The statistics of spikes trains: a stochastic calculus approach

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Jonathan Touboul, Olivier Faugeras

To cite this version:

Jonathan Touboul, Olivier Faugeras. The statistics of spikes trains: a stochastic calculus approach.

[Research Report] RR-6224, INRIA. 2007, pp.46. �inria-00156557v3�

a p p o r t

d e r e c h e r c h e

9-6399ISRNINRIA/RR--6224--FR+ENG

Thème BIO

The statistics of spikes trains: a stochastic calculus approach

Jonathan Touboul and Olivier Faugeras

N° 6224

June 21, 2007

Jonathan Toubouland OlivierFaugeras

ThèmeBIOSystèmesbiologiques

ProjetOdyssée

∗

Rapportdereherhe n°6224June21,200746pages

Abstrat: Wedisussthestatistisofspikestrainsfordierenttypesofintegrate-and-re

neuronsanddierenttypesofsynaptinoisemodels. Inontrastwiththeusualapproahes

in neurosiene, mainly based on statistial physis methods suh as the Fokker-Plank

equationorthemean-eld theory,wehosethepointoftheviewof thestohasti alulus

theoryto haraterizeneuronsin noisyenvironments. Wepresentfour stohasti alulus

tehniques that an be used to nd the probability distributions attahed to the spikes

trains. We illustrate the power of these tehniques for four types of widely used neuron

models. Despitethefatthatthesetehniquesaremathematiallyintriatewebelievethat

they an be useful for answering questions in neurosiene that naturally arise from the

variabilityofneuronalativity. Foreahtehniqueweindiateitsrangeofappliationand

itslimitations.

Key-words: neuronmodels,stohastiproesses,spikestrainsstatistis,rsthittingtime

∗

OdysséeisajointprojetbetweenENPC-ENSUlm-INRIA

Résumé: Nousnousintéressonsauxstatistiquesdetrainsdespikespourdiérentstypesde

neuronsintègre-et-tireetdiérentstypesdemodèlesdebruitsynaptique. Aladiérenedes

approheslassiquesenneurosienes,prinipalementbaséessurdesméthodesdephysique

statistiquetellesquel'équationdeFokker-Plankoulathéorieduhampmoyen,noushoi-

sissonslepointde vuede lathéoriedualul stohastiquepourharatériserlesneurones

dansdesenvironnementsbruités. Nousprésentonsquatre méthodesdealul stohastique

qui peuvent être utilisées pour haratériser la distribution de probabilité des trains de

spikes. Nousillustronslapuissanedeestehniquespourquatretypesdemodèlesdeneu-

rones ouramment utilisés. Bien que es tehniques soient mathématiquement omplexes,

nous royonsqu'elles peuvent êtreutiles pourrépondre auxquestions de neurosienequi

seposentnaturellement quand ons'intéresse àlavariabilité del'ativité neuronale. Pour

haunedestehniquesproposées,nousindiquonssondomained'appliationet seslimites.

Mots-lés: modèlesdeneurones,proessusstohastiques,tempsdespikes,premiertemps

d'atteinte

Introdution

Duringthe past thirty years, modelling and understanding the eets of noisein ortial

neuronshas been aentral and diult endeavorin neurosiene. Many approahes have

beenusedinordertoharaterizethespikestrains,mostofthemborrowedformstatistial

physis. Atthelevelofthe ell,theeets of noisehavebeenstudied rst byKnight[33℄

whointrodued andstudied the rstnoisy integrate-and-reneuron model. His work has

beengeneralizedbyGerstner[26℄. Brunelused theFokker-Plankequationtoharaterize

the eet of noiseat the level of the ell [12, 11℄ and of the network [10, 9℄. Samuelides

andhis olleaguesusedthemeaneld andlargedeviationsframeworkto haraterizelarge

sets of randomly onneted neurons driven by noise [56℄. In the present paperwe adopt

the point of view of the theory of stohasti alulus in an attempt to haraterize the

stohastipropertiesofneuronmodels andthestatistisofthespikestrainstheygenerate.

Weillustratethesetehniqueswithfourtypesofwidelyusedneuronmodels.

The tehniquesare mathematially quite intriate. Nevertheless, we believe that they

anbeusefulforansweringquestionsinneurosienethatnaturallyarisefromthevariability

of neuronal ativity. For instane, they an give aess to the probability distribution of

the spikes trains, while other methods only givepartial informations onthis distribution.

Moreover, the use of stohasti alulus methods enables us to get rid of suh tehnial

hypothesesasthestationarityoftheproess,thesparsityofthenetworksorthetimesales

approximations,whih aregenerallyrequired. Foreahtehniquepresentedweindiateits

rangeofappliabilityanditslimitations.

Intherstsetion, wedisusstheoriginofthevariabilityin ortialneuronsandtheir

mathematialmodelling,andjustifytheuseoftheBrownianmotion. Intheseondsetion,

wepresentdierentlassialmathematialmodels,whihdierintheirintrinsidynamisor

inthenoisemodelsused. Thethirdsetionisdediatedtothepresentationoffourimportant

stohasti methods for omputing spikes trains statistis, and to their appliation to the

dierent types of neurons presented in theseond setion. A largeappendix summarizes

briey the main mathematial notions that are needed in order for the paperto be self-

onsistentforreaderswhosestohastialulusis abitrusty.

1 Noise in neurons: soures and models

Invivoreordingsof neuronalativityareharaterizedbytheirhighvariability. Dierent

studiesofthespikestrainsofindividualneuronsindiatethattheringpatternsseemtobe

random. The originof the irregularityin the eletrialativityof ortial neuronsin vivo

hasbeenwidely studied and hasreeived nosatisfatory answersofar. Neverthelessit is

ommonlyadmittedthata) partofthis variabilityanbeonsideredasnoise[60, 58℄,and

b)thatalargepartofthenoiseexperienedbyaortialneuronisduetotheintensiveand

randomexitationofsynaptisites.

Wedesribesomeofthebiologialevidenethatsupportsthesestatementsandpropose

mathematialmodelsofthesynaptinoise.

1.1 Soures of variability

Itisgenerallyagreedthatalargepartof thenoiseexperiened byaortial neuronisdue

totheintensiveandrandomexitationofsynaptisites.

It has been observed from in vivo reordings of ortial neurons, in awake [15℄ and

anesthetizedanimals[18℄thataspontaneousativityexistsandthattherelatedspikeproess

anbeonsideredasPoisson. ThisPoissonmodelofindependentsynaptiinputs,orrather

itsdiusionlimitapproximation,isthemodelweusehere.

The origin of irregularitiesis still poorlyknown. Gerstner and Kistler in [26℄ disuss

this question at length. They obtain an interesting lassiation, and show that we an

distinguish between intrinsi noisesoures that generates stohasti behavior at the level

of the neuronal dynamis and extrinsi soures arisingfrom network eets and synapti

transmission. Webrieysummarizethemainpoints:

ˆ A permanent noise soure is the thermal noise linked with disrete nature of ele-

trihargearriers. Flutuationslinked withthis phenomenonare howeverof minor

importane omparedtoothernoisesouresin neurons.

ˆ The nite number of ion hannels is another noise soure.Most of the ion hannel

haveonlytwostates: theyareopenorlosed. Theeletrialondutivityofapath

of membrane is proportional to thenumber of open ion hannels. The ondutivity

thereforeutuatesandsodoesthepotential 1

ˆ Noise is alsodue to signaltransmissionand networkeets (extrinsi noise): synap-

ti transmission failures, randomness of exitatory and inhibitory onnetions, for

instane, and global networks eets (see for instane [10℄) where random exita-

tory/inhibitory onnetivity an produe highly irregular spikes trains even in the

abseneofnoise.

Intermofneuronmodelsweonentrateonseverallassesofintegrate-and-respiking

neuronmodels beausetheybringtogether arelativemathematial simpliityand agreat

power of expression. In this eld, Knight [33℄, pioneered the study of the eet of noise

withasimpliedmodelinwhihthethresholdwasdrawnrandomlyaftereahspike. Ger-

stner[26℄ extendedthese resultsand studied bothslownoise models, in whih either the

thresholdortheresetisdrawnrandomlyaftereahspike,andfastesaperatenoisemodels.

In theontext of synhrony in neuronal networks, Abbott et al [1℄ studied a phase noise

model. However,noneofthesemodelsanrepresentin arealistiwaythesynaptinoiseas

experienedbyortialneurons.

We onentrateon theeet of synapti urrents. Synapti urrentsan be desribed

byasimplesystemof ordinarydierentialequations(see forinstane [17℄). We studythe

impat of noise originating from realisti synapti models on the dynamis of the ring

probabilityofaspikingneuron.

1

Thereexistsmodelstakingintoaountthenitenumberofionhannel,andthattheyanreprodue

theobservedvariabilityinsomeases(seeforinstane[16℄)

Beause of spae onstrains we only explore twolevels of omplexity for the synapti

urrents,1) instantaneous(desribed by deltafuntion) synaptiurrents,and2) synapti

urrents desribed by an instantaneous jump followed by an exponential deay. The dy-

namisoftheringprobabilityofaneuronreeivingabombardmentofspikesthroughsuh

synaptiurrentsis studied in the framework of the diusion approximation (in the neu-

ronalontext,see[66℄). Thisapproximationisjustiedwhenalargenumberofspikesarrive

throughsynapsesthatareweakomparedtothemagnitudeoftheringthreshold,whihis

therelevantsituationintheortex. Inthediusionapproximation,therandomomponent

in thesynaptiurrentsanbetreatedasaBrownianmotionin theaseof instantaneous

synapses. Ontheotherhand,whensynapseshaveanitetemporalresponse,asinthemore

realisti models, synapti noise has a nite orrelation time and thus beomes olored

noise. Thanksto thediusionapproximation,thedynamisoftheringprobabilityanbe

studiedin theframeworkofthestohastialulustheory(seeforinstane[31℄).

1.2 Synapti noise modeling

ManymathematialdesriptionsofthesynaptiurrentI^{synhavebeenproposed(seeDes-}

texhe et al [17℄ or [26℄). Weonsider twotypes of inreasingly omplexsynapti urrent

models:

(i). Instantaneous synapses: if we neglet the synaptiintegration, onsidering that the

synaptitimeonstantsaresmallwithrespettothemembraneinteration,thepost-

synaptiinputanbedesribedbyaBrownianmotion,whihisthediusionapprox-

imationofaresaledsumofPoissonproesses. Forthisweassumethat thesynapti

inputsarespikesarrivingatN^synapsesi∈ {1, . . . , N}^{,eahwithasynaptieieny} ωi^{,atthespikestimes}t^k_i^{. Theinputsynaptiurrentanbewritten:}

dI_t^syn=

N

X

i=1

ωi

X

k

δ(t−t^k_i)^def=

N

X

i=1

ωidSi(t), ^(1.1)

where the Si(t)^{s are point proesses} representing the spikes trains arriving in eah synapse.

Neurons are onneted to thousand of neurons (in general, N ≈ 10³−10^{4). If we}

assumethatthesynaptiinputspikestimesfollowaprobabilitylawwithmeanµi^and

varianeσ²_i ^{(forinstane Poissonproesses,}σ_i²=µi^{)and arepairwise} independent 2

,

I^{syn is the sum of} N independent Poisson proesses, of mean ωiµi ^{and of variane}

2

Theindependenehypothesisisakeyhypothesisandisquitediulttojustifybiologially.Nevertheless,

thesameresultwouldholdunderverytehnialandstrongonditionsonthedeorrelation ofthe proess.

Itisaveryintriatetheoryandwewillnotdealwithithere.

ω_i²µi^{. Weassumethatthe}ωi^{saresuhthatthereexist} µ, σⁱⁿ(0,∞)^suhthat3:





 PN

i=1ωiµi_N−→

→∞µ PN

i=1ω_i²µi −→

N→∞σ²

ByDonsker'stheorem[5℄

N

X

i=1

ωi

Si(t)−µit _L

−→σWt ^(1.2)

where(Wt)t≥0^{isastandardBrownianmotion(seeAppendixA.1foradenition),and}

thesymbol

−→L ^{indiatesthattheproessonthelefthandsideonvergesinlawtothe}

proessontherighthandsidewhenN → ∞^.

Thediusionapproximationonsistsinapproximatingthesynaptijumpproess(1.1)

bytheontinuousproess:

I_t^syn=µt+σWt ^(1.3)

(ii). Exponentiallydeaying synaptiurrent: beausethe postsynapti interationhas a

niteintegrationtime, sayτs^{,thefollowingequationarisesnaturally}

τsdI_t^syn=−I_t^syndt+

N

X

i=1

ωi

X

k

δ(t−t^k_i) ^(1.4)

Note that wehaveassumed that τs ^{wasthesamefor allsynapsesand negletedthe}

rise time of thesynapti urrent. The seond assumption is justied on theground

thattherisetimeofasynapseistypiallyveryshortomparedtotherelaxationtime.

A diusion approximation similar to the one in the previous paragraph yields the

followingdiusion approximationofthesynaptinoisewithexponentialdeay:

τsdI_t^syn= (−I_t^syn+µ)dt+σdWt ^(1.5)

2 Neuron Models

In this paper, a neuron model is dened by (i) a membrane potential dynamis and (ii)

a synapti dynamis. The neuron emits a spike when its membrane potential reahes a,

possiblytime-varying, thresholdfuntion θ(t)^{. We are interestedin} haraterizing the se- quene {ti}, i = 1,· · ·^, ti > 0^, ti+1 > ti ^{when the neuron emits spikes. We present four}

3

Ingeneralthisonditionanbeahievedbyaresalingandahangeoftimeappliedtotheproess

simplemodelsofspikingneuronssubmittedtonoisysynaptiinput,disusstheirbiologial

relevane and perform abasi stohasti analysis of the spikestimes. In detail, aneuron

modelisdened byanequation:

τmdVt=f(t, Vt)dt+Ie(t)dt+dI^syn(Vt, t) ^(2.1)

wheref(t, v)^{governsthefreemembranepotentialdynamis,}Ie(t)^{istheinjetedorexternal}

urrent and the deterministi term of synapti integration, and I_t^{syn representsthe noisy}

synaptiinputsduetobakgroundsynaptiativity.

Inthefollowingsetions,wereviewdierentmodelsofneuronaldynamisinwhihthe

synaptiurrentanbedesribedbyoneofthemodelsdisussedin setion1.2.

2.1 Model I: The noisy leaky integrate-and-re model with instan-

taneous synapti urrent

Thesimplest model weonsider istheintegrate andrewhere themembranepotentialV

followsthefollowingstohastidierentialequation:

(τmdVt= (Vrest−Vt+Ie(t))dt+σdWt

V0= 0 ^(2.2)

whereτm^{isthetimeonstantofthemembrane,}Vrest ^{therestpotentialand}Wt^aBrownian

proessrepresentingthesynaptiinput. ThisequationistheOrnstein-Uhlenbekequation.

The neuron emits a spike eah time its membrane potential reahes a threshold θ ^{or a}

thresholdfuntion θ(t)^{. Whenaspikeisemitted,themembranepotentialis}reinitializedto theinitialvalue,e.g. 0.

Thisisthesimplestontinuousnoisyspikingmodel. Theleakyintegrate-and-reneuron

wasrstintroduedbyLapique[37℄inadisussiononmembranepolarizability. Itidealizes

theneuronasaapaitorinparallelwitharesistoranddrivenbyaurrentIe^{(seee.g. [26℄).}

The noisy integrate-and-re neuron with instantaneous synapti urrent (2.2) has re-

entlyreeivedalotofattentiontoinvestigatethenatureoftheneuralode[42,65,13,57℄.

As shown in setion 1.2, equation (1.3), it anbe seen as the diusion approximation of

Stein'smodel[25, 61℄wherethesynaptiinputsareonsideredasPoissonproesses.

Itisoneofthefewneuronalmodelsforwhihanalytialalulationsanbeperformed.

Indeed,equation(2.2)anbesolvedinalosedform:

Vt=Vrest(1−e⁻

t

τm) +_τ¹_m Rt 0e

s−t

τmIe(s)ds+_τ^σ_m Rt 0e

s−t

τmdWs ^(2.3)

Thestohasti proessVt ^is Gauss-Markov. It is the sumof a deterministipartand the produtofe^{−t/τmwiththerandomproess}Rt

0e^s/τmdWs^{denedbyastohastiintegral(see}

appendixA.1). ThankstoahangeoftimesalethroughtheDubins-Shwarz'theoremA.6

it anbe turned into aBrownian motion. It is easy to show that it is aentered Gauss-

Markovproesswithovarianefuntionρ(t) =^τ₂^m e2_τ^t_m

−1

. Thisfuntion isusedinthe

appliationoftheDubins-Shwarz'theoremto hangethetimesaletoobtainaBrownian

motion:

Rt

0e^s/τmdWs=L Wρ(t)^.

Thespiking ondition ofthis neuron, Vt =θ(t)^{, anbewritten in termof this simpler}

stohastiproess:

Z t 0

e

s

τmdWs=Wρ(t)=

τm

σ

(θ(t)−Vrest)e

t

τm +Vrest−τ¹m

Rt 0se

s

τmIe(s)ds def

= a(t) ^(2.4)

Inorder to fulll ourprogram weare thus naturallyled to study the rsthitting time of

theBrownianmotionWρ(t) ^{totheurvedboundary}a(t)^.

2.2 Model II: The noisy leaky integrate-and-re model with expo-

nentially deaying synapti urrent

Wemodify themodelof setion2.1 to inlude anexponentiallydeayingsynaptiurrent

asdesribedin setion1.2,equation(1.5):

τmdVt = (Vrest−Vt)dt+Ie(t)dt+I_t^syndt τsdI_t^syn = −I_t^syndt+σdWt

Thismodelis amorepreise desriptionof thesynaptiurrentand isstill simpleenough

to be analyzedmathematially. Nevertheless, itsanalytial study isquite hallengingand

onlyafewresultsareavailable.

We integrate this system of two stohasti dierential equations as follows. The rst

equationyields

Vt=Vrest(1−e⁻

t

τm) +_τ¹_mRt 0e

s−t

τmIe(s)ds+_τ¹_mRt 0e

s−t τm I_s^synds,

andtheseondequationanbeintegratedas

I_t^syn=I₀^syne^−τts + σ τs

Z t 0

e^sτ−stdWs,

whereI₀^{synisagivenrandomvariable.}

Wedene

1

α = _τ¹_m −τ¹s

. Replaing intherst equationIt^{syn byitsvaluein theseond}

equationweobtain

Vt=Vrest(1−e⁻

t

τm) +_τ¹_mRt 0e

s−t

τmIe(s)ds+

I₀^syn 1−^ττ^ms

(e⁻

t τs −e⁻

t

τm) + σ τmτs

e⁻

t τm

Z t 0

e^αs Z s

0

e

u τsdWu

ds