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Journal of Physiology - Paris 98 (2004) 452–466

Inferring network activity from synaptic noise

Michael Rudolph, Alain Destexhe *

Integrative and Computational Neuroscience Unit (UNIC), CNRS, 1 Avenue de la Terrasse (Bat. 33), 91198 Gif-sur-Yvette, France

Abstract

During intense network activity in vivo, cortical neurons are in a high-conductance state, in which the membrane potential (Vm) issubject to a tremendous fluctuating activity. Clearly, this ‘‘synaptic noise’’ contains information about the activity of the network,but there are presently no methods available to extract this information. We focus here on this problem from a computational neuro-science perspective, with the aim of drawing methods to analyze experimental data. We start from models of cortical neurons, in whichhigh-conductance states stem from the random release of thousands of excitatory and inhibitory synapses. This highly complex systemcan be simplified by using global synaptic conductances described by effective stochastic processes. The advantage of this approach is thatone can derive analytically a number of properties from the statistics of resulting Vm fluctuations. For example, the global excitatory andinhibitory conductances can be extracted from synaptic noise, and can be related to the mean activity of presynaptic neurons. We showhere that extracting the variances of excitatory and inhibitory synaptic conductances can provide estimates of the mean temporal cor-relation—or level of synchrony—among thousands of neurons in the network. Thus, ‘‘probing the network’’ through intracellular Vm

activity is possible and constitutes a promising approach, but it will require a continuous effort combining theory, computational modelsand intracellular physiology. 2005 Elsevier Ltd. All rights reserved.

Keywords: Computational model; Membrane potential fluctuations; High-conductance states; Fokker–Planck

1. Introduction

What does intracellular activity tell us about the activityof the embedding network? Can a link be drawn betweenthe cellular and network dynamics based on the apparentlystochastic aspects of neuronal activity? Indeed, neurons aresubject to many different sources of noise [58], which incombination with their excitability may significantly altertheir integrative properties [29,47,56,78,97,105,108] (reviewedin [23]). Of these different sources of noise, synaptic noise isclearly the most prominent one in activated states. In thecortex, synaptic noise originates from the seemingly ran-dom and sustained firing of individual cortical neurons

0928-4257/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.jphysparis.2005.09.015

* Corresponding author. Tel.: +33 1 69 82 34 35; fax: +33 1 69 82 34 27.E-mail address: [email protected] (A. Destexhe).

(5–40 Hz in awake animals; see [30,60,87,90,92]), combinedwith their remarkably dense connectivity (tens of thousandsynaptic inputs per neuron; see [18,20,21,39,93]). Underthese conditions, the membrane potential (Vm) of corticalneurons is depolarized (average V 65 mV comparedto V 80 mV in quiescent states without synaptic noise)and shows high-amplitude fluctuations (Vm standard devi-ation rV around 4 mV). In addition, this intense synapticactivity sets the cellular membrane into a high-conductancestate, in which the input resistance is reduced by 3 to 5times compared to quiescent states [2,3,8,23,67].

To describe the impact of synaptic noise on neuronaldynamics, a variety of models were proposed. Such modelscover a broad range of complexity, starting with integrate-and-fire models [7,11,28,53,54,98,104], stochastic mem-brane equations [51,58,99–101], up to biophysically detailedmodels of single neurons with release activity at thousandsof distributed individual synaptic terminals, each describedby Poisson processes [5,22,43,52,70,75–77,96,100]. An

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M. Rudolph, A. Destexhe / Journal of Physiology - Paris 98 (2004) 452–466 453

intermediate approach is to model synaptic noise with twoglobal conductances described by stochastic processes[27]. This effective representation was shown to provide acompact description of the total conductance resulting fromsynaptic inputs. It captures the amplitude of the conduc-tances, their standard deviation and spectral structure, aswell as reproduces response properties typical of corticalneurons in vivo [27,31,69]. Another advantage of this repre-sentation is that synaptic noise can be simulated in realtime, and the calculated stochastic synaptic current can beinjected in real neurons recorded in vitro [27]. This way,by using the dynamic-clamp technique [74,85], cellular stateswith in vivo-like electrophysiological properties can be rec-reated in vitro [16,27,31,69].

In addition to having a possibly important impact onintegrative properties, synaptic noise may also be used asa tool to extract information about network activity. Thisis due to the high interconnectivity of cortical neurons,for which most of the synapses originate in the cortex itself[9,21]. Synaptic noise, therefore, contains informationabout the firing of thousands of other cortical neurons.This information should, in principle, be extractable fromanalyzing Vm fluctuations, but no such method has beenproposed yet.

In the present paper, we focus on different means ofextracting information from network activity based onanalyzing synaptic noise. We start by reviewing the charac-teristics of high-conductance states in cortical neurons, aswell as different modeling approaches that reproduce thesestates. We next review models of synaptic noise that cantreated analytically, leading to methods for estimating thestatistical properties of the conductances underlying synap-tic noise, thereby providing a way to estimate the level ofactivity of presynaptic neurons. We show that similarmethods may also be used to estimate the temporal corre-lation in network activity, thereby providing estimates ofthe level of synchrony in the network from analyzing intra-cellular recordings of single neurons.

2. Materials and methods

Two types of models were numerically and analytically investigated.Both models simulated a single-compartment neuron, described by the fol-lowing membrane equation:

dV ðtÞdt

¼ 1

smðV ðtÞ ELÞ

1

CI synðtÞ þ

1

CIext. ð1Þ

Here, V(t) denotes the membrane potential, sm = C/GL the membranetime constant, C = aCm and GL = a gL, where Cm is the specific membranecapacity and a is the membrane area. Iext denotes a constant stimulationcurrent, gL and EL are the leak conductance and reversal potential, respec-tively. The term Isyn describes synaptic inputs to the neuron, and is givenby

I synðtÞ ¼ geðtÞðV ðtÞ EeÞ þ giðtÞðV ðtÞ EiÞ; ð2Þ

where ge(t) and gi(t) are global excitatory and inhibitory synaptic conduc-tances, respectively, with respective reversal potentials Ee and Ei. Theseconductances were modeled either by thousands of individual synapsesor by effective stochastic processes, as detailed below.

2.1. Biophysical model

In what will be referred to below as the biophysical model, synapticinputs were modeled by a large number of individual synaptic conduc-tances. In this case, the global synaptic conductances were written as

geðtÞ ¼XNn¼1

gAMPAmðnÞe ðtÞ;

giðtÞ ¼XMm¼1

gGABAmðmÞi ðtÞ;

ð3Þ

whereN andM are the total number of excitatory and inhibitory synapses,modeled by a-amino-3-hydroxy-5-methyl-4-isoxazolepropionic (AMPA)and c-aminobutyric acid (GABAA) postsynaptic receptors with quantalconductances gAMPA and gGABA, respectively [25]. The terms mðnÞ

e ðtÞ andmðmÞ

i ðtÞ represent the fractions of postsynaptic receptors in the open stateat each individual synapse, and were described by the following pulse-based two-state kinetic equations:

dmeðtÞdt

¼ aeXk

T ðt te;kÞð1 meðtÞÞ bemeðtÞ;

dmiðtÞdt

¼ aiXk

T ðt ti;kÞð1 miðtÞÞ bimiðtÞ;ð4Þ

where te,k and ti,k denote the release times at each individual synapse, ae,iand be,i are forward and backward rate constants for opening of theexcitatory and inhibitory receptors, respectively. T(t) denotes the concen-tration of released neurotransmitter in the synaptic cleft at time t, and isconsidered to be a step-function with T(t) = Tmax for a short time periodtdur after a release and T(t) = 0 afterwards (pulse kinetics). The parametersof these kinetic models of synaptic currents were obtained by fitting themodel to postsynaptic currents recorded experimentally [26].

To simulate synaptic background activity, all synapses were activatedrandomly according to independent or temporally correlated Poisson pro-cesses with mean rates kexc and kinh for AMPA and GABA synapses,respectively, and correlation measure c [76]. The activation of N-methyl-D-aspartate (NMDA) receptors is minimal at the voltage levels inves-tigated here, and were not included for simplicity. Correlation, orredundancy, in the activity of the synaptic inputs was generated fromN0 independent Poisson trains with a fixed rate k describing the releaseactivity at N0 independent synaptic terminals. At each time step, the activ-ity state (release or no release) of these N0 terminals was uniformly redis-tributed across N target terminals (N > N0) with an average probability ofN/N0. Here, N and N0 are related by N 0 ¼ N þ

ffiffiffic

pð1 NÞ with 0 6 c 6 1.

For c > 0, the number of independent channels N0 < N, meaning that sev-eral synapses (at average N/N0 times the release probability at a single ter-minal) co-release at the given time, whereas each individual terminal stillreleases according to a Poisson process with rate k.

The model was accessed analytically and simulated numerically with amembrane area of a = 34,636 lm2 (corresponding to the layer VI neocor-tical pyramidal cells from cat parietal cortex used in [27]). Passive param-eters were gL = 0.0452 mS/cm2, EL = 80 mV, Cm = 1 lF/cm2 [22,67].The reversal potentials for synaptic conductances were Ee = 0 mV andEi = 75 mV. Synaptic parameters were: gAMPA = 1.2 nS, gGABA = 0.6 nS,ae = 1.1 · 106 M1 s1, be = 670 s1 for AMPA receptors, and ai = 5 ·106 M1 s1, bi = 180 s1 for GABA receptors, with Tmax = 1 mM, tdur =1 ms in both cases. The number of synaptic terminals and their releaserates were chosen to obtain an average membrane potential of about65 mV with standard deviation around 4 mV characteristic for in vivostates of cortical neurons [22], and were N = 4472, M = 3801, kexc =2.16 Hz, kinh = 2.4 Hz and c = 0.7 (referred to as ‘‘standard parametersetup’’). To investigate the robustness of the presented findings againstchanges in the synaptic activity, N and M were varied between 500 and10,000. The release rates covered a range of 0.21 Hz 6 kexc 6 4.32 Hz,0.24 Hz 6 kinh 6 4.80 Hz, and 0 6 c 6 0.99. In all cases, simulations of100 s neural activity with a temporal resolution of 0.01 ms wereperformed.

454 M. Rudolph, A. Destexhe / Journal of Physiology - Paris 98 (2004) 452–466

2.2. Point-conductance model

In the second model, referred to in what follows as the point-conduc-

tance model, synaptic inputs were described as resulting from two globalsynaptic conductances [27]. To that end, the excitatory and inhibitory syn-aptic conductances in Eq. (2) were formulated in terms of independentone-variable stochastic processes similar to the Ornstein–Uhlenbeck pro-cess [102]

dgeðtÞdt

¼ 1

seðgeðtÞ ge0Þ þ

ffiffiffiffiffiffiffi2r2

e

se

sneðtÞ;

dgiðtÞdt

¼ 1

siðgiðtÞ gi0Þ þ

ffiffiffiffiffiffiffi2r2

i

si

sniðtÞ;

ð5Þ

where ne,i(t) stands for Gaussian white noise sources with zero mean andunit standard deviation. ge,i0 are the mean (static) excitatory and inhibi-tory conductances, re,i and se,i are their respective standard deviationsand time constants [37].

This model was accessed both analytically and numerically. In the lat-ter case, the membrane area, passive parameters and synaptic reversalpotentials were as in the biophysical model described above. Other synap-tic noise parameter values were chosen to yield a corresponding activitycharacteristic for in vivo states of cortical neurons (V 65 mV,rV 4 mV; see [22,67]), and were ge0 = 12.1 nS, gi0 = 57.3 nS, re = 12 nS,ri = 26.4 nS, se = 2.73 ms and se = 10.49 ms (referred to as ‘‘standardparameter setup’’). To investigate the robustness of the presented findingsagainst changes in the synaptic activity, all synaptic parameters (ge,i0,re,i, se,i) were varied between 0% and 250% of their standard values.Simulations of 100 s (some simulations up to 500 s) neural activity witha time resolution of 0.1 ms (some simulations with 0.01 ms) wereperformed.

All numerical simulations were performed in the NEURON simulationenvironment [42] and were run on PC-based workstations under theLINUX operating system.

3. Results

With an aim to provide a link between intracellular andnetwork activity (Fig. 1), we first review the properties ofhigh-conductance states in cortical neurons, as well ascompare those properties with biophysical models and sim-plified models. We next review an analytic treatment of

Fig. 1. Linking intracellular and network activity. The activity of themembrane potential (intracellular activity) results from the firing ofthousands of excitatory and inhibitory neurons in the network (networkactivity). Using stochastic neuronal models, both types of activity can belinked, thus providing a way to extract network activity from intracellularrecordings.

these simplified models based on global conductances,which yields a method for estimating the mean and vari-ance of synaptic conductances from the sole knowledgeof membrane potential amplitude distributions. We thenlink statistical properties of the presynaptic discharge activ-ity to the characterization of synaptic noise in terms of themean and variance of synaptic conductances. Finally, wecombine both methods and demonstrate how the temporalcorrelation and average firing rate in the presynaptic neu-ronal population can be deduced from intracellularrecordings.

3.1. High-conductance states

It has long been known that neurons in the central ner-vous system are not quiescent but are continuously solic-ited by synaptic inputs of various sources (Fig. 2A, left).One of these sources is associated with asynchronous andirregular firing in the cortical network. During such activ-ity, the EEG, as an indicator of population activity, showslow-amplitude fast fluctuations (‘‘desynchronized EEG’’)[91]. Intracellular recordings in the cortex of awake animalsor under anesthesia (such as ketamine–xylazine or barbitu-rate) paralleled with EEG recordings, aiming at the charac-terization of the cellular activity during such EEG-activated

states, date back to over 30 years ago [2,3,6,8,60,67,91,92,110] (for reviews see [23,90]). They found, that theintense synaptic inputs due to network activity result inseemingly random fluctuations of the membrane potential.The latter contrasts the relative quiescence usually seen inintracellularly recorded neurons in cortical slices keptin vitro (Fig. 2A, right).

This subthreshold activity caused by the sustained dis-charge of a large number of presynaptic neurons, alsocalled synaptic noise [5,22,55], has been further character-ized in several experimental [1,13,17,22,50,64,67] and theo-retical studies [4,5,22,45,49,70]. It was found that synapticnoise maintains the membrane in a markedly depolarizedstate compared to resting or quiescent conditions (depolar-ization by about 15 mV up to an average Vm of 65 mV;Fig. 2B and D, left). The persistent Vm fluctuations result-ing from the synaptic bombardment are fast and of largeamplitude (characterized by a Vm standard deviation rVbetween 2 and 6 mV; Fig. 2B and D, middle), and causehighly irregular and tonic discharges in the 5–40 Hz fre-quency range [90] (Fig. 2A, right). Furthermore, the largenumber of synaptic inputs maintains the membrane in ahigh-conductance state with fluctuating and low input resis-tance [2,3,6,60,90,110] (typically, about 12 MX comparedto about 60 MX in quiescent conditions; Fig. 2C and D,right), resulting in small membrane time constants of theorder of about 4 ms [22,23]. Although differences in theabsolute value of the input resistance are reported (e.g.depending on the behavioural state such as wakefulnessor sleep [90]), in most cases the input resistance are rela-tively low compared to those reported in vitro [7,67], irre-spective of the cortical area.

-60 mV

A

B C

D

AMPA

GABA

20 mV

500 ms-80 mV

High-conductance state

Quiescent

ρ(V)

V (mV)-80 -70 -60 -50

0.05

0.1

0.15

High-conduc-tance state

Quiescent

-60 mV

High-conductance state

Quiescent

200 ms

5 mV

-80 mV


V (mV)

Quiescent

-80

-75

-70

-65

-60


σ(mV)

Quiescent

1

2

3

4


R (MΩ)

Quiescent

in

10

20

30

40

50

Fig. 2. High-conductance states in cortical neurons. (A) Characteristics of high-conductance states. Left: scheme of synaptic inputs to cortical pyramidalneurons. Cortical neurons are embedded in a dense network and receive several thousand of excitatory and inhibitory synapses. Two types of synapses areshown: excitatory glutamatergic (AMPA) synapses and inhibitory GABAergic (GABA) synapses. Assuming that each neuron displays spontaneous‘‘background’’ firing in the 5–40 Hz range, these synapses are actively releasing, thereby activating large sustained synaptic conductances in soma anddendrites. Right, top: illustration of the membrane potential (Vm) activity during high-conductance states, with a depolarized and highly fluctuatingactivity. Right, bottom: same neuron after suppression of synaptic activity, close to the conditions seen in most in vitro preparations. (B) Vm distributionduring the two states depicted in A. (C) Illustration of the difference of input resistance (Rin) in both states. Injection of a hyperpolarizing current pulseshows several-fold difference of Rin between high-conductance (top; Rin = 11 MX) and quiescent (bottom; Rin = 57 MX) states (0.2 nA hyperpolarizingcurrent injected in this example). (D) Differences of electrophysiological properties between high- and low-conductance states. From left to right: averageVm (V ), Vm standard deviation (rV) and input resistance (Rin).


Functionally, these high-conductance states may havedrastic consequences for the subthreshold dynamics ofcortical neurons [70] and may shape their superthresh-old response [4,12,35,43,45,68,75–77,82,96]. A first obviousconsequence is the inherent stochasticity in the dynamics ofneurons in high-conductance states, which results in a con-siderable degree of variability in their response [19,43,46].This stochastic aspect may form the functional basis foreffects like enhanced responsiveness and gain modulation[16,31,43,69,88,89]. Second, during high-conductancestates, the complex interplay between conductance noise,voltage-dependent currents and dendritic action potentialsmay result in an modulable equalization of the efficacy ofindividual synapses [77]. This equalization, accompaniedby a more localized integration of synaptic inputs, mayfoster parallelized computations in dendritic subunits[62,107]. Third, modeling studies have predicted thatneurons in high-conductance states should have sharpertemporal processing capabilities [4,5,45,77]. This includes

a reduced site-dependence of the timing of somaticresponses, the resolution of high-frequency synaptic stimu-lation, as well as the detection of rapid (2 ms) and faintchanges of temporal correlation among thousands ofsynaptic input sources [75].

The peculiarities of neuronal dynamics in high-conduc-tance states can be viewed as a fingerprint of the underlyingsynaptic activity and, therefore, the activity in the embed-ding network. In what follows, we will restrict to the linkbetween the subthreshold dynamics and network activity.We will address the question to which extend Vm record-ings can be used to infer the statistical features of corticalnetwork activity.

3.2. Models of high-conductance states

To describe the impact of the sustained and intense net-work activity in the cortex in vivo on neuronal dynamics, avariety of models of synaptic noise in cortical neurons were


proposed. In what follows, we will restrict to two types ofsuch models, which allow an analytical treatment and,thus, allow to establish a more exact link between intracel-lular activity and network dynamics.

By combining intracellular recordings with biophysicallydetailed computational models of single neurons, it was pos-sible to estimate parameters characterizing synaptic activity(such as synaptic densities, quantal conductances andrelease rates) and, this way, to capture some of the proper-ties characterizing synaptic noise [22,67]. In such models,synaptic background activity is commonly described bythousands of individual synaptic terminals which releaserandomly according to Poisson processes [5,22,43,52,58,59,70,75,77,96,100]. Here, conductance changes duringrelease at single synaptic terminals are described by kineticmodels of postsynaptic receptors for excitatory (glutamateAMPA and NMDA receptors) and inhibitory (GABAergicreceptors; Fig. 3, top) synapses. The combined activity ofmultiple channels reflects the statistical properties of dis-charge activity in the embedding network, such as the aver-age rate, temporal correlation or synchrony in the activity ofindividual or populations of neurons in the network. Thepostsynaptic membrane, either part of a detailed three-dimensional morphological structure (multi-compartmentmodels; see e.g. [77] and references therein) or viewed as asingle compartment [82,96], integrates these conductancechanges (Fig. 3, middle) and translates them into a changeof the membrane potential (Fig. 3, bottom).

Fig. 3. Relation between single-synapse activity, global synaptic conductancneuron subject to randomly releasing excitatory (AMPA) and inhibitory (GAinhibitory synapses (top right; in both cases the synaptic conductance is showtraces) resulting from the sum over all excitatory and inhibitory synaptic conduThe combined action of AMPA and GABAergic synapses was to set the neuronamplitude Vm fluctuations and irregular firing (bottom trace).

The integration of the activity at thousands of individualsynaptic terminals results in global excitatory and inhibi-tory conductances which fluctuate around a mean value(Fig. 4A, left). These global conductances are characterizedby nearly Gaussian amplitude distributions q(g) (Fig. 4B,gray), and their power spectral density S(k) follows aLorentzian behaviour [27] (Fig. 4C, gray). One of the sim-plest stochastic processes which captures these statisticalproperties is the Ornstein–Uhlenbeck stochastic process,which was introduced at the beginning of the last centuryto describe Brownian motion [40,102,106]. This modelhas long been used as a effective stochastic model of synap-tic noise [53,71]. Indeed, it can be shown that incorporatingsynaptic conductances described by OU processes into sin-gle-compartment models provides a compact stochasticrepresentation that captures in a broad parameter regimetheir temporal dynamics (Fig. 4A, right), amplitude distri-bution (Fig. 4B, black) and spectral structure (Fig. 4C,black) [27], as well as subthreshold and response propertiesof cortical neurons characteristic for in vivo conditions[31,69].

3.3. Analytic methods to analyze membrane potential

fluctuations

The relative simplicity of the description of high-con-ductance states by effective stochastic processes, such asthe OU stochastic process, allows to investigate analy-

es and membrane potential. Simulations of a single-compartment modelBAA) synapses. The activity at single excitatory synapses (top left) and

n for 10 synapses), together with the global synaptic conductance (middlectances (here from 4472 excitatory and 3801 inhibitory channels) is shown.into a high-conductance state, with an average Vm around 65 mV, high-

Fig. 4. Simplified models of synaptic background activity. (A) Comparison between detailed and simplified models of synaptic background activity. Left:biophysical model of a cortical neuron with realistic synaptic inputs (similar model as in Fig. 3). Synaptic activity was simulated by the random release of alarge number of excitatory and inhibitory synapses (4472 and 3801 synapses, respectively; see scheme on top) in a single-compartment neuron. Individualsynaptic currents were described by two-state kinetic models of glutamate (AMPA) and GABAergic (GABAA) receptors (see Section 2). The traces showrespectively the membrane potential (Vm) as well as the total excitatory (AMPA) and inhibitory (GABA) conductances. Right: simplified ‘‘point-conductance’’ model of synaptic background activity produced by two global excitatory and inhibitory conductances (ge(t) and gi(t) in scheme on top),which were simulated by stochastic processes [27]. The traces show the Vm as well as the global excitatory (ge(t)) and inhibitory (gi(t)) conductances. (B)Distribution of synaptic conductances q(g) (gray: biophysical model; black: point-conductance model). (C) Comparison of the power spectral densitiesS(k) of global synaptic conductances obtained in these two models (gray: biophysical model; black: point-conductance model). The two models sharesimilar statistical and spectral properties, but the point-conductance model is more than two orders of magnitude faster to simulate.


tically membrane equations within the framework ofstochastic differential calculus [36,63,103]. Here, we willfocus on the passive stochastic membrane equation, Eq. (1),with a total synaptic current Isyn(t), Eq. (2), described byexcitatory and inhibitory stochastic conductances ge(t)and gi(t), Eq. (5), respectively.

The probability that the membrane potential takes avalue V at time t is given by the probability density func-tion q(V, t). The dynamics of q(V, t) is described by a Fok-ker–Planck equation which, in the steady-state limitt ! 1, can be solved analytically [78]. The solution yieldsan asymmetric steady-state probability distribution q(V)for the membrane potential V

qðV Þ ¼ N exp½A1 ln½B1ðV EeÞ2 þ B2ðV EiÞ2þ A2 arctan½B3ðV EeÞ þ B4ðV EiÞ. ð6Þ

Here,

A1 ¼kLþ keþ kiþ ueþ ui2ðueþ uiÞ

;

A2 ¼ ½2Cðge0ui gi0ueÞðEeEiÞGLðueðEeELÞþ uiðEiELÞÞþ Iextðuiþ ueÞ=½ðEeEiÞ

ffiffiffiffiffiffiffiffiueui

p ðueþ uiÞ;

B1 ¼ ue=C2;

B2 ¼ ui=C2;

B3 ¼ue

ðEe EiÞffiffiffiffiffiffiffiffiueui

p ;

B4 ¼ui

ðEe EiÞffiffiffiffiffiffiffiffiueui

p ;

where kL = 2CGL, ke = 2Cge0, ki = 2Cgi0, ue ¼ r2e~se and

ui ¼ r2i ~si, denote voltage-independent coefficients which de-

pend on the passive membrane as well as synaptic noiseparameters, and N is a normalization constant such thatR11 dV qðV Þ ¼ 1. Examples of distributions q(V) describedby Eq. (6) are shown in Fig. 5A (black) and compared withcorresponding numerical results (gray) for a rather sym-metric (Fig. 5A, left) and an asymmetric (Fig. 5A, right)case. The apparent asymmetry of the distribution must beviewed as a result of the multiplicative coupling of the sto-chastic conductances to the membrane potential in Isyn(t).

The solution in the steady state limit, Eq. (6), is equiva-lent to a model in which excitatory and inhibitory con-ductances are described by effective Gaussian whitenoise processes with variances proportional to r2

fe;ig~sfe;ig.Here, ~sfe;ig denote effective noise time constants given by

A

B C

NumericalAnalytic

0.2

0.15

0.1

0.05

-70 -65 -60 -55V (mV)

ρ(V)

0.12

0.08

0.04

-70 -65 -60 -55V (mV)

ρ(V)

-50 -45

NumericalAnalytic

2 641 53 7

-68

-66

-64

-62

τ (ms)0

V(m

V)

2 641 53 7

1

2

3

τ

σ

(ms)0

(mV

)V

Fig. 5. Vm distributions in high-conductance states. (A) Representative example of symmetric (left) and asymmetric (right) Vm distributions. Numericalsolutions (gray) are compared with the corresponding analytic distributions (black). (B,C): Mean V and standard deviation rV of the Vm distribution as afunction of membrane time constant s0 = C/(GL + ge0 + gi0). Numerical solutions (dots) are compared with results obtained by numerical integration ofthe analytic solution (gray). Parameter values: gL = 0.0452 mS cm2, Cm = 1 lF cm2, EL = 80 mV, ge0 = 12 nS, gi0 = 57 nS, re = 3 nS, ri = 6.6 nS,a = 30,000 lm2 (A, left), a = 40,000 lm2 (A, right), 50 lm2 P aP 100,000 lm2 (B,C).


~sfe;ig ¼ 2sfe;igs0=ðsfe;ig þ s0Þ, where s0 = C(GL + ge0 + gi0) isthe membrane time constant characterizing the high-con-ductance state [79]. These effective noise time constantswere introduced to enable the expression to match the volt-age distribution for several orders of magnitude of param-eter values (see details in Ref. [79]).

The validity of this extended expression Eq. (6) wasassessed by comparison with numerical solutions of thecorresponding stochastic point-conductance model (seeSection 2). Here, the characterization of the membranepotential distributions in terms of their means V and stan-dard deviations rV showed an excellent agreement betweennumerical result and analytic description (Fig. 5B and C;see also [78,79]). Further tests were performed with bio-physically more complex models, such as one-compartmentmodels with thousands of synaptic noise sources (see Sec-tion 2). This analytic description of the Vm distributionwas also validated in cases where thousands of synaptic ter-minals were distributed across spatially extended dendriticstructures, and in cases were voltage-dependent conduc-tances for spike generation were incorporated into themodel [78,80]. Taken together, these results suggest thatEq. (6) is an excellent description of the subthreshold Vm

distribution in high-conductance states. It, moreover, pro-vides a direct link between the characterization of intracel-lular dynamics and the effective parameters describing thestatistical properties of network activity and, therefore,

may provide a method to estimate the parameters of synap-tic noise from intracellular Vm recordings.

To make this link more explicit, we make use of thenearly symmetric form of the steady-state membranepotential distribution q(V) given in Eq. (6). Replacing theexponent by its power expansion up to second orderaround the maximum of the probability distribution V ,yields a Gaussian approximation of q(V)

qGaussðV Þ ¼1ffiffiffiffiffiffiffiffiffiffiffi2pr2

V

p exp ðV V Þ2

2r2V

" #; ð7Þ

which allows a characterization in terms of its mean (ormaximum)

V ¼ C1

C2

ð8Þ

and standard deviation

r2V ¼ 1

C32

½C22ðueE2

e þ uiE2i Þ 2C1C2ðueEe þ uiEiÞ

þ C21ðue þ uiÞ; ð9Þ

where

C1 ¼ kLEL þ keEe þ kiEi þ ueEe þ uiEi þ 2CIext;

C2 ¼ kL þ ke þ ki þ ue þ ui.


The apparent dependence of V and rV on synaptic noiseparameters allows us to link this statistical properties ofthe Vm distribution with the mean ge,i0 and standard devi-ation re,i of the excitatory and inhibitory synaptic con-ductance, respectively. Indeed, combining and invertingtwo sets of Eqs. (8) and (9), each of which characterizesthe steady-state membrane potential distributions at agiven injected constant current (Iext1 and Iext2, respectively),we obtain

gfe;ig0 ¼DIext12ðr2

V 2DE2fi;eg1 r2

V 1DE2fi;eg2Þ

ðDEe1DEi2 þ DEe2DEi1ÞDEfei;iegDV2

12

DIext12DEfi;eg2 þ ðIext2 GLDEfi;egLÞDV 12

DEfei;iegDV 12

; ð10Þ

r2fe;ig ¼

2CDIext12ðr2V 1DE

2fi;eg2 r2

V 2DE2fi;eg1Þ

~sfe;igðDEe1DEi2 þ DEe2DEi1ÞDEfei;iegDV2

12

; ð11Þ

where DEfe;igf1;2g ¼ Efe;ig V f1;2g; DV 12 ¼ V 1 V 2 andDIext12 = Iext1 Iext2.

Eqs. (10) and (11) define a method, the VmD method, forestimating the mean and variance of excitatory and inhibi-tory synaptic conductances from the sole knowledge ofintracellular activity at two different injected constant cur-rents. The core of the method is shown in Fig. 6A: Intracel-lular recordings (Fig. 6A, upper left) at two differentinjected current levels with amplitude Iext1 and Iext2 yieldtwo membrane potential time courses (Fig. 6A, upperright). After removing spikes, these recordings capturethe subthreshold intracellular activity, which is primarilydetermined by synaptic activity. Gaussian fits (Fig. 6A,bottom right, black) of the corresponding Vm distributionsq(V) (Fig. 6A, bottom right, gray) yield values for the mean(V 1 and V 2) and standard deviation (rV1 and rV2) of themembrane potential at the two different current levels.These values serve as input parameters for Eqs. (10) and(11), yielding estimates for the means (ge0 and gi0) and stan-dard deviations (re and ri) of excitatory and inhibitory syn-aptic conductances (Fig. 6A, bottom left).

This approach was successfully tested by characterizingthe excitatory and inhibitory conductance distributions instochastic neuronal models of various complexity [80], inin vitro recordings in combination with dynamic-clampexperiments [80,24], and in vivo intracellular recordings[81]. By averaging intracellular activity over repetitions ofsimilar network states, an extension of the method evenallows to estimate the time course of synaptic conduc-tances, in particular their mean and variance (not shown).Taken together, these results suggest that the presenteddescription of synaptic noise in terms of effective synapticconductances, along with the analytic solution given inEq. (6), which characterizes the subthreshold intracellularactivity, may open a new way for estimating effective statis-tical properties of excitatory and inhibitory conductancesin different network states, or even their time course duringresponses to stimuli.

3.4. Inferring network activity

So far, we described stochastic neuronal dynamics interms of effective stochastic processes for synaptic con-ductances, specifically the OU process defined in Eq. (5).Although such an effective representation allows to charac-terize synaptic noise in terms of the mean, variance and timeconstant of the total conductances resulting from synapticbombardment, information about the firing statistics ofthe presynaptic neurons and, hence, the statistical propertiesof the activity in the surrounding network can not directly bededuced. However, as pointed out in Section 3.2, the effec-tive stochastic conductances are a result of the combinedactivity at thousands of synaptic terminals releasing accord-ing to independent or temporally correlated Poisson pro-cesses. Thus, in principle, the statistical characteristics ofthe total synaptic conductances are linked to the statisticalproperties of the discharge activity of presynaptic neurons.

To draw this link more explicitly, we make use of thenotion of shot noise stochastic processes [72,73]. Originallytermed ‘‘Schroteffekt’’ [14,15,84], shot noise processesdescribe the output of a dynamical system (e.g. the cellularmembrane) characterized by a quantal response function(e.g. the conductance time course of a single release event)activated by a sequence of singular impulses occurring atrandom times [66], e.g. in a Poisson-distributed fashion[38,41,65]. This provides a direct link between discreteevents in time (namely the presynaptic release activity), thequantal kinetics characterizing a dynamical system (namelythe synaptic kinetics), and effective statistical parametersdescribing the systems output (namely the total synapticconductances). In neuroscience, however, the applicationof this powerful notion remains so far sparse, and restrictsmostly to the characterization of the behaviour of cells inthe early visual system [33,34,83,94, 95,109], at neuromus-cular junctions [32], the discharge behaviour in simpleneuron models [57], or information transfer in integrate-and-fire neuron models [44].

To which extent the notion of shot noise can be utilizedfor deducing statistical properties of the network activitydepends on how faithful the effective model of synapticnoise (Fig. 4A, right) represents the combined activity atthousands of synaptic terminals. As already outlined inSection 3.2, investigations of models with multiple synapticinputs channels impinging on a single compartment showthat the statistical (Fig. 4B) and spectral (Fig. 4C) charac-teristics of the total synaptic conductances provided byAMPA and GABAergic synapses indeed resemble, in abroad parameter regime, those of appropriate OU stochas-tic processes for excitatory and inhibitory conductances,respectively. In particular, the amplitude probability distri-butions q(g) of the total excitatory and inhibitory conduc-tances, which are Gaussian in the case of an OU stochasticprocess (Fig. 4B, black), take a nearly symmetric Gaussianform for multiple synaptic input channels (Fig. 4B, gray) aswell. Therefore, q(g) can be sufficiently well characterizedby its mean g and standard deviation rg.

Fig. 6. The ‘‘VmD method’’ for estimating synaptic conductances from active states. (A) Top left shows a scheme of intracellular recordings in corticalneurons in current-clamp configuration. The spontaneous activity during active states is recorded at two different injected current levels (Iext1, Iext2; topright), yielding two Vm distributions (bottom right; gray). The mean and variance obtained by Gaussian fits of these distributions (bottom right; black)serve as input to Eqs. (10) and (11), from which the mean (ge0 and gi0) and standard deviation (re and ri) of excitatory and inhibitory synapticconductances can be estimated (bottom left). (B) Test of the VmD method using biophysical models. Left: Distributions of the global synapticconductances q(g) obtained in a biophysical model of cortical neuron in a high-conductance state. The model was a single-compartment neuron whichreceived thousands of excitatory and inhibitory synapses (4472 AMPA and 3801 GABAA synapses) releasing randomly according to independent Poissonprocesses (same model as in Fig. 4). Right: Estimates of synaptic conductances from voltage fluctuations. Numerical simulations at two different injectedconstant currents Iext1 and Iext2 yielded membrane potential distributions (dark and light gray), which were used to estimate the mean and standarddeviation of synaptic conductances. The estimated values were used to reconstruct the distributions for inhibitory and excitatory conductances (left,black). These estimates were in remarkable agreement with those obtained numerically, both at the level of global conductances (left) and of the Vm (right).


From a more mathematical point of view, explicitexpressions for g and rg of single-channel Poisson-drivenshot noise process with integrable quantal response func-tions g(t), e.g. the two-state kinetic models considered here(see Section 2), are given by Campbells theorem [14,15,72,73]

g ¼ kZ 1

1dtgðtÞ; ð12Þ

r2g ¼ k

Z 1

1dtgðtÞ2. ð13Þ

A prerequisite for the applicability of Campbells theoremis the additive nature of the quantal processes. This is, ingeneral, not true for the two-state kinetic process consid-ered (see Section 2). Here, the change in the fraction of

channels in the open state me(t) and mi(t) at a new releasedepends on their actual value and, thus, on their releasehistory (see Eq. (4)). However, as numerical studies showed,additivity is well fulfilled for multiple synaptic channelswith low average release rates k and small effective synaptictime constants s 1/k for each individual channel. Theseconditions are, in general, met by realistic models ofAMPA and GABAergic synapses (see Section 2).

In order to extend Campbells theorem to the case ofmultiple correlated input channels as described above, weconstruct a new shot noise process based on the followingtwo assumptions: First, the time course of k co-releasingidentical quantal events gk(t) equals the sum of k quantaltime courses: gk(t) = kg1(t), g1(t) g(t). Second, the outputdue to N correlated Poisson processes equals the sum over


the time course of k (k = 1, . . . ,N0) co-releasing identicalquantal events gikðtÞ stemming from N0 independent Pois-son trains i = 1, . . . ,N0. For each of the N0 independentPoisson trains, gikðtÞ is weighted according to a binomialdistribution, which describes the probability that the activ-ity state of one of the N0 independent synaptic channels isredistributed across k out of the N total synaptic channels.With this, we obtain a direct extension of Campbells theo-rem to the case of multiple synaptic input channels. Specif-ically, the mean g and variance r2

g of multiple correlatedshot noise processes are given by

g ¼ kN 0

XNk¼0

qkðN ;N 0ÞZ 1

1dtgkðtÞ; ð14Þ

r2g ¼ kN 0

XNk¼0

qkðN ;N 0ÞZ 1

1dtgkðtÞ

2; ð15Þ

where

qkðN ;N 0Þ ¼N

k

pkð1 pÞNk

denotes the binomial weighting factor.Application of Eqs. (14) and (15) to the synaptic inputs

described by two-state kinetic models (see Section 2), weobtain

g ¼ D1kN ; ð16Þ

r2g ¼ D2kN 1þ N 1

N þ ffiffiffic

p ð1 NÞ

ð17Þ

for the mean g and variance r2g of the total synaptic con-

ductance, respectively. Here,

D1 ¼gmaxaTmax

bA2b tdurAþ etdurA 1

þ A 1 etdurA

;

D2 ¼g2maxa

2T 2max

2bA3fb 2tdurAþ 4etdurA e2tdurA 3

þ Ae2tdurA 1 etdurA

2g;with A = b + aTmax and the quantal conductance of thesynaptic channels gmax, are coefficients which only dependon the kinetic parameters of the synaptic model. Moreover,the spectral structure of the total conductances provided bythousands of synaptic input channels resembles for low-frequency m < 500 Hz that of an OU stochastic process.Therefore, the power spectral density S(m) can be describedby a Lorentzian

SðmÞ SLorentzðmÞ ¼2Ds2

1þ ð2psmÞ2; ð18Þ

where the total power D and effective time constant s arefunctions of the kinetic parameters.

Similar analytic expressions can be obtained for otherstatistical measures, such as correlation functions andamplitude distribution, and for a variety of synaptic kineticmodels, ranging from simple exponential models up tomulti-state Markovian models. In all cases, it can be shown

that the mean g, Eq. (16), and variance r2g, Eq. (17), of the

total synaptic conductance are monotonic functions of thetemporal correlation c and average single channel rate k.Here, an increase in k leads to a proportional increase ing without change in r2

g (Fig. 8B, compare black and gray).On the other hand, c is the determinant of the noise vari-ance, with an increase in the fluctuation amplitude forincreasing c (Figs. 7 and 8B, compare gray). From Eq.(17) we see that at zero correlation, r2

g takes its smallestvalue proportional to 2kN for N! 1. For fixed N, rgreaches its upper bound N2k for c = 1 and all N. These ana-lytic results were in excellent agreement with the numericalsimulations at various correlation levels and for differentsynaptic kinetics (not shown).

Interestingly, the sensitivity of conductance fluctuationsand, thus, the amplitude of resulting Vm fluctuations totemporal correlation among thousands of synaptic inputsprovides a way to detect changes in the statistics of the syn-aptic input activity. Previously it was shown that changesin the correlation led to a detectable change in the cellularbehaviour [75] (Fig. 7), even for faint (Pearson correlationcoefficient of about 0.05) and brief (down to 2 ms) correla-tion changes. Indeed, the monotonic dependence of themean, Eq. (16), and variance, Eq. (17), of the total conduc-tance on the channel firing rate k and temporal correlationc among multiple channels provide a method for character-izing presynaptic activity in terms of k and c based on thesole knowledge of g and r2

g (Fig. 8A)

k ¼ gD1N

; ð19Þ

c ¼ðD2gð2N 1Þ D1Nr2

gÞ2

ðN 1Þ2ðD2g D1r2gÞ

2. ð20Þ

Experimentally, values for g and r2g can either be obtained

using the voltage-clamp protocol, from which distributionsfor excitatory and inhibitory conductances can be esti-mated (Fig. 8A, top left), or by using the VmD method(Section 3.3; Fig. 8A, bottom left). Both approaches yieldestimates for the mean (ge0 and gi0) and standard deviation(re and ri) of excitatory and inhibitory conductances, fromwhich the correlation and individual channel rates for bothexcitatory and inhibitory subpopulations of synaptic inputscan be estimated using Eqs. (19) and (20), see Fig. 8A(right).

We tested this paradigm in numerical simulations of sin-gle compartmental models, in which the temporal correla-tion and average release rate at single synaptic terminalswere changed in a broad parameter regime (see Section2). Synaptic conductances were obtained using the volt-age-clamp [27] protocol (data not shown) or the VmDmethod based on current-clamp recordings [80] (see Section3.3). For single-compartment models, the estimated valuesfor the average release rate k matched nearly exact theknown input values for both excitatory and inhibitorysynapses (Fig. 8C, left). A good agreement was alsoobtained for estimates of the correlation c (Fig. 8C, right),

Fig. 7. Correspondence between network activity and global synaptic conductances. A biophysical model of cortical neuron in high-conductance state wassimulated (same model as in Fig. 4, with 4472 AMPA and 3801 GABAergic synapses releasing randomly). The graph shows, from top to bottom: rasterplots of the release times at excitatory (AMPA) and inhibitory (GABA) synapses, the global excitatory (ge(t)) and inhibitory (gi(t)) synaptic conductances,and the membrane potential (Vm(t)). The synapses were initially uncorrelated for the first 250 ms, resulting in rasters with no particular structure (top), andglobal average conductances which were about 15 nS for excitation and 60 nS for inhibition (bottom). At t = 250 ms, a correlation was introduced betweenthe release of individual synapses for a period of 500 ms (gray box). The correlation increases the probability of different synapses to co-release, whichappears as vertical bands in the raster. Because the release rate was not affected by correlation, the average global conductance was unchanged. However,the amount of fluctuations (i.e. the variance) of the global conductances increased as a result of this correlation change, and led to a change in thefluctuation amplitude of the membrane potential. Such changes in the Vm can affect spiking activity, thus allowing cortical neurons to detect changes in therelease statistics of their synaptic inputs.


independent on the protocol used. Similar investigationsperformed with multi-compartment models, however,showed that the suggested method yields an underestima-tion of k and c, especially for small c due to the filtering ofsynaptic inputs in spatially extended dendrites. Here, fur-ther research is needed to assess the applicability of the pro-posed shot noise approach.

4. Discussion and conclusion

What does the intracellular activity of a single neuron inan intact and active brain tell us about the activity in theembedding network? Can a link be drawn between the cel-lular and network dynamics based on the apparently sto-chastic aspects of neuronal activity? In this paper, wepropose an answer to these questions based on theoretical

and computational considerations. First, we show thatsubthreshold Vm fluctuations can be modeled by globalconductances (described by Ornstein–Uhlenbeck typestochastic processes) and that this type of effective modelprovides an excellent representation of the statistical prop-erties of more realistic—but at the same time also morecomplex—models of in vivo neuronal dynamics. Propertiessuch as the power spectral density and the distributions ofsynaptic conductances are remarkably well captured. Sec-ond, we reviewed a recent approach [78–80] in which fittingan analytic expression of the steady-state Vm distributionto (experimentally or numerically obtained) subthresholdVm distributions yields estimates of the best effective modelthat describes this subthreshold activity. Application of thisapproach to models described by thousands of individualsynaptic conductances leads to accurate estimates of themean and variance of the global synaptic conductances.

Fig. 8. Characterization of network activity from the mean and variance of synaptic conductances. (A) Method for estimating statistical properties of theactivity at multiple synaptic inputs terminals and, hence, of the activity in the embedding network. A statistical characterization of the total synapticconductances can either be obtained using the voltage-clamp protocol (top left) or the VmDmethod utilizing the current-clamp protocol (bottom left). Themean (ge0,gi0) and standard deviation (re,ri) of excitatory and inhibitory synaptic conductances (middle) yield estimates for the correlation c and averagerelease rate k at synaptic terminals (right) by using Eqs. (19) and (20). (B) Example of the dependence of the mean and variance of the total synapticconductances on the frequency and correlation of multiple synaptic input channels. Changes in the release rate (k, from 1 Hz to 5 Hz, from black to darkgray lines, respectively) leads to a proportional change in the mean and variance of the global conductance (GABAergic synapses were used here; forparameter values see Section 2). However, changes in the temporal correlation alone (from c = 0 to c = 0.7; from dark to light gray), did not impact on themean but only affected the variance. (C) Estimation of the rate k (left) and temporal correlation c among the multiple synaptic input channels (right) fromthe mean and variance of the global conductance distribution for different levels of background activity. The values estimated from the conductancedistribution obtained by using the VmD method are shown as functions of the actual values used for the simulations.


Finally, we established the link between this effectivedescription with global synaptic conductances and mean-ingful physiological parameters. As expected, the meanconductances are related to the average release rate at syn-aptic terminals. More interestingly, estimating the varianceof global conductances gives access to the temporal corre-lation among presynaptic neurons. Thus, this approachyields means to characterize statistical properties of net-work activity based on the sole knowledge of intracellularactivity, and should be applicable to experimental data.

Before application to experimental data, however, anumber of problems must be solved. The main problemis the possible contamination by voltage-dependent cur-rents. Several ionic conductances may be active at sub-threshold levels, such as the hyperpolarization-activatedcurrent Ih or various K+ currents. These currents are sus-ceptible to significantly alter the voltage distribution, possi-bly resulting in errors in the estimate of conductances.There are several ways to cope with this problem. First,one could try to integrate these currents explicitly in themembrane equation, e.g. by using linearizations of Hodg-kin–Huxley based voltage-dependent conductances forspike generation [48,61], and obtain expressions of mem-brane potential distributions that include their effect. Sec-ond, these currents may be blocked pharmacologically

(for example using cesium and QX314 in the recordingpipette), resulting in a more linear membrane. Third, thecontamination of voltage-dependent currents could be mini-mized by recording the subthreshold activity in the linearregime of the current–voltage relation of the membrane(as in [22]), which presumably contains few or no voltage-dependent contribution. Fourth, the synaptic filtering inspatially extended dendritic structures, which yields anunderestimation of the average release rate and temporalcorrelation, has to be incorporated. A possible solutionmight be to modify the shot noise description by usingquantal response functions whose properties depend onthe location in the dendritic tree.

The proposed approach should be tested in experimentsby comparing it to more standard methods, such asvoltage-clamp, ideally using the same cells. This type ofexperiment is feasible in vitro, where a switch from volt-age-clamp to current-clamp conditions can be easily real-ized in the same recording (see e.g. [86]; see also Ref. [10]in vivo). The estimates of conductance could then be com-pared from one method to another, thereby assessing thereliability of these estimates. However, it must be pointedout that voltage-clamp methods suffer from errors due tothe electrode series resistance. The current produced inthe membrane must be compensated by the voltage-clamp


amplifier. This current flows through the electrode seriesresistance, and the resulting ohmic drop causes an under-estimate of the conductances, although this drop can bepartially corrected (see [8]). The present VmD method isentirely based on current-clamp, and there is no such prob-lem because the currents are integrated by the membraneitself, while the electrode only reads the ‘‘results’’ of thisintegration. An in-depth theoretical and experimentalcharacterization of these aspects is needed to adequatelycompare the methods and possibly refine the presentapproach. In any case, the approach outlined here bringsus a step further towards our goal to obtain means toextract properties of network activity through analysis ofintracellular recordings, a goal that is only reachablethrough a tight combination of experimental and theoreti-cal approaches.

Acknowledgements

This research was supported by the Centre National dela Recherche Scientifique (CNRS, France), Future andEmerging Technologies (FET, European Union), theHuman Frontier Science Program (HFSP) and theNational Institutes of Health (NIH, USA). Additionalinformation is available at http://cns.iaf.cnrs-gif.fr.

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