF descriptionof spiking dynamics with an absolute refractory periodτr. The solution of the subsystem (x, y) with the initialconditions x(0) = 0 and y(0) = α2 is the so called al-pha function: x(t) = α2te−αt. The ith spike-time ti isdefined by V (ti) = 1. Every time V (t) spikes, an alphafunction of intensity γ excites the membrane dynamicsafter a time delay τd; this models the dendritic DAP’sinfluence on somatic spike generation. In contrast to[22] the DAP dynamics are linear without the possi-bility of potentiation or refractory dynamics. WhenI(t) = I is static the system given by Eqs. (1)-(3) givessimple tonic discharge for all values of I. Thus (1)-(3) is not an autonomous burst mechanism. There is

FIG. 5: ISI statistics. ISI histograms shown in A and ISI se-rial correlations in B for both slice experiment data and themodel Eqs. (1)-(3) computed from 100 seconds of spikingactivity. The model results show simulations when the den-drite is active (γ = 1.2), and when the dendrite is passive(γ = 0). Other details are identical to Fig. 4

no saddle-node bifurcation of limit cycles or chaos forany value of I. We have thus removed (or busted) the‘ghosting’ behaviour so crucial for our understandingof burst response to static inputs. However, when I(t)is a stochastic process with Gaussian spectra between0-60 Hz the dynamics of Eqs. (1)-(3) reproduces thefirst and second order statistics of both the spike trainand ISI sequence, as well as the linear signal processingof a broadband input stimulus, as they are measuredfrom slice experiments. We outline these results below.

Figure 4B shows numerical simulations of Eqs. (1)-(3) when driven with broadband Gaussian input. Notethe presence of both isolated spikes and bursts of twospikes. Both the dendritic DAP x(t) and the stimulusI(t) are also shown. Figure 5A shows bimodal inter-spike interval (ISI) histograms of pyramidal cell spiketrain data. The model shows similar behaviour withactive dendrites (γ > 0) yet only unimodal ISI his-tograms when the dendrite is passive (γ = 0). Further-more, both the model and data ISI serial correlationsare near zero for all lags except lag zero, as shown inFig. 5B. This shows that no significant patterning inthe ISI sequence occur in both the model and the data.

To measure action potential patterning we computethe spike train autocorrelation A(τ) [13] for both themodel and the data in Fig. 6A. After a short depressedprobability of spiking for approximately 4 ms (due torefractoriness) there is an enhanced probability of fir-ing a spike at 5 − 7ms which decays rapidly back tochance probability level by 10ms. The passive dendrite

ARTICLE III 796

FIG. 6: Spike train statistics. Spike train autocorrelationshown in A and power spectra in B for both slice experi-ment data and the model Eqs. (1)-(3) computed from 100seconds of spiking activity. The model results show simula-tions when the dendrite is active (γ = 1.2), and when thedendrite is passive (γ = 0). Other details are identical toFig. 4

model (γ = 0) shows that this increased probability isdue to the DAP current promoting spike discharge af-ter a spike for only a short period. For completenesswe also show the power spectrum, S, of the model anddata spike trains in Fig. 6B They both show a largeoscillation at 200 Hz indicative of the intra-burst ISI.The combination of both the bimodal ISI distributionsas well as the distinct non-Poissonian spike statisticsas observed from A(τ) and S set the stochastic behav-iour of Eqs. (1)-(3) as bursting. Similar statistics wereused to characterize the burst discharge of in vivo ELLpyramidal cells [1]. The reproduction of these statisticsby the model given in Eqs. (1)-(3) shows how a lineardendritic event can account for the burst properties ofELL pyramidal cells in response to broadband inputs.

Spike Train Processing

To investigate the processing done by pyramidal cellswe introduce the coherence C(f) [33] between the inputstimulus S and the spike train response R:

C(f) =SSR(f)2

SSS(f)SRR(f).

SSR(f) is the cross spectrum between S and R, SSS(f)is the power spectrum of S and SRR(f) is the powerspectrum of R. Quite simply C(f) is the frequency-dependent linear correlation between S and R; it is a

FIG. 7: Parallel processing of dynamic stimuli by bursts andisolated spikes. A An ISI threshold (10 ms) partitions thespike train into isolated spikes (spikes bordered by ISIs bothgreater than 10 ms) and bursts (spikes involved in an ISIless than 10 ms). The burst time is taken to be the timingof the first spike in the burst. B The coherence between thefull spike train (thin black line), isolated spike train (grayline), and burst train (thick black line) with the stimulus(Gaussian noise with power between 0-60 Hz). The resultsfor the data and the model are both shown. Other detailsare identical to Fig. 4

number bounded between 0 and 1 with 0 representingno correlation and 1 being perfect correlation at a givenfrequency f . The coherence between the full spike trainand the stimulus for both the data and the model sys-tem Eqs. (1)-(3) is relatively broadband (slightly low-pass), as shown in Fig. 7B. This is similar to in vivoresults when the stimulus is applied to a small fractionof the total receptive field [4].

In order to further dissect pyramidal cell coding Os-wald et al. [30] partitioned the full spike train into twocomponent trains: a train of isolated spikes and a bursttrain. This partitioning is schematically shown in Fig.7A, and briefly described here. Every spike in a spiketrain is part of two ISIs, the ISI that it begins and theone it terminates. An ISI threshold that corresponds tothe local minimum of the ISI probability distribution isset. If both ISIs that a given spike defines are greaterthan this threshold then that spike is considered an iso-lated spike. In a complementary sense, if one or bothof the ISIs are less than the threshold then that spike isinvolved in a burst. The burst train is further simpli-fied by removing all but the first spike in a burst; the

ARTICLE III 807

FIG. 8: Event triggered averages. The stimulus triggeredaverages for the full spike train (thin black line), burst train(thick black line), and the isolated spike train (grey line) areshown for both data (A) and the model (B

results presented are not sensitive to this reduction.Interestingly, the coherence between the so-

constructed burst train and the stimulus is low passwhile the coherence between the isolated spike trainand the stimulus is high pass (sometimes band pass).This occurred in a statistically significant fashion in ex-periment, the reduced model given in Eqs. (1)-(3), anda modified version of the full Ghostburster system [6](see Fig. 7B and [30] for results not shown). Thus,a partitioning of the spike train reveals the potentialto selectively code for both low frequency componentsvia bursts and high frequency components via isolatedspikes. Furthermore, this coding occurs in parallel sincethe full spike train simultaneously produces both burstsand isolated spikes to a dynamic stimulus.

Analysis of the event-triggered averages gives an in-tuitive understanding of this parallel coding. An eventtriggered average is defined as

〈I(τ)〉σ =1N

N∑i=1

I(ti − τ) (4)

where τ extends in both positive and negative direc-tions and ti is an event time in a given event trainσ =

∑i δ(t− ti). The sum runs over all events. Figure

8 gives 〈I(τ)〉σ where σ is either the full spike train, theburst train, or the isolated spike train. The model anddata results qualitatively match one another. A promi-nent feature of all event triggered averages is that theyare upstrokes within the stimulus (positive slope lead-ing to τ = 0). However, the burst triggered averagetime scale is longer and the intensity is larger as com-pared to the isolated spike triggered average. Phrasedanother way, the results of Fig. 8 show the simple re-sult that the stimulus must be depolarized at a largeintensity for a long period of time to allow for multiplespike events (bursts) to occur. This long period of the

upstroke translates to a low frequency coherence of theburst train, leading naturally to the results in Fig. 7.

This explanation of the parallel coding suggests thatburst dynamics are not required to reproduce theprocessing results of Fig. 7; this is indeed the case. If‘high’ and ‘low’ frequency ISI events in the spike trainfrom a simple LIF system (Eqs. (1)-(3) with γ = 0) aredefined and analysed in the same way as above a quali-tatively similar picture is obtained (results not shown).High-frequency spiking codes for low-frequency stimuliwhile low frequency spiking codes for high frequencystimuli. However, for a bursting system the ISI his-tograms are bimodal as compared to the unimodal ISIhistograms from a simple spiking neuron. Bimodalityin the ISI sequence defines an appropriate ISI thresh-old from which to separate ‘high’ frequency spike events(bursts) from ‘low’ frequency events (isolated spikes).This well-defined segregation of ISIs could be impor-tant for the reliability of downstream burst decodingmechanisms [17, 25].

DISCUSSION

Thresholds in bursting systems

The theoretical study [6] predicted a threshold sep-arating tonic and bursting behaviour in ELL pyrami-dal cells. There it was shown to be the result of asaddle-node bifurcation of limit cycles that destroyedthe stable tonic firing solution and gave way to a chaoticburst discharge. In this study we showed experimentalresults from in vitro ELL pyramidal neurons that in-dicated both a threshold in spike/burst dynamics andslow passage effects within a burst; both results areconsistent with a saddle-node bifurcation.

Thresholds between spiking and bursting behav-iour have been theoretically studied in several systems[2, 39, 40, 44]. In all of the above studies the burstthreshold corresponded to the creation of a homoclinictrajectory in phase space (a closed loop connecting thestable and unstable manifolds of a single saddle point).This loop serves as both a burst propagation (one sideof the loop) and a reinjection mechanism (other sideof the loop). There are several topologically distinctscenarios by which this can happen; [2] gives a detailedstudy of four of them for a generic 3 dimensional burst-ing system. This may also be the case for the ghost-burster system: a distinct reinjection mechanism is ob-served in the ISI return maps and homoclinic trajecto-ries are quite common in chaotic systems [15]. However,the higher dimensionality of the ghostburster system ascompared to the models studies in [2, 39, 40, 44] com-plicates a detailed analysis along those lines.

The slow passage effects shown in Fig. 3 are indica-

ARTICLE III 818

tive of a saddle-node bifurcation. Such slow passageeffects were recently shown in another bursting systemthat exhibits what is called a blue sky catastrophe [37].A blue sky catastrophes is a bifurcations where a asaddle-node bifurcation of periodics leads to an singlelimit cycle of arbitrarily long period. This effect resultsfrom the creation of a trapping region along a stablemanifold of the disappeared saddle limit cycle. Thecreation of ghostbursting may well be similar to such ascenario, however the ghostburst attractor is strange asopposed to the well defined periodic orbit that emergesfrom a blue sky catastrophe.

Sensory processing by bursts

The second part of this paper studied the responseof ELL pyramidal cells to broadband stochastic inputs.An LIF model with a simple linear dendritic responsewas sufficient to account for the first and second or-der statistics of both the spike and inter-spike intervalstrains. The model is much simpler than all other previ-ous models of ELL pyramidal cell dynamics [6, 7, 22].There does not exist a saddle-node bifurcation of limitcycles that causes the slow passage effects that were thefocus of the first section of this paper. The price for thissimplicity is that the model is inadequate when inputsare static or contain only low frequencies. However, inresponse to broadband inputs it shows that ELL pyra-midal cells may act similarity to a large number of cellsthat support active dendritic backpropagation (see [34]for a review).

The model also reproduces the ELL pyramidal cellparallel processing by bursts and isolated spikes. Thisparallel coding presupposes that the decoding mecha-nisms of higher brain centers are sensitive to this sortof encoding. Several burst decoding schemes have beenproposed; membrane resonance in the postsynaptic cell[17] as well as short-term synaptic plasticity [25]. ELLpyramidal cell output is decoded in the torus semicircu-laris (TS) of the midbrain. Fortune and Rose [11] haveshown that different TS cells have distinct tuning prop-erties to external in vivo inputs, specifically some showlow pass characteristics while others have a high passnature. Interestingly, low pass TS cells show a markedpresynaptic facillitation to high frequency afferent in-puts (< 10 ms) [12]. Their low frequency preferenceof external electrosensory inputs and high frequencyselectivity of synaptic driving suggests that an earlierprocessing stage is responsible for a frequency inversion- turning low frequency inputs into high frequency ISIs.We suggest that bursting ELL pyramidal cells may beresponsible for this inversion.

Acknowledgements

We would like to thank our many ghostbursting col-laborators whose ideas have helped to shape the re-sults presented here: Carlo Laing, Andre Longtin, RayTurner, and Maurice Chacron.

[1] J. Bastian and J. Nguyenkim. J. Neurophysiol. 85 10(2001).

[2] V.N. Belykh, I.V. Belykh, M. Colding-Jorgensen, andE. Mosekilde. Euro. Phys. J. E. 3, 205 (2000).

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[4] M.J. Chacron, B. Doiron, L. Maler, A. Longtin, and J.Bastian. Nature 423, 77 (2003).

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[6] B. Doiron, C. Laing, A. Longtin and L. Maler. J. Com-put. Neurosci. 12, 5 (2002).

[7] B. Doiron, A. Longtin, R. W. Turner and L. Maler. J.Neurophysiol. 86, 1523 (2001).

[8] B. Doiron, L. Noonan, N. Lemon, and R. W. Turner.J. Neurophysiol. 89, 324 (2003).

[9] B. Doiron, B. Lindner, A. Longtin, L. Maler, and J.Bastian. Phys. Rev. Lett. 93, 048101 (2004)

[10] B. Ermentrout. Neural Comp. 8, 979 (1996).[11] E.S. Fortune and G. Rose. J. Neurosci. 17, 3815 (1997).[12] E.S. Fortune and G. Rose. Trends Neurosci. 24, 564

(2001).[13] F. Gabbiani F, and C. Koch in Methods in Neuronal

Modelling: From Ions to Networks Ed. C. Koch andI. Segev, 313, (MIT Press, Cambridge MA, 1998).

[14] F. Gabbiani, W. Metzner, R. Wessel and C. Koch. Na-ture 384, 564 (1996).

[15] J. Guckenheimer and P. Holmes. Nonlinear Oscilla-tions, Dynamical Systems, and Bifurcations of VectorFields (Springer, Berlin, 1990).

[16] W. Heilengenberg Neural Nets in Electric Fish (MITPress,Cambridge MA, 1991)

[17] E. M. Izhikevich, N. S. Desai, E. C. Walcott and F. C.Hoppensteadt. Trends Neurosci. 26, 161 (2003).

[18] J. Keat, P. Reinagel, R.C. Reid, and M. Meistner. Neu-ron 30, 803 (2001).

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[20] R. Krahe and F. Gabbiani. Nature. Rev. Neurosci. 513 (2004).

[21] C. R. Laing, B. Doiron, A. Longtin, L. Noonan, R. W.Turner and L. Maler. J. Comput. Neurosci. 14, 329(2003).

[22] C. R. Laing and A. Longtin, Bull. Math. Biol. 64, 829(2002).

[23] C. R. Laing and A. Longtin, Phys. Rev. E, 67, 051928(2003).

[24] N. Lemon and R. W. Turner, J. Neurophysiol. 84, 1519(2000).

[25] J. E. Lisman Trends Neurosci 20, 38 (1997).[26] S. Martinez-Conde, S. Macknik, and D. Hubel. Proc.

Natl. Acad. Sci. USA. 99, 13920-13925 (2002).

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[27] M. E. Nelson, Z. Xu and J. R. Payne J. Comp. Physiol.A 181 523 (1997).

[28] L. Noonan, B. Doiron, C. R. Laing, A. Longtin and R.W. Turner, J. Neurosci. 23(4), 1524 (2003).

[29] A-M. M. Oswald, B. Doiron, and L. Maler. submitted(2004).

[30] A-M. M. Oswald, M.J. Chacron, B. Doiron, J. Bastian,and L. Maler. J. Neurosci 24 4351-4362 (2004).

[31] J. Rinzel and G. B. Ermentrout, in Methods in Neu-ronal Modeling: From Ions to Networks, Ed. C. Kochand I. Segev, (MIT Press, Cambridge MA, 1998).

[32] P. Reinagel, D. Godwin, M. Sherman, and C. Koch. J.Neurophysiol. 81, 2558 (1999).

[33] F. Rieke, D. Warland, R.R. de Ruyter van Steveninck,and W. Bialek Spikes:exploring the neural code (MITPress, Cambridge MA, 1997).

[34] A. Reyes. Ann Rev. Neurosci. 24, 653 (2001).[35] F. Rieke, D.A. Bodnar, and W. Bialek. Proc. R. Soc.

Lond. B 262, 259 (1995).

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Applications to Physics, Biology, Chemistry, and En-gineering. (Addison-Wesley, Reading, MA., 1994).

[39] D. Terman, SIAM J. Appl. Math. 51, 1418 (1991).[40] D. Terman, J. Nonlinear Sci. 2, 135 (1992).[41] R. W. Turner, L. Maler, T. Deerinck, S. R. Levinson,

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ARTICLE IV

B. Doiron and A. Longtin.Oscillations and Synchrony with Delays in Neural Networks.In part an exert from the journal article:

N. Masuda, B. Doiron, A. Longtin and K. AiharaCoding of oscillatory signals by globally coupled networks of spiking neurons.Submitted to Neural Computation.

83

ARTICLE IV 84

Oscillations and Synchrony with Delays in Neural Networks

Brent Doiron1,2 and Andre Longtin1

1 Physics Department, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, Canada, K1N 6N52 Department of Cellular and Molecular Medicine, University ofOttawa, 451 Smyth Rd., Ottawa, Ontario, Canada, K1H 8M5

The influence of delayed interactions in globally inhibitory coupled networks of spiking neurons isstudied. Computational work shows how global synchronous oscillatory activity occurs when eitherthe interaction strength between cells or the single unit excitability is increased. Using a spikeresponse formalism [12] we derive, and analytically study, a reduced delay differential equation forthe network activity. The transition to synchronous oscillations is shown to be mediated via a Hopfbifurcation in network activity.

INTRODUCTION

Synchronous oscillatory activity in networks of neu-rons is a well documented phenomena. It occurs in avariety of sensory systems [27], most famously in ol-factory [17] and cortico-thalamic networks [29]. Thiscollective neural behaviour has been linked to featurediscrimination [28, 30] as well as object binding [29]. Ina complementary fashion, oscillation and synchroniza-tion mechanisms are also classic topics in the theory ofcoupled oscillators and neural networks [16, 22, 25, 31].Traditionally there have been two important questionsfor both experimentalists and theorists investigatingcollective oscillations: 1) which system parameters de-termine the time scale of network oscillations, and 2)which parameters regulate synchronous oscillatory ac-tivity.

In classic theoretical works by Winfree [33] and laterKuramoto [16] the synchronization of networks of phasecoupled oscillators was addressed. They found that thestability of synchronous network behaviour was depen-dent on both the coupling parameter and network het-erogeneity. In particular, they also found that the time-scale of network oscillatory behaviour was dependenton the distribution of time scales of the individual os-cillators. Similar synchronization effects have also beenextended to pulse coupled units, a more relevant frame-work in theoretical neuroscience [1, 2, 12, 22, 24]. How-ever, the theoretical picture changes somewhat whenthe interaction between cells is delayed in time.

The interaction between real neurons is inevitablydelayed since both axonal transmission and synapticevents can not occur instantaneously. Also, recurrentor closed loop architectures are ubiquitous in the brain,and are found in invertebrate retina [15], thalama-cortical networks [28, 29], olivo-cerebellar system [18],and electrosensory networks [4]. It is now well estab-lished that time delays can produce oscillatory effectsin biological systems [5, 20]. Several studies of neuralnetworks have implemented delayed interactions andobserved that network synchrony and oscillatory be-

haviour are still dependent on interneuronal coupling[2, 9, 21]. However, they have shown that oscillationsare most stable for inhibitory interactions; this mirrorsexperimental observations that network oscillatory of-ten require inhibitory coupling [17, 27]. Finally, andperhaps most distinctive, the oscillation time scale isnow heavily dependent on the delay time scale [21].

In this paper we study synchronization and oscilla-tions in networks of pulse coupled leaky integrate-and-fire (LIF) neurons with delayed interaction. In thefirst section we computationally show that both thefeedback strength and the single cell excitability regu-late the intensity of a synchronous network oscillation.In the second section we derive dynamical equationsthat govern the population activity of large networksof delay-coupled neurons. This permits a theoreticaltreatment of delay-induced synchronous network oscil-latory activity. In particular, we will show how the re-sults in the first section occur via a supercritical Hopfbifurcation in network activity.

GLOBAL NETWORK OSCILLATORYACTIVITY

Model Description

Consider a non-autonomous and homogeneous net-work of N LIF neurons [12]. Let the membrane po-tential of the ith neuron at time t be given by Vi(t).The dynamics of Vi(t) evolve according to the follow-ing equations:

d

dtVi(t) = µ− Vi(t) +

√2Dξi(t) + gKτd

∗A(t). (1)

Spiking dynamics are represented in the network model(1) by introducing the reset discontinuity Vi(t+) = vR

when Vi(t) = vT . This gives a discrete set of spiketimes Ti = {ti1, . . . , tip} with ti1 < . . . < tip wheretim ∈ Ti is defined as the mth time t such that Vi(t) =vT . The spike train from neuron i is thus defined as

ARTICLE IV 852

xi(t) =∑

tim∈Tiδ(t− tim). The stochastic dynamics of

neuron i are given by the process ξi(t) where 〈ξi(t)〉 =0 and 〈ξi(t)ξj(t′)〉 = δijδ(t − t′). The parameter D isthe intensity of this intrinsic fluctuation. µ sets theexcitability or equivalently the time-independent inputcurrent given to a neuron; in this study µ < vT sothat the neurons are within the excitable regime. Anabsolute refractory period of length τR is assumed.

The connectivity in (1) is all to all and uniform. Thisbeing the case the introduction of a mean field is appro-priate. We call the mean field the population activityA(t) [1, 12, 16] and define it as follows:

A(t) = lim∆t→0

1∆t

nact(t; t + ∆t)N

=1N

N∑i=1

∑m

δ(t− tim). (2)

nact(t; t + ∆t) is the number of spikes that occur overthe network during the period (t, t + ∆t) where ∆t issmall. The double sum runs over the network and spikehistory. In the limit of large N A(t) is the instantaneousfiring probability of a single neuron in the network. De-viations in A(t) from its mean value (over time) indicatesynchronous activity in the network - a collective riseor fall in firing probability.

The network (1) is coupled via the common termgKτd

∗A(t) where

Kτd∗ f(t) =

∫ ∞

τd

f(t− τ)α2τe−ατΘ(τ)dτ. (3)

Θ(t) is the standard Heaviside function. The couplingterm is conceived as a composite process by which neu-rons project their output to a separate brain centrewhich integrates this input (standard ‘alpha’ function[12]) and then projects uniformly back to the originalnetwork via a common feedback pathway. This indi-rect interaction between cells via Kτd

also involves asignificant minimal delay term, τd, modelling both theintegration time of the distant brain regions and finiteaxonal conduction velocity. The parameter α thus rep-resents both a fast synaptic time scale and sets a dis-tribution of delays. In this study we confine g < 0to model only inhibitory interactions. Unless other-wise stated model parameters are set to τd = 1, τR =0.2, D = 0.1, α = 3, and N = 100. All simulations ofEqs. (1)-(3) are done with an Euler-Maruyama integra-tion routine [26] with a time step of 10−3 units. Time ismeasured in units of the membrane time constant τm;in the figures that follow we assume that τm =5 ms. Inthe next two subsections we study the synchronizationand oscillatory properties of the system given by Eqs.(1)-(3) as we vary both g and µ.

FIG. 1: Network synchrony and coupling. 100 ms segmentsof network activity A(t) are shown for connectivity strengthg = 0 (top), g = −2.5 (middle), and g = −3.5 (bottom).Note the oscillatory pattern as |g| increases. A(t) is numeri-cally approximated as in Eq. (2) with ∆t = 2.5 ms. µ = 0.8for all these simulations.

Feedback strength - g

When g = 0 Eqs. (1)-(3) is a disconnected networkof neurons. Time series of A(t) for g = 0 are shown inFig.1: A(t) fluctuates slightly about a constant value.This behaviour is reflective of asynchronous populationactivity; in any time interval ∆t we have roughly anequal probability of finding NA(t) spikes across thenetwork. The fluctuations in Fig. 1 are due to finitenetwork size. When g becomes increasingly negative(inhibitory) A(t) begins to show an oscillatory struc-ture, as evident in Fig. 1. Figure 2 shows that thepower spectrum of A(t) has a peak at approximately50 Hz, indicative of synchronous network oscillatory be-haviour. This peak grows in intensity as the inhibitoryconnectivity in the network grows. This synchronousnetwork oscillation even shapes the spike train patternof a single neuron in the network. Spectral analysisof a single spike train also shows a significant peak atapproximately 50 Hz, as shown in Fig. 2.

Before we analyse these results further, we show thatsimilar results can be obtained by changing the ex-citability of the single cell.

Excitability - µ

µ < vT represents the excitability of the single cell;the smaller µ is the more intense an input need be toelicit a spike event. Changing this parameter is physi-ologically quite distinct from a shift in g. Nevertheless,when µ is increased it induces oscillatory synchronousnetwork behaviour in a similar fashion as when g is de-

ARTICLE IV 863

FIG. 2: Network and single cell oscillations increase withnetwork coupling. A. Power spectral density of a singlespike train from the network (labelled S) for g = 0 (lightblack), g = −2.5 (grey), and g = −3.5 (thick black). Allthe spectra are normailzed so that limf→∞ S(f) = 1. B.Power spectral density of the network activity A(t) (labelledP )for g = 0 (light black), g = −2.5 (grey), and g = −3.5(thick black). In both A and B an oscillation grows with|g|. µ = 0.8 for all these simulations.

creased (see Fig. 3). Thus changes in µ and g can befunctionally equivalent with respect to a synchronousnetwork oscillation. The next section formalizes thisnumerical observation.

HOPF BIFURCATION IN NETWORKACTIVITY

In this section we derive dynamic equations for A(t).This will allow for a dynamical systems analysis of thecomputational results shown in Fig.s 1- 3.

From integral to differential systems

To ease the presentation of derivations for a popu-lation density dynamics we choose our single neurondescription to be that of the simplified spike responsemodel (SRM) (see [12] for a general introduction to

FIG. 3: Network and single cell oscillations increase withexcitability. A Power spectral density of a single spike trainfrom the network for µ = 0.5 (light black), µ = 0.8 (grey),and µ = 0.98 (thick black). B Power spectral density ofthe network activity A(t) for µ = 0.5 (light black), µ = 0.8(grey), and µ = 0.98 (thick black). In both A and B anoscillation grows with µ. g = −2.5 for all these simulations.

SRM neurons and a more complete derivation of pop-ulation density dynamics). Let the population consistof N neurons where the membrane potential of the ith

neuron is given as

ui(t) = ηi(t− t) + hi(t). (4)

h(t) is the component of the membrane potential thatis due to external inputs to the cell. Eq. (4) is sup-plemented with a spiking rule : the mth spike timeof neuron i, tim, is given by the mth time t such thatui(t) = θ. θ represents the threshold value of membranepotential where nonlinear ionic currents typically pro-duce action potentials in more realistic models of neuralbehaviour. η(t) depends on the time interval t− t wheret is the most recent past spike time for neuron i. Typ-ically η(t) is sufficiently negative for small t so as toeffectively ’reset’ the membrane after spike discharge.

The input potential, hi(t), is given by the followingintegral equation

hi(t) = gi

∫ ∞

0

εi(s)A(t−s−τd)ds+∫ ∞

0

κi(s)µi(t−s)ds

(5)

ARTICLE IV 874

where A(t) is the network activity introduced in Eq.(2). Eq. (5) separates hi(t) into two parts. First, anexternal input µi(t) that is convolved with a linear re-sponse kernel κi(s) (note that µi(t) is different thanthe membrane potential ui(t)). Second, we have net-work interactions whereby the population activity A(t)is convolved with a response kernel εi(s) that modelsboth a synaptic and membrane response. The para-meter gi denotes a network connectivity strength. Mo-tivated by the past section we set gi = g < 0 andµi(t) = µ. Note that the network interactions are de-layed by a time τd, similar to the network simulationspresented earlier. In addition, we consider a homoge-neous neural description with ηi(t) = η(t), εi(t) = ε(t),and κi(t) = κ(t). All these conditions set our networkto be a globally coupled inhibitory network of identi-cal neurons driven by a shared static stimulus µ. SRMneurons differ somewhat from LIF neurons. Most im-portant is that their dynamics are determined by anintegral equation rather than a differential equation.Also, a hard spike reset is replaced with a refractoryfunction. However, with the appropriate choice of η(t),ε(t), and κ(t) an SRM neural network can be identicalto network of coupled LIF neurons [12].

Gerstner and Kistler [12] have shown that for a fixednetwork of sufficiently simple model neurons (no adap-tive or bursting properties) we have, in the limit ofN →∞, the conservation law:

A(t) =∫ t

−∞Ph(t|t)A(t)dt, (6)

where Ph(t|t) is the probability that a neuron with aninput potential h(t) will fire again at time t given thatt < t is the time of the most recent past firing. Ph(t|t)is thus an input dependent inter-spike interval distrib-ution. We supplement Eq. (6) with the normalizationcondition ∫ t

−∞Sh(t|t)A(t)dt = 1, (7)

where Sh(t|t) is the survivor function, defined as theprobability that a neuron does not emit a spike duringthe time interval (t,t). Equation (7) simply states thatall neurons have fired at least one spike over their his-tory (−∞, t). We can relate the survivor function tothe interval distribution via:

Sh(t|t) = 1−∫ t

t

Ph(t|t). (8)

Upon inspection, clearly Sh(t|t) = 1 and Sh(t|t) decaysto zero as t → ∞, (this assumes that

∫∞t

Ph(t|t) = 1).A standard practice in renewal theory [6] is to consider

the rate of Sh(t) decay denoted by fh(t|t):

fh(t|t) = − d

dtSh(t|t)/Sh(t|t). (9)

f is commonly called the hazard function or the ‘death’rate (i.e. the decay of the ‘survivor’ function). Com-bining Eqs. (8) and (9) gives the simple result

Ph(t|t) = fh(t|t)Sh(t|t). (10)

Observing a particular interspike interval of length t− tis thus conditional upon having a neuron that fired attime t that both does not fire (or survives) up untiltime t and also fires (or dies) at time t.

Equations (5) and (6) in conjunction with an appro-priate choice of fh(t|t) and η(t− t) give a system of in-tegral equations which determine network activity. Inorder to obtain a manageable delay differential systemwe subsume all stochastic activity (membrane fluctua-tions, synaptic release times, etc.) within a noisy spikethreshold. We model this with a standard spike escaperate given by the hazard function,

fh(t|t) = τ−10 exp[β(h(t) + η(t− t)− θ)]. (11)

Here β characterises the threshold fluctuations. Asβ → ∞, f becomes a Heaviside function centered atθ, effectively giving a deterministic firing rule. In con-trast, β → 0 gives a uniform firing probability of 1/τ0

for all u producing a completely stochastic firing rule.Next we choose our refractory function to be η(t −

t) = −∞ for t < t < τr + t and η(t − t) = 0 fort > τr + t. This gives an absolute refractory periodτr, with no relative refractory period. Using the factthat Ph(t|t) = 0 for all t satisfying t < t < t + τr inconjunction with the normalization condition (7) andthe definition of η(t− t) reduces Eq. (6) to

A(t) = f [h(t)][1−

∫ t

t−τr

A(t′)dt′]

(12)

This is the Wilson-Cowan equation [32] for a popula-tion of neurons with only absolute refractoriness. If wefurther assume that A(t) varies with time-scales muchlarger than τr then we can use a coarse graining of timeto approximate Eq (12) as

A(t) =f [h(t)]

1 + τrf [h(t)]≡ q[h(t)] (13)

Finally if we assume that the response kernels are givenby ε(s) = κ(s) = τ−1

m exp(−s/τm) then we can trans-form the integral Eq. (5) into the following scalar delaydifferential equation

τmdh

dt= −h(t) + gA(t− τd) + µ (14)

ARTICLE IV 885

The advantage of these approximations and transfor-mations is that the system given by Eqs. (13) and (14)is more tractable than the original system of (5)-(6).The system (13)-(14) is similar to other delay systemspreviously studied [11, 19]. For a more detailed deriva-tion of non-delayed versions Eqs. (13) and (14) see [12].

Hopf Bifurcations

In this section we give a dynamical systems analysisof Eqs. (13) and (14). Inserting the approximation(13) in (14) and performing the transformation h →(h + I0)/θ and t → t/τm gives the non-dimensionalsystem

dh(t)dt

= −h(t) + gq[(h(t− τd)]. (15)

The new non-dimensional parameters are g ≡ g/θ andτd ≡ τd/τm. A(t) is now given by q[h(t)] with q as inEq. (13) but with τr being replaced by τr ≡ τr/τm andfh(t) being replaced by fh(t) ≡ τ0exp[β(h(t) + µ− 1)],with β ≡ βθ, µ ≡ µ/θ, and τ0 ≡ τ0/τm. For simplicitywe hereafter we drop the tilde notation.

Fixed points of (15), labelled h∗, are given by theroots of the following transcendental equation

h∗ = gq(h∗) (16)

Because of the form of q Eq. (16) only admits one realroot for all values of β, τ0, τr > 0 and g < 0. Linearizingabout the fixed point gives the following local dynamics

dh(t)dt

= −h(t) + Dh(t− τd), (17)

where D ≡ gdq/dh|h=h∗ = qβexp(β(h∗+µ−1))τ0/(τ0+τrexp(β(h∗ + µ − 1)))2. We trust that the D used inthis section will not be confused with the noise intensityused in the previous section.

The ansatz h(t) = h0eλt (with h0 some constant)

gives the following characteristic equation for Eq. (17):

λ + 1−De−λτd = 0 (18)

In the appendix we analyse Eq. (18) and show that Eq.(15) admits a countably infinite number of Hopf bifur-cations that are characterised by the sequence {Dk}.However, we are only interested in the case when h∗

changes stability. This occurs when two conjugateeigenvalues cross the imaginary axis as D goes throughDk for some k ∈ N and all Re(λ) < 0 for all the otherroots of Eq. (18). We contend that this occurs onlyfor k = 0 in the series given in Eqs. (25) and (26) (see[11]). Fixing τd, substituting b0 from Eq (25) into Eq.

(26), and rearranging gives that D0 is the root of thefollowing transcendental equation:

arccos1

D0= τd

√1−D2

0. (19)

Given a fixed τd, β, τ0, and τr, Eqs. (16) and (19)determine the set of values Γ = {g, µ} for which the sys-tem (15) is at a Hopf bifurcation. Γ defines a smoothcurve in the g/µ parameter space which effectively par-titions this space into stable and unstable (or oscilla-tory) regimes. Using standard root finding methods wesolve the transcendental systems (16) and (19) and plotthe curve defined by Γ in Fig. 4.

FIG. 4: The bifurcation set Γ defines a curve of Hopf bifur-cations (supercritical) in g/µ parameter space. Equations(16) and (19) were solved using a bisection root finding al-gorithm with a tolerance of 10−5. The solid curve was com-puted for τd = 1 and partitions the parameter space intostable and unstable regimes, as labelled. The dashed curveis the curve of Hopf bifurcations for τd = 2. The unstableregion for this curve is below it and the stable is above, sim-ilar to the solid curve. Other parameters were τr = 0.3 andβ = 1.

It is immediately apparent that for a fixed µ thereis only one Hopf bifurcation as g is varied; howeverthe converse is not true. For fixed g stable oscillatorynetwork firing develops and then is lost through a re-verse Hopf bifurcation as µ grows large. This is shownin the bifurcation diagram of A(t) shown in Fig. 5.This is intuitively understood as the saturation of in-hibitory feedback for large µ (given by g). Since in-hibitory strength is effectively fixed, large inputs canovertake any network inhibitory effects.

Center manifold reduction and normal form calcu-lations for functional retarded differential systems arepossible [10], however such an analysis is beyond thescope of this paper. Nevertheless, it has been per-

ARTICLE IV 896

FIG. 5: Hopf Bifurcation in population dynamics. Bifur-cation diagram for A(t) with µ as a bifurcation parameterand g = −3. Solid lines represent stable fixed points whiledashed lines are unstable. Open circles are the minimumand maximum of the oscillatory solution for A(t) born outof the Hopf bifurcations. These were determined by inte-gration of Eq. (15) using an Euler approximation schemewith a time step of 10−3. Other parameters were τd = 1,τr = 0.3, and β = 1.

FIG. 6: Oscillations in population activity. Numerical in-tegration of Eq. (15) is shown for g = 0 (top), g = −3.8(middle), and g = −4.0 (bottom). Compare these resultsto those of Fig.1.

formed for a general class of retarded functional dif-ference equations of which Eq. (15) is an example [11].This has produced an expression for the coefficient ofthe third order term in the normal form of the Hopfbifurcation; this coefficient determines the criticality ofthe bifurcation (sub versus supercritical). Applicationof this expression to the exact form of Eq. (15) showsthat for all (µ, g) ∈ Γ we have that the Hopf bifurcationis supercritical, disallowing any bistability (not shown).

A supercritical Hopf bifurcation produces a stablelimit cycle. The amplitude of this limit cycle grows asthe bifurcation parameter grows away from the criti-

FIG. 7: Delay induced oscillations. The curve of Hopf bi-furcations is shown in g, τd parameter space. Oscillatorybehaviour occurs when the delay is long.

cal point. An appearance and growth of an oscillationin A(t) as Eqs. (13) and (14) pass through the Hopfbifurcation is shown in Fig. (6). This oscillation canoccur as either g decreases or µ increases; this mirrorsthe computational results shown earlier, evident upona comparison of Figs. 1 and 6.

With the conditions for a Hopf bifurcation now de-tailed we can also see that the oscillatory behaviour isdependent on the delay time τd. This is shown in Fig.7. For small delays h∗ is stable for large intervals ofboth µ and g. When the delay term becomes compa-rable to the evolution time scale of the system (τm)then h∗ loses stability via the Hopf bifurcation. Thisphenomena is often called delay-induced instability.

DISCUSSION

Delay induced oscillatory activity is common in bi-ological systems [5, 20]. Oscillatory mechanisms thatdepend on collective network activity are often synony-mous with network synchronization. In this study weshow how networks of homogeneous LIF neurons pro-duce oscillatory spike behaviour in a synchronous fash-ion. This behaviour is dependent on both the strengthof an inhibitory network connectivity (g) and the sin-gle unit excitability (µ). Furthermore, the oscillation isobservable from either a macroscopic network measure,such as population activity, or the microscopic singleneuron spike train. Using a spike response formalism[12] we derived dynamic equations for the populationfiring rate. Analysis of these equations showed thatthere exists a curve of Hopf bifurcations in g, µ spacethat separates asynchronous network and synchronousoscillatory network dynamics. This provided a deeperunderstanding of our computational observations.

Detailed studies by Brunel and Hakim [3] and later

ARTICLE IV 907

Brunel [2] outline similar results as those presentedhere. However, their analysis uses a diffusive noiseframework in which they study a delayed Fokker-Planckequation. They show that a Hopf bifurcation in theprobability density gives way to a collective networkoscillation. Our study differs in that we use an escapenoise formalism and a renewal hypothesis to derive dy-namic population equations. A Hopf bifurcation in ourdynamics mirrors the results in [2, 3].

The LIF system and its population dynamic reduc-tion is meant to model recurrent closed-loop neuralarchitectures. This connectivity is common in manysensory systems, most prominently in cortico-thalamicnetworks [29]. Indeed, collective oscillatory behaviouris well documented in many sensory brain areas, and ithas been speculated that it may be delay-induced [27].Our study, like many others, explores how network con-nectivity and single cell properties regulate collectiveoscillatory behaviour. However, an interesting questionremains unaddressed: how do such oscillatory mecha-nisms interact with, and potentially code, a naturalisticstimuli? This is the focus of recent work by the authorsin the electrosensory system [7, 8].

APPENDIX

In this appendix we present the analysis of the char-acteristic equation

λ + 1−De−λτd = 0 (20)

In general, λ ∈ C; let λ = a + ib where a, b ∈ R.Substitution into Eq. (20) gives

a = De−τda cos(τdb)− 1 (21)b = −De−τda sin(τdb) (22)

Consider the case when a = 0. From Eqs. (21) and(22) we then have:

1 = D cos τdb (23)

b =√

D2 − 1. (24)

For b ∈ R we have the restriction that |D| ≥ 1. Eqs.(23)-(24) admit a countably infinite number of solutionsthat are given by the following sequence:

bk =1τd

[arccos

1Dk

+ 2πk

](25)

Dk =√

b2k + 1 (26)

with k ∈ N. For each k we have a pair of conjugateeigenvalues, λk = ±ibk, that lie on the imaginary axis.We will show that for each k we have that the system(15) is at a Hopf bifurcation.

The Hopf bifurcation theorem for delayed systemsis similar to the case for non-delayed systems [13]. Inorder for a system to be at a Hopf bifurcation we requirethat (1) a = 0, (2) bk 6= 0, and (3) da

dDk

∣∣∣a=0

6= 0.

Condition (1) is satisfied by assumption and condition(2) by substituting |Dg| > 1 into Eq. (25). It is simpleto show that condition (3) is satisfied for any k ∈ N.

Differentiating Eqs. (21) and (22), rearrangingterms, and setting a = 0 gives

da

dDk

∣∣∣∣a=0

=cos τdbk + τDk

2Dkτd cos τdbk + τ2d D2

k + 1. (27)

Consider the converse of condition (3), i.e. thatda

dDk

∣∣∣a=0

= 0. Then we have from Eq. (27) the relation

cos τdbk + τdDk = 0. (28)

Substituting b from Eq. (25) and rearranging terms inEq. (28) gives τd = −1/D2

g . This implies that τd < 0which is a contradiction to the original assumptions,thus da

dDk

∣∣∣a=0

6= 0. This argument holds for all k ∈ N.

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B. Doiron, M. J. Chacron, L. Maler, A. Longtin and J.BastianInhibitory feedback required for network oscillatory responses to communication butnot prey stimuli.Nature. 421: 539-543, 2003.

92

ARTICLE V 93

the use of laboratory animals. Briefly, monkeys were anaesthetized with sufentanil citrate(6–12 mg kg21 h21; intravenous (i.v.) injection) and paralysed with Pavulon(pancuronium bromide; 0.05 mg kg21 h21; i.v.). CO2 and body temperature weremaintained at 4.0% and 37.5 8C, respectively. Each eye was brought into focus on thescreen of a Trinitron monitor using custom-fit contact lenses. On two animals withapparent binocular divergence, the two eyes were brought into convergence by a prism.The whole screen occupied a visual field of 198 £ 148 that covered the visual field of therecorded portions of V2 (2–78 along the vertical meridian). In one animal, anatomicaltracers were injected into V2. Its post-mortem occipital operculum was sectioned in thetangential plane and stained for cytochrome oxidase and tracers. A detailed description ofthese methods has been published previously3.

Optical recordingThe intrinsic optical signal was recorded using a slow-scan CCD (charge-coupled device)array camera (Photometrics, CH 250 0206) that was focused 0–300 mm below the surfaceby a tandem lens system. The cortical surface was illuminated by light at 630 ^ 15 nmwavelength. In each series of frames, four 103-ms-long frames taken 1.3–2.8 s after theonset of each stimulus were used as response frames. Two background frames were taken at0.7 and 0.2 s before stimulus onset. Each stimulus, viewed binocularly by the animal, lastedfor 3 s and was separated by 10 s of uniform grey on the screen. In each trial, 50 repetitionsof each stimulation condition were interlaced pseudo-randomly with other conditions,and the corresponding frames were averaged. A differential image between twostimulation conditions was calculated by subtracting the response frame of one conditionfrom that of the other condition, divided by the first background frame. A single-condition image of one stimulation condition was calculated by dividing thecorresponding response frame with the first background frame. For each stimulationcondition, a control image was calculated by dividing the second background frame withthe first background one. In each trial, four control images derived from four differentstimulation conditions were used for the statistical analysis. In each functional image, apair of gaussian spatial filters (s.d. ¼ 49 and 326 mm, respectively) was used to remove thehigh and low spatial frequency noises. The four single-condition images corresponding tofour different time points under the same stimulation condition were then averaged toobtain the final single-condition image for each condition. For each pixel, the intensityvalues in these four single-condition images were compared with those in the four controlimages by the Student’s t-test, to determine the significance level of the activation. In twoanimals, a series of frames was taken when the screen was kept at a constant uniform grey(blank presentation). The blank presentations were pseudo-randomly interlaced with thestimulus presentations. When the single-condition images during the blank presentationswere used as controls, the regions significantly activated by each stimulus were largely thesame as those derived from the above analysis, which used the second background imagesas controls (Supplementary Fig. 3).

To identify the colour-preferring modules, we used isoluminant red/green gratings(sinusoidal, 0.25 cycles degree21, drifted at 1 cycle s21), high-contrast achromatic gratings(sinusoidal, 2.0 cycles degree21, 2.0 cycles s21, 100% luminance contrast) and low-contrast achromatic gratings (sinusoidal, 0.25 cycles degree21, 2.5 cycles s21, 7%luminance contrast). They all had an average luminance of 10 cd m22 and fourorientations (08, 908, 458 and 1358). The two colours in the red/green gratingscorresponded to the colours of the red and green phosphors of the cathode-ray tube,respectively (see Fig. 1a). The colour-preferring modules that we identified appeared darkin the differential image that subtracted the responses to high-contrast achromaticgratings from those to isoluminant gratings, but not in the differential image thatsubtracted the responses to high-contrast gratings from those to low-contrast achromaticgratings.

To study the cortical activation evoked by a single colour, we used static, colour/greygratings (square wave, 0.25 cycles degree21, two alternating spatial phases, 08, 908, 458,1358), or uniform colours covering the full screen (10 cd m22). CIE-xy chromaticitycoordinates of the tested colours are: red (0.55, 0.33), orange (0.54, 0.40), yellow (0.45,0.47), lime (0.35, 0.54), green (0.27, 0.49), aqua (0.23, 0.36), blue (0.16, 0.08), purple(0.23, 0.11), pink (0.38, 0.27), and white and grey (0.31, 0.31). The uniform grey betweenstimuli and the grey in each grating had a luminance of 10 cd m22. In isoluminantgratings, all colours had a luminance of 10 cd m22. In luminance-varying gratings, theluminance of different colours were (in cd m22): red, 14; orange, 18; yellow, 28; green, 19;aqua, 22; blue, 8; purple, 11. These values are similar to those used by ref. 9 so that theyreflect the difference in luminance of the colours described by different names of a naturallanguage.

Electrode recordingElectrode penetrations were made vertically with the guide of functional maps. All unitswere recorded between 0 and 1 mm below cortical surfaces. After the receptive field of asingle- or multi-unit was plotted, flashed colour bars or drifting colour/grey gratings (1.5cycles degree21, 1 cycle s21) were presented in the receptive field for 1.28 s every 3.28 s. Netneuronal responses were obtained by subtracting the spontaneous firing rate during 1 sbefore the presentation of each stimulus from that during the presentation. Ten repetitionsof each stimulus were interlaced pseudo-randomly with other stimuli, and their netneuronal responses were averaged to obtain the mean response to the stimulus.

Received 30 September; accepted 26 November 2002; doi:10.1038/nature01372.

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12. Peterhans, E. & von der Heydt, R. Functional organization of area V2 in the alert macaque. Eur.

J. Neurosci. 5, 509–524 (1993).

13. Gegenfurtner, K. R., Kiper, D. C. & Fenstemaker, S. B. Processing of colour, form, and motion in

macaque area V2. Vis. Neurosci. 13, 161–172 (1996).

14. Kiper, D. C., Fenstemaker, S. B. & Gegenfurtner, K. R. Chromatic properties of neurons in macaque

area V2. Vis. Neurosci. 14, 1061–1072 (1997).

15. Moutoussis, K. & Zeki, S. Responses of spectrally selective cells in macaque area V2 to wavelengths and

colors. J. Neurophysiol. 87, 2104–2112 (2002).

Supplementary Information accompanies the paper on Nature’s website

(ç http://www.nature.com/nature).

Acknowledgements We thank L. Cleary, E. Kaplan, J. Maunsell and J. Krauskopf for their

suggestions on the manuscript. We also thank A. Zych and X. Huang for computer programming,

Q. Huang for technical support and J. Chen for discussion. This research was supported by an

individual grant from the National Eye Institute to D.J.F. and a core grant from the National Eye

Institute to the University of Texas, Health Science Center at Houston.

Competing interests statement The authors declare that they have no competing financial

interests.

Correspondence and requests for materials should be addressed to Y.X.

(e-mail: youping@camelot.mssm.edu).

..............................................................

Inhibitory feedback required fornetwork oscillatory responses tocommunication but not prey stimuliBrent Doiron*†, Maurice J. Chacron*†, Leonard Maler†, Andre Longtin*& Joseph Bastian‡

* Physics Department, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada† Department of Cellular and Molecular Medicine, University of Ottawa, Ottawa,Ontario K1H 8M5, Canada‡ Department of Zoology, University of Oklahoma, Norman, Oklahoma 73019,USA.............................................................................................................................................................................

Stimulus-induced oscillations occur in visual1,2, olfactory3–6 andsomatosensory7 systems. Several experimental2,3,5 and theoreti-cal8–13 studies have shown how such oscillations can be generatedby inhibitory connections between neurons. But the effects ofrealistic spatiotemporal sensory input on oscillatory networkdynamics and the overall functional roles of such oscillations insensory processing are poorly understood. Weakly electric fishmust detect electric field modulations produced by both prey(spatially localized)14 and communication (spatially diffuse)15

signals. Here we show, through in vivo recordings, that sensorypyramidal neurons in these animals produce an oscillatoryresponse to communication-like stimuli, but not to prey-likestimuli. On the basis of well-characterized circuitry16, we con-struct a network model of pyramidal neurons that predicts thatdiffuse delayed inhibitory feedback is required to achieve oscil-

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ARTICLE V 94

latory behaviour only in response to communication-like stim-uli. This prediction is experimentally verified by reversibleblockade of feedback inhibition that removes oscillatory beha-viour in the presence of communication-like stimuli. Our resultsshow that a sensory system can use inhibitory feedback as amechanism to ‘toggle’ between oscillatory and non-oscillatoryfiring states, each associated with a naturalistic stimulus.

Organisms must discriminate between sensory inputs with vary-ing spatiotemporal structure. Electric fish offer a clear examplebecause prey and communication signals differ greatly in theirspatial extent. Sensory systems have evolved architectures to processsuch signals efficiently. Feedback connections are common in neuralsystems, especially at the thalamo-cortical level, where they controlsensory transmission2. Feedback is also a prominent feature ofbrainstem electrosensory circuitry, and its organization parallelsthat of cortico-thalamic pathways16. We show here how inhibitoryfeedback allows electrosensory lateral line lobe (ELL) pyramidalneurons16 to switch between non-oscillatory and oscillatory states,each associated with prey-like or communication-like stimuli,respectively.

The electric fish Apteronotus leptorhynchus produces an electricfield through a rhythmic electric organ discharge (EOD). Electro-receptors respond to amplitude modulations of this field producedby conspecifics or objects. These afferents encode amplitude modu-lations17 and synapse onto ELL pyramidal neurons16,18. Two stimu-lus geometries, ‘local’ and ‘global’, were used to activate receptorafferents and pyramidal neurons (see Methods). Local geometryapplies the stimulus to a small fraction of the fish’s skin, stimulatingonly a portion of the centre of the receptive field of a pyramidalneuron from which we record (Fig. 1a, left). This produces localizedamplitude modulations of the fish’s own EOD that are spatiallysimilar to those produced by small aquatic invertebrate prey14.Global geometry, however, applies a bilateral stimulus homoge-nously to the whole fish, stimulating both the centre and surroundreceptive fields of pyramidal neurons (Fig. 1b, left). This producesamplitude modulations that are spatially similar to communicationsignals15. Large objects (obstacles, root masses) will producespatially extensive amplitude modulations, but these are expectedto be heterogeneous. These fish routinely give communicationresponses15 in response to global stimuli with gaussian temporalstructure (data not shown), showing that these effectively mimiccommunication stimuli.

Figure 1 shows a typical pyramidal cell response recorded in vivowhen band-limited gaussian EOD amplitude modulations (0–40 Hz) were applied locally or globally (n ¼ 15). With local stimu-lation, the spike time autocorrelation19, A(t), was positive at smallvalues of the lag, t (after a negative correlation owing to refractori-ness), and histograms of the interspike interval (ISI) showed a singlepeak (Fig. 1a, middle and right). A(t) shows that the spikes are notindependent of each other; rather, there is a tendency to producehigh-frequency (,100-Hz) clusters of spikes20. By contrast, whenthe stimulus was applied globally, a damped oscillation in A(t) andtwo distinct peaks in the ISI histogram appeared, suggestingoscillatory (and/or bursting) behaviour (Fig. 1b, middle andright); no such structure is observed for local stimulation. It isthis transition from non-oscillatory to oscillatory behaviour con-tingent on the switch from local to global stimulation that is thefocus of our study.

To verify the presence of oscillatory behaviour and to comparequantitatively the discharge patterns under both stimulus geome-tries, we computed the power spectral densities (PSDs) of the spiketrain during both global and local stimulation (Fig. 1d). There was asignificant peak in the PSDs during global stimulation(32.41 ^ 3.92 Hz; n ¼ 15) that was absent during local stimulation,demonstrating that the spike train is oscillatory only during globalstimulation. Using the PSDs, we defined an ‘oscillation index’ (seeMethods); this index was larger under global ð10:12^

3:68 spikes2 s21Þ than under local ð2:28^ 0:62 spikes2 s21Þ geome-tries (pair-wise t-test, P ¼ 0.0034; n ¼ 15). In addition, jointinterval histograms computed from responses to global and localstimuli differed markedly. Under local stimulation, the joint intervalhistograms showed no clear structure; however, with global stimuli,short ISIs (,15 ms) were followed preferentially by long ISIs(.15 ms) and vice versa (Fig. 1c). Oscillatory discharge was notobserved with local stimulation, even with a contrast up to six timesthat of the global stimulus, although the peak firing rates weresimilar to those during global stimulation. These results show thatan oscillatory discharge occurs for global but not local stimulation.

The requirement of a global stimulus for oscillatory dischargesuggests that the mechanism involves the integration of spatiallydiffuse inputs. The negative component of A(t) at t < 15 mssuggests that delayed inhibitory processes are also involved. ELLcircuitry is well described and indicates that feedforward pathwayssatisfying these criteria are unlikely to exist16. But ELL pyramidalneurons project to an isthmic structure (n. praeminentialis dorsa-lis), which in turn provides several feedback projections to theELL16. This includes a spatially diffuse delayed (,15 ms) inhibitorypathway emanating from a small group of bipolar cells thatterminates exclusively on ELL pyramidal cells21–23. Theoreticalanalysis of models of biological and chemical systems with delayednegative feedback have shown how delays can often supportoscillatory dynamics24; however, little is known about how thespatial extent of a driving stimulus can regulate a feedback-mediated oscillation. Thus, we constructed a neural networkmodel of the ELL to test the hypothesis that delayed inhibitoryfeedback is sufficient for global stimulus-induced oscillations.

Figure 1 ELL pyramidal neurons show differential responses to local (prey-like) and global

(communication-like) stimuli (n ¼ 15). a, Local stimulation. Left, local stimulus model; a

dipole was placed near the skin to stimulate only a part of the pyramidal cell receptive field

centre. Middle and right, A(t) and ISI histograms of a typical pyramidal cell response to a

local stimulus. b, Global stimulation. Left, global stimuli model; two Agþ/AgCl wire

electrodes were placed transverse to the fish to produce spatially extensive stimuli. Middle

and right, A(t) and ISI histograms of pyramidal cell response to a globally applied stimulus.

The mean firing rates of the stimulated cell were 21.8 spikes s21 and 22.8 spikes s21 for

local and global stimulus geometries, respectively. Note that the same cell was used for

both stimuli. c, Joint interval histograms under local (left) and global (right) stimulation.

d, The PSDs of the spike train under local and global stimulus geometries. Broken lines

surrounding the PSDs are the 95% confidence bands (^2 s.d.).

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ARTICLE V 95

In brief, we modelled the pyramidal neurons as a network of leakyintegrate-and-fire (LIF) neurons (ref. 19 and see Methods). Allneurons transmit their spikes to a central cell population, G. Foreach spike that it receives, G feeds back an inhibitory synaptic(a-function19) response to all pyramidal neurons in the networkafter a fixed time delay, td. The physiological interpretation of G isthe integration of pyramidal cell output by bipolar cells of then. praeminentialis dorsalis and their GABA (g-aminobutyric acid)-mediated projection back to the ELL. A diagram of the networkmodel connectivity is shown in the left panel of Fig. 2a.

Each LIF neuron was driven by two sources. The first was a noiseinput, h i(t), with positive bias (the subscript i denotes spatialposition), with h being uncorrelated across space. This modelledstochastic spontaneous activity is similar to that seen in vivo20. Thesecond input source was band-limited (0–40 Hz) gaussian noise,S(t), mimicking the temporal nature of the electrosensory stimulusused in the experiments. To model local geometry, S(t) was appliedto a single neuron, whereas global geometry was modelled byapplying S(t) homogeneously to all neurons. With local stimu-lation, A(t) and ISI histograms measured from the neuron receivingS(t) were similar to those observed experimentally, indicating anon-oscillatory state (compare Figs 1a and 2a, middle and right).With global stimulation, however, a transition to an oscillatoryresponse was observed, similar to that observed experimentally(compare Figs 1b and 2b, middle and right). The oscillation index ofthe model was 8.66 spikes2 s21 during local stimulation and15.0 spikes2 s21 during global stimulation. Joint interval histogramsconstructed from model spike trains during both local and globalconditions were qualitatively similar to those constructed from thedata.

These model results were insensitive to parameter heterogeneitiesin the network resulting from the known variability of pyramidalneurons (ref. 20 and see Methods). In addition, a sixfold increase inthe contrast of a local stimulus did not induce oscillatory discharge,similar to experimental findings. We have previously proposed atwo-compartmental model of ELL pyramidal cells25 that simulatesthe single-cell bursting dynamics observed during current injectionin vitro26. We have also qualitatively reproduced the simulationresults presented above (Fig. 2) using a network of these morerealistic pyramidal cells (see Supplementary Information).

During local stimulation, the only shared input among neurons isfrom the global inhibitory feedback, G. In spite of this commoninput, the intrinsic noise sources, h i(t), were of sufficient strength tosuppress significant correlations between neurons in the network.This is demonstrated in the model network raster plots (Fig. 3a,top), which show a lack of coincident firing between individualneurons during local input, and by network stimulus histograms(summed activity of all neurons) that weakly fluctuate about aconstant value (Fig. 3a, middle). The feedback response, G (Fig. 3a,bottom), is determined by the network firing pattern, as illustratedby the network spike histograms, and is thus also roughly constant.

During global conditions, however, S(t) was distributed homo-geneously over the whole network, evoking stimulus-inducedcorrelations between neurons. This allowed for increased networksynchronization, as shown in network raster plots (Fig. 3b, top) andnetwork stimulus histograms (Fig. 3b, middle). As a result of thisincreased correlation, G received near-coincident spikes from acrossthe network, yielding a significant summation of a-functionresponses. This enhanced response is fed back to all neurons inthe network after a delay time, td, causing ‘waves’ of inhibition thatsuppress network firing (Fig. 3b, bottom). As an inhibitory waveabates, subsequent stimulus-induced coincident firing across thenetwork develops. Thus, the interplay between stimulus-inducedcorrelations and an associated delayed increase of feedback inhi-bition creates an oscillation in the network activity with perioddetermined by both the axonal delay, td, and the synaptic timeconstant a.

The simulation results shown in Figs 2 and 3 suggest that thediffuse delayed inhibitory feedback is an integral component of thepyramidal cell oscillatory response to global stimuli. To verify thisexperimentally, we used local injection of xylocaine, a sodiumchannel antagonist, to specifically block the inhibitory feedbackcomponent of the tractus stratum fibrosum (StF) pathway (n ¼ 10;see Fig. 4 and Methods). The effect of this block was to removetemporarily feedback inhibition to the ELL, effectively opening theloop from the n. praeminentialis dorsalis to ELL. Xylocaine injec-tion reversibly blocked oscillatory responses to globally appliedstimuli: A(t) showed no damped oscillations and the ISI histogramswere unimodal (Fig. 4b, c, black traces). However, these neurons

Figure 2 ELL neural network simulations involving global inhibitory feedback show

differential responses to local and global stimuli. a, Local stimulation. Left, network

showing the pyramidal cell layer (PCL) projecting spikes to a kernel, G(t), which feeds back

inhibitory responses to the PCL after a delay t d. For local stimuli only one cell receives the

stimulus (arrow). Middle and right, A(t) and ISI histograms of the simulated LIF neuron

during local stimulation. b, Global stimulation. Left, identical network to that in a, but all

LIF neurons receive the same stimulus. Middle and right, A(t) and ISI histograms of a LIF

neuron response to global stimulation. The mean firing rates of the cell were

17.5 spikes s21 and 16.5 spikes s21 for local and global stimulus geometries,

respectively.

Figure 3 Simulated asynchronous firing and synchronized oscillations during one

presentation of a local or global stimulus, respectively. Top, network raster plots under

local (a) and global (b) stimulus with network stimulus histograms shown underneath.

Bottom, the total value of the feedback inhibition, G(t), is plotted for the same simulations

that gave the spike times.

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ARTICLE V 96

showed clear oscillatory behaviour before and after recovery fromStF (inhibitory component) blockade (Fig. 4b, c, red traces, control;green traces, recovery). The oscillation indices for control andrecovery were similar (pair-wise t-test, P ¼ 0.353; n ¼ 10) andsignificantly different from those observed during feedback inhibi-tory blockade (pair-wise t-test, P ¼ 0.02; n ¼ 10). These experimentsverify the model’s prediction that diffuse inhibitory feedback isrequired for ELL pyramidal neurons to show differential responsesto local versus global stimuli.

Our network simulations predict that the synchronization of ELLpyramidal cell firing is an aspect of the oscillatory mechanisminvolved during global stimulation (Fig. 3). Field potential record-ings from the ELL show oscillations during global stimulation (datanot shown), consistent with synchronous activity; however, inter-neuron populations in the ELL16 and the expected stimulus-drivenincrease of synchronization during global stimulation complicatethe interpretation of this observation.

Inhibition is necessary for synchronization in the olfactorysystem3,5 and thalamocortical circuits2,8. Modelling studies haveshown that synchronous oscillatory solutions exist when inhibitoryfeedback gain is strong and inhibitory synaptic timescales arelong8,9,11,13, or when there are sufficiently long axonal delays9,10,13.Other studies have shown that network oscillations can occur withconstant depolarizing input12 or with increased mean firing rate ofuncorrelated excitatory Poisson inputs13. In the latter study, oscil-lations also occurred with increased spatial spread of uncorrelatedinputs distributed to the network. In this study, by contrast,synchronous oscillations were produced not by varying feedbackgain, intrinsic cellular properties or mean network depolarization,but by simply changing the spatial extent of a stimulus.

It is known that pyramidal neurons can estimate the time courseof prey-like stimuli applied under local, but not global, geome-tries27. But these cells do produce bursts of spikes in response tospecific features of a global stimulus28. These studies suggest thatpyramidal cell coding strategies may also depend on the spatialextent of stimuli. We have confirmed that the stimulus encodingabilities of pyramidal cells can be altered during the oscillatorydischarge induced by a global stimulus (see Supplementary Infor-mation).

We have described a mechanism whereby the spatial extent ofstimuli determines the firing behaviour of sensory neurons in theelectrosensory system. In addition, these stimuli were spatiallysimilar to those that the fish would encounter in either prey locationor communication situations, giving a clear behavioural signifi-cance to our results. We have shown that this mechanism requiresdelayed inhibitory feedback—a very general feature of the nervoussystem2,29. To our knowledge, this is the first demonstration of howdelayed inhibitory feedback enables neurons in a sensory channel totoggle between two distinct firing states, with each state connectedto a behaviourally relevant stimulus. A

MethodsPhysiologyExtracellular single-unit recordings from ELL pyramidal neurons were identified on thebasis of recording depth and response patterns27. Random amplitude modulations(RAMs) of the EOD were produced by adding an amplitude-modulated sinusoidalwaveform to the continuing discharge of the fish. The sinusoid was phase locked to theEOD with a period roughly equal to that of the EOD. RAMs were produced by multiplyingthe sinusoid by zero-mean band-limited gaussian noise (0–40 Hz, eighth orderButterworth filter). For local geometry, we applied stimuli through a small dipole, tipspacing of 2 mm, positioned at a site in the cell’s receptive field centre 2–3 mm lateral to theskin. Global geometry stimuli were applied homogeneously through two electrodes 19 cmlateral to either side of the fish (Fig. 1b, left). The typical global stimulus amplitude was250 mV cm21, and the magnitude of the local stimulus was adjusted so that electroreceptorafferent responses were equivalent for either geometry27. Histograms and autocorrelationbin widths were 1 ms and were computed from about 4,000 spikes in all cases. ELLpyramidal cells occur in both basilar and non-basilar varieties18; our results include bothsubtypes. We limited recordings to pyramidal neurons from lateral and centrolateral ELLsegments with mean spontaneous firing frequencies greater than 15 Hz.

Reversible block of the tractus StF, which conveys descending excitatory and inhibitoryinputs to pyramidal neurons21, was achieved through micropressure ejection of the localanaesthetic xylocaine. A xylocaine filled micropipette (5–10-mm tip diameter) carrying abipolar pair of 25-mm stainless steel stimulation wires was positioned rostral to the ELL at adepth where electrical stimulation (0.1 ms, 15–25 mA current pulses) evoked thecharacteristic StF field potential in the ELL22. Because the inhibitory component of the StFlies at its ventral-most aspect just dorsal to the lateral lemniscus (ELL efferents)21, thepipette was positioned just dorsal to a site where antidromic activation of ELL pyramidalneurons occurred. Ejection of xylocaine (2% concentration; 100-ms pulses at 50 p.s.i.)reduced the StF field potential, verifying blockade; the loss of pyramidal cell oscillatoryresponses paralleled this reduction. Blockade of the dorsal StF, which conveystopographical excitatory feedback16,22, eliminated the StF field potential but did notprevent global stimulus-induced pyramidal cell oscillations (data not shown). Allprocedures were in accordance with the University of Oklahoma animal care and useguidelines. The oscillation index was defined as the difference between the maximum andthe minimum value of the PSD between 20 and 40 Hz.

Network simulationsWe model the ELL pyramidal cell layer as a network of N LIF model neurons19. Themembrane potential of the ith neuron is labelled Vi(t). When Vi(t) crosses a spikethreshold v, it is reset to the potential (Vreset) and a spike is emitted from neuron i. Betweenspike times (Vi(t) , v), the dynamics are governed by the following stochastic delay-differential system:

dVi

dt¼

2V i

tmþ hiðtÞ þBi þ SiðtÞ2 Gðt 2 tdÞðVi 2 V rÞ ð1Þ

GðtÞ ¼g

N

XN

j¼1

XMj ðtÞ

m¼0

ðt 2 tjmÞ

aexp 1 2

ðt 2 tjmÞ

a

� �ð2Þ

tm is the passive membrane time constant, and a constant depolarizing bias B i is set sothat, for zero input ðhi ¼ g ¼ S¼ 0Þ; each neuron fires periodically ðBi . v=tmÞ: h i(t) is azero mean Ornstein–Uhlenbeck process30, with time constant th. The Ornstein–Uhlenbeck processes are independent (h i(t) and h j(t) are not correlated if i – j) and theautocorrelation of h i(t) is given by

khiðtÞhiðtþ tÞl¼j2

2th

e2t=th ð3Þ

RAMs were introduced through a zero mean band-limited (0–40 Hz) gaussian noise, S(t),with variance W. Local stimulation was simulated by applying S(t) to only one LIF neuron,whereas for global stimulus geometry S(t) was applied to all neurons.

The direct inhibitory feedback pathway16 is modelled by the kernel G(t) in equation(2). G(t) receives spikes from the jth neuron, t jm, over the history of neuron j(0 # m # M j (t), where M j (t) is the total spike count of neuron j at time t). G(t)transforms a spike input, t jm , into an a-function conductance19 with time constant a. G(t)then projects the summation of all inputs uniformly to all neurons in the network after afixed delay, td. The gain of the pathway is set by the constant g . 0, and G(t) multiplies abattery term (Vi 2 Vr) with reversal potential Vr . The LIF neurons do not have localconnections among themselves, as observed for ELL pyramidal cells16.

The intrinsic model parameters were set to tm ¼ 10 ms, j ¼ 5.5 nA, B i ¼ 0.84 nA (we

Figure 4 Blockade of the inhibitory component of the StF pathway. a, Pyramidal neurons

in the ELL project excitatory connections to bipolar cells (BP) of the n. praeminentialis

dorsalis through the lateral lemniscus pathway (LL). In turn, the BP cells project back to

the ELL through the StF, providing GABA-mediated feedback to the pyramidal neurons.

StF axons were blocked by application of xylocaine. b, c, ISI histograms (b) and A(t) (c) are

plotted for control cells (red), during the block (black) and after recovery from xylocaine

(green). Histograms and A(t) were constructed as in Fig. 1 and smoothed (gaussian-

filtered, j ¼ 5 ms) to facilitate comparison between states.

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ARTICLE V 97

assumed a capacitance of 1 nF) and th ¼ 15 ms. These were chosen so that for S(t) ¼ 0,

A(t) and ISI histograms were similar to those observed under spontaneous conditions

in vivo20. We set N ¼ 100 to make local and global stimulation significantly distinct. The

RAM stimulus variance was W ¼ 0.238 nA2, which resulted in firing statistics comparable

to those of stimulated pyramidal neurons (Fig. 2). The feedback parameters were set to

g ¼ 390 S F21, a ¼ 3 ms, V r ¼ V reset ¼ 0 mV; to model the feedback as GABAA shunting

inhibition, previously reported for interactions between bipolar and pyramidal cells23. The

loop delay was set to td ¼ 12 ms, allowing a fit to experimental data (Figs 1 and 2), and is

also comparable to previously estimated values22. Equation (1) was integrated by a simple

Euler–Maruyama30 scheme with Dt ¼ 0.025 ms. The effects of parameter heterogeneities

were tested by choosing B i and tm from gaussian distributions with mean values as given

above. The results were qualitatively similar to the homogeneous case for distributions

with coefficient of variations of 0.82 and 1.5 for the B i and tm distributions, respectively.

Received 20 August; accepted 26 November 2002; doi:10.1038/nature01360.

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Acknowledgements We thank J. Lewis, C. Laing, R. W. Turner and M. Higley for reading the

manuscript. Funding was provided by the National Science and Engineering Research Council

(B.D., M.J.C., A.L.), the Canadian Institutes of Health Research (A.L., L.M.) and the National

Institutes of Health (J.B.).

Competing interests statement The authors declare that they have no competing financial

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Correspondence and requests for materials should be addressed to B.D.

(e-mail: bdoiron@science.uottawa.ca).

..............................................................

Fidelity in planar cell polaritysignallingDali Ma*, Chung-hui Yang†, Helen McNeill‡, Michael A. Simon†& Jeffrey D. Axelrod*

* Department of Pathology, Stanford University School of Medicine, Stanford,California 94305-5324, USA† Department of Biological Sciences, Stanford University, Stanford, California94305, USA‡ Cancer Research UK, London Research Institute, 44 Lincoln’s Inn Fields, LondonWC2A 3PX, UK.............................................................................................................................................................................

The polarity of Drosophila wing hairs displays remarkablefidelity. Each of the approximately 30,000 wing epithelial cellsconstructs an actin-rich prehair that protrudes from its distalvertex and points distally. The distal location and orientation ofthe hairs is virtually error free, thus forming a nearly perfectparallel array. This process is controlled by the planar cellpolarity signalling pathway1–4. Here we show that interactionbetween two tiers of the planar cell polarity signalling mechan-ism results in the observed high fidelity. The first tier, mediatedby the cadherin Fat5, dictates global orientation by transducing adirectional signal to individual cells. The second tier, orche-strated by the 7-pass transmembrane receptor Frizzled6,7, alignseach cell’s polarity with that of its neighbours through the actionof an intercellular feedback loop, enabling polarity to propagatefrom cell to cell8. We show that all cells need not respondcorrectly to the presumably subtle signal transmitted by Fat.Subsequent action of the Frizzled feedback loop is sufficient toalign all the cells cooperatively. This economical system is there-fore highly robust, and produces virtually error-free arrays.

A group of signalling molecules, including Frizzled (Fz)7, Dishev-elled (Dsh)9, Flamingo (Fmi)10,11, Van Gogh12,13 and Prickle (Pk)14

mediates planar cell polarity (PCP) signalling in various developingDrosophila tissues1–4. In wing cells, these proteins participate in anintercellular feedback loop that generates subcellular asymmetryand directs prehair location8,15,16. Frizzled on the distal side of onecell recruits Dsh to the membrane, thus stabilizing Fz localization,and simultaneously recruits Pk to the proximal side of the adjacentcell, where it prevents Dsh localization (Fig. 1a). A competitiontherefore occurs between Fz on either side of the intercellularboundary, amplifying small differences in Fz levels to all-or-nonedifferences8. Coupled with the observation that cells do notaccumulate high Fz levels and construct prehairs on two sides,this mechanism enables polarity to propagate from cell to cell(Fig. 1b). For example, removing Fz from a clone of cells producesa difference in Fz at the distal boundary of the clone that is reversedfrom the wild type, and this reversal propagates, resulting in the‘domineering non-autonomy’ first described by ref. 6 (Fig. 1b). Thequestion remains, however, of what initiates the imbalance ofFz activity along the proximal–distal axis of each cell in a wild-type wing. We have proposed previously that, in the eye, two

letters to nature

NATURE | VOL 421 | 30 JANUARY 2003 | www.nature.com/nature 543© 2003 Nature Publishing Group

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Supplementary material - Doiron et al. (2003) Nature 421:539-544, 2003.

Brent Doiron1,2, Maurice J. Chacron1,2, Leonard Maler2, Andre Longtin1 and Joseph Bastian3

1 Physics Department, University of Ottawa, Ottawa, Ontario, Canada, K1N 6N52 Department of Cellular and Molecular Medicine, University of Ottawa, Ottawa, Ontario, Canada, K1H 8M5

3 Zoology Department, Univeristy of Oklahoma, Norman, Oklahoma, 73019.

INTRINSIC BURST MECHANISMS AND NETWORK OSCILLATIONS

LIF model neurons were used to represent ELL pyramidal cells in the network simulations presented in themain results of the letter (Figs 2,3 of [1]). This type of model neuron makes no assumptions upon the intrinsicionic capabilities of the cell membrane and as such is a simplistic representation of cell behavior. The advantageof using this description to explore network behavior is that it allows for a clear distinction between emergentnetwork properties and intrinsic cellular dynamics. Furthermore, the lack of assumptions of specific ionic channelsincreases the generality of any results obtained. However, the disadvantage of using such simple neurons is thatthey inevitably under represent the true complexity of the intrinsic behavior of the real cells. The is indeed the casefor ELL pyramidal cells where extensive in vitro, in vivo, and modeling studies have described complex burstingfiring patterns mediated by several nonlinear ionic currents. In this section we will first review our past results onthe intrinsic bursting dynamics of ELL pyramidal cells, summarizing a previously presented model that reproducesthe bursting nature of cells recorded in vitro. Next we will show how the oscillatory discharge described in themain letter is not dependent upon this dynamics and can be achieved when using the more realistic bursting modelneurons. Finally, we hypothesize upon interactions between the network and intrinsic cell dynamics.

ELL pyramidal cells are spatially extended neurons possessing a long apical dendritic arborization [2] (up to800µm). This dendritic shaft does not branch until approximately 200µm from the soma and immunohistochemicalstudies show that a patched distribution of active Na+ channels exist over this proximal dendritic area [2]. In vitroexperiments have shown that the Na+ distribution supports an active dendritic action potential backpropagationthat establishes a depolarizing after potential (DAP) at the soma subsequent to somatic spike replolarization[2, 3]. This DAP transiently affects the excitability of the somatic membrane, facilitating the production of furthersomatic action potentials. A novel form of burst discharge has been shown to occur in these pyramidal cells wherebythe width of the dendritic action potential is dynamically regulated resulting in an increased DAP amplitude [3, 4].A recent modeling study have shown that a cumulative inactivation of the dendritic K+ current responsible fordendritic spike repolaraization is sufficient to account for this effect [4]. This increase in DAP amplitude causes areduction in somatic ISI, which then through increased dendritic K+ inactivation causes a further increase in DAPamplitude. This positive feedback process establishes a continual and accelerated reduction of somatic ISI. Thissomatic-dendritic interplay is abruptly halted when the somatic ISI is lowered below the refractory period of thedendritic action potential resulting in failure of dendritic backpropagation. This removes the subsequent somaticDAP and the somatic ISI is lengthened and AHP is hyperpolarized. The entire process groups action potentialsinto small clusters (typically 3-6), which we label as bursts. Figure 1a gives a schematic of this burst process whilea complete description of this mechanism is given in detail elsewhere [3, 4].

The experiments and modeling described above gave sufficient understanding of the burst mechanism so as toconstruct reduced models [5, 6] which are computationally less involved than large scale compartmental models[4] yet retain the essence of the cellular dynamics which support this form of burst discharge. Specifically one ofthese descriptions [5], entitled the Ghostburster, was set of 6 coupled nonlinear differential equations detailing atwo compartment description of an ELL pyramidal cell. We have extended our mathematical model of the ELLpyramidal cell network so as to replace the simple LIF description of pyramidal cells with the more realistic, albeitmore complex, Ghostburster description. The model is as follows:

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2

CdVs,i

dt= gNa,s[m∞,s(Vs,i)]2(1− ns,i)(VNa − Vs,i) + gdr,sn

2s,i(VK − Vs,i)

+gL(VL − Vs,i) +gc

κ(Vd,i − Vs,i) + Si(t) + Bi + ηi(t)−G(t− τd(Vs,i − Vr) (1)

dns,i

dt=

n∞,s(Vs,i)− ns,i

τn,s(2)

CdVd,i

dt= gNa,d[m∞,d(Vd,i)]2hd,i(Vna − Vd,i) + gdr,dn

2d,ipd,i(VK − Vd,i)

+gL(VL − Vd,i)−gc

1− κ(Vs,i − Vd,i) (3)

dhd,i

dt=

h∞,d(Vd,i)− hd,i

τh,d(4)

dnd,i

dt=

n∞,d(Vd,i)− nd,i

τn,d(5)

dpd,i

dt=

p∞,d(Vd,i)− pd,i

τp,d(6)

G(t) =g

N

∑i

∑m

t− timα

exp−(t− tim)

α(7)

Equations (1)-(6) replace the LIF equation shown in the Methods section of [1]. Vs,i represents the somaticpotential while ns,i is the activation gate of a somatic K+ rectifier current. Together these two variables supportthe spike dynamics of the somatic compartment of the ith Ghostburster Vd,i models the dendritic potential whereasnd,i and hd,i are respectively the dendritic K+ activation and Na+ inactivation gates; these variables allow fordendritic action potential backpropagation in the ith Ghostburster neuron. pd,i is the cumulative inactivation ofdendritic K+ current; this variable evolves with a timescale slower the spiking currents and has been shown to becrucial for this form of burst discharge[4, 5]. Physiological justification for equations (1-6) as well as parametervalues are given elsewhere [5]. The feedback kernel G is given by equation (7) and is identical to the one used in[1], except that the spiketimes tim are now defined as the mth time that Vs,i > 0 and dVs,i/dt > 0. Note thatequation (1) shows that the feedback G(t), the stimulus Si(t), intrinsic noise source ηi(t), and cell bias Bi allinfluence only the somatic compartment directly. Separating these inputs amongst both the somatic and dendriticcompartments adds additional complexity and was not considered in this study. Some parameters describing theinputs and interaction are changed from those given in the methods section of [1] due to the switch of neuronmodels. For the simulation results presented below they are as follows: Bi = 9, σ = 0.9, W = 36, and g = 0.004.The remaining parameters are identical to those presented in the methods section of [1]. The intrinsic modelparameters are set so that the model cell is in a chaotic bursting state with timescales similar to that observed invitro, an example burst is shown in Fig 1a.

Figures 1b and 1c show the spike time autocorrelation, A(τ), and inter spike interval (ISI) histograms for themodel described by equations (1)-(7). These results show behavior during local and global stimulation qualitativelysimilar to that of the LIF simulations (Fig. 2 of [1]) and the ELL pyramidal cell data (Fig 1 of [1]). Specifically, theghostburster cells show oscillatory behavior only during global stimulation, as marked by the damped oscillationin A(τ) and bimodality in the ISI histogram (Fig. 1c). The power spectra density (PSD) of the ghostbursterspike trains also show a characteristic peak at approximately 30 Hz only for global stimulation. The PSDs duringlocal and global geometries yield oscillation indices of 75.41 spk2/s and 142.06 spk2/s respectively. These resultsall show that the general phenomena of network oscillatory behavior during global stimulus is possible when ELLpyramidal cells are modeled with a realistic description based on extensive in vitro characterization. The spikefrequencies of the bursting neurons are similar to those observed in vitro during burst discharge ( 70 Hz); this islarger than those observed in vivo. This remains a discrepancy between the Ghostburster model network and boththe experimental results and the LIF model network. This is to be expected since the Ghostburster was modeledso as to fit in vitro and not in vivo data.

Definitive proof that active action potential backpropagation occurs during in vivo conditions and leads to burstdischarge as described in Fig. 1a has not yet been established. This would require intra-dendritic recordingsshowing dendritic action potentials and their subsequent failure at high firing rates. However, pyramidal cell in

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3

FIG. 1: A network of Ghostburster neurons involving global inhibitory feedback shows differential responses to both localand global stimuli. a) Schematic of Ghostbursting dynamics. Dendritic (top) and somatic (bottom) membrane potentialsduring a single model burst. Note the increasing spike frequency culminating at a high frequency somatic spike doubletcausing failure of dendritic backpropagation. The traces were produced by integrating equations (1)-(7) with parametervalues given elsewhere [5]. b) Local stimulation of Ghostburster network gives non-oscillatory network dynamics as evidentby the lack of an oscillation in A(τ) and unimodal characteristic of the ISI histogram. c) Global stimulation of Ghostbursternetwork gives oscillatory network dynamics as seen by the damped oscillation in A(τ) and bimodal ISI histogram.

vivo firing statistics do show ISIs that are within the range reported for somatic doublet ISIs connected to dendriticfailure as observed in vitro (5-8 ms) [3], thus it is possible that dendritic failures do occur. Indeed, there is a higherincidence of these low ISIs during global stimulation suggesting that Ghostbursting dynamics occurs preferentiallyduring this stimulus geometry (see Fig. 1 of [1] and unpublished observations). This would be consistent witha potential feedback-mediated mechanism that would gate burst dynamics specifically during global stimulation.Interestingly, a threshold for burst discharge has been theoretically and experimentally observed [5, 8], and thethreshold can be modulated by a variety of intrinsic ionic currents suggesting that such gating mechanism would befeasible [7, 8]. However, much further work is required to verify and expand upon these preliminary speculations.

INFORMATION TRANSFER AND NETWORK OSCILLATIONS

Establishing a relation between the dynamics of networks of neurons and their information processing abilities isa current area of interest. In this section we will begin to address how the feedback mediated oscillatory dynamicsdescribed in this paper relate to the previously established information processing abilities of ELL pyramidal cells.ELL pyramidal cells have been shown to estimate Gaussian stimuli poorly under global stimulus conditions [9–11].In contrast, these same cells have performed adequate estimation of locally applied stimuli [9]. Thus, it is shownthat not only are the discharge patterns of pyramidal cells dependent upon the spatial extent of stimuli, buttheir coding strategies are as well. To explore a potential connection we have also used linear stimulus estimationtechniques [9, 12] to measure the ability of ELL pyramidal neurons to estimate a time varying stimulus duringoscillatory and non oscillatory firing patterns. We note that the stimulus used is identical to that presented inthe methods section, zero mean band limited (0-40 Hz) Gaussian noise. Briefly, we reconstructed the stimulus byconvolving the spike train with the Wiener-Kolmogorov filter that minimized the mean square error ε2 betweenthe actual stimulus and the reconstructed stimulus. A measure of this accuracy of the reconstruction is the codingfraction γ = 1 − ε/σ , where σ is the standard deviation of the stimulus. A spike train which constructs thestimuli perfectly gives γ = 1, while a spike train that gives a chance estimate gives γ = 0. These measuresare described in detail in a number of studies [11–13] and the associated MATLAB algorithms are available athttp://www.klab.caltech.edu/ gabbiani/singproc.html.

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4

There are two categories of ELL pyramidal cells, basilar (E-cells) and non-basilar (I-cells) [14]; oscillatoryfiring dynamics were observed for both types. However, the stimulus estimation analysis that we present here isconfined to spike trains only from I-cells. The interpretation of the results for E-cells spike trains is more difficult,potentially due to additional interneuronal interactions (unpublished observations); these results will be presentedelsewhere. As previously reported [9] the coding fraction for pyramidal cells was significantly reduced under global(γ = 0.0547±0.037) as compared to local (γ = 0.170±0.061) geometry (p=0.004, n=6, pairwise t-test). However,the StF blockade partially restored the coding fraction under global geometry (γ = 0.136 ± 0.018, which is notsignificantly different from local geometry, p=0.5, t-test, n=6). The γ values presented here are lower than thoseshown previously [9], this is due to the broad band (0-40 Hz) nature of the stimulus used. Nevertheless, a correlationbetween oscillatory dynamics and a reduction in estimation ability was observed for a subset of pyramidal cells(I-cells). To our knowledge, this preliminary result is the first successful pharmacological manipulation of neuralcoding.

[1] Doiron B., Chacron M.J., Maler L., Longtin A. and Bastian J.. Nature 421, 539-543, (2003).[2] Turner R. W., Maler L., Deerinck T., Levinson S. R. and Ellisman M. H. TTX-sensitive dendritic sodium channels

underlie oscillatory discharge in a vertebrate sensory neuron. J. Neurosci. 14 6453-6471 (1994).[3] Lemon N. and Turner R. W. Conditional spike backpropagation generates burst dischrage in a sensory neuron. J.

Neurophysiol. 84 1519-1530 (2000).[4] Doiron B., Longtin A., Turner R. W. and Maler L. Model of gamma frequency burst discharge generated by conditional

backpropagation. J. Neurophysiol. 86 1523-1545 (2001).[5] Doiron B., Laing C., Longtin A. and Maler L. Ghostbursting: a novel neuronal burst mechanism. J. Comput. Neurosci.

12 (2002).[6] Laing C. R. and Longtin, A. A Two-Variable Model of a Somatic Dendritic Interactions in a Bursting Neuron. Bul.

Math. Biophysics 64 829-860 (2002).[7] Rashid A. J., Morales E., Turner R. W. and Dunn R. J. Dendritic Kv3 K+ channels regulate burst threshold in a

sensory neuron. J. Neurosci. 21 125-135 (2001).[8] Doiron B., Noonan L., Lemon N. and Turner R. W. Persistent Na+ current modifies somatic and dendritic spike

dischrage to regulate gamma frequency burst output. J. of Neurophysiol. 89 324-334 (2003).[9] Bastian J., Chacron M. J. and Maler L. Receptive field organization determines pyramidal cell stimulus encoding

capability and spatial stimulus selectivity. J. Neurosci. 22 4577-4590 (2002).[10] Gabbiani F., Metzner W., Wessel R. and Koch C. From stimulus encoding to feature extraction in weakly electric fish.

Nature 384 564-568 (1996).[11] Metzner W., Koch C., Wessel R. and Gabbiani F. Feature extraction by burst-like spike patterns in multiple sensory

maps. J. Neurosci. 18 2283-300 (1998).[12] Rieke F., Warland D., de Ruyter van Steveninck R. R. and Bialek W. Spikes: Exploring the Neural Code (MIT,

Cambridge, MA, 1997).[13] Gabbiani F and Koch C. in Methods in Neuronal Modeling: From Ions to Networks Ed. C. Koch and I. Segev, (MIT

Press, Cambridge MA, 1998). pp. 313-360.[14] Berman N. J. and Maler L. Neural architecture of the electrosensory lateral line lobe: adaptations for coincidence

detection, a sensory searchlight and frequency-dependent adaptive filtering. J. Expt. Biol. 202 1243-1253 (1999).

ARTICLE VI

B. Doiron, B. Lindner, A. Longtin, L. Maler and J. BastianOscillatory Activity in Electrosensory Neurons increases with the spatial correlationof the stochastic input stimulus.Physical Review Letters. 93: 048101, 2004.

102

ARTICLE VI 103

Oscillatory Activity in Electrosensory Neurons Increases with the Spatial Correlationof the Stochastic Input Stimulus

Brent Doiron,1,2 Benjamin Lindner,1 Andre Longtin,1 Leonard Maler,2 and Joseph Bastian3

1Physics Department, University of Ottawa, 150 Louis Pasteur, Ottawa, Ontario, Canada K1N 6N52Department of Cellular and Molecular Medicine, University of Ottawa, 451 Smyth Road, Ottawa, Ontario, Canada K1H 8M5

3Department of Zoology, University of Oklahoma, Norman, Oklahoma 73019, USA(Received 6 August 2003; published 20 July 2004)

We present results from a novel experimental paradigm to investigate the influence of spatialcorrelations of stimuli on electrosensory neural network dynamics. Further, a new theoretical analysisfor the dynamics of a model network of stochastic leaky integrate-and-fire neurons with delayedfeedback is proposed. Experiment and theory for this system both establish that spatial correlationsinduce a network oscillation, the strength of which is proportional to the degree of stimulus correlationat constant total stimulus power.

DOI: 10.1103/PhysRevLett.93.048101 PACS numbers: 87.18.Hf, 02.30.Ks, 02.50.Ey, 87.19.Bb

The nontrivial effects of noise on the dynamics ofphysical, chemical, and biological systems are beginningto be uncovered [1]. In particular, stochastic forcing cansometimes produce qualitative dynamics that are absentin the deterministic system. Neural networks [2–5] areideal for the study of stochastic dynamics in spatiallyextended systems, since noise constitutes a significantcomponent of neural activity. However, while specificdynamics have been linked to combinations of networkarchitecture and stochastic forcing, direct functional in-terpretation of the noise-induced dynamics are oftenlacking. A common feature of neural network architec-ture is that higher-order stages of processing influencelower-order stages through delayed feedback projections[6–8]. Much theoretical work has focused on the dynam-ics that such a recurrent connectivity produces, both with[4,9,10] and without [11] explicit axonal delays. Never-theless, the functional role that delayed feedback plays inbiological neural networks is poorly understood. Thus,the combination of stochastic forcing and delayed net-work interactions is a timely yet relatively unexploredarea of study.

In a recent study, we showed how diffuse delayedinhibitory feedback caused oscillatory spike trains fromelectrosensory neurons in response to stochastic com-municationlike, but not preylike, stimuli [12]. It washypothesized that the key distinguishing feature betweenthese two inputs was that communication stimuli corre-late the activity of many neurons while prey stimuli donot. However, in [12], we were unable to separate thiscorrelation from the fact that communication stimulihave a greater degree of spatial power since they coverlarge portions of the sensory field [13]. This is in contrastto prey which are spatially compact stimuli. In this Letter,we show through a combination of novel experimentaland theoretical analyses that it is indeed the spatial cor-relation in stimuli and not the total spatial power that isessential for oscillatory network spiking.

Experiments.—The weakly electric fish Apteronotusleptorhynchus has an electric organ that discharges aquasisinusoidal electric field (between 600 and1000 Hz) surrounding its body [14]. Nearby objects orcommunicating fish distort this field so that an effectiveelectric image is projected on the surface of the skin.These distortions are amplitude modulations (AMs) ofthe carrier field and are recorded by arrays of electro-receptors that line the surface of the skin. The electro-receptor afferents project to pyramidal cells of theelectrosensory lateral line lobe (ELL) which process in-put and then project to higher brain centers [8]. Thesecenters then output back to the ELL pyramidal cell layerproducing an effective closed loop architecture for pe-ripheral electrosensory processing.

To explore the effects of stochastic stimuli with vari-able spatial correlation, an array of four dipoles wasconstructed and placed near the surface of the skin[Fig. 1(a)]. The jth dipole in the array (j � 1; . . . ; 4)emitted a random AM, Ij�t�, consisting of two distinctstochastic processes, one intrinsic to the dipole, �i�t�, andone common to all dipoles, �G�t�. Specifically, we write

Ij�t� � ��������������1� c

p�j�t� �

���c

p�G�t��; (1)

where � is the total intensity of the applied stimulus Ijmeasured in units of Vcm�1 (� is kept constant for ourstudy). Both �i�t� and �G�t� are zero mean and Gaussianwith a frequency content that was uniformly distributedbetween 0–60 Hz (eighth order Butterworth filter). We seth�j�t��G�t�i � 0 and h�j�t��k�t�i � �jk. This implies thathIj�t�Ik�t�i=hIj�t�

2i � c, where c is the normalized cova-riance between any two dipoles in the array. Stimuli arespatially uncorrelated for c � 0, similar to that producedby a root mass or rocky substrate, while c near onemimics a communication signal. We remark that the totalspatial power of the input, 4�2, is independent of c.

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048101-1 0031-9007=04=93(4)=048101(4)$22.50 2004 The American Physical Society 048101-1

ARTICLE VI 104

Single unit in vivo extracellular recordings from ELLpyramidal cells were obtained for various c values; ex-perimental techniques are as in [12]. Figure 1(b) showsthe power spectrum of a spike train from a typical pyr-amidal cell (n � 7) to stimuli with spatial correlationc � 0 and c � 1.

A weak low-frequency oscillation is apparent whenc � 0. This oscillation is reduced and replaced with ahigh-frequency oscillation when c � 1. To quantify thisshift in power with c, we introduce the statistic f1;f2;c �Rf2f1S�f� df, where S�f� depends on c. A shift in S�f�

between c � 0 and c � 1 gives a nonzero value in�f1;f2 � f1;f2;1 � f1;f2;0. Over our data ensemble (n �7), we have that for low frequencies �2;22 � �55:8�20:7 (spikes2=s2) and for high frequencies �40;60 �53:7� 30:8 (spikes2=s2). These statistics are both signifi-cantly different from zero (p2;22 � 3:9 � 10�4 andp40;60 � 7:3� 10�3; pairwise t test). We note that therelative difference in spike rate for c � 0 andc � 1 was only 0:61� 0:43%, indicating that the shiftin power cannot be accounted for by changes in inputintensity. In total, these results show that ELL pyramidalneurons redistribute spike train power from low to highfrequencies as the spatial correlations of a global inputincrease.

Model.—Consider a nonautonomous homogeneous net-work of N leaky integrate-and-fire (LIF) neurons [10].The dynamics of the membrane potential Vj�t� of the jthneuron (j � 1; . . . ; N) evolves according to

_V j � �Vj ��� �j�t� � Ij�t� �gN

XNk

K�d xk�t�: (2)

Time is measured in units of the membrane time constant(6 ms). The dynamics is complemented by the usualspike-and-reset rule: Whenever Vj�t� reaches the thresh-old vT , a spike is fired (the mth firing time of the jthneuron is denoted by tj;m), and the neuron is in an absoluterefractory state for time �R followed by a reset of Vj to thevalue vR. The spike train of the jth neuron reads xj�t� �P��t� tj;m� (the sum is taken over all spike times). In

Eq. (2), the single neuron properties are described by�Vj, �, and �j�t� standing for a leakage term, a constantbase current, and an internal Gaussian white noise ofintensity D, respectively. This intrinsic noise leads tospontaneous activity even in the absence of stimuli, asalso observed in experiments [15]. Furthermore, we as-sume that the external stimulus I�t� used in the experi-ments according to Eq. (1) is transduced by the afferentsinto an input current I�t� of the same shape; instead ofband limited noise, however, we use white noise in themodel, for simplicity. The last term in Eq. (2) represents amean-field-like feedback of the spike trains of all neuronswhich is convolved with a standard function [10] anddelayed by a time �d:

K�d x�t� �Z 1

�dd� x�t� �� 2�e� �����; (3)

where ��t� is the Heaviside function [16]. The kernel K�dis conceived as a composite process by which pyramidalcells first project their output to a ‘‘higher’’ brain centerwhich integrates this input and then projects uniformlyback to the original network via a common feedbackpathway. This effective indirect interaction between cellsvia K�d also involves a significant minimal delay term,�d, modeling both the integration time of the distant brainregions and finite axonal conduction velocity. The timescale of �d for the inhibitory pathway of interest is of theorder of the membrane time constant (6 ms) [8,12]. Theparameter thus both represents a fast synaptic timescale and a distribution of delays. In this study, we confineg < 0 to model the inhibitory interactions of a specificfeedback pathway previously shown to cause oscillatorynetwork behavior [12]. Other known feedback pathways[8] are not treated (see [12] for biophysical justification).In order to allow for an analytical treatment, our modeldiffers slightly from the one in Ref. [12]: Inhibition entersas an additive current term and the internal noise is white.

Theory.—In the steady state, we split the input currentsinto two parts: (i) base and leak currents, internal noise ofstrength D, and time-independent mean of the feedback��gr0�[r0 � hx�t�i is the stationary firing rate of a singleLIF neuron]; (ii) external signal and time-dependent partof the feedback. The first part constitutes a network ofuncoupled white-noise driven LIF neurons with an effec-tive base current �0 � �� gr0��

0� that has to be self-consistently determined from the well-known formulafor the spike rate of a single LIF neuron r0��0� � ��r �����$

p R��0�vR�=���D

p

��0�vT �=���D

p dx ex2erfc�x���1 [erfc�x� is the comple-

mentary error function [16] ]. With respect to the

b)

c=1c=0

a)

I (t)1

I (t)4

I (t)3

I (t)2

frequency (Hz)0 40 80 120

30

25

20

15

10

S(s

pike

/s)

2

FIG. 1. Spatial correlation of stimuli determines oscillatoryactivity in electrosensory neurons. (a) Schematic of experimen-tal stimulation. Single unit activity was recorded from ELLpyramidal cells while one side of the fish was stimulated withan array of four dipoles. Each dipole had a tip spacing ofapproximately 1.2 cm and the centers of adjacent dipoleswere separated by 2.2 cm. Each dipole was 1.5 cm from theskin surface and stimulated roughly 2 cm2 of the skin fortypical stimulus contrasts (250 �Vcm�1). This ensured thatthere was limited overlap of the distinct dipole’s electricimages on the skin. (b) Spike train power spectra, S, whenthe spatial correlation between the dipoles, c, was set to 0 (lightcurve) and one (dark curve).

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ARTICLE VI 105

time-dependent part of the feedback and the externalinput stimulus, we treat the system as a linear one char-acterized by the susceptibility A�!� with respect to theinput stimulus. Expressed by ~x�0;�j � F�x�0;�;j � r0�=

����T

p

(F �RT0 dt e

i!t denotes the Fourier transform; xj�t� andx0;j�t� are the spike train in the presence and absence ofstimulus and feedback, respectively), this ansatz reads

~x j�!� � ~x0;j�!� � A�!��~Ij�!� �

gN

~K�d�!�Xk

~xk�!��;

(4)

where ~Ij�!� � FIj�t�=����T

pand ~K�d�!� � FK�d�t�=

����T

p.

Our choice of K�d gives ~K�d�!� � ei!�d=�1� i! �1�2.Equation (4) provides a system of equations relating

h~xj~x j i (spike train power spectrum), h~xj~x ki (j � k; crossspectrum between distinct spike trains), and h~xj~I

j i (cross

spectrum between the stimulus and a spike train), where denotes complex conjugation. We further assume thath~x0;j~x

0;ki � h~x0;j~I

j i � h~xj ~�

ki � 0 �j � k� and that N !

1 so as to neglect terms of order 1=N and higher.Finally, we replace the spectrum of the transmittedstimulus S0�!;D� � �2jA�!;D�j2, as it arises from purelinear response theory, by S0�!;Q�, where Q � D��2=2, representing the total noise intensity. Con-sequently, the susceptibility A is also evaluated at Q.

With these assumptions it can be shown [17] that thepower spectrum of the spike train from a representativeneuron, S � h~xj~x

j i (with T ! 1), is given by

S � S0 � c�2jAj22<�g ~K�dA� � jg ~K�dAj

2

j1� g ~K�dAj2

: (5)

We let S0 � h~x0~x 0i and have dropped both the j notation

and the ! dependence for S. Equation (5) shows that csimply sets the strength of the deviation of S from theuncoupled case (S0). The precise form of the deviation isdetermined only by the internal (A) and feedback (K�d)dynamics.

Equation (5) is applicable to a variety of neuron mod-els; the special form of the LIF neuron allows for ana-lytical expressions for both S0 and A [18]:

S0�!;Q� � r0jDi!�

�0�vT���Q

p �j2 � e2�jDi!��0�vR���

Qp �j2

jDi!��0�vT���

Qp � � e�ei!�RDi!�

�0�vR���Q

p �j2;

A�!;Q� �r0i!=

����Q

p

i!� 1

Di!�1��0�vT���

Qp � � e�Di!�1�

�0�vR���Q

p �

Di!��0�vT���

Qp � � e�ei!�RDi!�

�0�vR���Q

p �;

where � � �v2R � v2T � 2��vT � vR��=4Q and Da�z� de-notes the parabolic cylinder function [16].

In Fig. 2(a), we show S computed via both simulationsof the network model (2) and the theory as given by (5)when c � 0 and c � 1 . The simulation and theory resultsmatch quite well for the chosen parameters. As c isincreased, we see qualitative agreement with the experi-ments in several respects. For c � 1, the deviation term in

(5) introduces a 50 Hz oscillation and a suppression ofpower at low frequencies, as compared to the c � 0 case.However, we note that the experimental results also con-tain a low-frequency oscillation when c � 0; this is notreproduced by either the simulations or the theory. Theorigins of this low-frequency oscillation are currently notknown; however, our mathematical model of the ELLdoes not include other known feed forward and feedbackpathways [8]. Figure 2(b) shows spike time raster plotsfrom the simulations of Fig. 2(a). They show that forc � 0 there is relatively asynchronous behavior, whereasfor c � 1 a degree of network synchrony is apparent.However, computing the average correlation coefficientbetween any two cells shows only an increase from ap-proximately 10�4 to 0.065. The results of Fig. 2 were notsensitive to model parameters; the model presented in[12] shows similar behavior as does the present modelfor a range of � values spanning both the sub- and supra-threshold (�< vT and �> vT). Network oscillationsemerging from relatively asynchronous sparsely con-nected stochastic delayed networks have been reportedfor a similar system [4], however, spatial correlation ofthe input was not considered; also in contrast, our net-work is not sparse. The effects of spatial correlation havebeen studied numerically in locally and electricallycoupled networks of FitzHugh-Nagumo neurons [3] tobe compared to the globally and inhibitory coupled net-work studied here. In [3], it was observed that regularizedactivity occurred for low input spatial correlations, incontrast to the results presented here.

A consequence of a linear response treatment of net-work (2) is that the shift in power is linear in c. To show

25

20

15

150100500frequency (Hz)

40

30 ms

a) b)c=0

c=1

S(s

pike

s\s

)2

FIG. 2. Network model simulations and theoretical results forspatial correlation c � 0 and c � 1. (a) Spike train powerspectra S for the network model given by (2) (c � 0, opencircles; c � 1, closed circles) and the theoretical result (5) (c �0, thin line; c � 1, thick line). (b) Network raster plots for thesimulations shown in (a). The top plot is for c � 0 and thebottom plot is for c � 1. The parameters for both the simula-tions and theory were vR � 0, vT � 1, �R � 0:1, � � 0:5, g ��1:2, � 3, �d � 1, D � 0:08, �2 � 0:16, and �0 � 0:3286.All simulations were integrated via a Euler integration schemewith a time step of 10�3. The plots were rescaled so that themembrane time constant was 6 ms; note that ! � 2$f.

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this, we plot in Fig. 3(a) f1;f2;c as a function of c for bothlow-frequency and high-frequency bands. There is goodagreement between obtained from the theory given by(5) and simulation of the network (2). The validity of ourlinear response ansatz for ELL pyramidal cells is sup-ported experimentally when is measured from ELLpyramidal neurons for a range of c values, shown inFig. 3(b).

We finally study the dependence of the network oscil-latory dynamics on the axonal delay �d. Figure 3(c) showsS for �d � 0:5 and �d � 3 (or in real time 3 and 18 ms,respectively). As expected, the frequency of oscillationdecreases when �d increases, due to the ei!�d term in K�d .However, of interest is that the oscillation coherence(height of peak in S divided by peak half width) issignificantly larger for larger �d. This is understoodfrom the low pass nature of both the susceptibility A(specifically jAj2) and the kernel K�d (due to ). Whenwe take A! 1 and ! 1 in Eq. (5), the coherence ofthe oscillation in S shows no dependence upon �d (resultsnot shown).

Inhibitory feedback, with or without axonal delays,giving rise to network oscillations is a well studied phe-nomenon [4,9–11]. The effect of spatiotemporal stimulion the dynamics of networks of noisy neurons, and, inparticular, on the presence of oscillatory spiking, ispoorly understood. Here we have shown using novelexperimental and theoretical methods that the spatialcorrelation of the stimulus alone can induce such oscil-

lations; this effect does not require an increase in thestimulus power integrated over space. Oscillatory net-work behavior in response to specific stimuli are nowbeing catalogued in a variety of sensory systems[6,12,19]. Accounting for input-dependent dynamicalphenomena, as shown here, will provide a deeper under-standing of sensory brain function and, more generally, ofnets of excitable elements.

We would like to thank J. Lewis, M. J. Chacron,J. Middleton, and A.-M. Oswald for discussions. Thisresearch was funded by NSERC (B. D., B. L., and A. L.),CIHR (L. M.), OPREA (B. L. and A. L.), and NIH (J. B.).

[1] J. Garcıa-Ojalvo and J. M. Sancho, Noise in SpatiallyExtended Systems (Springer-Verlag, New York, 1999);V. K. Vanag et al., Nature (London) 406, 389 (2000);A. Sanz-Anchelergues et al., Phys. Rev. E 63, 056124(2001).

[2] P. Jung et al., Phys. Rev. Lett. 74, 2130 (1995); P. Tass,Phase Resetting in Medicine and Biology (Springer-Verlag, Berlin, 1999); A. B. Neiman et al., Phys. Rev.Lett. 88, 138103 (2002); M. P. Zorzano et al., Physica D(Amsterdam) 179, 105 (2003).

[3] C. Zhou et al., Phys. Rev. Lett. 87, 098101 (2001);H. Busch et al., Phys. Rev. E 67, 041105 (2003).

[4] N. Brunel et al., Neural Comput. 11, 1621 (1999).[5] B.W. Knight, Neural Comput. 12, 473 (2000).[6] A. M. Sillito et al., Nature (London) 369, 479 (1994).[7] E. Ahissar et al., Cereb. Cortex 13, 53 (2003); Le Masson

et al., Nature (London) 417, 854 (2002).[8] N. J. Berman et al., J. Exp. Biol. 202, 1243 (1999).[9] M. Mackey et al., J. Math Biol. 19, 211 (1984); U. Ernst

et al., Phys. Rev. Lett. 74, 1570 (1995); J. Foss et al.,Phys. Rev. Lett. 76, 708 (1996).

[10] W. Gerstner and W. M. Kistler, Spiking Neuron Models(Cambridge Univerisity Press, New York, 2002).

[11] X.-J. Wang et al., Neural Comput. 4, 84 (1992); P. C.Bressloff et al., Neural Comput. 12, 91 (2000); D. J.Mar et al., Proc. Natl. Acad. Sci. U.S.A. 96, 10 450(1999); J. Rinzel et al., Science 279, 1351 (1998);J. White et al., J. Comput. Neurosci. 5, 5 (1998).

[12] B. Doiron et al., Nature (London) 421, 539 (2003).[13] G. K. H. Zupanc et al., Can. J. Zool. 71, 2301 (1993);

W. Metzner, J. Exp. Biol. 202, 1365 (1999).[14] W. Heiligenberg, Neural Nets in Electric Fish (MIT

Press, Cambridge, MA, 1991).[15] J. Bastian et al., J. Neurophysiol. 85, 10 (2001).[16] Handbook of Mathematical Functions, edited by

M. Abramowitz and I. A. Stegun (Dover, New York,1970).

[17] Eqs. (4) and (5) are due to Benjamin Lindner. Detailswill be given elsewhere.

[18] B. Lindner et al., Phys. Rev. Lett. 86, 2934 (2001);B. Lindner et al., Phys. Rev. E 66, 031916 (2002).

[19] M. Stopfer et al., Nature (London) 390, 70 (1997);H. Kashiwadani et al., J. Neurophysiol. 82, 1786(1999); B. van Swinderen et al., Nat. Neurosci. 6, 579(2003); K. MacLeod et al., Science 274, 976 (1996).

450

400

350

1.00.50.0

350

3001.00.50.0

400

Γ(s

pike

s/s

)2

2Γ

(spi

kes

/s)

22

c

c

b)

a)

30

25

20

15

150100500

S(s

pike

s\s

)2

frequency (Hz)

c)τ d=3

τ d=0.5

FIG. 3. Oscillation dependence on stimulus correlation andaxonal delay. (a) f1 ;f2;c plotted as a function of c for both�f1; f2� � �2; 22� Hz (theory, thin line; simulations, opencircles) and �f1; f2� � �40; 60� Hz (theory, thick line; simula-tions, filled circles). Numerical integration of S computed from(5) was done with a fixed interval of �f � 0:1 Hz, whileintegration of S determined from simulation of (2) used thefast-Fourier transform discretization step of �f � 0:5 Hz.(b) obtained from spike trains of ELL pyramidal cells fora range of c values. Data were obtained from same experimentsas those for Fig. 1. (c) S shown for �d � 0:5 (theory, thin curve;simulations, open circles) and �d � 3 (theory, thick curve;simulations, filled circles). c � 1 for both curves. All otherparameters are as given in Fig. 2.

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107

CONCLUSION 108

This thesis has two distinct sections:

I. Chapters 1-3 introduced and explored a novel form of burst discharge exhibited by in vitro ELL

pyramidal cells.

II. Chapters 4-6 studied stimulus-induced network oscillations that requires delayed inhibitory con-

nectivity.

I will first discuss how the chapters within a section relate to one another. Then I will outline how

the results from both sections potentially interact. I end with some final thoughts on the impact of the

results on sensory science.

7.1 Ghostbursting: large, medium, and small models

Ghostbursting is a bursting dynamic that is dependent on soma-dendrite interactions. The first chapter

of this thesis presented an extremely detailed model of an ELL basilar pyramidal neuron. This model

included multiple nonlinear ionic conductances, each with differing spatial distributions over the somatic-

dendritic axis. The computational approach was to use a finite element scheme to discretize the dendritic

tree and simulate the cable equation. This is now standard in modern computational studies of dendritic

processes [1, 5]. Active dendritic channels were shown to be an important component of ELL pyramidal

cell bursting since the burst termination mechanism was the failure of dendritic backpropagation - a

truly nonlinear spatial effect of single neuron dynamics.

The second chapter, even though it modelled the same bursting behaviour as Chapter 1, revealed

results quite distinct from Chapter 1. Chapter 2 introduced a two-compartment model of an ELL

pyramidal cell entitled the Ghostburster. From the naive perspective, a two-compartmental model is a

horrible approximation of the cable equation. However, it is not proper to conceive of the Ghostburster

model as trying to accomplish this, there is no space constant nor the possibility of realistic action

potential propagation down a cable. Space is only loosely modelled by a diffusive coupling between the

soma and the ‘dendrite’. Despite these simplifications the model was able to capture the main features

of the burst discharge observed in both the large model and experiment.

The success of the Ghostburster was because even though true action potential backpropagation is a

spatial effect, it is not a subtle one. The parameters of both the large model and the Ghostburster were

such that, if the soma fires an action potential, then it will induce a definitive response in the dendrite:

either the full backpropagation of the action potential in the large model or a dendritic spike in a single

1

CONCLUSION 109

dendritic compartment of the Ghostburster model. Regardless, the effect on the soma is the same - a

depolarizing after-potential (DAP) appears after somatic spike repolarization. The dynamic regulation

of the DAP is key for Ghostbursting to occur. In fact, a simple LIF model with recurrent excitation

modelling a dynamically regulated DAP can also produce the full Ghostbursting scenario; there are no

dendrites in this model [6].

The low-dimensionality of the Ghostburster model allowed for a proper dynamical systems analysis

of the burst mechanism. It was shown that the burst mechanism was truly novel and did not conform

to known burst classification schemes [4, 3]. A bifurcation study of the full burst system (slow and fast

subsystems) produced two concrete predictions:

I. Two distinct dynamic behaviours should be observed in a single ELL pyramidal cell: tonic and

burst discharge. Furthermore, when the intensity of a static input is increased, tonic firing should

give way to bursting, contrary to most other burst mechanisms.

II. The threshold separating tonic and burst firing is a saddle-node bifurcation of limit cycles. This

bifurcation imparts a slow passage effect, or ‘ghosting’, to the burst discharge. Near the saddle-

node bifurcation bursts should have a long duration, while far away from the bifurcation, they

should have a short duration.

Chapter 3 gave experimental evidence for both of these predictions. Thus it was shown that a single

sensory neuron can have two distinct modes of activity, tonic and bursting, and a wide variety of burst

time-scales. Both of these properties could be useful for the coding of stimuli that can change in both

intensity and time-scale, as are routine for weakly electric fish environments.

The final section of Chapter 3 showed that when broadband time-varying inputs were presented to in

vitro pyramidal neurons, a working mathematical model for these cells became much simpler. A LIF-like

neural model with only non-potentiating DAPs and no dendritic refractoriness could reproduce the first

and second order statistics of both the spike and ISI trains observed in experiment. Thus the non-

autonomous Ghostburster system is in some sense simpler than the autonomous one that was studied in

chapters 1 and 2. The exact dependence on the bandwidth of the driving input for such a reduction in

model complexity is still not known; a future study is planned (A-M. Oswald, personal communication).

2

CONCLUSION 110

7.2 Hopf bifurcations and stimulus induced oscillations

Chapter 4 introduced the phenomenon of synchronous network oscillations in a network of LIF neurons

coupled with diffuse delayed inhibition. In particular, it showed two distinct biological parameters con-

trolled synchronous network oscillatory behaviour - single cell excitability (µ) and network connectivity

(g). Studying generalized equations for network activity revealed that a Hopf bifurcation is the mathe-

matical mechanism by which oscillations were created. Further, a curve of Hopf bifurcations existed in

µ/g parameter space, showing how shifts in either µ or g can be in some sense equivalent with respect

to network oscillations. The delayed connectivity was essential for both allowing the possibility of this

phenomenon (delay-induced instability) and setting the time-scale of the oscillation.

The peripheral electrosensory system has diffuse delayed inhibitory connectivity and thus has the

potential to oscillate synchronously. Indeed, the fifth chapter showed a synchronous oscillatory dynamic

within the ELL network. In addition, the oscillation mechanism requires delayed inhibition and so may

be similar to the dynamic discussed in Chapter 4. However, this behaviour occurred only in response

to communication-type stimuli. Thus, unlike the results of Chapter 4 oscillations in the ELL are not an

autonomous behaviour.

Stimulus specificity is sensible if a sensory system is to use synchronous network oscillations as

a coding mechanism; responding to all stimuli with network oscillations in effect codes for nothing.

A relevant question is then :what are the features of communication stimuli that elicit a synchronous

oscillatory response in the ELL? Considering the results of Chapter 4 a natural hypothesis would be that

the presence of communication stimuli modulates network connection strength or single cell excitability.

This would in effect push the network activity through the Hopf bifurcation and into an oscillatory state.

However, the results presented in Chapter 6 offer a different explanation. Communication calls

synchronize the ELL since they are spatially correlated inputs. A synchronous discharge from the ELL

pyramidal cell network causes a recurrent ‘wave’ of inhibition that stops firing activity in the ELL after

a delay. Synchronous firing is renewed after the inhibition abates and the process begins again. Thus

the synchronization enforced by communication stimuli sets the system into an oscillatory state. Prey

inputs or background scenes can not induce a synchronization of the ELL and hence did not elicit an

oscillatory response. The effect did not require a shift of single cell or network properties, which could

in principle take time; this oscillatory response may be instantaneous. However, measures such as power

spectrum and autocorrelation require sufficient data to compute. It is not currently know how long must

communication stimuli be presented to observe an oscillatory ELL.

It is possible that the oscillation mechanisms in Chapters 4 and 5/6 operate simultaneously. Os-

3

CONCLUSION 111

cillatory behaviour could promote a potentiation of the bipolar cell/pyramidal cell synapse giving an

effective increase in feedback strength (g). Also, as eluded to in the supplementary materials section of

chapter 5, the oscillatory dynamic could promote a burst response in the ELL. This may be viewed as

an effective increase in the cell’s excitability (µ) since the DAP could now be potentiated during a burst.

Either scenario would lower the threshold (the Hopf bifurcation) that gates the oscillation.

Stimulus-specific synchronous oscillatory behaviour is becoming a classic theme in a variety of sensory

systems [8]. Most notable of these are oscillations in the cortex [10] and thalamo-cortical circuits [9]

elicited by large moving bars over the visual field, or net work oscillations in the olfactory systems of

insects that require specific scents [7, 11]. In addition, feedback [9] and inhibition [11] has been shown to

crucial for these effects. However, what has not been done is to relate how the features of the moving bars

or the scents interact with a neural mechanism responsible for an oscillation. This is the major scientific

achievement of Chapters 5 and 6. They show that it is both the spatial extent and the spatial correlation

of communication calls that interacts with a diffuse, delayed, closed loop inhibitory architecture in the

sensory brain to give rise to an oscillation. This is then an example of how the architectural organization

of the sensory brain is appropriate for the coding of a natural stimuli.

7.3 Oscillatory networks of Ghostbursters.

A natural extension of this thesis would be to explore the network oscillations shown in Chapters 4-6

with the model neurons not being simple LIF models but the more appropriate Ghostbursting neurons

of Chapters 1-3. This was briefly done in the supplementary materials section of Chapter 5. There it was

shown that Ghostbursting dynamics does not preclude the ability for the network to oscillate selectively

to communication inputs. However, consider the following two facts:

I. When the driving stimulus was broadband, in vitro pyramidal cell responses were modelled without

the complicated Ghostburster model, as shown in chapter 3. Thus ‘true’ Ghostbursting (complete

with the saddle-node bifurcation of limit cycles) is blocked in the presence of high frequency stimuli.

II. The network oscillatory response of Chapters 5 and 6 requires high-frequency inputs (data not

shown). This is because a resonance between the time-scales of the stimulus and of the delayed-

induced network oscillations must be satisfied for oscillations to happen.

The combination of I and II leads to the conclusion that even though the dynamics of Ghostbursting

does not preclude network oscillations, the required stimulus conditions for network oscillations preclude

4

CONCLUSION 112

true Ghostbursting. However, the simple doublet generation mechanism that was presented at the end

of Chapter 3 may interact with the network oscillatory dynamic. Unfortunately, the theory developed

in Chapter 6 would not directly apply to such a network since the analytic background power spectrum

S0(ω) and the susceptibility A(ω) are applicable to only classic LIF neurons and not the modified LIF

model used in chapter 3. Numerical estimation of both S0(ω) and A(ω) would need to be done and used

in a similar fashion as Chapter 6. However, it is not known if linear response is even appropriate for a

system that can give two distinct outputs, i.e. isolated spikes and bursts. Nevertheless, such a study

would indeed be of interest since it has now long been know that bursts code for specific features in the

stimulus [2]; how a network-induced oscillation could modify such a code is a timely but open question.

7.4 Final Thoughts

This thesis dealt with the sensory brain at both the cellular and systems level. One commonality in the

research is that the sensory coding strategies all involved a temporal ordering via interaction. Specifically,

we saw a temporal ordering of spike times into bursting packets. This required interactions between the

soma and dendrite in a single neuron. We also detailed the creation of a synchronous network oscillation;

this is some sense a double ordering - synchrony and oscillation are both temporal behaviours. This effect

was dependent on the widescale interaction of many pyramidal cells in the ELL.

The fact that the brain codes stimuli with a temporal ordering of its responses is not surprising.

What is interesting is that this required interactions. In both the single neuron bursting and network

oscillations the respective system interactions are what gave the specific sensory code the ability to be

selective for a particular input. The bursts were selective for low-frequency components - dendritic DAPs

must coincide with the low-frequency components in order to elicit a burst. The synchronous oscillations

were selective for communication stimuli - the oscillatory inhibitory circuit required a synchronizing input

to be present. Having selective responses is a prime requirement for any code. Perhaps then, the simplest

message that can be given is that ”Biology is in the interactions”. I am sure someone has said that, but

it deserves repetition.

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CONCLUSION 113

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