Development of a methodology for the examination of conductance densities and distributions of hippocampal

oriens-lacunosum/moleculare interneurons using ensemble modelling

by

Vladislav Sekulić

A thesis submitted in conformity with the requirements for the degree of Master of Science

Department of Physiology University of Toronto

© Copyright by Vladislav Sekulić, 2013

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Development of a methodology for the examination of conductance densities and distributions of hippocampal oriens-lacunosum/moleculare interneurons using ensemble modelling

Vladislav Sekulić Master of Science, 2013

Department of Physiology University of Toronto

Abstract

The hippocampus is a brain region that is critically involved in memory formation. Stratum

oriens-lacunosum/moleculare (O-LM) interneurons have been shown to modulate incoming

sensory information onto principal cells in CA1. Multi-compartment computational models of

O-LM cells have been developed to better understand their functional roles in network contexts.

Due to the variability and incompleteness of experimental details, however, a population of

models that collectively captures intrinsic O-LM cell behavior is needed. We generated a

database of O-LM models with physiologically plausible ranges for conductance densities using

NEURON simulations on a supercomputer cluster. A subset of models that best represented

O-LM cell electrophysiological output was subsequently extracted from the database and

analyzed in order to determine correlations in conductance densities. Three major co-regulatory

balances were found, which provide specific hypotheses for experimental investigations and

point to the possibility of identifying a “signature” of conductance density balances for particular

neuronal cell types.

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Acknowledgments

I would like to acknowledge the many contributors without whom the research presented in this

work would not have been possible. First and foremost of all, I would like to thank my

supervisor, Dr. Frances Skinner, for providing me with the opportunity to pursue research in

neuroscience. Her enthusiasm, scientific integrity and rigour, and myriad helpful discussions

have been a constant source of encouragement and inspiration in my work and growth as a

research trainee. Our mutual collaborator, Dr. Josh Lawrence, also deserves special gratitude not

just for providing the experimental data that was the sine qua non for this project, but also for his

high enthusiasm for collaborative work that has proven to be infectious. Along these lines,

special thanks goes to Dr. Cengiz Günay for developing the PANDORA toolbox as well as for

helpful discussions and troubleshooting sessions, without which this project would have been far

more difficult and time-consuming to complete.

I would furthermore like to thank the members of my supervisory committee, Dr. Zhong-

Ping Feng and Dr. Peter Pennefather, for the time and effort they have taken to help me along the

path to completing this Master’s project. Their suggestions, fresh insights, and criticisms have

provided the corrective impetus that proved to be essential for the maturation and completion of

this work.

Likewise, I would like to thank the past and present members, both official and

unofficial, of Dr. Skinner’s lab, for their helpful comments and discussions. They include Katie

Ferguson, Ernest Ho, Nathan Insel, Phillip Tse-Chiang Chen, Felix Njap, and Owen Mackwood.

My deep thanks go to all the loved ones in my life. My parents have been two of my

biggest supporters from the very beginning, and encouraged me every step of the way. Particular

gratitude goes to Mihaela Poca for her support and encouragement.

Finally, I would like to acknowledge and thank the various funding sources that provided

the necessary support for my research and studies. These include the Department of Physiology

at the University of Toronto, the U.S. National Institutes of Health (NIH), and the National

Sciences and Engineering Research Council of Canada (NSERC).

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Table of Contents

Acknowledgments  ..............................................................................................................................  iii  

Table  of  Contents  .................................................................................................................................  iv  

List  of  Tables  .........................................................................................................................................  vi  

List  of  Figures  ......................................................................................................................................  vii  

List  of  Appendices  ................................................................................................................................  ix  

  Introduction  .....................................................................................................................................  1  1

1.1   The hippocampus ............................................................................................................. 1  1.1.1   Structural  organization  and  behavioural  correlates  ........................................................................  1  1.1.2   Hippocampal  rhythms  and  the  role  of  interneurons  ........................................................................  2  

1.2   The oriens-lacunosum/moleculare interneuron ................................................................ 3  1.2.1   Morphology  and  connectivity  .....................................................................................................................  3  1.2.2   The  O-­‐LM  cell  as  putative  pacemaker  neuron  .....................................................................................  3  1.2.3   Role  in  hippocampal  CA1  microcircuit  functioning  ..........................................................................  4  

1.3   Mathematical and computational modelling ..................................................................... 5  1.3.1   The  need  for  mathematical  and  computational  modelling  ............................................................  5  1.3.2   Biological  variability  and  ensemble  modelling  ...................................................................................  6  

1.4   Previously developed computational models of O-LM cells ............................................. 7  1.4.1   Multi-­‐compartment  models  ........................................................................................................................  7  1.4.2   Uncertainty  in  the  characterization  of  currents  in  O-­‐LM  cell  models  .......................................  7  1.4.3   Rationale  for  ensemble  modelling  of  O-­‐LM  cells  ................................................................................  9  

1.5   Thesis organization ........................................................................................................ 10  1.5.1   Hypothesis  .......................................................................................................................................................  10  1.5.2   Outline  ...............................................................................................................................................................  11  

  Methods  and  Materials  ..............................................................................................................  12  2

2.1   Experimental data .......................................................................................................... 12  2.1.1   Hippocampal  slice  preparation  ...............................................................................................................  12  2.1.2   Electrophysiological  recordings  .............................................................................................................  12  

2.2   Simulation and analysis software ................................................................................... 13  

  Results  ............................................................................................................................................  15  3

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3.1   Development of a robust methodology for ensemble modelling .................................... 15  3.1.1   Outline  of  results  ...........................................................................................................................................  15  3.1.2   Performing  model  simulations  (Step  1)  ..............................................................................................  16  3.1.3   Generating  model  database  (Step  2)  ....................................................................................................  27  3.1.4   Determining  database  of  acceptable  models  (Step  3)  ..................................................................  33  3.1.5   Finding  conductance  density  balances  (Step  4)  ..............................................................................  38  

3.2   Application of the methodology to O-LM cells ................................................................ 40  3.2.1   Outline  of  results  ...........................................................................................................................................  40  3.2.2   Models  successfully  ranked  according  to  appropriate  matching  against  physiological  O-­‐

LM  cells  .............................................................................................................................................................................  41  3.2.3   Discovering  critical  balances  in  conductance  densities  ...............................................................  51  

  Discussion  ......................................................................................................................................  63  44.1   Summary ........................................................................................................................ 63  4.2   Predictions ..................................................................................................................... 63  4.3   Co-regulations between conductance densities critically determine O-LM cell output .. 65  4.4   Limitations and future work ............................................................................................ 68  

References  ............................................................................................................................................  71  

Appendix  A  –  Rationale  for  active  conductance  density  ranges  ........................................  78  

Appendix  B  –  Electrophysiological  measures  in  PANDORA  ................................................  84  

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List of Tables

Table 3.1. Fitted passive properties for the two O-LM model morphologies …………………. 21

Table 3.2. Summary of maximum conductance density values used in the model database

construction ……………………………………………………………………………………. 22

Table 3.3. The four databases constructed as part of the ensemble modelling approach ……... 33

Table 3.4. Model parameters for the two highest-ranked per-morphology models with

somatodendritic h-current ……………………………………………………………………... 36

Table 3.5. Passive properties for both the reference model and the highly-ranked morphology 1

and morphology 2 models …………………………………………………………………….. 37

Table 3.6. High-order parameters as determined by dimensional stacking analysis …………. 54

Table A1. Summary of Ih currents and conductances reported in neocortical and hippocampal

pyramidal cells ………………………………………………………………………………... 80

Table B1. The measures used for analyzing and comparing −90 pA current clamp traces of both

model and experimental O-LM cells in PANDORA …………………………………………. 85

Table B2. The measures calculated from the –90 pA current clamp experimental dataset …... 86

Table B3. The measures used for analyzing and comparing +90 pA current clamp traces of both

model and experimental O-LM cells in PANDORA …………………………………………. 87

Table B4. The measures calculated from the +90 pA current clamp experimental dataset ….. 90

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List of Figures

Figure 1-1. Drawing of a transverse slice of the rabbit hippocampus by C. Golgi ……………… 1

Figure 1-2. Preferential firing of various interneuron subtypes in CA1 during different phases of

the theta rhythm …………………………………………………………………………………. 2

Figure 2-1. Characteristic +90 and –90pA current clamp somatic voltage response of an

experimental O-LM cell …………………………………………………………………………13

Figure 3-1. The methodology developed in this work for the analysis of conductance densities

and distributions of multi-compartment models using ensemble modelling ………………..…..16

Figure 3-2. Morphological reconstructions of the two experimental O-LM cells provided by

collaborators …………………………………………………………………………………… 17

Figure 3-3. Voltage traces of highly-ranked models corresponding to original and re-fit passive

properties ………………………………………………………………………………………. 37

Figure 3-4. The methodology for ensemble modelling applied to O-LM multi-compartment

models …………………………………………………………………………………………. 41

Figure 3-5. Ranking of models according to distance from experimental dataset …………….. 43

Figure 3-6. Gradual transition in models’ goodness-of-fit to experimental O-LM cells………. 44

Figure 3-7. Discrete transitions in models’ goodness-of-fit to experimental O-LM cells …….. 45

Figure 3-8. The general criterion for determining a cutoff of a subset of appropriate O-LM

models …………………………………………………………………………………………. 46

Figure 3-9. The first failure-to-fire model in the ranked database subsets of morphology 1 and

morphology 2 models …………………………………………………………………………. 48

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Figure 3-10. First failure-to-fire models compared to more poorly-ranked, but otherwise

appropriate O-LM models ……………………………………………………………………. 49

Figure 3-11. The time constant values of the hyperpolarization-induced sag for highly-ranked

models ………………………………………………………………………………………… 50

Figure 3-12. Dimensional stack image of the subset of ranked O-LM models extracted using the

general cutoff criterion ……………………………………………………………………….. 52

Figure 3-13. Parameter correlation histograms of gh and gh distribution parameters in the highly-

ranked model subsets …………………………………………………………………………. 55

Figure 3-14. Characteristic model from version 1 of the database …………………………… 57

Figure 3-15. No correlations exhibited with low-order conductances ……………………….. 58

Figure 3-16. Correlation with local preference ……………………………………………….. 59

Figure 3-17. Co-regulation between gNad and gKDRf conductances …………………………… 60

Figure 3-18. Co-regulation between gh and gKDRs conductances ……………………………... 61

Figure 3-19. Co-regulation between gh and gKA conductances ……………………………….. 62

Figure 4-1. Correlation with local preference as an interaction between two independent

conductance density preferences ……………………………………………………………… 66

Figure 4-2. Co-regulatory balances as mutually-dependent distributions of conductance

densities ……………………………………………………………………………………….. 67

Figure 4-3. Inadequacy of highly-ranked models to match the sag characteristic of experimental

O-LM cells ……………………………………………………………………………………. 69

Figure B-1. Example histograms of electrophysiological measures extracted from the

experimental O-LM data ……………………………………………………………………… 95

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List of Appendices

Appendix A – Rationale for active conductance density ranges……………………………….. 78

Appendix B – Electrophysiological measures in PANDORA………………………………… 84

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Introduction 1

1.1 The hippocampus

1.1.1 Structural organization and behavioural correlates

The hippocampal formation is a complex of brain regions located bilaterally within the medial

temporal lobe of humans and other mammalian species. The formation itself consists of various

structures including the hippocampus proper as well as adjacent regions, such as the entorhinal

cortex, that perform crucial roles in the gating of information to and from the hippocampus and

the rest of the brain. The hippocampal structure itself is typically partitioned into three

subdivisions: CA1, CA2, and CA3 [1]. The synaptic chain from dentate gyrus, to CA3, and then

to CA1, forms the classical “trisynaptic loop” that constitutes the major pathway for information

flow in the hippocampus. The clearly delineated laminar formation of the hippocampal regions is

one reason why it has been widely studied. Since the pyramidal cells, constituting the dominant

excitatory cell type in the CA regions, reside within a single layer, it is easy to access and record

from these cells using electrodes (Fig. 1-1).

Figure 1-1. Drawing of a transverse slice of the rabbit hippocampus by C. Golgi. Source: [2].

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There is a large body of evidence demonstrating that brain rhythms coordinate the

patterns of neuronal activity underlying sensory processing, motor control, cognition and

behaviour [3-6]. Several distinct rhythms have been detected in the hippocampus and correlated

with behavioural states. The theta rhythm (4-10Hz) is one prominent rhythm observed in

hippocampus [7]. It has been associated with various activities such as learning and memory [8,

9], spatial navigation [10, 11], exploratory movements [12, 13], and REM sleep [14]. There is

therefore ample reason to study the molecular and structural neurobiology of the hippocampus in

order to better understand how its activities contribute to behaviour and how it may contribute to

pathological conditions such as epilepsy [15] and Alzheimer’s disease [16, 17].

1.1.2 Hippocampal rhythms and the role of interneurons

The majority of cells in the hippocampus consist of the excitatory pyramidal or principal cells,

with fairly uniform morphological and functional properties. On the other hand, there is a great

diversity of inhibitory cell types in the hippocampus that all release the inhibitory

neurotransmitter GABA. This variety is expressed in the heterogeneity of

interneuron morphologies, connectivity patterns, and laminar locations

[18]. Although both pyramidal cells and inhibitory interneurons fire

during hippocampal rhythms, it has been shown that interneurons play a

dominant role in the temporal structuring of these rhythms [19, 20]. The

morphological and synaptic heterogeneity of interneuron effects on

pyramidal cells allows for preferential activity during different oscillatory

phases, pointing to the distinct functional roles of these cells in shaping

rhythmic network activity [21, 22] (Fig. 1-2). Therefore, in order to

understand the functioning of the hippocampus, it is critical to examine

the contributions to overall network behaviour of interneuron

electrophysiological characteristics and synaptic properties.

Figure 1-2. Preferential firing of various interneuron subtypes in CA1

during different phases of the theta rhythm. Adapted from [19].

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1.2 The oriens-lacunosum/moleculare interneuron

1.2.1 Morphology and connectivity

One interneuron type in the CA1 region of the hippocampus is the oriens-lacunosum/moleculare

(O-LM) cell. The O-LM cell somata are located in the oriens layer and their dense axonal

arborizations project to the lacunosum/moleculare layer [23, 24]. O-LM cells receive excitatory

glutamatergic input primarily from adjacent CA1 pyramidal cells [25] and, in turn, form

inhibitory GABAergic synapses onto the apical dendrites of these same pyramidal cells and other

interneurons [26, 27]. These synapses between pyramidal and O-LM cells therefore form a

classic inhibitory feedback loop and point to a crucial role of O-LM cells in regulating pyramidal

cell output. O-LM cells have also been shown to possess intrinsic pacemaker activity implicated

in the maintenance of population activities in the theta frequency range in CA1 [28, 29].

Additionally, O-LM cells have been demonstrated to be capable of the induction of long-term

potentiation (LTP) [30]. Thus, studying the intrinsic and synaptic properties of O-LM cells is

critical in helping us understand hippocampal network output.

1.2.2 The O-LM cell as putative pacemaker neuron

O-LM cells possess a diversity of voltage-gated ion channels within their somatic, axonal, and

dendritic compartments that work together to give rise to the particular characteristics of O-LM

cell output. For instance, O-LM cell dendrites possess elevated densities of voltage-gated sodium

and potassium ion channels, double that of pyramidal cells [31, 32]. Another noteworthy aspect

of O-LM cells is the presence of hyperpolarization-activated cation channels (Ih) that have been

conjectured to contribute to the spontaneous firing behind O-LM cell pacemaker activity [29, 33,

34]. Recent work, however, has suggested that Ih does not play a role in the generation of theta-

paced output in O-LM cells. Rather, O-LM cells may act as transmitters of theta-modulated input

onto downstream pyramidal cells, a feature regulated by afterhyperpolarization characteristics,

not Ih [35]. This was demonstrated by holding O-LM cells in a high-conductance state closer to

threshold, a regime that is regarded to be more relevant to in vivo conditions due to the constant

barrage of combined incoming excitatory and inhibitory synaptic input [36]. As a result, the O-

LM cells were too depolarized for Ih to be activated. However, the observed inability of the Ih

current in O-LM cells to produce theta-frequency spiking activity neither spontaneously nor in

response to theta-modulated inputs may have been due to the limitation of injecting conductance-

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based synaptic input via dynamic clamp onto the cell soma only [35]. If Ih is present in dendrites,

it may be the case that the integrative properties of the dendritic tree in response to synaptic input

may yet allow for a role of Ih in modulating synaptic inputs in order to contribute to population

theta-frequency output. Unfortunately, the distribution of Ih channels in the soma and dendrites

of O-LM cells is not known, and it is particularly difficult to perform recordings from

interneuron dendrites. Thus, multi-compartment modelling approaches would be beneficial in

exploring possible somatodendritic Ih channel locations.

1.2.3 Role in hippocampal CA1 microcircuit functioning

The role of the CA1 microcircuit in hippocampal functioning is not yet fully elucidated, though

there is experimental and computational evidence that it is important in the formation of new

memories. A recent computational model of the CA1 microcircuit that incorporated multi-

compartment models of several known interneuron types in CA1, as well as pyramidal cells, all

synaptically interconnected in anatomically known ways, demonstrated a proof-of-principle of

the encoding and retrieval of information during different phases of the hippocampal theta

rhythm [8, 37, 38]. The model demonstrated that during the theta peak or encoding phase, axo-

axonic and basket cells inhibit axonal output from pyramidal cells, while synaptic plasticity is

invoked on pyramidal cell dendrites. During the theta trough or retrieval phase, bistratified cell

inhibition contributes to the raising of threshold to fire of pyramidal cells whereas O-LM cells

inhibit distal dendritic input arriving from entorhinal cortical regions. These inhibitory processes

during retrieval are required for the pyramidal cells to accurately fire in patterns corresponding to

previously stored information. Recently, there has been direct experimental evidence for this

computational model, and for the particular role of O-LM cells proposed by the model. First,

recordings from freely moving rats demonstrated that with cholinergic receptor antagonism by

scopolamine, pyramidal cell firing shifting towards the theta peak was disrupted while in novel

environments and shifted towards the theta trough while in familiar environments. Since

cholinergic input to hippocampal regions facilitates memory encoding, as seen by the

impairment of information encoding during blockade of muscarinic acetylcholine receptors [39-

41], these results “unmask” the physiological phase preference for firing of pyramidal cells

during the theta peak during encoding processes and theta trough during retrieval processes, a

key predictor of the previously mentioned computational models of CA1 microcircuit function.

Second, a recent optogenetic study in mice showed that O-LM cells inhibited pyramidal cell

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regions in distal stratum lacunosum/moleculare regions, the site of arrival of entorhinal cortical

information, while also inhibiting stratum radiatum interneurons that synapse onto proximal

pyramidal cell dendrites and provide feedforward inhibition of Schaffer collateral information

from CA3 [42]. Entorhinal cortical input mediates novel sensory information whereas CA3 input

facilitates retrieval of previously stored information, which has been correlated with the timing

of the input arriving from these regions during theta peak and trough, respectively [42-45]. The

confirmation of the gating of information by O-LM from CA3 and entorhinal cortical regions

into the CA1 provides corroborative evidence for the role of O-LM cells during theta-modulated

encoding and retrieval of information in CA1. Additional experimental investigation as well as

computational modelling is therefore needed to further elucidate the intrinsic and synaptic

mechanisms of the putative roles of O-LM cells in CA1 microcircuit functioning.

1.3 Mathematical and computational modelling

1.3.1 The need for mathematical and computational modelling

Ever since Hodgkin and Huxley mathematically quantified the action potential by describing the

ionic flows across the membrane in terms of differential equations [46], the contribution of

mathematical models to the understanding of neuronal systems has grown. Mathematical and

computational modelling serves to explicate and quantify theories of neuronal function, as well

as to provide novel predictions for experimental verification. For example, as discussed in

Section 1.2.3, computational modelling of the CA1 microcircuit area played a key role in

clarifying the role of theta phase-modulated encoding and retrieval by providing specific

predictions for experimental confirmation. Furthermore, mathematical and computational models

of O-LM cells have proven to be central in the debate as to its role as a pacemaker cell in

hippocampal functioning (Section 1.2.2). Modelling can also provide for the testing of

predictions and hypotheses that are very difficult, if not impossible, to perform experimentally.

For example, the presence of dendritic Ih in O-LM cells is a crucial question that is difficult to

answer experimentally due to the challenging nature of performing dendritic recordings from the

very small and narrow processes of O-LM cells [47]. Multi-compartment computational models

that incorporate morphological details and physiologically known complements of ionic

conductances can provide principled predictions for the presence or absence of dendritic Ih by

comparing the model’s output in both of these cases with respect to experimental data. Due to

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the demonstrated importance of O-LM cells in hippocampal functioning (Section 1.2), it is

important to continue developing computational models of them in order to further explore and

clarify the means by which their intrinsic properties contribute to the physiological role of O-LM

cells in network functioning.

1.3.2 Biological variability and ensemble modelling

It is becoming increasingly apparent that different combinations of ion channel conductance

densities can lead to similar neuronal and network output, highlighting the role of homeostatic

and compensatory factors [48-51]. For instance, several ion channel types across multiple

vertebrate and invertebrate species have been demonstrated to have varying conductances, up to

2-6-fold [52-54]. In order for cells of a single type to maintain consistent output across this

variability in intrinsic conductances, it would be reasonable to suppose that different

conductances actively balance against each other in a homeostatic way so as to maintain typical

cell output required for network functioning [51]. A recent study demonstrated this conclusively,

by showing that neurons of a single type within the same network, even within the same animal,

modulate conductance density balances so as to maintain the required cell output [55]. The

implications of these findings are several-fold. Perhaps the most clinically relevant is that it is

important to characterize the variability in conductances inherent in a cell type so as to be able to

develop effective drugs. Without taking into account how ion channel conductances can vary

between neurons of an identified target cell type, even within a single individual, a drug that

targets these channels may have varying levels of efficacy, no effect, or even reversal of effect.

There is furthermore a critical implication of this body of work towards computational

modelling. In particular, the traditional approach of building hand-tuned, biophysically based

computational models of neurons has been proven to be inadequate to capture compensatory

effects of ion channel conductance densities across individual neurons of the same type. An

alternative is to build a population, or ensemble of models with systematically varying

conductance densities that can then be used to determine the critical balances of conductances in

producing the neuronal output [56, 57].

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1.4 Previously developed computational models of O-LM cells

1.4.1 Multi-compartment models

Detailed, multi-compartment models of O-LM cells have been previously developed [47, 58, 59].

They were used to explore the dendritic localization of various ionic conductances that are

known to be present in O-LM cells but whose presence in dendrites, or even distributions, are not

known. The effect of synaptic input location and strength on the generation of forward- and

back-propagating action potentials was also examined [47].

A more recent multi-compartment model of O-LM cells was developed in conjunction

with experimental data from O-LM cells in order to examine the role of the M-current (IM), a

muscarinic voltage-activated potassium conductance, in the excitability of O-LM cells [58]. The

O-LM model developed as part of the study incorporated the morphologies of two O-LM cells

using resectioned 70um slices imaged and reconstructed using a Leica DMRB bright-field light

miscroscope and Neurolucida software [58]. The two morphologies were chosen for

reconstruction so as to obtain a variety of dendritic spread and thus account for the observed

variability in O-LM dendritic trees. Passive properties for the O-LM model morphologies were

adjusted to reproduce the transient membrane responses observed experimentally. The model

also included nine active voltage-gated ionic conductances known to be present in O-LM cells.

These are the sodium current as described by Hodgkin and Huxley, INa, fast and slow delayed

rectifier potassium currents, respectively IKDRf and IKDRs, the transient or A-type potassium

current, IA, the L- and T-type calcium currents, respectively ICaL and ICaT, the calcium-activated

potassium current, IAHP, the hyperpolarization-activated mixed cation current, Ih, and the

muscarinic current, IM [58, 59].

1.4.2 Uncertainty in the characterization of currents in O-LM cell models

Although the computational model described in Section 1.4.1 (above) represents the most

complete O-LM model prior to this work [59], there are several aspects of the model that are

unknown, or have not been experimentally measured in O-LM cells, and therefore required

approximation, assumption, or the adapting of data from other cell types. First there is the fact

that conductances are known to exist in O-LM cells that are not present in current O-LM models,

such as the calcium-dependent non-selective cation current (ICAT) [61]. Second, regarding the

8

conductances that are present in the O-LM cell models, there are three general aspects of these

particular ionic currents for which experimental data might be missing: the distribution of the

channels along the cell’s morphological extent, the conductance density of the current, or the

details of the kinetics of the channels. For any given ionic current included in the present model,

one or all of these aspects have either not been measured in O-LM cells directly or have been

measured but in an incomplete fashion [58, 59]. Therefore, in all cases, there are uncertainties as

to the distributions, conductances, and kinetics of the various ionic currents. Even for the

currents that have been studied the most extensively in O-LM cells, there is some incomplete

knowledge. What is known and not known about the various currents in O-LM cells which were

included in the models are outlined in the following subsections.

1.4.2.1 Sodium currents

The sodium currents in O-LM cells have been well characterized, with dendritic distributions and

kinetics obtained from [31]. In the case of the kinetic data, the forward and backward rate

constants of the Hodgkin-Huxley formalism were fitted to the steady-state activation and

inactivation curves for the sodium current [47]. However, even though dendritic sodium currents

were found to exist, they were only measured to within 100 µm of the soma or axon-bearing

dendrite that, although constituting a large portion of the dendritic tree, does not cover it entirely

(See [31], Fig. 1C).

1.4.2.2 Potassium currents

For the IKDRf, IKDRs, and IA potassium currents included in the model, the kinetics were obtained

from [60]. The conductance densities for these channels were not measured directly but rather

expressed as a percentage of total outward (potassium) current. The distributions for IKDRf and

IKDRs were set to the somatic and dendritic compartments as per [31], whereas the IA current was

also distributed dendritically but based on simulations of somatic-only or somato-dendritic IA

performed during the the early development of the O-LM model [47]. The muscarinic current,

IM, was measured directly in O-LM cells in the study that led to the development of the latest O-

LM model [58, 59].

9

1.4.2.3 Hyperpolarization-activated non-selective cation current (Ih)

In the case of Ih, although it is known to be present in O-LM cells, and the steady-state activation

curve was obtained by fitting directly against O-LM experimental data from [34], the

conductance density was estimated by fitting the resulting hyperpolarization-induced “sag” that

is produced by Ih, to the experimentally observed sag amounts. The voltage-dependent time

constant of activation of Ih was measured in [34] by fitting exponential curves to the voltage

clamp data for two data points, one at –70 mV and the other at –120 mV [47]. The dendritic

distribution of Ih in O-LM cells is furthermore unknown. Since the role of Ih in O-LM cell

activity is of interest in hippocampal interneuron research [34, 35], the question of localization of

dendritic Ih in O-LM cells was therefore of particular focus for this work.

1.4.2.4 Calcium currents and calcium dynamics

Though there is evidence for the presence of L- and T-type calcium currents in O-LM cells [59],

these have not been studied at all in any interneuron type, much less O-LM cells in particular. As

a result, the conductances and kinetics were obtained from models that were developed for CA3

pyramidal cells, with calcium dynamics modelled from those in thalamic relay cells [59].

1.4.3 Rationale for ensemble modelling of O-LM cells

The conductance densities, distributions, and kinetics for the ionic currents included in the O-LM

model described above were obtained using principled means from experimental data, and as

much as possible from O-LM cells directly. However, even though the resulting O-LM model is

the most detailed model to date, there are still many uncertainties as to its intrinsic properties, as

described above. Some of these properties, such as dendritic and axonal distributions, are not

known largely because it is technically difficult to perform recordings on the thin neurites of O-

LM cells and other interneurons [47]. As a result, the previous modelling work explored some

possibilities for dendritic distributions; for example, eight different cases of various

combinations of dendritic distributions for the IA and Ih channels [47]. The distributions that

were able to lead to the model demonstrating tonic firing in absence of depolarizing current

injection – as per in vitro findings for O-LM cells [34] – were preferred in the subsequent model

analysis. However, it is entirely possible that due to the particular values used for the

conductance densities and kinetics of the channels, only certain combinations of dendritic

distributions led to tonic firing of the model, and that other, equally feasible combinations of

10

channel kinetics, conductances, and distributions might also have led to tonic firing. These

considerations, along with increasing evidence of homeostatic regulation of conductances to

maintain idiosyncratic cell output (Section 1.3.2), suggest that it is important to explore a large

array of possible combinations of distributions and conductance density values that can only be

afforded by an ensemble modelling approach. This can lead to a better understanding of the

intrinsic mechanisms that lead to physiological O-LM output and that can then be tested

experimentally.

It is important to emphasize that even if the details of all the intrinsic properties were

measured in a small number of O-LM cells, due to the known variability in neuronal intrinsic

properties as outlined in Section 1.3.2, “Biological variability and ensemble modelling”, a single

model built to include this data would not necessarily capture all of the dynamics possible for O-

LM cells to exhibit. On the other hand, an ensemble modelling approach could examine multiple

physiologically plausible combinations of intrinsic properties and hence potentially uncover

interesting balances between subsets of ionic conductances that are necessary for the

conservation of the O-LM cell intrinsic electrophysiological phenotype.

1.5 Thesis organization

1.5.1 Hypothesis

The goal of this work is to apply the ensemble modelling approach to build biologically realistic

models of hippocampal O-LM cells. The main hypothesis of this work is that building a database

of computational models with systematically varied ion channel maximal conductance densities

will reveal critical balances underlying O-LM cell output. We further hypothesize that the model

database analysis can be expanded, in conjunction with additional experimental data, to

understand additional elements of O-LM cell function that are presently unknown, such as the

presence of dendritic Ih currents and the role and mechanisms of cholinergic modulation of O-

LM cells. A primary prediction is that there will be particular conductance density balances that

reflect physiological compensatory balances; that is, they can be tested by further experimental

investigations of O-LM cells.

11

1.5.2 Outline

The rest of the thesis is organized as follows. The experimental data and computational tools

used for building and analyzing ensembles of O-LM models is described in Section 2, “Methods

and Materials”. The development of an ensemble modelling approach as well as its application

for analyzing the variability in conductances in the model database is given in Section 3,

“Results”. Summary and implications of the results are given in Section 4, “Discussion”.

12

Methods and Materials 2

2.1 Experimental data

2.1.1 Hippocampal slice preparation

The experimental data used in the present work was obtained by collaborator Dr. Josh Lawrence

from the University of Montana, who had collected it as part of a previous work [61]. Mice aged

fourteen to twenty-one days, half of which were of strain 129J1 and half CF1 were deeply

anesthetized by isoflurane volatile inhalation and sacrificed. The brain was rapidly removed and

placed in ice-cold artificial cerebrospinal fluid (ACSF) with the following composition (mM): 87

NaCl, 2.5 KCl, 1.25 NaH2PO4, 25 NaHCO3, 25 glucose, 75 sucrose, 7 MgCl2, 0.5 CaCl2

saturated with 95% O2 and 5% CO2, pH 7.4. Hippocampal slices of thickness 300 µm in the

transverse plane were cut using a VT1000S (Leica Microsystems, Bannockburn, IL, USA) or

Vibratome 3000 Deluxe (Vibratome, St. Louis, MO, USA) and placed in a continuously

oxygenated, warm (36°C) ACSF bath for at least 30 min before use.

2.1.2 Electrophysiological recordings

The set of recordings used in the present work consisted of those obtained during whole cell

current clamp conditions of O-LM cells. The cells were maintained at approximately −60 mV,

which resulted in a membrane potential of approximately −73.8 mV after a junction potential

correction of −13.8 mV. This was done using a small negative bias or holding current applied

through the somatic recording pipette (−8.0 ± 4.0 pA, n = 11). Injection of a −5 pA current

applied similarly would periodically be done in order to verify a tight seal with the cell

membrane. Depending on experimental protocol, an additional depolarizing (+90 pA) or

hyperpolarizing (−90 pA) would then be applied for a duration of 1 s, after which the additional

depolarizing or hyperpolarizing current injection would cease. The depolarizing current stimulus

would result in regular action potential firing in the observed O-LM cells, with a “saw-tooth”

firing profile typically seen in O-LM cells [62] and a long-lasting afterhyperpolarization. The

hyperpolarizing current stimulus would result in a characteristic “sag” back towards the resting

potential, attributable to the presence of the hyperpolarization-activated current, Ih. The

recordings obtained from this experimental protocol consisted of ten identified O-LM cells in

13

total, with several +90 pA and −90 pA recording for each cell, resulting in a total of 56 usable

experimental traces. See Fig. 2-1 for characteristic voltage traces taken from a single O-LM cell

as well as examples of the above electrophysiological characteristics of O-LM cells. These

recordings were used in the current work as constraints on the models generated for the ensemble

modelling approach as outlined in Sections 3.1.2 and 3.1.3.

Figure 2-1. Characteristic +90 and –90pA current clamp somatic voltage response of an

experimental O-LM cell. Data from the set of recordings used for this work [58].

2.2 Simulation and analysis software

The NEURON Simulation Environment [63, 64] was used to simulate the O-LM models.

NEURON allows for the development, manipulation, and simulation of Hodgkin-Huxley based

multi-compartment models whose current flows are governed by the cable equation [64, 65]. The

previously developed O-LM model used as the reference model in this work [58, 59] was

obtained from the ModelDB database of neuronal computational models [66], accession number

102288.

The ensemble modelling approach requires a principled method of comparing model

outputs with one another and with experimental data in order to obtain a subset of models that

best correspond to the experimental data. PANDORA’s Toolbox is a set of software functions

implemented in the MATLAB programming environment that facilitates the statistical analysis

0 0.5 1 1.5 2−120

−100

−80

−60

−40

−20

0

20

40

Experimental cell 4405#3Mem

branepotential(m

V)

Time (s)

+90pA−90pA

14

of model and experimental neuronal voltage traces [67]. It specifies classes, or conceptual

groupings of functions, to import voltage traces from files, extract quantitative

electrophysiological measures from the traces, calculate various statistical analyses such as

histograms, correlations and covariances of measures, and calculate “distances” between any two

traces. PANDORA was used extensively in this work for constructing and analyzing the

developed ensemble of O-LM models.

Finally, the MATLAB software environment for numerical computation and visualization

was used extensively for the analysis of data, using and interacting with various PANDORA

classes and functions, applying the CBDR visualization technique (Sections 3.1.5.1,

“Dimensional stacking” and 3.2.3.1, “Dimensional stacking can be used to determine ‘high-

order’ conductances”), and visualizing the conductance density correlation images (Sections

3.1.5.3, “Determining conductance density correlations and co-regulations” and 3.2.3,

“Discovering critical balances in conductance densities”).

15

Results 3

3.1 Development of a robust methodology for ensemble modelling

3.1.1 Outline of results

The framework for building and analyzing an ensemble of computational models developed in

this work follows an iterative, or cyclical pattern as shown in Figure 3-1. The approach begins

with the development and simulation of an ensemble of models (top of figure), and moves

clockwise. The reason that bidirectional arrows are shown between each step is that a move back

to a previous step may be necessary before continuing onto the next (clockwise) step in

sequence. For example, the development of “version 1” and “version 2” of the model database

was necessitated due to uncertainties in Ih dendritic distributions and conductance densities (See

Section 3.2.3.2, “Observed tradeoff between Ih conductance densities and distributions”, and

Appendix A).

The “Performing model simulations” step as applied to this work is covered in Section

3.1.2. The “Generating model database” step is covered in Section 3.1.3. The next step,

“Determining database of acceptable models” is covered in Section 3.1.4. The final step,

“Finding conductance density balances” is described in Section 3.1.5. The results of applying

this ensemble modelling approach is detailed in Section 3.2, “Application of the ensemble

modelling approach”. As seen in Fig. 3-1, the final step in the approach can then lead to another

iteration of the model database analysis, by generating a new set of questions to be addressed by

further refinement and simulation of the model, thus leading back to the first step, “Perform

model simulations”. See Section 4.3, “Limitations and future work”, for an example of how the

next cycle of ensemble modelling might take shape for this work.

16

Figure 3-1. The methodology developed in this work for the analysis of conductance densities

and distributions of multi-compartment models using ensemble modelling. The cyclical nature of

the approach is emphasized.

3.1.2 Performing model simulations (Step 1)

The first required step in the methodology developed here was to obtain a set of simulation

outputs of O-LM models that collectively covered a wide range of varied intrinsic properties, in

this case corresponding to maximum conductance densities and distributions (Fig. 3-1). Note that

some reference or canonical mathematical O-LM model must have first existed, i.e., have been

previously developed. Then, an appropriate compartmentalization for the cell morphologies used

in the models needed to be found (Section 3.1.2.1). Second, the passive membrane properties of

the model needed to be adjusted so that differences in passive membrane response properties

could be controlled for when using the provided electrophysiological data as constraints for the

subsequent selection of appropriate O-LM models (Section 3.1.2.2). Then, the reference O-LM

model needed to be prepared for the systematic variation of its intrinsic properties and for

simulation on a large scale. For the former, physiologically plausible ranges of conductance

densities needed to be selected (Section 3.1.2.3), and an automated method of running a model

with custom conductance densities and distribution parameters needed to be specified (Section

3.1.2.4). For the latter, a way of fully automating the generation of the model output in the

17

NEURON simulation environment needed to be developed. This was crucial for the ability to run

hundreds of thousands of individual O-LM models on a supercomputer cluster (Section 3.1.2.5).

Finally, in evaluating these simulations, we extracted models that were consistent with how the

experiments were performed (Section 3.1.2.6). The details for each of these steps are described

in the following subsections.

3.1.2.1 Morphological reconstructions and compartmentalization

The reference model used in this work is based on a previous detailed multi-compartment model

of O-LM cells [47, 58, 59] (See Section 1.4, “Previously developed computational models of O-

LM cells”). The O-LM model includes two morphological specifications that were obtained from

mouse hippocampal O-LM cells and reconstructed using Neurolucida software. The

reconstructions were then imported into a three-dimensional vector format compatible with the

NEURON software for multi-compartment neuronal modelling [63]. See Fig. 3-2 for a visual

depiction of the two morphological reconstructions: morphology 1, or “Richy”, and morphology

2, or “Starfish”.

Figure 3-2. Morphological reconstructions of the two experimental O-LM cells provided by

collaborators. (A) Morphology 1, or “Richy”. (B) Morphology 2, or “Starfish”. CA1 strata are

labelled as follows: “o” – stratum oriens; “p” – stratum pyramidale; “r” – stratum radiatum;

“lm” – stratum lacunosum/moleculare. Adapted from [58].

The imported morphological data specified somatic, dendritic, and axonal processes in

cable form; that is, complex three-dimensional morphologies were reasonably simplified to a set

of connected one-dimensional cables of various lengths named sections in the NEURON

A B

18

software. Each section is modelled using the cable equation that describes flow of current

through the axial and transmembrane dimensions of the cable [64, 65]. In order to maintain

accuracy in the simulation of current flow through a cable, however, the transmembrane current

through any given point of membrane for a particular section must be adequately approximated

by the transmembrane current through the centre of that section. Depending on the length of the

cable section, this assumption may or may not hold, so a section generally needs to be divided

into smaller compartments. This is accomplished in the NEURON simulation environment by

specifying the nseg parameter, which determines how many segments of equal length a given

section is divided into. The equations for calculating the total capacitive and ionic components of

the membrane current are then resolved for each segment, which has the overall effect of

increasing the spatial resolution of the cylindrical representation of the neurite in question. One

common strategy of increasing the spatial resolution, and hence the accuracy, of the multi-

compartment model is by increasing the number of compartments uniformly across the entire

morphological extent of the model – that is, increasing the nseg parameter, or number of

segments, equally across all sections of the model. Although this can lead to sufficiently accurate

simulations of the current flow through the model’s various segments, it has an additional effect

in that certain sections of the model may have more compartments than are needed in order to

maintain appropriate results. This is due to the observation that certain branches of a given

model’s morphology may require a more fine-grained structure of compartments than other

branches [68]. By specifying an nseg parameter that is consistently applied across all segments,

however, the computational time of the model is increased more than is necessary.

An alternative to the approach described above is to only increase the amount of

compartments required on an individual per-segment basis in order to maintain accurate

simulation results. The developers of the NEURON software proposed the “d_lambda rule” as

such an alternative method for determining a spatial grid of compartments for model

morphologies [68]. The rule is based on the observation that transient, or high-frequency

components, of current signals decay more rapidly with distance than slowly changing signals

due to the action of cytoplasmic resistance and membrane capacitance as low-pass filters [68].

Consequently, a principled criterion for determining an appropriate spatial grid relies on

computing the frequency-dependent length constant λf at a high enough frequency that will

preserve the propagation of transient signals. Hines and Carnevale [68] suggest that because

19

most neuronal cells have membrane time constants greater than 8ms – corresponding to a

membrane frequency of approximately 20Hz – the frequency-dependent length constant of λ100,

at 100Hz, should act as a sufficient upper bound. Therefore, accurate simulation of transient

signals will require that compartments be separated by a distance that is some fraction of λ100, to

be determined by hand.

For the two model morphologies used in this work, the appropriate fraction of λ100 was

determined by setting up the “rig” in NEURON for fitting the passive properties of one model

and morphology to experimental O-LM data (see Section 3.1.2.2, “Fitting of passive properties

to experimental data”, below). An initial fraction of λ100 was assigned as the parameter to

determine the number of segments within each section, and therefore the total number of

compartments in the model. A trial run was then initiated in the Multiple Run Fitter (MRF) to

determine the error of the model’s membrane response to a −5 mV voltage clamp step compared

to the experimental O-LM cell average. The passive properties were held fixed at the values

defined in the reference model. Afterwards, the fraction of λ100 was lowered, resulting in a model

with more compartments and hence greater simulation accuracy, and the MRF trial was re-run.

This was continued until the error value for the model did not change appreciably, thus

indicating that the model output was being simulated with sufficient accuracy. The final values

determined for the Richy and Starfish λ100 fractions were, respectively, 0.0101 and 0.00465.

3.1.2.2 Fitting of passive properties to experimental data

The model morphologies used in this work were obtained from a previous study where they were

reconstructed from experimental O-LM cells using Neurolucida [58]. However, the experimental

data used in the current work was obtained from a different study on O-LM cells [61]. In

particular, the recordings were made from cells for which morphological reconstructions were

not available. Rather than creating arbitrary model cell morphologies, however, the previously

created reconstructions were adapted for this work by re-fitting the passive properties of the

models using the provided experimental voltage-clamp recordings. The passive membrane

properties for the O-LM model consist of the axial resistivity (Ra), the specific membrane

capacitance (Cm), the membrane resistance (Rm), the passive leak reversal potential (EL), and the

potassium leak conductance (gKL) [58, 47]. Together, these properties determine the passive

electrical activity of the membrane in response to small charging transients [69]. An appropriate

20

set of experimental recordings that reflect the passive membrane response of the O-LM cells was

extracted. This consisted of “seal test” recordings, of which nine were available, each from a

different O-LM cell. The seal test protocol used was that of stepping a voltage clamp to −5 mV

from the holding potential, waiting for 100 ms, and then stepping back to 0 mV, observing the

resulting current response throughout the entire 200 ms run time of the experiment. The resulting

traces were averaged for the nine cells and imported into the Multiple Run Fitter (MRF), a built-

in optimization tool in NEURON. Each of the Richy and Starfish morphologies were loaded into

NEURON in turn, and a virtual voltage clamp was inserted into the somatic region of each

model to duplicate the experimental seal test setup. The resulting current response of the models

were compared to the experimental average, and an error value computed by the MRF software.

The MRF would then automatically adjust the passive parameters until the model’s seal test trace

more closely matched the experimental average. Various parameters were adjusted individually

depending on whether the steady-state portion of the trace or the charging portion was in error.

For the steady-state portion, EL and gKL were allowed to be varied as they primarily determine

the cell response during steady-state; the other parameters were held constant. For the charging

portion of the trace, only Rm was allowed to be varied. The specific membrane capacitance Cm

was allowed to vary for both the Richy and Starfish models but was taken out of the optimization

procedure altogether as soon as the values started to differ too much from 0.9 µF/cm2, which is

the average value observed nearly universally across various neuronal types [70]. Therefore, all

further fitting of the model’s membrane response to charge transients, described by the

membrane time constant τm, was determined solely by Rm since 𝜏! =  𝑅!𝐶! [69]. This resulted

in the alternation of fitting Rm for the transient responses of the membrane with fitting EL and

gKL for the steady-state membrane response. Note that Ra was not allowed to vary as 300 Ω ⋅ 𝑐𝑚

is the typical, approximate value found in neurons [71]. This was continued until the model

current response did not appreciably improve compared to the averaged experimental trace. At

this point, the reconstructed morphologies, in combination with the passive properties adjusted to

the provided experimental dataset, provided a reasonable match to the experimental O-LM cells.

See Table 3.1 for the resulting passive properties which were used for the rest of the model

simulation runs. For a discussion of the re-fitting of the passive properties after the highly-ranked

model database subset was acquired, see Section 3.1.2.3, below.

21

Passive property Model morphology 1 Model morphology 2

Ra (Ω ⋅ 𝑐𝑚) 300 300

Cm (µF/cm2) 0.96857 0.9

Rm (Ω ⋅ 𝑐𝑚!) 59,156 39,038

EL (mV) −73.588 −73.8424

gKL (S/cm2) 9.9005 × 10-10 1.00115 × 10-9

Table 3.1. Fitted passive properties for the two O-LM model morphologies.

3.1.2.3 Active conductances and density ranges

The O-LM model adapted for this work has nine different voltage-gated active conductances

defined in the model specification. These include the sodium current (INa), the fast- and slow-

delayed rectifier potassium currents (respectively, IK-DRf and IK-DRs), the transient A-type

potassium current (IA), the L- and T-type calcium currents (ICaL and ICaT), the calcium-activated

potassium current (IK(Ca) or IAHP), the hyperpolarization-activated mixed cation current (Ih), and

the muscarinic potassium current (IM). For the ensemble modelling approach, the maximum

conductance densities were varied for the model, thus constituting a way for the models to

exhibit different ion channel expression levels. The values that the maximum conductance

densities for the various ion channels were allowed to take were determined on a case-by-case

basis depending on what was previously known about that ion channel type, and specifically

about its presence and somato-dendritic densities in the O-LM cell. The reasoning for these

choices described in Appendix A. Table 3.2, below, lists the final maximum conductance density

values used in the model database construction (Section 3.1.3).

22

Active Conductances

Maximum conductance density values (pS/µm2)

Compartmental locations

gNa,s (somatic) 60, 107, 220 Soma

gNa,d (dendritic) 70, 117, 230 Dendrites, axon

gKDRf 6, 95, 215, 506 Soma, dendrites, axon

gKDRs 2.3, 42, 92, 222 Soma, dendrites, axon

gA 2.5, 32, 72, 169 Soma, dendrites

gh (version 1) 0.5, 16, 53, 90 Soma only or soma and dendrites

gh (version 2) 0.02, 0.05, 0.1, 0.3, 0.5 Soma only or soma and dendrites

gCaL 12.5, 25, 50 Dendrites

gCaT 1.25, 2.5, 5 Dendrites

gAHP 2.75, 5.5, 11 Dendrites

gM 0.375, 0.75, 1.5 Soma, dendrites

Table 3.2. Summary of maximum conductance density values used in the model database

construction.

3.1.2.4 Automating the model

The reference model used in this work was originally designed to be run manually in the

NEURON simulation environment. The construction of a large population or database of these

models, however, necessitated two changes to the model’s specification in order to eliminate the

need to evaluate each model’s output manually: (1) the maximum conductance density values

needed to be specified as parameters to the invocation of the NEURON instance in which the

model would execute, rather than as hard-coded parameters in the model code itself; (2) the

simulation and output of the model needed to be fully automated. These requirements were

handled by writing “wrapper” code in NEURON that would in an automated fashion generate a

model instance via command-line arguments from the shell environment in which the parameters

of the model were specified. These included the cell morphology (Richy or Starfish), maximum

23

conductance densities, the Ih distribution parameter, the current injection step (−90 or +90 pA),

and a verbosity parameter to be used for troubleshooting the model. When the wrapper code is

executed, a function iteratively sets the per-section maximum conductance densities equal to the

values specified as arguments in the invocation of the model from the command-line. The

morphology of the cell is also loaded and the passive properties are assigned to the model’s

segments (See Section 3.1.2.2 for a discussion on the values of the passive properties). The name

of the output file for this particular model instance is generated based on the collection of mode

parameters. This allows for the identification of a model based on the filename of its output,

which is needed by PANDORA in order to extract the model parameters from the filename itself.

These adjustments to the model code facilitated its use for this work, where hundreds of

thousands of model instances with different parameter combinations would have to be evaluated

in an efficient manner. The software framework for the evaluation of the ensemble of models on

a supercomputer cluster is discussed further in Section 3.1.2.5.

3.1.2.5 Simulation framework

The ensemble modelling approach used here is the variant sometimes referred to as the “brute-

force” [72] approach because it depends on systematically varying all of the parameters and

generating model output for each possible combination of parameters. By varying the maximum

conductance densities of the O-LM model in this work, of which there are ten (treating the

somatic and dendritic sodium conductances separately), as well as the distribution of 𝑔! along

somatic only versus somatodendritic compartments and, finally, the morphology of the model

used, there are a total of 933,120 possible models. Considering that experiments for both −90 pA

and +90 pA current injections need to be applied to each model, this results in 1,866,240 total

simulations that need to be evaluated. Each model takes about 3 minutes to evaluate on a modern

computer with 8 cores, thus allowing 8 simultaneous models to be evaluated at any given time.

Given this, it would still take approximately 486 days to evaluate all of the required models –

much more if the bias current for each model must also be fit (see Section 3.1.2.5.3).

Furthermore, such an ensemble of models often has to be re-evaluated as details of the model are

modified, for example, changing the range of possible maximum conductance density values.

Running these simulations on a single computer, or even distributed amongst two or three lab

computers, is a prohibitive task. Therefore, the use of high-performance computing (HPC) is

required.

24

For this work, the SciNet HPC supercomputer cluster was used for evaluating the model

outputs. The SciNet General Purpose Computing (GPC) cluster consists of 3,780 nodes with 8

cores each [73]. Considering the priority of tasks given for this work, dozens of nodes were

usable simultaneously. This reduced the total time to evaluate all 1,866,240 models to

approximately 48 hours. However, being able to handle all of the model simulations required

significant automation. Three tools needed to be implemented in order to meet this criterion: (1)

a script to generate the command-line invocations of all of the models; (2) fully automated

NEURON code to evaluate the output of each (discussed in Section 3.1.2.4); (3) an efficient

system for finding missing models. These tools were constructed from scratch as part of this

work and were tested extensively on the GPC cluster; the details of points (1) and (3) are

provided in the following two subsections.

3.1.2.5.1 Generating jobs containing all model invocations

The process of generating jobs – a collection of code to be executed on a single supercomputer

cluster node – for all of the models was handled by a single script, gen-jobs.sh. The logic for

the script consisted of a series of nested loops that would range over the possible values for all of

the model parameters – maximum conductance densities, morphology, 𝑔! distribution, and

current injection step – and write the resulting command-line invocation of the model to a file

that would be included as part of a single job on the GPC cluster. Each job’s data file contained

up to 2,500 such model invocations, and the gen-jobs.sh script furthermore ensured that a

startup script for each job was generated with references to the appropriate data file containing

model invocations. Finally, gen-jobs.sh facilitated the submitting, and subsequent

monitoring, of all jobs by grouping the resulting job submission scripts and data files into

subdirectories with the morphology, 𝑔! distribution, and current injection steps as branch points.

3.1.2.5.2 Finding and generating jobs for missing models

Each job running on the GPC cluster would run for a variable amount of time since each model’s

simulation evaluation time differed. Differences in simulation runtimes were accountable to

differences in the model’s intrinsic channel properties, which introduced variability in the

model’s output. For example, a specific combination of ion channel maximum conductance

densities could give rise to a higher firing frequency for one model as compared to another, with

the output of the former taking longer to evaluate due to the greater number of spikes. One

25

prominent source of variability in simulation runtimes did not arise due to the simulation itself,

but rather to the amount of time required to fit the bias current against a particular model (see

Section 3.1.2.5.3 below for full details). Exacerbating this variability in runtimes was the fact

that the method for generating jobs using nested loops (as outlined in Section 3.1.2.5.1 above)

ensured that the collection of 2,500 models in a job contained only slightly varied parameters

from each other. This is because the parameters specified towards the beginning of the nested

loop would not be incremented, and would thus remain the same, for large consecutive

evaluations of the nested loops specified further down. Thus in any given collection of 2,500

models, the models would most likely have the same values for most of the parameters. As a

result, if a particular combination of parameters would produce a model that would need a

greater amount of time to evaluate, the likelihood that the other models in the same job as the

model with that particular combination would also require a longer amount of time to evaluate

since they would only differ very slightly. Therefore, the time to evaluation of any given job

varied significantly. A complicating factor for was that the SciNet cluster enforces a maximum

runtime of 48 hours to any given job so as to prevent failed jobs from running indefinitely and

thus unnecessarily consuming node resources that could be used in allocating the execution of

other jobs. Consequently, although some jobs in this project would run within the 48 hour

allotted maximum time, others would extend past the 48 hour limit. These jobs would then be

terminated in an automated fashion by the SciNet scheduler system. Once all jobs were

completed, successfully or not, it was therefore necessary to find out in an efficient manner

which jobs were “missing” and needed to be re-evaluated in order that all models that were

specified had their outputs generated. This was done by writing a script, named, O-

LM_get_overtime_jobs.sh, that would iterate over all the resulting job logs and determine

which ones ran overtime, based on the record of time spent on a particular job. If any jobs hit

their time limit and were therefore terminated, they were then split into two smaller jobs of 1,250

models and resubmitted to the SciNet scheduler. This latter step was performed by a separate

script, gen-final.sh. The newly created jobs could then be re-submitted. A beneficial result

of this system of finding missing jobs and generating new jobs was that it could be repeated.

That is, if a particular job containing 1,250 models also ran past the time limit specified by

SciNet, it could then be processed once more and turned into two more, smaller jobs of 625

models each. This process was continued until all model outputs were successfully evaluated.

26

3.1.2.6 Adjusting O-LM models for bias current

In the experimental dataset, each O-LM cell’s membrane potential had to be held constant in

order to maintain consistency across experiments and, in particular, the state of the voltage-gated

ion channels present in the membrane so as to minimize the amount of experimentally-induced

variability. This was accomplished by dynamically varying the amount of bias current which

was injected prior to, and concurrently with, the subsequent ±90 pA hyperpolarizing or

depolarizing current injection step in order to maintain a Vm of approximately −74 mV prior to

the current injection step (see Section 2.1.2, “Electrophysiological recordings”). The resulting

bias, or holding current, varied between experimental cells, from approximately −11 pA to +7

pA. In order for the O-LM models to be comparable to the experimental cells, the models’

somatic Vm also had to be held at −74 mV. However, analogously to the situation with the

experimental O-LM cells, each model could vary in the amount of holding current required to

keep it at −74 mV due to the differing maximum conductance densities used in each model. It

was therefore necessary to implement a procedure that would “fit” the amount of holding current

required for each model so as to keep its somatic Vm at −74 mV prior to the current injection

step. The basic procedure for a given model was simply to begin with a greater amount of

hyperpolarizing current than would be anticipated, −15 pA, and a 1 second-long simulation was

run for the model in order to evaluate its somatic Vm. If the Vm was more negative than −74 mV,

the holding current would be incrementally decreased, i.e., made slightly less negative. As the

membrane potential approached −74 mV on subsequent iterations of the fitting procedure, the

size of the increment was gradually decreased so as to make the changes in Vm between iterations

smaller and thus ensure that it would converge smoothly to −74 mV. The fitting procedure

became more complicated as more special cases of model outputs were found. For example,

some combinations of parameters resulted in models that were not capable of firing action

potentials and were clearly not appropriate representations of O-LM cells. These models would

inevitably require large amounts of holding current to bring them to −74 mV. Additionally, many

models exhibited firing activity even before Vm was brought close to −74 mV; that is, their

threshold to fire was more negative than the desired −74 mV holding potential. Unfortunately,

the original holding current fitting algorithm would take the average Vm of the entirety of the 1

second-long simulation, which would “hide” the presence of the spikes, and thus the algorithm

would often fail to converge with these models. Due to the presence of both of these classes of

27

models, an initial spike threshold-finding phase was implemented prior to the holding current

fitting algorithm proper. That is, the amount of holding current was first slowly decreased –

made more positive – until one of three conditions was met: (1) either the model cell would start

firing action potentials prior to reaching the −74 mV target Vm, where the presence of values

greater than, or equal to, –10mV indicated that a spike was present, or (2) the holding current

reached +15 pA, in which case the model was likely to not fire very easily or, finally, (3) the

model cell started to fire action potentials somewhere between the first two cases, i.e., with a Vm

greater than −74 mV and a holding current less than +15 pA. In the lattermost case, the holding

current fitting procedure was then initiated to find the precise amount of holding current

required, and the full simulation for the model was then evaluated and recorded. In the first two

cases, as such models were not appropriate representation of an O-LM cell, they were discarded,

and no further simulations were performed for those models. By following this procedure,

609,143 out of a total of 933,120 models were found to be inadequate, with 323,977 models

being considered acceptable and retained for further analysis of conductance density balances.

3.1.3 Generating model database (Step 2)

The next step in the methodology developed for this work necessitated importing the set of

model simulations with systematically varied intrinsic properties into specialized software for the

analysis of model and experimental traces so as to produce the model database for further

analysis of conductance density balances (Fig. 3-1). This required adapting previously existing

software for the statistical analysis of electrophysiological and model traces (Section 3.1.3.1).

Then, the model and experimental traces needed to be imported into this software in such a way

so as to allow appropriate comparison of their respective traces (Section 3.1.3.2). Once the model

and experimental outputs were analyzed and their electrophysiological properties quantified, the

resulting model database could be sorted, or ranked, according to the models’ goodness-of-fit to

the experimental dataset (Section 3.1.3.3). This would allow for the subsequent step of

determining acceptable models that best reproduced the electrophysiological dataset (Section

3.1.4). The details for the generation of the model database are described in the following

subsections.

28

3.1.3.1 Framework for analysis of model and experimental traces

3.1.3.1.1 Modifications required for PANDORA

The original code for PANDORA used for this project was provided by Cengiz Günay at Emory

University and contained most of the required functionality. The experimental paradigm used in

this work specified a ±90 pA, 1 s-long current injection period. As a result, the trace was

logically divided into three separate periods corresponding to (1) the period prior to the current

injection, (2) the period of the current injection itself, and (3) the period following the current

injection. In PANDORA, these periods are named, respectively, the spontaneous (or spont),

“pulse” or current injection (or cinj), and recovery (or recov) periods. It was important to

separate the trace into three periods so that quantitative metrics could be extracted and analyzed

on a per-period basis. For example, the firing frequency of the model or cell during the current

injection period must be treated separately from the firing frequency of the model or cell after the

current injection is terminated as these reflect different states of ionic channels and membrane

potential changes. Therefore, when comparing a model with an experimental trace, the firing

frequencies of the model or experimental cells should be evaluated against each other during the

appropriate periods under consideration.

Although PANDORA contained the basic framework for processing current clamp traces,

the codebase for this functionality was not fully implemented by the original author. Therefore,

with assistance from the original author, changes to PANDORA were made as part of this work

in order to fully develop the code that could handle current clamp traces. These changes were

ultimately folded back into the main release of PANDORA so that others could take advantage

of functionality for analyzing current clamp traces [67].

3.1.3.1.2 O-LM cell electrophysiological features

PANDORA contains the capacity to analyze many more electrophysiological measures than are

useful, or even applicable, for the voltage traces used in this work. Therefore, a crucial aspect of

the ensemble modelling approach was to apply the various measures that PANDORA can

support to both the −90 pA and +90 pA experimental current clamp traces, and then see which

ones actually provided useful information that could be used as a way to quantitatively compare

model and experimental traces. This was done by importing the experimental traces into

PANDORA and plotting histograms for the values of each of the provided measures. If

29

PANDORA could not calculate a meaningful value for a particular measure, it would return the

MATLAB value of NaN (“not a number”, i.e., a non-value) for the evaluation of that measure for

the trace. As a first pass, any measures for which there were any NaN values were not included

for further use in the ensemble modelling analysis. Of the resulting measures, those that only had

zeros for all of the traces were furthermore not included. Finally, of the remaining measures, a

subset containing the most useful information was selected. For example, many measures that

calculated the statistical modes of other measures were not included. Furthermore, minimizing

the number of measures used would serve to facilitate and simplify the interpretation of

subsequent database analysis. For the −90 pA traces, there were far fewer useful measures than

for the +90 pA traces. This was partly because many of the measures included in PANDORA are

for analyzing spiking behaviour, and none of the −90 pA traces had any spiking activity. See

Appendix B for the list of measures used for both the +90 pA and –90pA traces, including

descriptions of their meaning in PANDORA. See Fig. B-1 in Appendix B for example

histograms of some of the electrophysiological measures of the experimental data.

3.1.3.2 Importing model and experimental traces into PANDORA

Each job on SciNet, responsible for running up to 2,500 models, would result in a file that

consisted of the packaged results of all the models that were run on the cluster node that

executed the jobs. The resulting packaged files were then transferred to a local lab computer and

processed in order to be imported into PANDORA for further analysis. The same directory

structure used on the SciNet cluster for job specification – consisting of directory branch points

for the cell morphology, 𝑔! distribution, and current injection parameters (see Section 3.1.3.1,

“Framework for analysis of model and experimental traces”) – was also maintained on the local

lab computer so as to facilitate the loading of models into PANDORA. This is because by

dividing the models into subdirectories, several MATLAB instance could be run in parallel so as

to import separately the contents of each subdirectory. In order to perform the actual import, the

params_cip_trace_fileset function had to be called for each set of models so as to create a

fileset. The fileset in PANDORA is used as a reference to both the model and experiment voltage

outputs that can be passed to subsequent functions that analyze the traces and create database of

the resulting measures. In order for the traces for both models and experiment to be comparable,

however, various accompanying metadata needed to be given to the

params_cip_trace_fileset function. The reason for this was that the provided

30

experimental data used for this work varied in the duration of recording that was present prior to

the depolarizing or hyperpolarizing current injection steps. Since this pre-current injection period

(or “spontaneous” period – see terminology introduced in Section 3.1.3.1.1) was used to

calculate certain measures including Vm, it was necessary to keep the duration of this period

constant for all experimental and model traces so as to remove any resulting

discrepancies between traces that were due to the different lengths of the spontaneous

periods. Similarly, the experimental traces would end their recordings at different

times. As a result, the trace_time_start and trace_time_end parameters to the

params_cip_trace_fileset function were specified on a case-by-case basis for the

experimental traces so as to maintain a duration of 180 ms for the spontaneous period (the

minimum duration for this period across all experimental traces), and 1,000 ms for both the

current injection and recovery periods of all experimental and model traces. The model and

experimental traces could then be imported into PANDORA. The resulting filesets from the

separate MATLAB instances were then combined into one large fileset for the model trace data.

Another fileset for the experimental data was also generated. These were then used for the

analysis and ranking operations described in Section 3.1.3.3, below.

3.1.3.3 Ranking models against experimental data

In order to study the intrinsic properties of the models that best match experimental O-LM cell

output and therefore examine their conductance density balances that give rise to stereotypical

O-LM cell behaviour, a way of comparing and ranking models against experimental O-LM cells

is needed. The framework introduced in Section 3.1.3.1, above, consisting of PANDORA’s

Toolbox, provides the quantitative basis for principled comparisons between model and

experimental voltage traces. This consists of a collection of electrophysiological measures which

PANDORA automatically extracts from all model and experimental traces, as they are loaded

into a database (see Section 3.1.3.2, “Importing model and experimental traces into PANDORA).

In fact, a database in PANDORA is simply an object that consists of a two-dimensional matrix

with rows corresponding to individual traces – whether of model or experimental cells – and

columns corresponding to the parameters of the models or experiments as well as the

electrophysiological measures calculated for that particular trace by PANDORA. The columns

corresponding to the parameters represent either the maximum conductance densities, cell

morphology, 𝑔!distribution, or current injection step in the case of model traces; in the case of

31

experimental traces, the parameters correspond to the cell number, date of recording, and

recording episode. The parameters are required for collating information for a particular model

or experimental cell that might be spread out across several rows of the database matrix, since

each model and experimental cell has several voltage traces associated with it. The following

subsections describe, respectively, the method of ranking all model traces against a single

experimental trace, and the method for computing an aggregate distance metric of all models

against all experimental data.

3.1.3.3.1 Calculating the distance between model and experimental traces

In order for a comparison between model and experimental cells to be made, a method for

quantifying an aggregate measure of the “closeness” between a model and experimental trace is

needed. PANDORA provides a ranking function, rankMatching, that takes as its arguments a

database consisting of model traces as well as a critical database consisting of one experimental

trace to compare all of the models against. The critical database is a matrix with only two rows;

the first row is just the row for the target experimental trace copied from the original database

containing all traces, and the second row consists of calculated standard deviations for the

measures of the experimental trace against the full experimental database. The ranking function

then calculates the Euclidean distance between all the models in the provided model database

and the single experimental trace, as per the following equation in the supplementary methods of

[67]:

𝑑!,! =  

|𝑥! −  𝑦!|𝑁𝜎!

!

!!!

(3.1)

where xi and yi represent the ith measure, out of N total measures, of the model and experimental

traces, respectively, 𝜎! is the standard deviation of the measure in the experimental database, and

dx,y is the “distance” or error between model trace x and experimental trace y. The denominator is

implemented using the second row of the critical database. The importance of the 𝜎!

normalization term is to penalize models whose measures differ significantly when those

measures are tightly constrained in the experimental database – that is, when 𝜎! is small. On the

other hand, models with measures that vary significantly in the experimental database – that is,

when 𝜎! is large – will not be penalized by the distance calculation if they differ significantly

32

from the experimental measure, yi. Effectively, the equation calculates the standard score, or z-

score, of all of the model’s measures against the experimental measures. The resulting distance

value, dx,y, represents how close of a match a model trace is to an experimental trace. Larger

distance, or error, values correspond to models that are “further away”, or worse matches against

the experimental trace. On the other hand, lower distance, or error, values correspond to models

that are “closer”, or better matches against the experimental trace.

3.1.3.3.2 Calculating the aggregate distance between all models and the experimental dataset

The rankMatching function described above returns a database of models sorted by their

distances from a single experimental trace. In order to compare the models against all the

experimental data, a method of combining the distance values of all model traces against all

experimental traces is required. For this work, a new MATLAB function, OLM_rank, was

written to implement the straightforward procedure of summing the distance values of a model

against all experimental traces in the following fashion:

𝑑! =  1𝑁!

𝑑!,!!

(3.2)

where dx,y is the distance of model trace x against experimental trace y, Ny is the total number of

experimental traces, and dx is the distance of model trace x against all of the experimental data. It

is important to normalize the total distance dx with the number of experimental traces Ny so that

distances between databases with different number of experimental traces can be meaningfully

compared to one another. In the case of the current work, there are four per-morphology and per-

current injection step databases that are constructed to rank the models against the experimental

data (Table 3.3).

33

Database name Morphology of

models in

database

Current

injection step of

models in

database

Number of

experimental

traces (Ny in

equation 3.2)

myrdb1_neg90 Cell 1 (Richy) −90 pA 16

myrdb1_pos90 Cell 1 (Richy) +90 pA 40

myrdb2_neg90 Cell 2 (Starfish) −90 pA 16

myrdb2_pos90 Cell 2 (Starfish) +90 pA 40

Table 3.3. The four databases constructed as part of the ensemble modelling approach.

The consolidated database can then be sorted by the aggregate distance value, resulting in

a ranking of models with monotonically increasing distance values. Models that are better

representations of O-LM cells are towards the front of the list with low distance values and

defined to be highly ranked. On the other hand, models that are poorer representations of O-LM

cells reside further down the list of ranked models, and have larger distance values, thus defined

as poorly ranked. The problem arises of which models to consider as acceptable representation

of O-LM cells, and which to consider as unacceptable representations, for subsequent analysis of

conductance density correlations. This problem of selecting a subset of acceptable models from

the database of ranked models is discussed in Section 3.1.4, below.

3.1.4 Determining database of acceptable models (Step 3)

Once the available set of models were organized into a ranked database that sorted the models’

goodness-of-fit to the experimental dataset along quantified electrophysiological criteria, a

subset of the models that best represented the experimental O-LM cells from which the

electrophysiological data was acquired needed to be found (Fig. 3-1). Two general approaches

were developed. One was inherently objective – that is, agnostic to any particular

electrophysiological characteristic – that depended on the rate of change of the error, or

“distance”, of each model from the experimental dataset (Section 3.1.4.1). In order to test

34

whether the ensuing cutoff in the ranking of models in the database was appropriate, two

electrophysiological characteristics representative of the set of experimental data were used as

checks of the objective criterion (Section 3.1.4.2). The highest ranked models were also used to

validate the fitting of the passive properties, performed early on in the work prior to the

simulation of the models (Section 3.1.4.3). The objective criterion now being confirmed as being

appropriate for use in determining representative O-LM models, a cutoff of acceptable

O-LM models was then selected for further analysis of conductance density balances (Section

3.1.5). The implementation of these analyses of the database are described in the following

subsections.

3.1.4.1 Using an objective criterion

As part of the ensemble modelling approach, a database of ranked models is constructed as

described in Section 3.1.3. The problem then arises of finding a principled criterion of selecting

a subset of the models that best represent O-LM cells. One solution is to implement an objective

criterion, that is, one that does not take any particular individual electrophysiological measures

of the database into account but rather directly utilizes the distance, or error, values. The

objective criterion used for this work was to calculate the first-order derivative of the error

values so as to compare the rate of change of the error as a function of the ranking of the models

in the database. This was implemented using the built-in MATLAB function diff, which

calculates the differences between adjacent values of a vector. Nearly identical results were

obtained using the multivariate gradient function. Indeed, since the distance, or error, of a

model is a function of one variable – the model’s rank in the database – the multivariate gradient

therefore reduces to the univariate differences of the distances along the rank, just like diff.

3.1.4.2 Using electrophysiological criteria

For more restrictive criteria in selecting a subset of appropriate O-LM models from the database,

two electrophysiological measures as extracted by PANDORA were selected. The two

electrophysiological criteria chosen were the firing frequency during the current injection pulse,

or the PulseSpikeRate PANDORA measure, and the time constant of the hyperpolarization-

induced sag, or the PulsePotTau PANDORA measure. The PulseSpikeRate measure is calculated

from the firing frequency of the model or cell during the 1s-long +90 pA current injection period

by obtaining the number of spikes and dividing by the duration of the period. The PulsePotTau

35

measure is obtained by fitting a single exponential to the voltage sag response of the membrane

during the 1s-long −90 pA current injection period. These two measures were chosen to be

representative of the +90 pA and −90 pA model and experimental traces as discussed in the

Results, Section 3.2.2.3.

3.1.4.3 Validating passive properties fit

The work of fitting the passive membrane properties of the model to the experimental data, in

setting up the developed mathematical models to perform the model simulations in Step 1, was

done using the reference model, as discussed in Section 3.1.2.2. The set of passive properties

were then held constant for all of the models subsequently obtained by varying the maximum

conductance densities of their voltage-gated ion channel models. Once the models were ranked

against the experimental data as per Section 3.1.3.3, a set of highly-ranked models was obtained.

However, active conductances play a role in the current/voltage dynamics of the experimental

protocol used in fitting the passive properties. It is thus conceivable that a model with different

maximum conductance densities may result in a different fit of the passive properties if the

maximum conductance densities of its ion channel models are allowed to vary. Given that the

reference model possessed maximum conductance densities that were fit by hand, it is possible

that the set of parameters for the reference model does not allow the model to rank highly in the

subset of appropriate O-LM models. As a result, it was important to check whether the maximum

conductance densities of the highly-ranked models resulted in very different fits of the passive

membrane properties or not. If so, the simulation outputs of the entire ensemble of models

should probably be recreated using the newly fit passive properties.

To ascertain this, two highly-ranked models of each model morphology were taken from

the database subset of appropriate O-LM models. In particular, the most highly-ranked model of

each morphology with somatodendritic h-currents were found and used to re-fit the passive

properties. This is because h-current in both somatic and dendritic compartments cover a much

greater surface of the cell’s membrane and can therefore affect more strongly affect the model’s

membrane response to the passive properties experimental protocol. For more on this, see the

discussion of version 1 versus version 2 of the database in Section 3.1.2.3, “Active conductances

and density ranges.” The two models obtained were rank 1 from the cell 1-specific subset of the

36

aggregate distance ranking and rank 3 from the cell 2-specific subset of the aggregate distance

ranking. The parameters for these two models are shown in Table 3.4.

The same protocol described in Section 3.1.2.2, “Fitting of passive properties to

experimental data,” was used in re-fitting the passive properties using these two models. After

the fitting procedure was completed, the passive membrane properties of the two models were

compared to those obtained from the reference model. Table 3.5 compares the passive properties

of the reference model to the highly-ranked cell 1 and cell 2 models.

Conductances 𝒈 values for model morphology 1, rank 1 (pS/µm2)

𝒈 values for model morphology 2, rank 3 (pS/µm2)

gNad 117 230

gNas 220 107

gKDRf 215 506

gKDRs 2.3 2.3

gKA 2.5 32

gh (soma + dendrites) 0.02 0.02

gCaL 50 25

gCaT 5 2.5

gAHP 5.5 11

gM 0.375 0.75

Table 3.4. Model parameters for the two highest-ranked per-morphology models with

somatodendritic h-current.

37

Passive properties

Reference model

Re-fit values for model morphology 1, rank 1

Re-fit values for model morphology 2, rank 3

Ra 300 300 300

Cm 0.9 0.96857 0.9

Rm 39037.7 61117 40397

EL −73.84 −71.37 −68.637

gKL 9.9005-10 9.9137-10 9.9256-10

Table 3.5. Passive properties for both the reference model and the highly-ranked cell 1 and cell 2

models.

Note that the passive properties do not seem to vary appreciably. To verify that the

differences in passive properties did not significantly affect the model behaviour, the voltage

traces of the two highly-ranked models before and after fitting of the passive properties were

compared (Fig. 3-3).

Figure 3-3. Voltage traces of highly-ranked models corresponding to original and re-fit passive

properties. The blue traces show the model response to +90 pA current injection and with

original passive properties, whereas the red traces show the model response to +90 pA current

injection and with the re-fit passive properties, for the models with morphology 1, rank 1 (A) and

morphology 2, rank 3 (B).

0 0.5 1 1.5 2

−100

−80

−60

−40

−20

0

20

40

60Morphology 1, rank 1

Mem

branepotential(m

V)

Time (s)

Original (+90 pA)Re−fit (+90 pA)

0 0.5 1 1.5 2

−100

−80

−60

−40

−20

0

20

40

60Morphology 2, rank 3

Mem

branepotential(m

V)

Time (s)

Original (+90 pA)Re−fit (+90 pA)A B

38

As can be seen, the voltage responses are very similar regardless of whether the original

or re-fit passive properties were used. Therefore, it was determined that the passive properties

obtained by fitting the reference model against the experimental data were adequate for the

ensemble of models subsequently obtained, and that it was not necessary to re-evaluate the

simulations using the newly fit passive properties. In principle, however, it may be preferable to

use the re-fit passive properties for future work.

3.1.5 Finding conductance density balances (Step 4)

The final step in the methodology developed here pertained to the investigation of the subset of

acceptable O-LM models in order to find any patterns in their conductance densities that would

reveal putative compensatory balances (Fig. 3-1). First, a technique for the visualization of high-

dimensional spaces such as the space of conductances in the model database was applied to

determine which conductances were most crucially contributing to the acceptable models’

goodness-of-fit to the experimental dataset (Section 3.1.5.1). Then, correlation images were

created that showed the number of models in the highly-ranked subset as a function of any two

of the high-order conductances (Section 3.1.5.2). These were then categorized into pairwise

conductances that either showed no correlation, correlation with local preference, or correlation

with co-regulatory relationship (Section 3.1.5.3). The third correlatory relationship, that of co-

regulatory balance, was found in a limited subset of all pairwise high-order conductance

correlation images, thus resulting in the striking finding that only a small number of

compensatory balances were critical for the production of idiosyncratic O-LM cell

electrophysiological output. Details for determining conductance density balances are described

in the following subsections.

3.1.5.1 Dimensional stacking

The high dimensionality of the models, each consisting of ten active conductances, one

morphology parameter, and one h-current distribution parameter, results in a 12-dimensional

model whose output changes as a function of parameters is difficult to visualize. One technique

developed to visualize a high-dimensional space of parameters is named clutter-based dimension

reordering (CBDR) and was applied to neuronal modelling by Taylor and Marder in 2006 [74].

CBDR reduces a multi-dimensional data space, in this case one of neuronal model parameters,

into two dimensions that can be readily visualized with full accuracy, i.e., with no averaging or

39

approximations. The basic technique involves building a nested montage of two-dimensional

images where the x- and y-axes correspond to the values of two of the model parameters such as

maximum conductance density values, and the data points correspond to some model output that

is of interest. In this present work, this corresponds to the model’s distance value. This image is

then embedded into a higher-order two-dimensional image where two other parameters are

varied. For each combination of values of this second pair, the entire first image is then re-

computed. This continues until all model outputs (distance values) are visualized in a two-

dimensional image. An optimal ordering of conductances in the construction of the stack can be

achieved by minimizing the “edginess” of the image, that is, maximizing the amount of

clustering of models outputs (in this case, of distance metric values). This technique is useful for

determining which parameters are more important for determining model outputs, as shown in

the Results, Section 3.2.2.1. The high-order conductances are used for generating conductance

density correlation images.

3.1.5.2 Conductance density correlation images

One technique for determining which conductances in the O-LM models affect each other to give

rise to the model outputs is to generate conductance density correlation images. These are three-

dimensional histograms with the x- and y-axes corresponding to the maximum conductance

density values of two high-order conductances (as determined by CBDR dimensional stacking

analysis, cf. Section 3.1.5.1), with the z-axis corresponding to the number of models in the

highly-ranked subset of appropriate O-LM models that take on any particular combination of the

two maximum conductance densities in the image. This was calculated in MATLAB using both

subsets of highly-ranked O-LM models determined by the general or objective criterion as well

as the restricted criterion described in Sections 3.1.4.1 and 3.1.4.2. A script,

OLM_cond_correl.m, was written to take a given subset of O-LM models and generate an (m,

n) two-dimensional matrix where m is the index into the vector of possible maximum

conductance density values for one conductance, and n is the index into the vector of possible

maximum conductance density values for the other conductance. Then, the (m, n)-th entry of the

matrix corresponds to the number of models in the given subset that take on the mth and nth

maximum conductance density values for the two conductances. The resulting three-dimensional

histogram is then visualized using the bar3 function in MATLAB.

40

3.1.5.3 Determining conductance density correlations and co-regulations

Conductance density correlations and co-regulations were determined by examining the

conductance density correlation histograms (cf. Section 3.1.5.2) for each pairwise combination

of high-order conductances. Classification into three categories was then performed by visual

inspection. The three categories corresponded to three types of relationships observed in the

conductance density correlation images: (i) no correlation, (ii) correlation with local preference,

or (iii) correlation with co-regulation. With no correlation, the histograms show a “flat”

landscape, meaning that any given conductance density combination is equally likely to be

present in the highly-ranked model database subset. For the case of correlation with local

preference, there is a “peak” in the histogram such that one particular combination of

conductance densities has the most models in the highly-ranked database subset. Finally, with

the co-regulatory case, there is a characteristic “ridge” in the histogram corresponding to a range

of conductance density values that accompany highly-ranked models in the database subset. The

ridge follows the pattern where changing density values of one conductance are matched with

changing density values of the other conductance in order for the models to remain in the highly-

ranked subset.

3.2 Application of the methodology to O-LM cells

3.2.1 Outline of results

The methodology developed in Section 3.1 was applied to previously developed, multi-

compartment hippocampal oriens-lacunosum/moleculare (O-LM) cell models. The four steps in

the methodology as shown in Fig. 3-1 took on the specific details illustrated in Fig. 3-4, below.

These include the particular incoming data to each step, as well as the data produced as a result

of that step to feed into the next step, in a clockwise fashion (with possibility of retaking steps as

indicated by the bidirectional arrows).

41

Figure 3-4. The methodology for ensemble modelling applied to O-LM multi-compartment

models. The methodology developed in this work as applied to the analysis of conductance

densities and distributions of hippocampal O-LM cell models using ensemble modelling

(compare to Fig. 3-1).

The results of the methodology applied to the O-LM models developed in this work are

described in the following sections.

3.2.2 Models successfully ranked according to appropriate matching against physiological O-LM cells

3.2.2.1 Features of highly- versus poorly-ranked models

The ensemble of O-LM models whose initial Vm could be held at –74 mV using variable holding

current and whose outputs were therefore simulated were ranked against the experimental data

using electrophysiological characteristics as described in Section 2, “Methods and Materials”. As

described in Section 3.1.3.3, “Ranking models against experimental data”, a database in

PANDORA consisting of models of both morphologies was constructed with models sorted in

42

descending order according to the distance, or error, against the entire experimental dataset. The

highly-ranked models were those near the top, or beginning of the matrix representing the

database (with rows corresponding to individual models), and had smaller rank and distance, or

error, values. The poorly-ranked models were those whose rows were near the bottom, or end of

the matrix, and had larger rank and distance, or error, values. See Fig. 3-5 for a plot of the

distance values against the ranking of the models in the entire database. By comparing highly-

ranked models against poorly-ranked models, it is apparent from the voltage traces that more

highly-ranked models depict better representations than poorly-ranked models. See insets in Fig.

3-5 for comparison of a highly-ranked model’s ±90 pA output against a poorly-ranked model. It

is readily apparent that the highly-ranked model’s output exhibits features that the poorly-ranked

model lacks: the two immediately apparent ones is a pronounced hyperpolarization-induced sag

in the –90 pA current clamp trace, as well as appropriate firing characteristics in the +90 pA

current clamp trace. The highly-ranked models thus seem to capture important intrinsic

properties of O-LM cells that allow the models to produce stereotypical O-LM output, which the

poorly-ranked models do not. This property of the highly-ranked models stems from the use of

the aggregate distance metric, calculated from the collection of electrophysiological measures

chosen for both ±90 pA traces. The distance is therefore a reasonable quantitative metric that can

be used to extract appropriate O-LM representations from the model database.

43

Figure 3-5. Ranking of models according to distance from experimental dataset. Every point on

the x-axis corresponds to a model, whether of morphology 1 or 2. The distances achieved by the

most poorly-ranked models reaches a maximum value of 62.23 (truncated, not shown on y-axis).

Characteristic traces of both highly-ranked and poorly-ranked models shown (compare with

electrophysiological trace in Fig. 2-1), with dashed lines indicating location in ranking.

3.2.2.2 Discrete and gradual transitions in model rankings

The order of the models sorted according to their distance from the experimental dataset alone

already reveals important characteristics of the set of model parameters and their relation to the

production of appropriate O-LM cell output. In particular, the goodness-of-fit of models to the

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experimental dataset changes in different ways according to the ranking of the models, with two

general types of transitions. One is a gradual transition of models becoming less and less

acceptable representations of O-LM cells as their distance metric or error, and therefore rank,

increases. This can be seen in the smooth increase in distance metric across the middle range of

the ranked database. Two models that are close to each other in rank and simultaneously exhibit

gradual loss in goodness-of-fit to the O-LM cell dataset are shown in Fig. 3-6. Note that the two

models differ in their ranking value so that one is ranked more highly than the other, but

nevertheless exhibit similar electrophysiological characteristics. The second type of transition

seen in the ranked model database is discrete. Here, models rapidly lose important features of O-

LM cell electrophysiological characteristics, and the sharp loss in goodness-of-fit to the

experimental dataset is reflected by the sudden jump, or increase, in distance metric. An example

of two models that sit on opposite sides of such a transition point is seen in Fig. 3-7. Here, even

though the difference in rank between the two models is the comparable to those from Fig. 3-6,

the models nevertheless exhibit drastically different electrophysiological characteristics. In

particular, the loss of action potentials in the more poorly-ranked model is conspicuous. These

two types of transitions demonstrate that interesting properties of the underlying model database

can be investigated just by examining the order of ranked models in the database.

Figure 3-6. Gradual transition in models’ goodness-of-fit to experimental O-LM cells. Both

models are extracted from the middle range, around ranks 10,000 – 60,000, delineated by the

smooth increase in distance metric. Note the similarity in the electrophysiological characteristics

of their ±90 pA current clamp traces.

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Figure 3-7. Discrete transitions in models’ goodness-of-fit to experimental O-LM cells. The

models here are extracted from the higher range of ranks in the database

(greater than rank 100,000), corresponding to poorly-ranked models. Note the conspicuous loss

of important O-LM electrophysiological characteristics, in particular the rapid drop in spike

height, as well as failed spikes in the +90pA current clamp trace for the more poorly-ranked

model (right).

3.2.2.3 A principled criterion for extracting acceptable models can be found

It is clear from the ranking of models according to their distance from the experimental O-LM

dataset that some models in the database are better representations of O-LM cells than others. In

particular, models that are more highly-ranked tend to better capture important properties of O-

LM cell output than more poorly-ranked models. The distance metric can therefore provide for a

basis to separate acceptable O-LM models from those that are not inappropriate and should not

be used for further analysis of putative O-LM cell intrinsic properties. The difficulty lies in the

fact that although the distance metric does allow for a separation of good models from poor ones,

it is not clear from the ranking alone which models should be included in the subset of

appropriate O-LM cell representations and which should not (Fig. 3-4, “Determining database of

acceptable models”). This is especially problematic when considering that many of the models in

the ranked database exhibit gradual transitions in their goodness-of-fit to the experimental O-LM

dataset, as demonstrated in the previous section and Fig. 3-6. A criterion for separating the

models in the ranked database, thereby extracting a subset of appropriate O-LM models, is

required. The criterion was applied separately to two subsets of the ranked database, one

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corresponding to models of morphology 1 and the other to models of morphology 2. This was

done to allow for any potential differences in the location of the cutoff when extracting an

acceptable subset of O-LM models.

Figure 3-8. The general criterion for determining a cutoff of a subset of appropriate O-LM

models. The plots show the rate of change of the distance metric as a function of model rank in

the subset of ranked models according to (A) models of morphology 1 and (B) models of

morphology 2. The vertical lines show the proposed cutoff point for acceptable models.

The first proposed criterion a “general” criterion that is intended to be “objective” – that

is, not dependent on any particular electrophysiological measure alone. The rate of change, or

first-order derivative, of the distance metric was used for this purpose. In particular, a rapid rise

in the derivative could indicate at which ranking models become significantly worse and

therefore would be a putative cutoff point for including models in the subset of appropriate O-

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LM representations. When an approximation to the first-order derivative was calculated for the

entire ranked database, it was found that the rate of change of model errors, or distance values,

rose sharply at the first few thousand ranks of the database and then continued to rise at a very

slow rate for a large portion of the ranking. Then, at approximately ranks 60,000 and 90,000 of

the subsets of ranked models corresponding to morphologies 1 and 2, respectively, there was a

rapid increase in the rate of change of the distance as determined by eye (Fig. 3-8). The subset of

models from the ranked database corresponding to the first 60,000 models of morphology 1 and

the first 90,000 models of morphology 2 is therefore the putative subset of appropriate O-LM

models determined using this general criterion.

The second proposed criterion was developed to provide a foil or counterpoint to the

general criterion by considering a limited set of representative electrophysiogical measures for

the model traces. It was hypothesized that a reasonable cutoff point could be found for

determining acceptable models by comparing the values of these specific electrophysiological

measures in the ranked model database against the experimental O-LM dataset. For the +90 pA

traces, the firing frequency during the current injection period was selected as the representative

characteristic. For the –90pA traces, the time constant of a single exponential fit to the

hyperpolarization-induced sag was selected as the representative characteristic. The reason these

two characteristics were chosen, and not some other characteristics, was that these two reflect

important dynamics of the O-LM model operation that takes into account multiple voltage-gated

conductances and other properties. In particular, several conductances acting in concert are

important for determining the spike threshold, action potential width, etc., which are correlated

with, and reflected in, the firing frequency of the models. Similarly, several conductances,

especially the h-current, that are active at subthreshold Vm levels participate in the generation and

modulation of the hyperpolarization-induced sag. When considering the firing frequency of the

models in the +90 pA current injection protocol, it was found that the highly-ranked models

exhibited relatively reasonable firing frequencies with respect to the experimental data, with a

mean of 14Hz for morphology 1 models, 20Hz for morphology 2 models, and 22Hz for the mean

of experimental O-LM cells. However, a model with very low firing frequency was observed

fairly early on in the ranking. When visualizing the model’s voltage trace, it was apparent that

the particular combination of conductances for the model did not allow for sustained firing of

48

action potentials for the duration of the current injection period. This type of model was therefore

termed a “failure-to-fire” model (Fig. 3-9).

Figure 3-9. The first failure-to-fire model in the ranked database subsets of morphology 1 and

morphology 2 models. (A) The first model (left, arrow) at rank 13,613 of the subset of models of

morphology 1 to demonstrate failure-to-fire characteristics (right). (B) Similarly, the first model

(left, arrow) at rank 19,245 of the subset of models of morphology 2 to demonstrate failure-to-

fire characteristics (right).

The first failure-to-fire model for the morphology 1 subset of models occurred at rank

13,613 and the first failure-to-fire model for the morphology 2 subset was at rank 19,245. The

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location of these models in their respective per-morphology subsets of the entire ranked database

thus served as putative cutoff points for determining appropriate O-LM models. It is immediately

apparent that this constitutes a much more restricted subset of O-LM models. This is especially

clear considering that there were more poorly-ranked models relative to the first failure-to-fire

models that nevertheless seemed to otherwise exhibit appropriate O-LM electrophysiological

characteristics, including the firing frequency, within the more general subset of highly-ranked

models (Fig. 3-10).

Figure 3-10. First failure-to-fire models compared to more poorly-ranked, but otherwise

appropriate O-LM models. (A) The outputs of the first failure-to-fire model of morphology 1

(left) compared to a model very close, but poorer, in ranking (right). (B) The outputs of the first

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failure-to-fire model of morphology 2 (left) compared to a model very close, but poorer, in

ranking (right).

It is important to remember that the firing frequency, though an important

electrophysiological measure, is only one of many used for determining the rank of a particular

model. Therefore, even though the firing frequency measure might provide a more constrained

cutoff for determining a subset of appropriate models, it may at the same time prove to be too

restrictive and leave out many otherwise appropriate O-LM models.

The examination of the time constant of the hyperpolarization-induced sag for the models

compared to the experimental data revealed that a larger subset of highly-ranked models of both

morphologies proved to constitute appropriate O-LM models than the subset based on the firing

frequency measure. In particular, there did not seem to be a point in the ranking at which a

qualitatively poorer model arose, like there was with the failure-to-fire models when considering

the firing frequency measure. Rather, there was a smoothly decreasing time constant as a

function of decreasing model rank, with at least the first 50,000 ranked models in both per-

morphology database subsets exhibiting reasonable time constant measures (Fig. 3-11).

Figure 3-11. The time constant values of the hyperpolarization-induced sag for highly-ranked

models. (A) The sag time constant for the subset of models with morphology 1. (B) The sag time

constant for the subset of models with morphology 2. In both cases, the vertical line indicates the

cutoff when more poorly-ranked models begin to exhibit inappropriate time constant values.

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Importantly, using the time constant measure as a cutoff criterion led to more highly-

ranked models of both morphologies being part of a putative subset of appropriate O-LM models

than when using the firing frequency measure. Because of this, the first failure-to-fire models

uncovered when considering the firing frequency measure was used as the more restricted cutoff

point for considering appropriate O-LM models in the ranked database. This restricted criterion

represents a more conservative estimate of which subset of highly-ranked models should be

considered appropriate O-LM representations. On the other hand, the approximate derivative of

the distance metric represents a more general criterion that is more inclusive of O-LM models in

the ranked database as constituting appropriate O-LM models. These two cutoff points were both

considered when determining conductance density correlations. If the resulting correlations and

co-regulations found in the restricted subset of O-LM models did not differ appreciably from

those found in the more general subset, then the latter would be considered the set of appropriate

O-LM models. This analysis is described in Section 3.2.3, below.

3.2.3 Discovering critical balances in conductance densities

3.2.3.1 Dimensional stacking can be used to determine “high-order” conductances

In order to determine whether the active conductances in the subset of highly-ranked O-LM

models exhibit any correlations or homeostatic balances, a good first step is to visualize how the

model outputs change as a function of the changes in their parameters. This was done using

dimensional stacking, a technique for visualizing a high-dimensional space of models in two-

dimensions. Fig. 3-12 shows a dimensional stack image for the subset of ranked models

extracted using the general cutoff criterion.

52

Figure 3-12. Dimensional stack image of the subset of ranked O-LM models extracted using the

general cutoff criterion. Any given point in the image represents one combination of model

parameters, and therefore specifies a single model in the database. The rank of the model is

represented using a graded colour scheme where “hotter” colours, those closer to the red end of

the spectrum, correspond to more highly-ranked models whereas the “cooler” colours, those

closer to the blue end of the spectrum, correspond to more poorly-ranked models. The bars on

the x- and y-axes show the resolution of changes for the particular model parameter specified by

the label on each bar. The lengths of the bars show the extent of a single step of change in

maximum conductance density value of the given conductance, with the longest bars on the x-

and y-axes corresponding to the first-order parameters, the second-longest to the second-order

parameters, and so on.

KAIh distribution

IhMNasCaL

Nad

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53

The ordering of the parameters in a given “stack” is important in determining sensitivity

of model outputs to parameter changes. An optimal stack can be found where the models are

organized in the image such that highly-ranked models cluster together more tightly than lower-

ranked models. In this case, the high-order parameters (conductance densities and distributions)

in the stack reflect those conductances whose changes in value are associated with changes in

model ranking, or goodness-of-fit to experimental data. On the other hand, lower-order

conductances are those for which changes can be made and yet the ranking of the models are not

appreciably affected. Therefore, high-order parameters, or conductance densities and

distributions, are those to which the model distances are most sensitive and therefore are the

likeliest to demonstrate compensatory balances with each other. To determine this, the high-

order parameters were used to construct the conductance correlation images. However, although

the first-order parameters in the dimensional stack images are readily considered the highest-

order and the fifth-order parameters are readily considered the lowest-order, it is not entirely

clear whether the third- and fourth-order parameters should be considered high-order or low-

order. Therefore, for the conductance correlation images, third- and fourth-order parameters were

also used in order to check whether they showed correlations with other conductances. See Table

3.6 for the list of high-order conductances as determined by the per-morphology ranked database

subsets. It can be seen that most high-order conductances, especially of the first- and second-

orders, are largely shared between the four model database subsets. This is an indication that the

conductances that are important for determining O-LM model output do not critically depend on

morphology or distribution of h-current along soma or dendrites. Furthermore, the high-order

conductances do not appreciably change according to the cutoff criterion used for determining

the subset of appropriate O-LM models. This is the first indication that the general criterion,

corresponding to the more inclusive subset of highly-ranked O-LM models, is adequate for

delineating a set of appropriate O-LM models that can then be used in analyzing conductance

density balances.

54

Cell 1, gh soma only

Cell 1, gh soma and dendrites

Cell 2, gh soma only

Cell 2, gh soma and dendrites

Orders 1-2

Orders 3-4

Orders 1-2

Orders 3-4

Orders 1-2

Orders 3-4

Orders 1-2

Orders 3-4

Conductances in general

model subset

gNad gKDRf gKDRs

gA

gh gNas gM

gAHP

gNad gh

gKDRs gA

gKDRf gNas gM

gAHP

gNad gKDRf gKDRs

gA

gh gNas gM

gAHP

gNad gh

gKDRs gA

gKDRf gNas gM

gAHP

Conductances in restricted model subset

gNad gKDRf gKDRs

gA

gh gNas gM

gAHP

gNad gh

gKDRf gKDRs

gA gNas gM

gAHP

gNad gKDRf gKDRs

gA

gh gNas gM

gAHP

gNad gh

gKDRf gKDRs

gA gNas gM

gAHP

Table 3.6. High-order parameters as determined by dimensional stacking analysis. The ranked

database is subdivided into eight subsets according to lines of morphology, h-current

distribution, and cutoff criterion. The ordering of conductances is mostly preserved across all

cases.

3.2.3.2 Observed tradeoff between Ih conductance densities and distributions

Once the high-order conductances were determined, conductance correlation histograms were

constructed for each pairwise combination of high-order conductances (Fig. 3-4, “Finding

conductance density balances”). One of the first salient results of this analysis was the relation of

the h-current distribution parameter to the rest of the parameters in the model. It was found that

the distribution of highly-ranked models did not change appreciably depending on whether the

model subset corresponding to gh in the soma only was considered or the model subset

corresponding to gh in the soma and dendrites. In other words, most parameters’ contributions to

O-LM model outputs did not depend on h-current distributions along the somatodendritic extent

of the model. The exception was the h-current conductance itself, gh. In this case, highly-ranked

models with somatic gh only exhibited much higher gh maximum conductance density values

than highly-ranked models with somatodendritic gh (Fig. 3-13).

55

Figure 3-13. Parameter correlation histograms of gh and gh distribution parameters in the highly-

ranked model subsets. (A) Histograms of correlations between gh and gh distribution in the

general (left) and restricted (right) subsets of highly-ranked models of both morphologies. (B)

Histograms of correlations between gh distribution and a high-order parameter, gNad, in the

general (left) and restricted (right) subsets of highly-ranked models of both morphologies. Note

the change in gh maximum conductance density values depending on gh distribution in (A) as

well as the insensitivity of gNad parameter choices in the highly-ranked models to the changes in

gh distribution in (B).

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This relationship between gh maximum conductance densities and gh distributions is

straightforward to understand by considering the total amount of conductance across the model

cell’s membrane depending on gh distribution. In the case of somatic gh only, there are fewer

compartments in the model that have gh in their membrane, and therefore a smaller surface area,

so a greater amount of gh conductance is required in order for the model to maintain appropriate

O-LM cell characteristics (mostly localized to the hyperpolarization-induced sag measure). On

the other hand, with gh distributed in both the soma and dendrites, if the maximum conductance

densities of gh took on the higher values, there would be an overwhelming amount of inward

current originating from all somatodendritic compartments. In fact, this is the reason why two

different model databases had to be constructed. Originally, the parameter range for gh maximum

conductance densities in “version 1” of the database were determined based on experimental data

taken from pyramidal cells (see Table 3.2 in Section 3.1.2.3, “Active conductances and density

ranges”, as well as Appendix A). The resulting model database contained models that, for the

most part, did not exhibit appropriate O-LM electrophysiological characteristics. Only a small

subset of models demonstrated appropriate O-LM characteristics, and of those, most of them

took on the lowest value of 𝑔!, or gh maximum conductance density, of 0.5 pS/µm2. Therefore,

“version 2” of the database was subsequently created, with a lower range of 𝑔!, as can be seen in

Table 3.2. In this case, 0.5 pS/µm2 was set to be the highest possible value for 𝑔!. The reason

that the majority of models in version 1 of the database did not resemble O-LM cells was that,

for the most part, there was too much gh flowing into the cells, and so the balance of

conductances was thrown off and cells either fired action potentials at extremely high rates

uncharacteristic of O-LM cells, or exhibited “jittery” Vm characteristics without clear spiking

activity (Fig. 3-14). The conductance density correlation images of 𝑔! and gh distribution, as

well as the differences in model characteristics between versions 1 and 2 of the database,

collectively confirm the relationship of lower overall 𝑔!with somatodendritic distribution of gh,

and higher overall 𝑔! with a somatic gh distribution.

57

Figure 3-14. Characteristic model from version 1 of the database. Note the very high firing

frequencies that are uncharacteristic of O-LM cells, even with –90pA hyperpolarizing current

injection.

3.2.3.3 Conductance density correlations can be categorized

A central hypothesis in this work is that characterizing the relationships between conductances in

the an ensemble of models, corresponding to appropriate O-LM cell representations, will reveal

any compensatory balances that are required for determining idiosyncratic O-LM cell output. To

this end, conductance density correlation histograms were constructed in order to find any

correlations among the conductances deemed to be high-order by the dimensional stacking

analysis. The set of high-order conductances demonstrated in the database (cf. Table 3.6) were

the fast and slow delayed-rectifier potassium conductances (gKDRf and gKDRs, respectively), the

hyperpolarization-activated cation conductance (gh), the A-type potassium conductance (gA), the

muscarinic conductance (gM), and the dendritic sodium conductance (gNad). These were the first-

and second-order conductances revealed by dimensional stacking analysis. Two conductances of

the third- and fourth-order were included in the conductance correlation – the somatic sodium

conductance (gNas) and calcium-dependent potassium conductance (gAHP) – just to provide a

check that they did not exhibit any correlations with other conductances when considering the

subset of appropriate O-LM models. This would verify that the third- and fourth-order

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conductances can be considered low-order and thus not critical for obtaining characteristic O-LM

cell output.

When analyzing the histograms consisting of number of models from the subset of

appropriate models versus the conductance densities of two conductances, three types of

relationships were observed, similar to those reported in model database analysis of

computational models of neurons in the crustacean stomatogastric ganglion network [75], that of

(i) no correlation, (ii) correlation with local preference, or (iii) correlation with co-regulation.

The first was that there was no observed correlation. This was most apparent when the histogram

appeared “flat”, that is, when there was little difference in the number of highly-ranked models

that took on any particular combination of conductance density values (Fig. 3-15). The case of

no correlation was equally observed in the general and restricted subset of highly-ranked models.

Figure 3-15. No correlations exhibited with low-order conductances. The conductance density

correlation histograms for gNas and gAHP showed that there was no correlation between these

conductances, both in the general subset of highly-ranked models (A) as well as in the restricted

subset (B).

The second observed relationship was that there was a correlation but with a preference

for a local combination of conductance density values (Fig. 3-16). In this case, although the

conductances are correlated, they do not show any compensatory balance; there is simply a

particular combination of values that is most expressed in the highly-ranked model subsets.

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Figure 3-16. Correlation with local preference. The conductance density correlation histograms

for gKDRf and gh (both distributions of gh included) showed that there was a (weak) correlation

between these conductances, with the nature of the correlation being that a particular

combination of gKDRf and gh maximum conductance density values was preferred, as seen by the

“peak” in the middle. The histograms correspond to (A) the general subset of highly-ranked

models, and (B) the restricted subset.

A third correlatory relationship was observed in a limited number of cases, where

increase in one conductance was accompanied by an increase in the other conductance in order

for those combination of conductance density values to correspond to the appropriate subset of

highly-ranked models. This final type of observed correlation corresponds to a compensatory

balance, and therefore a co-regulation of the two conductances. The co-regulations found are

described in the next section.

3.2.3.4 Co-regulations between conductance densities are few but critical

Three major co-regulations between high-order conductances were observed. These applied to

both the general and restricted subsets of highly-ranked models, as well as across models of both

morphologies. The first consisted of a co-regulation between gNad and gKDRf conductances

(Fig. 3-17). This co-regulation was found regardless of gh distribution, that is, across all of the

models in the general subset.

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Figure 3-17. Co-regulation between gNad and gKDRf conductances. The conductance density

correlation histograms between gNad and gKDRf conductances show evidence for co-regulation.

(A) The general subset of highly-ranked models. (B) The restricted subset.

The other two co-regulations were observed, but only in the subset of models with gh

conductance distributed in the soma and dendrites as opposed to in the soma only. In this case,

models with somatodendritic gh exhibited additional co-regulations of gh and gKDRs (Fig. 3-18) as

well as gh and gA (Fig. 3-19). In all three co-regulatory cases, note that the majority of highly-

ranked models in the histograms reside on the “ridge” located in the direction of positive

correlation between each pairwise comparison of conductances. In other words, when one

conductance density is increased, the other must be increased as well in order for a model to be

highly ranked and thus lay on the “ridge”. Furthermore, note that only such positive correlations

were observed in the co-regulatory relationships. Note that in the presentation of the data, the

direction of increasing conductance densities are reversed in Figs. 3-18 and 3-19 as compared to

Fig. 3-17, so as to clearly show the “ridge” of positive correlation.

An interesting result is that of all possible pairwise combination of conductances, of

which there are !!! = 55, treating gh for somatic and somatodendritic distributions separately,

only three combinations, consisting of five different active conductances, exhibited co-regulatory

behaviour. It is therefore tempting to speculate that these co-regulations may form a “signature”

of ionic conductance relationships that is characteristic of O-LM cells.

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Figure 3-18. Co-regulation between gh and gKDRs conductances. (A) The general subset of

highly-ranked models, with (left) corresponding to the additional subset of models with gh in

soma only and (right) gh in soma and dendrites. (B) The restricted and subset of highly-ranked

models corresponding to the additional subset of models with gh in soma only (left) and gh in

soma and dendrites (right). Note that the co-regulation is observed only in the case of gh in the

soma and dendrites (A and B, right).

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Figure 3-19. Co-regulation between gh and gKA conductances. (A) The general subset of highly-

ranked models, with (left) corresponding to the additional subset of models with gh in soma only

and (right) gh in soma and dendrites. (B) The restricted and subset of highly-ranked models

corresponding to the additional subset of models with gh in soma only (left) and gh in soma and

dendrites (right). Note that the co-regulation is observed only in the case of gh in the soma and

dendrites (A and B, right).

16972

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Discussion 4

4.1 Summary

In this work, a robust methodology for the examination of conductance densities and

distributions in multi-compartment O-LM cell models in order to determine co-regulatory

balances in conductances was developed (Figs. 3-1, 3-4). A previously developed, hand-tuned

multi-compartment O-LM cell model was adapted to provide the basis for an ensemble of

models with individually varying ion channel maximum conductance densities and distributions.

A virtual experimental protocol, analogous to that applied to a set of physiological O-LM cells in

mice, was applied to the models. The resulting voltage traces were analyzed with respect to those

from the experimental data, and a ranking of models’ goodness-of-fit to the experimental data

was performed using a distance metric that was quantitatively derived from a collection of

electrophysiological measures. A subset of the models that were the best match to the

experimental dataset was extracted using a principled criterion that depended on the rate of

change of the models’ distance metric. The subset was analyzed to find any compensatory

balances between the conductances of the model. Dimensional stacking was used to determine

which conductances most contributed to whether models were highly-ranked, termed high-order

conductances. Creating histograms of pairwise conductance densities for the high-order

conductances revealed three co-regulatory balances. One consisted of a co-regulation between

fast delayed-rectifier potassium and dendritic sodium conductances, which applied to the entire

subset of models representing appropriate O-LM cells. The other two co-regulations were only

found in the subset of models corresponding to those with dendritic h-current. In this case, both

slow-delayed rectifier potassium as well as A-type potassium conductances were also co-

regulated with h-current.

4.2 Predictions

The main hypothesis described in Section 1.5.1, “Hypothesis”, that building and analyzing an

ensemble, or database of representative O-LM cell models would lead to uncovering of

conductance density balances that allow the models to produce characteristic O-LM cell output,

was confirmed. Although several conductances in the highly-ranked subset of O-LM models

64

exhibited a preference for a localized, or small range of maximum conductance density values,

there were nevertheless three co-regulations that were discovered, involving five different

conductances: gNad, gKDRf, gh, gKDRs, and gA. The primary prediction from Section 1.5.1, that co-

regulatory balances for particular combinations of conductances would be found that would be

critical for the models’ conformance to experimental data, thus holds. This work indicates that

we might expect to see co-regulatory balances of particular conductances in physiological O-LM

cells. This would require demonstrating that the conductances that exhibit co-regulations via the

correlation analysis done here actually balance against each other mechanistically. It is

compelling to assume that this is the case, given that the conductances shown to have co-

regulatory relationships in this work in every case mediate opposing current flows and are of

similar time scales of activation. However, this would have to be confirmed experimentally.

The question of whether Ih is present in O-LM cell dendrites was not conclusively

addressed in this work. In the subset of acceptable O-LM models as determined by the general

cutoff criterion, about half of the models contained Ih in somatic compartments only (77,806 out

of 154,000 total), and about half contained Ih in somatodendritic compartments (76,197 out of

154,000 total). However, as noted in Section 3.2.3.4, the observed co-regulatory balances

included that of gh and gKDRs as well as gh and gKA in the case of somatodendritic gh only. We

propose that these observed co-regulatory balances can provide a basis for an indirect way of

experimentally testing for the presence of dendritic Ih. Specifically, by measuring Ih, IKDRs, and

IKA currents in O-LM cells and comparing the latter two outward potassium to the inward mixed

cation h-current. If a co-regulatory balance were indeed observed experimentally, our present

work would predict it is likely that the h-current is present in the dendrites.

We further hypothesize that the model database can be expanded to address particular

questions regarding O-LM cell intrinsic properties. For example, the range of values that we

assigned to the maximum conductance densities of the h-current was initially found to be too

high, as seen from the resulting inappropriate O-LM models in the first version of the database

(Section 3.2.3.2). As a result, the second version was designed to use a lower range of values,

which indirectly serves as a prediction for the range of h-current maximum conductance

densities that are expected to be found experimentally.

65

Because the current work has resulted in an ensemble of appropriate O-LM models, the

parameters of this subset can be expanded to include additional characteristics that we wish to

explore. For example, specific spatial compartments of muscarinic conductances in O-LM cells –

corresponding to sites of postsynaptic acetylcholine release determined experimentally – can be

added to the models in order to better understand the substantial afterdepolarization (ADP) that is

known to be partly mediated by muscarinic acetylcholine receptors in O-LM cells [61]. The

present ensemble of appropriate O-LM models can then be expanded to include a variety of

parameter values for muscarinic conductance localization, as well as the inclusion of other

currents implicated in the ADP formation (such as the ICAT current [61]), thus generating

hypothesis for the mechanisms underlying the muscarinic ADP in O-LM cells.

4.3 Co-regulations between conductance densities critically determine O-LM cell output

Several correlations were found between conductances that were of the second type outlined in

Section 3.2.3.3, “Conductance density correlations can be categorized”, namely of the type that

shows a preference for a local combination of maximum conductance density values. However,

these correlations can be seen to be reflective of a preference for a particular subset of maximum

conductance density values on an individual, per-conductance basis, not dependent on any

particular combination of conductances. This can be seen most readily by comparing the

conductance density correlation histograms demonstrating the local preference type with the

histogram for the number of models in the subset of appropriate O-LM models that take on

particular maximum conductance density values for the conductances separately (Fig. 4-1).

66

Figure 4-1. Correlation with local preference as an interaction between two independent

conductance density preferences. (A) The conductance density correlation histogram for gNad and

gh in the subset of highly-ranked appropriate O-LM models in the database. (B) The histograms

of the number of models in the highly-ranked subset that have particular maximum conductance

density values for gNad and gh, considered separately.

If viewed from the sides, the pairwise conductance density correlation histogram reflects

the per-conductance histograms in Fig. 4-1B. This means that the number of models in the

highly-ranked subset of appropriate O-LM models for a particular value of an individual

conductance are simply spread out evenly along the values of the other conductance. In this case,

the distribution of highly-ranked models across the entire range of maximum conductance

density values for one conductance does not appreciably change when different maximum

conductance density values of the other conductance are considered. For example, by fixing the

𝑔!"# value to the lowest allowable one in Fig. 4-1A, the resulting histogram of models

distributed across 𝑔!resembles the per-conductance histogram of 𝑔! alone. The same holds true

the other way around; fixing a particular value of 𝑔! and comparing the resulting histogram of

𝑔!"# with the per-conductance histogram of 𝑔!"# alone. This means that although the two

conductances seem to be correlated, they in fact operate independently of each other. The

converse is seen when comparing the conductance density correlation histograms for the third

type of correlation observed, that of co-regulation, with the per-conductance histograms for the

conductances exhibiting the putative co-regulation (Fig. 4-2).

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Figure 4-2. Co-regulatory balances as mutually-dependent distributions of conductance

densities. (A) The conductance correlation histogram for gKDRs and gh in the subset of highly-

ranked appropriate O-LM models in the database that exhibit somatodendritic gh conductances.

(B) The histograms of the number of models in the highly-ranked subset with somatodendritic gh

considered separately for gKDRs and gh.

Here, unlike the case of correlation with local preference (Fig. 4-1), it is clear that the

distribution of highly-ranked models along the range of allowed maximum conductance density

values for one conductance strongly depends on the value of the other conductance. When taking

a “slice” of the histogram from one conductance but keeping the maximum conductance density

of the other fixed (Fig. 4-2A), the resulting histogram does not resemble the per-conductance

histograms (Fig. 4-2B). For example, the per-conductance histogram for 𝑔!"#$ (Fig. 4-2B, top)

demonstrates that there are decreasing numbers of highly-ranked models that are assigned the

middle and higher 𝑔!"#$ values. However, when comparing this with the pairwise conductance

correlation histogram for 𝑔!"#$ and 𝑔! (Fig. 4-2A), it can be seen that the models that have the

highest value of 𝑔!"#$ preferentially express the higher values of 𝑔!, and that there are no

models with the highest 𝑔!"#$ value that take on the lowest 𝑔! value. This demonstrates that, in

order for models to be included in the highly-ranked subset of O-LM models, they need to

possess a particular combination of values for 𝑔!"#$ and 𝑔! such that increases in the former are

accompanied by increases in the latter. Otherwise, the models are not included in the highly-

ranked subset, and are thus not appropriate O-LM cell representations. Therefore, the co-

regulations uncovered by database analysis shows that these critically determine behaviours that

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are required for O-LM cell electrophysiological output. Whether the co-regulations shown here

are reflective of actual compensatory balances between these ion channels physiologically

remains to be determined by experimental means, as mentioned in Section 4.1.2, “Predictions”.

However, the work here does indicate that it is likely that the conductances that exhibit co-

regulatory balances via correlation analysis share a physiological compensatory balance of ion

channel expression levels.

4.4 Limitations and future work

Although the ranking of models against the experimental electrophysiological dataset shows that

that highly-ranked models better conform to physiological cell behaviour than poorly-ranked

models (Section 3.2.2, “Models successfully ranked according to appropriate matching against

physiological O-LM cells”), visual inspection of the highly-ranked model current clamp traces

reveals that there are aspects of O-LM cell function that are not adequately captured by even the

highest-ranking models. As described in Section 1.4.2, there are three ways in which ionic

currents in the models might be inadequate or lacking: the distribution of the channels along the

cell’s morphological extent, the conductance density of the current, or the details of the kinetics

of the channels. Perhaps the most prominent feature of this kind that was observed in the highly-

ranked model subset is the hyperpolarization-induced sag that is known to occur as a result of the

h-current [34]. Even the highest-ranking models do not exhibit sags that are as pronounced as the

ones seen in the experimental dataset (Fig. 4-3).

69

Figure 4-3. Inadequacy of highly-ranked models to match the sag characteristic of experimental

O-LM cells. Voltage traces of experimental O-LM cell (red) and the highest-ranked model from

the database superimposed on the same voltage-time axis to show mismatch of sag amount and

time course of the models against the experimental cell.

The fact that the sag characteristics of the highly-ranked models do not adequately match

the experimental dataset indicates that it is not possible for the O-LM model itself to match the

sag characteristics of the physiological O-LM cells considered for this study. This implies that

perhaps the mathematical formulation of the h-current used in the model is itself not an accurate

reflection of h-current dynamics in physiological O-LM cells. This is possible given that the time

constant for the h-current used in the reference O-LM model was minimally constrained due to

lack of experimental data; as mentioned in Section 1.4.2.3, only two data points were used for

the fitting of the time constants for the h-current [47]. Alternatively, other conductances that are

active at subthreshold levels comparable to the activation range of the h-current might also play a

role to attenuate the sag seen in the highly-ranked models. This would indicate that a finer grid of

possible maximm conductance density values is required in order to find the regions in the model

parameter space where the combination of conductances allow for a more pronounced sag.

As mentioned in Section 4.2, both distributions of h-current used in this study – either

somatic gh only or somatodendritic gh – were equally represented in the general subset of

appropriate highly-ranked O-LM models. In particular, the number of highly-ranked models with

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somatic gh only was 77,806. The number of highly-ranked models with gh distributed in the soma

and dendrites was 76,194. This constitutes a nearly equal split of highly-ranked models between

these two distributions of gh. Although these results, from the perspective of the ensemble

modelling approach, indicate that it is possible for a detailed multi-compartment O-LM model to

exhibit appropriate O-LM electrophysiological characteristics either with dendritic gh or without

dendritic gh, the meaning of this result in terms of O-LM physiology is unclear at present. One

explanation might be that there is a bimodality in O-LM morphologies or even subtypes, where

one expresses Ih channels in somatic regions only, and the other expresses Ih channels in the

dendrites as well. Another is that an O-LM cell may be able to modulate levels of HCN

expression in dendrites depending on activity context, such that when h-current is needed in the

dendrites, the channels can be inserted into the membrane, and other conductances adjusted as

required to maintain the desired output. An alternative to these speculations regarding O-LM

morphology and physiology is to suppose that the distribution of gh along dendritic

compartments in the model was not appropriate with respect to actual O-LM cells. For

simplicity, the maximum conductance densities of gh for all of the dendritic compartments were

set to be equal; that is, a uniform distribution of gh in the case of dendritic h-current was

imposed. The rationale for this at the time was to provide a simple dichotomy of “presence”

versus “absence” of dendritic gh without considering more complicated dendritic spatial

distribution patterns. However, given the results in this work, it may be that this uniform

distribution is not physiologically plausible, and that rather a non-uniform distribution needs to

be considered. Indeed, previous cell-attached patch clamp studies of both hippocampal and

cortical pyramidal cells indicated a marked increase in h-current conductance densities in the

distal dendritic regions compared to the proximal ones [76-78]. Although these studies were

done in excitatory pyramidal cells and not in inhibitory interneurons, it is reasonable to suppose

that interneurons such as the O-LM cell might also exhibit such non-uniform dendritic h-current

distributions. This would be a potential avenue of investigation for future work.

Although these appear to be limiting factors of the model database results, when seen

from the point of view of model development and analysis, they indicate the very parts of the

models that need to be improved. Indeed, they reflect gaps in our understanding of O-LM cells

that need to be filled in with further experimental, in conjunction with modelling work. In this

light, these limitations are a strength of the ensemble modelling approach.

71

References

1. Andersen P, Morris R, Amaral D, Bliss T, O’Keefe J (2007) The Hippocampus Book.

New York: Oxford University Press, p. 37.

2. Golgi C, Bentivogli M, Swanson L (2001) On the fine structure of the pes hippocampi

major. Brain Res Bull 54: 461-483. Translated from Golgi C (1886).

3. Buzsáki G. (2006). Rhythms of the Brain. New York, NY: Oxford University Press, pp.

111-135.

4. Buzsáki G, Draguhn A (2004) Neuronal Oscillations in Cortical Networks. Science 304:

1926-1929.

5. Engel AK, Fries P, Singer W (2001) Dynamic predictions: oscillations and synchrony in

top-down processing. Nat Rev Neurosci 2: 704-716.

6. Lakatos P, Shah AS, Knuth KH, Ulbert I, Karmos G, et al. (2005). An Oscillatory

Hierarchy Controlling Neuronal Excitability and Stimulus Processing in the Auditory

Cortex. J Neurophysiol 94: 1904-1911.

7. Buzsáki G (2002) Theta Oscillations in the Hippocampus. Neuron 33: 325-340.

8. Hasselmo M, Bodelon C, Wyble B (2002) A proposed function of the hippocampal theta

rhythm: Separate phases of encoding and retrieval of prior learning. Neural Comput 14:

793-817.

9. Lisman JE, Idiart JA (1995) Storage of 7 ± 2 short term memories in oscillatory

subcycles. Science 267: 1512-1515.

10. Kahana MJ, Sekuler R, Caplan JB, Kirschen M, Madsen JR (1999). Human theta

oscillations exhibit task dependence during virtual maze navigation. Nature 399: 781-

784.

11. O’Keefe J, Recce ML (1993) Phase relationship between hippocampal place units and the

EEG theta rhythm. Hippocampus 3: 317-330.

12. Grastyan E, Lissak K, Madarasz I, Donhoffer H (1959) Hippocampal electrical activity

during the development of conditioned reflexes. Electroencephalogr Clin Neurophysiol

Suppl 11: 409-430.

13. Vanderwolf CH (1969) Hippocampal electrical activity and voluntary movement in the

rat. Electroencephalogr Clin Neurophysiol 26: 407-418.

72

14. Cantero JL, Atienza M, Stickgold R, Kahana MJ, Madsen JR, et al. (2003) Sleep-

dependent theta oscillations in the human hippocampus and neocortex. J Neurosci 23:

10897-10903.

15. Jefferys JGR (2010) Advances in understanding basic mechanisms of epilepsy and

seizures. Seizure 19: 638-646.

16. Scott L, Feng J, Kiss T, Needle E, Atchison K, et al. (2012) Age-dependent disruption in

hippocampal theta oscillation in amyloid- overproducing transgenic mice. Neurobiol

Aging 33: 1481.e13-23.

17. Babiloni C, Frisoni GB, Pievani M, Vecchio F, Lizio R, et al. (2009) Hippocampal

volume and cortical sources of EEG alpha rhythms in mild cognitive impairment and

Alzheimer disease. Neuroimage 44: 123-125.

18. Freund TF, Buzsáki G (1996) Interneurons of the hippocampus. Hippocampus 6: 347-

470.

19. Klausberger T, Somogyi P (2008) Neuronal diversity and temporal dynamics: The unity

of hippocampal circuit operation. Science 321: 53-57.

20. Klausberger T, Marton LF, O’Neill J, Huck JHJ, Dalezios Y, et al. (2005).

Complementary Roles of Cholecystokinin- and Parvalbumin- Expressing GABAergic

Neurons in Hippocampal Network Oscillations. J Neurosci 25: 9782-9793.

21. McBain CJ, Fisahn A (2001) Interneurons unbound. Nat Rev Neurosci 2: 11-23.

22. Somogyi P, Klausberger T (2005) Defined types of cortical interneurone structure space

and spike timing in the hippocampus. J Physiol 562: 9-26.

23. Ramón y Cajal S (1893) Estructura del asta de ammon y fascia dentata. Anal Soc Espan

Historia Natural 22: 53-114.

24. Sik A, Penttonen M, Ylinen A, Buzsáki G (1995) Hippocampal CA1 interneurons: an in

vivo intracellular labeling study. J Neurosci 15: 6651-6665.

25. Blasco-Ibáñez JM, Freund TF (1995) Synaptic input of horizontal interneurons in stratum

oriens of the hippocampal CA1 subfield: structural basis of feed-back activation. Eur J

Neurosci 7: 2170-2180.

26. Klausberger T (2009) GABAergic interneurons targeting dendrites of pyramidal cells in

the CA1 area of the hippocampus. Eur J Neurosci 30: 947-957.

73

27. Maccaferri G, Roberts JDB, Szucs P, Cottingham CA, Somogyi P (2000) Cell surface

domain specific postsynaptic currents evoked by identified GABAergic neurons in rat

hippocampus in vitro. J Physiol 524: 91-116.

28. Gillies MJ, Traub RD, LeBeau FEN, Davies CH, Gloveli T, et al. (2002) A model of

atropine-resistant theta oscillations in rat hippocampal area CA1. J Physiol 543: 779-793.

29. Rotstein HG, Pervouchine DD, Acker CD, Gillies MJ, White JA, et al. (2005) Slow and

Fast Inhibition and an H-Current Interact to Create a Theta Rhythm in a Model of CA1

Interneuron Network. J Neurophysiol 94: 1509-1518.

30. Perez Y, Morin F, Lacaille JC (2001) A hebbian form of long-term potentiation

dependent on mGluR1a in hippocampal inhibitory interneurons. Proc Natl Acad Sci U S

A 98: 9401-9406.

31. Martina M, Vida I, Jonas P (2000) Distal initiation and active propagation of action

potentials in interneuron dendrites. Science 287: 295-300.

32. Migliore M, Shepherd GM (2002) Emerging rules for the distributions of active dendritic

conductances. Nat Rev Neurosci 3: 362-370.

33. Lupica CR, Bell JA, Hoffman AF, Watson PL (2001) Contribution of the

Hyperpolarization-Activated Current (Ih) to Membrane Potential and GABA Release in

Hippocampal Interneurons. J Neurophysiol 86: 261-268.

34. Maccaferri G, McBain CJ (1996) The hyperpolarization-activated current (Ih) and its

contribution to pacemaker activity in rat CA1 hippocampal stratum oriens-alveus

interneurons. J Physiol 497: 119-130.

35. Kispersky TJ, Fernandez FR, Economo MN, White JA (2012) Spike Resonance

Properties in Hippocampal O-LM Cells Are Dependent on Refractory Dynamics. J

Neurosci 32: 3637-3651.

36. Destexhe A, Rudolph M, Paré D (2003) The High-Conductance State of Neocortical

Neurons In Vivo. Nat Rev Neurosci 4: 739-751.

37. Cutsuridis V, Cobb S, Graham BP (2010) Encoding and retrieval in the hippocampal

CA1 microcircuit model. Hippocampus 20: 423-446.

38. Cutsuridis V, Hasselmo M (2012) GABAergic Contributions to Gating, Timing, and

Phase Precession of Hippocampal Neuronal Activity During Theta Oscillations.

Hippocampus 22: 1597-1621.

74

39. Rogers JL, Kesner RP (2003) Cholinergic modulation of the hippocampus during

encoding and retrieval. Neurobiol Learn Mem 80: 332-342.

40. Antonova E, Parslow D, Brammer M, Simmons A, Williams S, Dawson GR, Morris R

(2011) Scopolamine disrupts hippocampal activity during allocentric spatial memory in

humans: an fMRI study using a virtual reality analogue of the Morris Water Maze. J

Psychopharmacol 25: 1256-1265.

41. Atri A, Sherman S, Norman KA, Kirchhoff BA, Nicolas MM, Greicius MD, Cramer SC,

Breiter HC, Hasselmo ME, Stern CE (2004) Blockade of central cholinergic receptors

impairs new learning and increases proactive interference in a word paired-associate

memory task. Behav Neurosci 118: 223-236.

42. Brankack J, Stewart M, Fox SE (1993) Current source density analysis of the

hippocampal theta rhythm: associated sustained potentials and candidate synaptic

generators. Brain Res 615: 310-327.

43. Colgin LL, Denninger T, Fyhn M, Hafting T, Bonnevie T, Jensen O, Moser MB, Moser

EI (2009) Frequency of gamma oscillations routes flow of information in the

hippocampus. Nature 462: 353-357.

44. Mizuseki K, Sirota A, Pastalkova E, Buzsa´kiG (2009) Theta oscillations provide

temporal windows for local circuit computation in the entorhinalhippocampal loop.

Neuron 64: 267-280.

45. Scheffer-Teixeira R, Belchior H, Caixeta FV, Souza BC, Ribeiro S, Tort AB (2012)

Theta phase modulates multiple layer-specific oscillations in the CA1 region. Cereb

Cortex 22: 2404-2414.

46. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its

application to conduction and excitation in nerve. J Physiol 117: 500-544.

47. Saraga F, Wu CP, Zhang L, Skinner FK (2003) Active dendrites and spike propagation in

multi-compartment models of oriens-lacunosum/moleculare hippocampal interneurons. J

Physiol 552: 673-689.

48. Edelman GM, Gally JA (2001) Degeneracy and complexity in biological systems. Proc

Natl Acad Sci U S A 98: 13763-13768.

49. Olypher AV, Calabrese RL (2007) Using constraints on neuronal activity to reveal

compensatory changes in neuronal parameters. J Neurophysiol 98: 3749-3758.

75

50. Olypher AV, Prinz AA (2010) Geometry and dynamics of activity-dependent

homeostatic regulation in neurons. J Comput Neurosci 28: 361-374.

51. Prinz AA, Bucher D, Marder E (2004) Similar network activity from disparate circuit

parameters. Nat Neurosci 7: 1345-1352.

52. Golowasch J, Goldman MS, Abbott LF, Marder E (2002) Failure of averaging in the

construction of a conductance-based neuron model. J Neurophysiol 87: 1129-1131.

53. Schulz DJ, Goaillard JM, Marder E (2006) Variable channel expression in identified

single and electrically coupled neurons in different animals. Nat Neurosci 9: 356-362.

54. Goaillard JM, Taylor AL, Schulz DJ, Marder E (2009) Functional consequences of

animal-to-animal variation in circuit parameters. Nat Neurosci 12: 1424-1430.

55. Ransdell JL, Nair SS, Schulz DJ (2013) Neurons within the Same Network Independently

Achieve Conserved Output by Differentially Balancing Variable Conductance

Magnitudes. J Neurosci 33: 9950-9956.

56. Marder E, Taylor AL (2011) Multiple models to capture the variability in biological

neurons and networks. Nat Neurosci 14: 133-138.

57. Günay C, Edgerton JR, Jaeger D (2008) Channel Density Distributions Explain Spiking

Variability in the Globus Pallidus: A Combined Physiology and Computer Simulation

Database Approach. J Neurosci 28: 7476-7491.

58. Lawrence JJ, Saraga F, Churchill JF, Statland JM, Travis KE, et al. (2006)

Somatodendritic Kv7/KCNQ/M Channels Control Interspike Interval in Hippocampal

Interneurons. J Neurosci 26: 12325-12338.

59. Skinner F, Saraga F (2010) Single Neuron Models: Interneurons. In: Hippocampal

Microcircuits: A Computational Modeler’s Resource Book. New York, NY: Springer

Science, pp. 399-422.

60. Lien CC, Martina M, Schultz JH, Ehmke H, Jonas P (2002) Gating, modulation and

subunit composition of voltage-gated K+channels in dendritic inhibitory interneurones of

rat hippocampus. J Physiol 538: 405-419.

61. Lawrence JJ, Statland JM, Grinspan ZM, McBain CJ (2006) Cell type-specific

dependence of muscarinic signaling in mouse hippocampal stratum oriens interneurons. J

Physiol 570: 595-610.

76

62. Pangalos M, Donoso JR, Winterer J, Zivkovic AR, Kempter R, et al. (2013) Recruitment

of oriens-lacunosum-moleculare interneurons during hippocampal ripples. Proc Natl

Acad Sci U S A 110: 4398-4403.

63. Hines ML, Carnevale NT (1997) The NEURON Simulation Environment. Neural

Computation 9: 1179-1209.

64. Carnevale NT, Hines ML (2006) The NEURON Book. Cambridge, UK: Cambridge

University Press, pp. 50-54.

65. Rall W (1977) Core conductor theory and cable properties of neurons. In: Kandel ER, ed,

Handbook of Physiology, vol. 1, part 1: The Nervous System. Bethesda, MD: American

Physiological Society, pp. 39-98.

66. Hines ML, Morse T, Migliore M, Carnevale NT, Shepherd GM (2004) ModelDB: A

Database to Support Computational Neuroscience. J Comput Neurosci 17:7-11.

67. Günay C, Edgerton JR, Li S, Sangrey T, Prinz AA, et al. (2009) Database Analysis of

Simulated and Recorded Electrophysiological Datasets with PANDORA’s Toolbox.

Neuroinform 7: 93-111.

68. Hines ML, Carnevale NT (2001) NEURON: a tool for neuroscientists. The

Neuroscientist 7: 123-135.

69. Hille B (2001) Ion Channels of Excitable Membranes. Sunderland, MA: Sinauer

Associates, Inc., p. 12.

70. Gentet LJ, Stuart GJ, Clements JD (2000) Direct measurement of specific membrane

capacitance in neurons. Biophys J 79: 314-320.

71. Dayan P, Abbott LF (2001) Theoretical Neuroscience: Computational and Mathematical

Modeling of Neural Systems. Cambridge, MA: The MIT Press, p. 155.

72. Van Geit W, De Schutter E, Achard P (2008) Automated neuron model optimization

techniques: a review. Biol Cybern 99: 241-251.

73. Loken C, Gruner D, Groer L, Peltier R, Bunn N, et al. (2010) SciNet: Lessons Learned

from Building a Power-efficient Top-20 System and Data Centre. J Phys: Conf Ser

256:012026.

74. Taylor AL, Hickey TJ, Prinz AA, Marder E (2006) Structure and Visualization of High-

Dimensional Conductance Spaces. J Neurophys 96: 891-905.

77

75. Smolinski TG, Prinz AA (2009) Computational Intelligence in Modeling of Biological

Neurons: A Case Study of an Invertebrate Pacemaker Neuron. IEEE Proc Intl Joint Conf

Neur Net 2964-2970.

76. Magee JC (1998) Dendritic Hyperpolarization-Activated Currents Modify the Integrative

Properties of Hippocampal CA1 Pyramidal Neurons. J Neurosci 18: 7613-7624.

77. Berger T, Larkum ME, Lüscher HR (2001) High Ih Channel Density in the Distal Apical

Dendrite of Layer V Pyramidal Cells Increases Bidirectional Attenuation of EPSPs. J

Neurophysiol 85: 855-868.

78. Kole MHP, Hallermann S, Stuart GJ (2006) Single Ih Channels in Pyramidal Neuron

Dendrites: Properties, Distribution, and Impact on Action Potential Output. J Neurosci

26: 1677-1687.

79. Angelo K, London M, Christensen SR, Häusser M (2007) Local and Global Effects of Ih

Distribution in Dendrites of Mammalian Neurons. J Neurosci 27: 8643-8653

80. Traub RD, Jefferys JGR, Miles R, Whittington MA, Tóth K (1994) A branching dendritic

model of a rodent CA3 pyramidal neurone. J Physiol 481: 79-95.

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Appendix A – Rationale for active conductance density ranges

Sodium current, INa

As per the original reference O-LM model, INa in the soma took on separate values from INa in

dendrites, but were distributed uniformly in both. For the conductance density ranges,

calculations were made from Fig. 2B in [31]. The “middle” values in the allowable range,

however, were taken directly from [60] and are the values from the hand-tuned O-LM model.

These nevertheless very closely correspond to the mean maximum sodium conductance density

values in [31].

Delayed rectifier potassium currents, IKDRf, IKDRs

Values were calculated from Fig. 2B in [31], just like with INa, above. However, as is the case

with the reference model, the conductance densities for each delayed rectifier channel type were

spread uniformly across both soma and dendrites [79]. This is despite the observation of an

increase in potassium channel conductance densities moving away from soma into dendrites as

reported in O-LM cells [31]. We decided to not differentiate between somatic and dendritic

conductance densities because (i) the upper ranges of the KDRf and KDRs channels are large

enough that the small increase which would be expected in dendritic compartments would

presumably not have made much of a difference in the model output; and (ii) differentiating

between somatic and dendritic conductance densities would have effectively added two “new”

ion channels with their own maximal conductance density ranges, thereby substantially

increasing the number of model simulations to run. Therefore, the maximum conductance

densities calculated from [31] for the dendritic values of both delayed rectifier channel types

were used throughout the model’s soma and dendrites. These were weighted by the fractions of

total outward potassium current that was previously reported for each of these channel types in

hippocampal stratum oriens-alveus interneurons [79]. For the axonal compartments, the same

values as the dendritic ones were used, because as shown in [31], O-LM cell axons attach to

proximal dendrites in most cells.

79

For the conductance density ranges, the discussion in [59] about 6-27 pS/µm2 being more

realistic was taken into account even though the values reported in [31] were substantially

higher. We took the middle ground by incorporating the lower values together with the higher

ones in the same range. It is noteworthy, however, that even though the higher values [31] were

not considered realistic in that study, probably due to the idiosyncrasies of the experimental

conditions, the sodium maximum conductance density values in the original hand-tuned model

were nevertheless given values taken from that same work (and therefore from the same

experimental conditions). In any case, one of the advantages of the model database approach is

that such uncertainties regarding appropriate maximum conductance density ranges can be

tested, and refined, over subsequent iterations of database development.

Transient potassium current, IA

The A-type potassium maximal conductance density ranges were taken from the values for

outward potassium current reported in [31] in dendrites (since they were higher than in soma)

and weighted by the fraction of total potassium current reported for the A-type transient current

in [60]. This channel type is only present in the soma and dendrites, as per [58, 59]. Just as with

the delayed rectifier potassium currents, the lowest end of the maximal conductance range for IA

was taken directly from the reference O-LM model [58], for the same considerations as

mentioned in the above section on the delayed rectifier potassium currents.

Hyperpolarization-activated cation current, Ih

The approach used in the hand-tuned reference O-LM model [60, 47] for determining the

maximum conductance densities for Ih was not appropriate for our present work since we wished

to examine the effects of varying Ih rather than using any particular hand-tuned value. Therefore

the estimated range was taken from literature on dendritic Ih distributions in various neuronal

types. Various studies reported Ih densities along dendrites in either hippocampal or cortical

pyramidal cells using cell-attached patch clamp recordings [76-78]. All three of these studies

showed markedly increasing conductance densities in distal dendrites, with sharp increases

beginning at about 300-400µm from the soma. In [78], the authors performed recordings most

distally of all these studies, up to 1000 µm away from the soma, and noted an exponential rise in

Ih conductance density as a function of distance from soma. In contrast to all of these, [79]

reported a linear distribution of Ih along the length of cerebellar Purkinje cell dendrites.

80

Although the dendritic distribution of Ih channel densities in pyramidal cell dendrites is

thus fairly well characterized, what is known of Ih distributions in O-LM cell dendrites is non-

existent. One approach to providing a biologically plausible dendritic maximum conductance

density range is to adapt the situation in cortical and hippocampal pyramidal cells. A

complicating factor is that O-LM cell dendrites are more spatially compact than those of

pyramidal cells, with the majority of dendrites not extending past ~300µm from the soma [47,

31]. As mentioned, the Ih conductance densities in pyramidal cell dendrites were within a

relatively constrained range up to about 300µm from the soma, and were deemed to be “tiny”

currents by [77]. Comparing the values obtained in the three studies that measured Ih

conductance densities, the values in this extent of the dendrites are described in Table A1.

Table A1. Summary of Ih currents and conductances reported in neocortical and hippocampal

pyramidal cells. The listed current density values are taken directly from the studies referenced

in the first column, and were converted to conductance density values using Ohm’s Law. The

calculated conductance density values are shown in bold. No current density values are listed for

study [78] since conductance density values were measured in the study. Question marks indicate

that the values were difficult to determine from the figures in the papers, but were nevertheless

deemed to be miniscule by visual inspection (< 5 pA/µm2).

Neuron type and reference to study

Approximate dendritic Ih maximal conductance density ranges within 300µm of soma

Layer 5 cortical pyramidal cells [77]

? – 9pA/µm2 (at a –125 mV hyperpolarizing step)

è ? – 72pS/µm2

Layer 5 cortical pyramidal cells [78]

? – 90pS/µm2 (lower range difficult to determine due to smallness of figure 1B and lack of explicit numbers in text of paper)

CA1 hippocampal pyramidal cells [76]

2 – 12pA/µm2 (at –125 to –130 mV hyperpolarizing

step)

è 16 – 94pS/µm2 (used –127.5 mV in calculation)

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Taking the minimum and maximum of all the ranges given in the above table, the range for

maximal Ih conductance density in the O-LM dendrites was therefore set to be 16 – 90pS/µm2 for

the first version of the database (see Table 2.2 in Section 3.1.2.3, “Active conductances and

density ranges”). This assumes that the Ih density distribution in O-LM dendrites is more or less

uniform. If we were to apply the distribution of Ih conductance densities in pyramidal cells as

outlined above in a proportional fashion, i.e., exponential rise throughout the length of O-LM

cell dendrites, we would get vastly increasing values. The exploration of nonuniform

distributions of Ih may be an option in a future iteration of the model database. For the present

work, however, we ventured to simply assess whether or not uniform somatodendritic Ih resulted

in models that better fit the experimental recordings of O-LM cells than somatic distributions of

Ih. Accordingly, the values used were applied simultaneously to all the compartments in the

dendrites for those models that had Ih included within them. Finally, it is worth remarking on the

value of Ih in the hand-tuned O-LM model. This was set to 0.5pS/µm2, which was lower than our

current estimated values from literature. The reason that this value was used is that in the

development of the reference O-LM model [58], the Ih conductance density was fit so as to

produce the same hyperpolarization-induced “sag” in the voltage output – presumably fit to the

electrophysiological recordings. With Ih in the soma only, the fit value became 0.5 pS/µm2,

whereas with Ih both in soma and dendrites (linearly distributed), the fit value became 0.0156

pS/µm2. As noted, in the first iteration of the database, we used 0.5 pS/µm2 as the lowest value

that Ih maximum conductance densities could assume, with the next highest value being 2

pS/µm2, i.e., the lowest of the range determined in the above literature review. This ultimately

turned out to be excessive because even the lowest conductance value was still too high for the

models that had Ih both in soma and dendrites. Furthermore, in the first iteration of the database,

most of the highly-ranked models had Ih in soma only, and furthermore took on the lowest value

of the conductance density range – 0.5 pS/µm2. Thus, the second iteration of the database

required lowering the range significantly, as described in Section 3.2.3.2, “Observed tradeoff

between Ih conductance densities and distributions”.

Calcium currents, ICaL and ICaT

Calcium currents have not been well characterized in hippocampal interneurons [59]. Most

somatodendritic recordings of calcium current densities have been performed on neocortical or

hippocampal CA1/CA3 pyramidal cells [36, 80]. From these studies, it seems that dendritic

82

calcium channels, particularly of the L-type variety, increase in density as a function of distance

from the soma. Values for conductance densities are typically reported to be around 10-30

pS/µm2, which is in line with the calculations performed for the reference O-LM model [59]. The

potential nonuniformity of channel densities was not taken into account in this current work due

to lack of evidence of nonuniform calcium channels in O-LM cell dendrites. Therefore, the

dendritic calcium maximum conductance densities were assigned uniformly across the dendritic

tree. Furthermore, because the literature did not seem to show much variety in calcium channel

conductance densities, we arbitrarily varied the assigned maximum conductance densities to

explore different values in the model database. Three values were incorporated into the initial

database construction, corresponding to 50%, 100%, and 200% of the reported average of

approximately 25 pS/µm2 for L-type calcium channels, which was also used in [59], and the

same range of 50%, 100%, and 200% was applied to the T-type calcium channel values, using

2.5 pS/µm2 as the average (100%) value.

Calcium-activated potassium current, IK(Ca) or IAHP

The kinetic model for the apamin-sensitive, calcium-dependent current responsible for the slow

AHP, or IAHP, was adapted from a CA3 pyramidal model for the reference O-LM model, as

described in [59]. As with the L- and T-type calcium currents described above, the maximum

conductance density of IAHP was determined by simulation of a calcium spike, while sodium

currents were blocked. The resulting value assigned in the reference O-LM model was 5.5

pS/µm2. This was in line with the value of 8 pS/µm2 measured in a guinea pig CA3 pyramidal

cell. Other experimentally-derived conductance density ranges for the IAHP current are not

present in the literature, to the best of our knowledge. Thus, just as with the calcium currents

above, we varied the conductance density ranges arbitrarily by using 50%, 100%, and 200% of

the hand-tuned value of 5.5 pS/µm2.

Muscarinic potassium current, IM

Direct electrophysiological measurements of the muscarinic potassium current in O-LM cells

were made in [58]. It was found that IM constituted approximately 20% of the total outward

current as measured in the soma. This was used in that study as a constraint for calculating

putative IM conductance densities with different distributions of IM channels across the soma,

dendrites, and axon. Various values were obtained in this manner, ranging from 0.75 to 6.4

83

pS/µm2 (the distribution of IM within axon only is not considered in the present work). Although

the conductance densities for IM in soma & primary dendrites were found to be around 2-3

pS/µm2 across two different cells and two calculation methods, up to 6.4 pS/µm2 was determined

to be the case if IM was located within the soma only. Nevertheless, minimal variance was

observed for a particular distribution of IM densities. Combined with an overall lack of IM

conductance densities reported in the experimental literature, this led us in the current study to

continue the use of the scaling of conductance density values as above, that is, of 50%, 100%,

and 200% of the reference O-LM model’s hand-tuned value of 0.75 pS/µm2.

84

Appendix B – Electrophysiological measures in PANDORA

Note on nomenclature

Throughout this appendix, the names of the periods of a voltage trace are indicated as per

Section 3.1.3.1.1, “Modifications required for PANDORA”. That is, the “initial” or

“spontaneous” period corresponds to the portion of the trace prior to the application of –90 pA

hyperpolarizing or +90 pA depolarizing current injection. The “pulse” period corresponds to the

length of the trace corresponding to the current injection application, being 1s in duration. The

“recovery” period corresponds to the remainder of the trace, after the additional current injection

has been removed.

85

Measure name Meaning

IniSpontPotAvg The average Vm value for the duration of the spontaneous (pre-current injection) period.

IniSpontPotRange The range of Vm values for the duration of the spontaneous period. It is important to differentiate between models that have the same average Vm but large variance, since most experimental traces have little variance in their initial Vm values prior to the current injection period.

RecSpontPotAvg The average Vm value for the duration of the recovery (post-current injection) period.

PulsePotMin The minimum value of Vm obtained during the current injection period. I.e., the minimum membrane voltage induced by the hyperpolarization.

PulsePotMinTime The time at which the minimum of Vm occurred during the current injection period.

PulsePotSag The amount of sag (in mV) exhibited by the trace as a result of the hyperpolarization. This is a measure of the depolarizing effects of the Ih current.

PulsePotTau The time constant for fitting an exponential curve to the decay of the hyperpolarization-induced sag.

RecSpont1SpikeRate The firing frequency of the model or experimental cell in the first half of the recovery period.

RecSpont1SpikeRateISI The inter-spike interval of the spikes in the first half of the recovery period.

RecovSpikes The number of spikes in the recovery period. It was not appropriate to use the RecovSpikeRate, or firing frequency for the entire duration of the recovery period, because most model and experimental cells exhibited a limited number of post-inhibitory rebound spikes, which were primarily confined to the initial portion of the recovery period.

Ihold The amount of holding current that was applied to keep the model or experimental cell’s membrane potential at −74 mV (Section 3.1.2.5.3).

Table B1. The measures used for analyzing and comparing −90 pA current clamp traces of both

model and experimental O-LM cells in PANDORA.

86

Measure Units Average Standard deviation

IniSpontPotAvg mV -74.0980 0.5766

IniSpontPotRange mV 0.4810 0.1379

RecSpontPotAvg mV -70.3880 1.4016

PulsePotMin mV -113.8694 3.6641

PulsePotMinTime ms 166.8000 54.6685

PulsePotSag mV 14.1929 3.0796

PulsePotTau ms 47.3813 8.0496

RecSpont1SpikeRate Hz 1.8748 1.9956

RecSpont1SpikeRateISI ms 2.3603 3.1874

RecovSpikes Number of spikes

0.9375 0.9979

Ihold pA -0.0016 0.0067

Table B2. The measures calculated from the –90 pA current clamp experimental dataset. The

average values across all voltage traces, as well as the standard deviation of the measures within

the dataset, are provided. There were 11 measures used in total.

87

Measure name Meaning

[Ini/Rec]SpontPotAvg Average Vm during the initial or recovery periods of the trace.

IniSpotPotRange Difference between the maximum and minimum Vm values during the initial period.

[Ini/Rec]SpontSpikeRate Firing rate of the given period of the trace.

[Ini/Rec]SpontSpikeRateISI Averaged inter-spike-interval (ISI) across the entire period of the trace.

PulseFirstISI First ISI in the pulse period; i.e., the time between the first and second spikes in the pulse period.

PulseFirstSpikeTime Amount of time between the start of the pulse period and the first spike (delay to first spike).

PulseIni100msISICV Coefficient of variation of inter-spike-intervals (ISI) within the first 100ms of the pulse period.

PulseIni100msRest1SpikeRate Firing rate of the first half of the remainder of the pulse period following the initial 100ms.

PulseIni100msRest1SpikeRateISI Averaged inter-spike-interval (ISI) of the first half of the remainder of the pulse period following the initial 100ms.

PulseIni100msRest2SpikeRate Firing rate of the second half of the remainder of the pulse period following the initial 100ms.

PulseIni100msRest2SpikeRateISI Averaged inter-spike-interval (ISI) of the second half of the remainder of the pulse period following the initial 100ms.

PulseIni100msSpikeRate Firing rate of the initial 100ms of the pulse period.

PulseIni100msSpikeRateISI Averaged inter-spike-interval (ISI) of the initial 100ms of the pulse period.

PulsePotAvg Average Vm for the entire pulse period.

PulseSFA Spike frequency accommodation (SFA) of the inter-spike-intervals (ISI) during the pulse period.

PulseSpikeRate Firing frequency during the pulse period.

PulseSpikeRateISI Mean inter-spike-interval (ISI) between spikes during the pulse period.

88

RecIniSpontPotRatio Ratio of the averaged Vm of the recovery period to the averaged Vm of the initial period.

RecIniSpontRateRatio Ratio of the firing rage of the recovery period to the firing rate of the initial period.

RecSpont1SpikeRate Mean firing rate of the first half of the recovery period.

RecSpont1SpikeRateISI Averaged inter-spike-interval (ISI) of the first half of the recovery period.

RecSpont2SpikeRate Mean firing rate of the second half of the recovery period.

RecSpont2SpikeRateISI Averaged inter-spike-interval (ISI) of the second half of the recovery period.

[Ini/Pulse/Recov]Spikes Number of spikes in the given period.

[Spont/Pulse/Recov] SpikeAmplitudeMean*

The mean amplitude of the spikes during the current injection period. Amplitude is calculated by taking the difference in spike height from the Vm at spike initiation. Spike initiation is determined by finding the point of maximum curvature in the V-dV/dt phase plane (PANDORA supports additional methods for spike initiation detection).

[Spont/Pulse/Recov] SpikeBaseWidthMean*

Width of the base of spikes, averaged across all spikes in a period.

[Spont/Pulse/Recov] SpikeDAHPMagMean*

Magnitude of the double afterhyperpolarization (AHP) peak, if any, where a double AHP indicates the presence of a second afterhyperpolarization (AHP) peak after an initial AHP following the termination of a spike.

[Spont/Pulse/Recov] SpikeFallTimeMean*

Time a spike takes to fall from its peak back to the spike initiation point, averaged across all spikes in a period.

[Spont/Pulse/Recov] SpikeFixVWidthMean*

Width of spikes at a particular Vm value (default of –10 mV), averaged across all spikes in a period.

[Spont/Pulse/Recov] SpikeHalfVmMean*

Vm equal to half of the spike height, averaged across all spikes in a period.

[Spont/Pulse/Recov] SpikeHalfWidthMean*

Width of spike at the point where Vm is equal to half of the spike height, averaged across all spikes in a period.

[Spont/Pulse/Recov] SpikeInitTimeMean*

Time at initiation of a spike.

89

[Spont/Pulse/Recov] SpikeInitVmBySlopeMean*

Vm at point of spike initialization, taken to be at the point where the first-order derivative of Vm exceeds a threshold (default dV/dt threshold of 15 mV/s).

[Spont/Pulse/Recov] SpikeInitVmMean*

Default measure finding the initialization of a spike as the point of maximum curvature in the VdV/dt phase plane.

[Spont/Pulse/Recov] SpikeMaxAHPMean*

The magnitude of Vm change from the spike initiation point to the minimum Vm, calculated during the abolishing of a spike.

[Spont/Pulse/Recov] SpikeMaxVmSlopeMean*

Maximum slope, or first-order derivative of Vm.

[Spont/Pulse/Recov] SpikeMinTimeMean*

Time at which the minimum value of Vm occurs during a spike, relative to the start of the spike.

[Spont/Pulse/Recov] SpikeMinVmMean*

Minimum Vm during a spike.

[Spont/Pulse/Recov] SpikePeakVmMean*

Vm at the spike height; i.e., maximum Vm.

[Spont/Pulse/Recov] SpikeRiseTimeMean*

Time from spike initiation to the maximum Vm, or spike height.

Table B3. The measures used for analyzing and comparing +90 pA current clamp traces of both

model and experimental O-LM cells in PANDORA. Measure names with square brackets

indicate different measures that are similarly calculated for two different periods in a given

voltage trace. For example, “[Ini/Rec]SpontPotAvg” indicates that there are two associated

measures: IniSpontPotAvg, and RecSpontPotAvg that both calculate the average membrane

potential (Vm) for, respectively, the initial and recovery periods of the trace. Measures with an

asterisk (*) appended to the end indicate that there is also an associated measure that calculates

the mode. For example, SpontSpikeAmplitudeMean* indicates that there are two associated

measures: SpontSpikeAmplitudeMean, the mean of the firing rate during the initial period; and

SpontSpikeAmplitudeMode, or the most common value of the firing rate (mode) during the

initial period.

90

Measure Units Average Standard deviation

IniSpontPotAvg mV -74.6496 0.9130

IniSpontPotRange mV 0.5763 0.2256

IniSpontSpikeRate Hz 0 0

IniSpontSpikeRateISI ms 0 0

PulseFirstISI ms 33.2600 8.6276

PulseFirstSpikeTime ms 19.5125 5.7147

PulseISICV Normalized (STD/mean)

0.0735 0.0467

PulseIni100msISICV Normalized (STD/mean)

0.0155 0.0150

PulseIni100msRest1SpikeRate Hz 29.6177 6.3746

PulseIni100msRest1SpikeRateISI ms 29.9273 6.6652

PulseIni100msRest2SpikeRate Hz 13.0058 7.1120

PulseIni100msRest2SpikeRateISI ms 26.1329 7.2306

PulseIni100msSpikeRate Hz 30.2500 8.6194

PulseIni100msSpikeRateISI ms 32.1269 8.0853

PulsePotAvg mV -63.0639 3.5556

PulseSFA Ratio of ISIs at end of current injection step period to ISIs at beginning

1.2062 0.1211

PulseSpikeRate Hz 22.2022 3.0063

PulseSpikeRateISI ms 29.4606 6.9570

RecIniSpontPotRatio Normalized 1.0489 0.0161

91

RecIniSpontRateRatio Normalized (from Hz)

1.0500 0.3162

RecSpont1SpikeRate Hz 0.0500 0.3162

RecSpont1SpikeRateISI ms 0.0500 0.3162

RecSpont2SpikeRate Hz 0 0

RecSpont2SpikeRateISI ms 0 0

RecSpontPotAvg mV -78.3013 1.4137

RecSpontSpikeRate Hz 0.0250 0.1581

RecSpontSpikeRateISI ms 0.0250 0.1581

IniSpikes Number of spikes

0 0

SpontSpikeAmplitudeMode mV 0 0

SpontSpikeBaseWidthMode ms 0 0

SpontSpikeDAHPMagMode mV 0 0

SpontSpikeFallTimeMode ms 0 0

SpontSpikeFixVWidthMode ms 0 0

SpontSpikeHalfVmMode mV 0 0

SpontSpikeHalfWidthMode ms 0 0

SpontSpikeInitTimeMode ms 0 0

SpontSpikeInitVmBySlopeMode mV 0 0

SpontSpikeInitVmMode mV 0 0

SpontSpikeMaxAHPMode mV 0 0

SpontSpikeMaxVmSlopeMode mV/ms 0 0

SpontSpikeMinTimeMode ms 0 0

SpontSpikeMinVmMode mV 0 0

SpontSpikePeakVmMode mV 0 0

92

SpontSpikeRiseTimeMode ms 0 0

PulseSpikeAmplitudeMean mV 62.2142 6.7792

PulseSpikeAmplitudeMode mV 60.4678 6.2002

PulseSpikeBaseWidthMean ms 2.4165 0.5318

PulseSpikeBaseWidthMode ms 2.3531 0.5910

PulseSpikeDAHPMagMode mV 2.1059 4.7729

PulseSpikeFallTimeMean ms 1.5278 0.4732

PulseSpikeFallTimeMode ms 1.4002 0.3889

PulseSpikeFixVWidthMode ms 0.8609 0.3033

PulseSpikeHalfVmMean mV -20.8595 2.4297

PulseSpikeHalfVmMode mV -21.0108 2.5291

PulseSpikeHalfWidthMean ms 1.1839 0.3072

PulseSpikeHalfWidthMode ms 1.1764 0.3214

PulseSpikeInitTimeMean ms 6.0641 0.1733

PulseSpikeInitTimeMode ms 6.0483 0.2072

PulseSpikeInitVmBySlopeMean mV -51.0073 1.6429

PulseSpikeInitVmBySlopeMode mV -50.5397 1.7625

PulseSpikeInitVmMean mV -52.0359 1.5669

PulseSpikeInitVmMode mV -51.7270 1.4142

PulseSpikeMaxAHPMean mV 17.9520 3.4551

PulseSpikeMaxAHPMode mV 17.3238 3.7842

PulseSpikeMaxVmSlopeMean mV 133.4409 29.5413

PulseSpikeMaxVmSlopeMode mV 121.6295 28.1033

PulseSpikeMinTimeMean ms 12.7524 1.6653

PulseSpikeMinTimeMode ms 12.0520 1.4580

93

PulseSpikeMinVmMean mV -69.6621 4.1244

PulseSpikeMinVmMode mV -69.2497 5.0529

PulseSpikePeakVmMean mV 10.1783 5.7317

PulseSpikePeakVmMode mV 9.2620 5.1281

PulseSpikeRiseTimeMean ms 1.0344 0.1741

PulseSpikeRiseTimeMode ms 0.9946 0.2147

PulseSpikes Number of spikes

22.2000 3.0060

RecovSpikeAmplitudeMode mV 1.5576 9.8511

RecovSpikeBaseWidthMode ms 0.0814 0.5147

RecovSpikeDAHPMagMode mV 0 0

RecovSpikeFallTimeMode ms 0.0500 0.3162

RecovSpikeFixVWidthMode ms 0.0360 0.2275

RecovSpikeHalfVmMode mV -0.4501 2.8470

RecovSpikeHalfWidthMode ms 0.0433 0.2740

RecovSpikeInitTimeMode ms 0.0206 0.1304

RecovSpikeInitVmBySlopeMode mV -1.2289 7.7725

RecovSpikeInitVmMode mV -1.2289 7.7725

RecovSpikeMaxAHPMode mV 0.6680 4.2245

RecovSpikeMaxVmSlopeMode mV 2.2079 13.9643

RecovSpikeMinTimeMode ms 0.6675 4.2216

RecovSpikeMinVmMode mV -1.8969 11.9970

RecovSpikePeakVmMode mV 0.3287 2.0786

RecovSpikeRiseTimeMode ms 0.0314 0.1984

RecovSpikes Number of spikes

0.0250 0.1581

94

Table B4. The measures calculated from the +90 pA current clamp experimental dataset. The

average values across all voltage traces, as well as the standard deviation of the measures within

the dataset, are provided. Of the full set of possible measures, 93 were used.

95

Histograms of measures for –90 pA experimental traces

Below are listed the histograms of electrophysiological measures extracted from the subset of

experimental O-LM data corresponding to the –90 pA current injection protocol. The histograms

are listed in order from left-to-right, top-to-bottom, of the measures listed in Tables B1 and B2.

−75.5 −75 −74.5 −74 −73.5 −730

0.5

1

1.5

2

2.5

3IniSpontPotAvg measure (−90pA)

Cou

nt

Vm (mV)0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

3IniSpontPotRange measure (−90pA)

Cou

ntChange in Vm (mV

−74 −73 −72 −71 −70 −69 −680

0.5

1

1.5

2

2.5

3

3.5

4RecSpontPotAvg measure (−90pA)

Cou

nt

Vm (mV)−120 −118 −116 −114 −112 −110 −108 −106

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5PulsePotMin measure (−90pA)

Cou

nt

Vm (mV)

100 120 140 160 180 200 220 240 260 2800

0.5

1

1.5

2

2.5

3

3.5

4PulsePotMinTime measure (−90pA)

Cou

nt

Time (ms)9 10 11 12 13 14 15 16 17 18

0

0.5

1

1.5

2

2.5

3

3.5

4PulsePotSag measure (−90pA)

Cou

nt

Change in Vm (mV)

96

35 40 45 50 55 60 650

0.5

1

1.5

2

2.5

3

3.5

4PulsePotTau measure (−90pA)

Cou

nt

Time constant, tau (ms)0 1 2 3 4 5 6

0

1

2

3

4

5

6

7RecSpont1SpikeRate measure (−90pA)

Cou

nt

Firing rate (Hz)

0 2 4 6 8 10 120

1

2

3

4

5

6

7RecSpont1SpikeRateISI measure (−90pA)

Cou

nt

ISI (ms)0 0.5 1 1.5 2 2.5 3

0

1

2

3

4

5

6

7RecovSpikes measure (−90pA)

Cou

nt

Number of spikes

−0.015 −0.01 −0.005 0 0.005 0.010

0.5

1

1.5

2

2.5

3

3.5

4Ihold measure (−90pA)

Cou

nt

Holding current (pA)

97

Histograms of measures for +90 pA experimental traces

Below are listed the histograms of electrophysiological measures extracted from the subset of

experimental O-LM data corresponding to the +90 pA current injection protocol. The histograms

are listed in order from left-to-right, top-to-bottom, of the measures listed in Tables B3 and B4.

Because the measures for the +90 pA traces were selected in programmatic fashion, some of the

measures were not applicable for all periods of the trace, thus resulting in some measures that

reported zero values for all traces. The histograms for such measures are not included below.

−78 −77 −76 −75 −74 −73 −720

2

4

6

8

10

12

14IniSpontPotAvg measure (+90pA)

Cou

nt

Vm (mV)0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

1

2

3

4

5

6

7

8

9

10IniSpontPotRange measure (+90pA)

Cou

nt

Change in Vm (mV)

15 20 25 30 35 40 45 50 550

2

4

6

8

10

12

14

16PulseFirstISI measure (+90pA)

Cou

nt

ISI (ms)10 15 20 25 30 350

1

2

3

4

5

6

7

8PulseFirstSpikeTime measure (+90pA)

Cou

nt

Time (ms)

98

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

5

10

15PulseISICV measure (+90pA)

Cou

nt

Covariance of ISI0 0.01 0.02 0.03 0.04 0.05 0.06

0

2

4

6

8

10

12

14PulseIni100msISICV measure (+90pA)

Cou

nt

Covariance of ISI

20 25 30 35 40 450

1

2

3

4

5

6

7

8

9

10PulseIni100msRest1SpikeRate measure (+90pA)

Cou

nt

Firing rate (Hz)20 25 30 35 40 450

1

2

3

4

5

6

7

8

9PulseIni100msRest1SpikeRateISI measure (+90pA)

Cou

nt

ISI (ms)

0 5 10 15 20 25 300

2

4

6

8

10

12PulseIni100msRest2SpikeRate measure (+90pA)

Cou

nt

Firing rate (Hz)0 5 10 15 20 25 30 35 40

0

1

2

3

4

5

6

7

8

9

10PulseIni100msRest2SpikeRateISI measure (+90pA)

Cou

nt

ISI (ms)

99

20 25 30 35 40 45 500

2

4

6

8

10

12

14

16

18

20PulseIni100msSpikeRate measure (+90pA)

Cou

nt

Firing rate (Hz)15 20 25 30 35 40 45 500

5

10

15PulseIni100msSpikeRateISI measure (+90pA)

Cou

nt

ISI (ms)

−68 −66 −64 −62 −60 −58 −56 −540

1

2

3

4

5

6

7

8PulsePotAvg measure (+90pA)

Cou

nt

Average Vm (mV)0.9 1 1.1 1.2 1.3 1.4 1.50

1

2

3

4

5

6

7PulseSFA measure (+90pA)

Cou

nt

ISIend/ISIbeginning

16 18 20 22 24 26 28 300

2

4

6

8

10

12

14PulseSpikeRate measure (+90pA)

Cou

nt

Firing rate (Hz)15 20 25 30 35 40 450

1

2

3

4

5

6

7

8

9

10PulseSpikeRateISI measure (+90pA)

Cou

nt

ISI (ms)

100

1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.1 1.110

1

2

3

4

5

6

7

8

9

10RecIniSpontPotRatio measure (+90pA)

Cou

nt

Vm,recov / Vm,spont1 1.5 2 2.5 3

0

5

10

15

20

25

30

35

40RecIniSpontRateRatio measure (+90pA)

Cou

nt

Firing rate ratio (recov/spont)

0 0.5 1 1.5 20

5

10

15

20

25

30

35

40RecSpont1SpikeRate measure (+90pA)

Cou

nt

Firing rate (Hz)0 0.5 1 1.5 2

0

5

10

15

20

25

30

35

40RecSpont1SpikeRateISI measure (+90pA)

Cou

nt

ISI (ms)

−83 −82 −81 −80 −79 −78 −77 −760

2

4

6

8

10

12RecSpontPotAvg measure (+90pA)

Cou

nt

Average Vm (mV) 0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40RecSpontSpikeRate measure (+90pA)

Cou

nt

Firing rate (Hz)

101

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40RecSpontSpikeRateISI measure (+90pA)

Cou

nt

ISI (ms)35 40 45 50 55 60 65 70 750

2

4

6

8

10

12

14

16

18

20PulseSpikeAmplitudeMean measure (+90pA)

Cou

nt

Spike height (mV)

35 40 45 50 55 60 65 700

2

4

6

8

10

12PulseSpikeAmplitudeMode measure (+90pA)

Cou

nt

Spike height (mV)1.5 2 2.5 3 3.5 40

1

2

3

4

5

6

7

8

9

10PulseSpikeBaseWidthMean measure (+90pA)

Cou

nt

Spike width at base (ms)

1.5 2 2.5 3 3.5 4 4.50

2

4

6

8

10

12PulseSpikeBaseWidthMode measure (+90pA)

Cou

nt

Spike width at base (ms)0 2 4 6 8 10 12 14 16 18

0

5

10

15

20

25

30

35PulseSpikeDAHPMagMode measure (+90pA)

Cou

nt

Double AHP magnitude after spike (mV)

102

0.5 1 1.5 2 2.5 3 3.50

1

2

3

4

5

6

7

8

9

10PulseSpikeFallTimeMean measure (+90pA)

Cou

nt

Time to termination of spike (ms)0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8

9PulseSpikeFallTimeMode measure (+90pA)

Cou

nt

Time to termination of spike (ms)

0 0.5 1 1.50

2

4

6

8

10

12

14PulseSpikeFixVWidthMode measure (+90pA)

Cou

nt

Spike width at fixed Vm = −10mV (ms)−30 −28 −26 −24 −22 −20 −18 −160

2

4

6

8

10

12PulseSpikeHalfVmMean measure (+90pA)

Cou

nt

Vm at half spike height (mV)

−30 −28 −26 −24 −22 −20 −18 −160

1

2

3

4

5

6

7

8

9

10PulseSpikeHalfVmMode measure (+90pA)

Cou

nt

Vm at half spike height (mV) 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

9PulseSpikeHalfWidthMean measure (+90pA)

Cou

nt

Spike width at half spike height (ms)

103

0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12PulseSpikeHalfWidthMode measure (+90pA)

Cou

nt

Spike width at half spike height (ms)5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.40

1

2

3

4

5

6

7

8

9

10PulseSpikeInitTimeMean measure (+90pA)

Cou

nt

Time at spike initialization (ms)

5.5 6 6.50

2

4

6

8

10

12PulseSpikeInitTimeMode measure (+90pA)

Cou

nt

Time at spike initialization (ms)−54 −53 −52 −51 −50 −49 −48 −47 −46 −450

2

4

6

8

10

12PulseSpikeInitVmBySlopeMean measure (+90pA)

Cou

nt

Vm at spike init using max slope method (mV)

−54 −53 −52 −51 −50 −49 −48 −47 −46 −450

2

4

6

8

10

12PulseSpikeInitVmBySlopeMode measure (+90pA)

Cou

nt

Vm at spike init using max slope method (mV)−55 −54 −53 −52 −51 −50 −49 −48 −47 −460

2

4

6

8

10

12PulseSpikeInitVmMean measure (+90pA)

Cou

nt

Vm at spike initialization (mV)

104

−56 −55 −54 −53 −52 −51 −50 −49 −480

1

2

3

4

5

6

7

8

9PulseSpikeInitVmMode measure (+90pA)

Cou

nt

Vm at spike initialization (mV) 10 12 14 16 18 20 22 240

2

4

6

8

10

12PulseSpikeMaxAHPMean measure (+90pA)

Cou

nt

Max AHP after spike (mV)

8 10 12 14 16 18 20 22 240

1

2

3

4

5

6

7PulseSpikeMaxAHPMode measure (+90pA)

Cou

nt

Max AHP after spike (mV)40 60 80 100 120 140 160 1800

1

2

3

4

5

6

7

8

9

10PulseSpikeMaxVmSlopeMean measure (+90pA)

Cou

nt

Max slope of Vm during spike (mV/ms)

40 60 80 100 120 140 160 1800

2

4

6

8

10

12PulseSpikeMaxVmSlopeMode measure (+90pA)

Cou

nt

Max slope of Vm during spike (mV/ms)10 11 12 13 14 15 16 170

1

2

3

4

5

6

7

8

9PulseSpikeMinTimeMean measure (+90pA)

Cou

nt

Time at minimum Vm during spike (ms)

105

10 11 12 13 14 15 160

1

2

3

4

5

6

7

8

9PulseSpikeMinTimeMode measure (+90pA)

Cou

nt

Time at minimum Vm during spike (ms)−75 −70 −65 −600

1

2

3

4

5

6

7

8

9PulseSpikeMinVmMean measure (+90pA)

Cou

nt

Minimum Vm during spike (mV)

−76 −74 −72 −70 −68 −66 −64 −62 −600

1

2

3

4

5

6

7

8

9

10PulseSpikeMinVmMode measure (+90pA)

Cou

nt

Minimum Vm during spike (mV)−15 −10 −5 0 5 10 15 200

2

4

6

8

10

12

14PulseSpikePeakVmMean measure (+90pA)

Cou

nt

Peak Vm during spike (mV)

−10 −5 0 5 10 15 200

2

4

6

8

10

12

14

16

18PulseSpikePeakVmMode measure (+90pA)

Cou

nt

Peak Vm during spike (mV) 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

1

2

3

4

5

6

7

8

9

10PulseSpikeRiseTimeMean measure (+90pA)

Cou

nt

Time from spike initialization to height (ms)

106

0.8 1 1.2 1.4 1.6 1.8 20

2

4

6

8

10

12PulseSpikeRiseTimeMode measure (+90pA)

Cou

nt

Time from spike initialization to height (ms)16 18 20 22 24 26 280

2

4

6

8

10

12

14PulseSpikes measure (+90pA)

Cou

nt

Number of spikes

0 10 20 30 40 50 60 700

5

10

15

20

25

30

35

40RecovSpikeAmplitudeMode measure (+90pA)

Cou

nt

Spike height (mV)0 0.5 1 1.5 2 2.5 3 3.5

0

5

10

15

20

25

30

35

40RecovSpikeBaseWidthMode measure (+90pA)

Cou

nt

Spike width at base (ms)

0 0.5 1 1.5 20

5

10

15

20

25

30

35

40RecovSpikeFallTimeMode measure (+90pA)

Cou

nt

Time to termination of spike (ms)0 0.5 1 1.5

0

5

10

15

20

25

30

35

40RecovSpikeFixVWidthMode measure (+90pA)

Cou

nt

Spike width at fixed Vm = −10mV (ms)

107

−20 −15 −10 −5 00

5

10

15

20

25

30

35

40RecovSpikeHalfVmMode measure (+90pA)

Cou

nt

Vm at half spike height (mV) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

5

10

15

20

25

30

35

40RecovSpikeHalfWidthMode measure (+90pA)

Cou

nt

Spike width at half spike height (ms)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

35

40RecovSpikeInitTimeMode measure (+90pA)

Cou

nt

Time at spike initialization (ms)−50 −40 −30 −20 −10 00

5

10

15

20

25

30

35

40RecovSpikeInitVmBySlopeMode measure (+90pA)

Cou

nt

Vm at spike init using max slope method (mV)

−50 −40 −30 −20 −10 00

5

10

15

20

25

30

35

40RecovSpikeInitVmMode measure (+90pA)

Cou

nt

Vm at spike initialization (mV) 0 5 10 15 20 25 300

5

10

15

20

25

30

35

40RecovSpikeMaxAHPMode measure (+90pA)

Cou

nt

Max AHP after spike (mV)

108

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

40RecovSpikeMaxVmSlopeMode measure (+90pA)

Cou

nt

Max slope of Vm during spike (mV/ms)0 5 10 15 20 25 30

0

5

10

15

20

25

30

35

40RecovSpikeMinTimeMode measure (+90pA)

Cou

nt

Time at minimum Vm during spike (ms)

−80 −70 −60 −50 −40 −30 −20 −10 00

5

10

15

20

25

30

35

40RecovSpikeMinVmMode measure (+90pA)

Cou

nt

Minimum Vm during spike (mV)0 2 4 6 8 10 12 14

0

5

10

15

20

25

30

35

40RecovSpikePeakVmMode measure (+90pA)

Cou

nt

Peak Vm during spike (mV)

0 0.2 0.4 0.6 0.8 1 1.2 1.40

5

10

15

20

25

30

35

40RecovSpikeRiseTimeMode measure (+90pA)

Cou

nt

Time from spike initialization to height (ms)0 0.2 0.4 0.6 0.8 1

0

5

10

15

20

25

30

35

40RecovSpikes measure (+90pA)

Cou

nt

Number of spikes