A markovian model for stochastic integrate-and-fire networks

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networks

Jonathan Touboul

To cite this version:

Jonathan Touboul. A markovian model for stochastic integrate-and-fire networks. [Research Report]

RR-6661, INRIA. 2008. �inria-00323643�

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a p p o r t

d e r e c h e r c h e

9-6399ISRNINRIA/RR--6661--FR+ENG

Thème BIO

A markovian model for stochastic integrate-and-fire networks

Jonathan Touboul

N° 6661

September 22, 2008

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Jonathan Touboul

∗

ThèmeBIOSystèmesbiologiques

ProjetOdyssée

Rapportdereherhe n°6661September22,200842pages

Abstrat: In this paper we introdue and study a mathematial framework in order to

haraterizeand simulate networks of noisyintegrate-and-reneurons. This framework is

basedonamarkovianmodelizationofthenetwork,similartotheevent-basedmodelization

ofdeterministinetworks. Inthese networksthevalueofinterestateah neuronisnotthe

membranepotentialitselfbuttherelatedountdownproess,whihisdenedlooselyasthe

time remaining to thenext spike if nothing ours meanwhile in the network. The main

issueofthismodelizationistoensurethatthedynamisofthisountdownproess,possibly

supplemented with other variables, is an autonomous Markov proess (i.e. that does not

depend onthemembrane'spotential).

We provethat awide rangeof integrate-and-re neuronmodels and dierent typesof

interationstintothisgeneralmathematialframework. Thisframeworkinvolvesrenewal

proessesandhasalreadybeenstudiedintheeldofrandomnetworksinamorerestrited

setting by Cottrell, Robert, Turova forinstane [6, 7,13, 27, 28℄, and from a mathemat-

ialviewpoint, ergodiity matters havebeen disussed Fayolle, Menshikov, Malyshev and

Borovkov[12,3℄.

Thismodelizationprovidesaveryeientalgorithmtosimulatelargenetworksofnoisy

integrate-and-reneuronmodels. Wedisussdierenttypesofimplementations,anddevel-

oppedtogetherwithRenaudKervienandAlexandreChariotaveryeientparalelsimulator

implementonGPU.

Key-words:

∗

jonathan.touboulsophia.inria.fr

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Résumé : In this paper we introdue and study a mathematial framework in order to

haraterizeand simulate networks of noisyintegrate-and-reneurons. This framework is

basedonamarkovianmodelizationofthenetwork,similartotheevent-basedmodelization

ofdeterministinetworks. Inthese networksthevalueofinterestateah neuronisnotthe

membranepotentialitselfbuttherelatedountdownproess,whihisdenedlooselyasthe

time remaining to thenext spike if nothing ours meanwhile in the network. The main

issueofthismodelizationistoensurethatthedynamisofthisountdownproess,possibly

supplemented with other variables, is an autonomous Markov proess (i.e. that does not

depend onthemembrane'spotential).

We provethat awide rangeof integrate-and-re neuronmodels and dierent typesof

interationstintothisgeneralmathematialframework. Thisframeworkinvolvesrenewal

proessesandhasalreadybeenstudiedintheeldofrandomnetworksinamorerestrited

setting by Cottrell, Robert, Turovafor instane [6, 7, 13, 27, 28℄, and from a mathema-

tial viewpoint, ergodiitymattershave beendisussed Fayolle,Menshikov,Malyshev and

Borovkov[12,3℄.

Thismodelizationprovidesaveryeientalgorithmtosimulatelargenetworksofnoisy

integrate-and-reneuronmodels. Wedisuss dierenttypesofimplementations,anddeve-

lopped togetherwithRenaudKervienand AlexandreChariotaveryeientparalelsimu-

latorimplementonGPU.

Mots-lés :

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Figure1: Ageneralneuralnetworkarhiteture: thenetworkisomposedofneurons(blue

irles)onnetedthroughadiretionalonnetivitymap(blak arrow)withsynaptie-

ienywij^. ^Theîntrinsi^dynamisând^theêet ôfânînoming^spikeôn^thepostsynapti neuronanbemodeledinmanyways

1 Theoretial framework

In this paper we build abridge between a wide range of biologial networks models and

a general mathematial framework. The type of network we onsider is omposed of N

stohastiintegrate-and-reneurons(seegure1). Classialy,neuron'sativityisdesribed

byitsmembranepotential. Themembranepotential'sdynamisweonsiderinthispaperis

stohasti: eahneuron reeivesathis synapsesnoisyinputsorrespondingto therandom

ativity ofionhannelsandat theexternalativityof thenetwork,asreviewed[24℄. This

randomspikeinomingis heremodelled asBrownianmotion,usingadiusionapproxima-

tion. Dierenttypesofintrinsidynamisandofsynaptiintegrationwillbeonsideredans

anoexistin agivennetwork.

Duringthe timeintervalswhereno spikeisemittedin the network,themembranepo-

tential of eah neuron evolves as independently to the other's, aording to its intrinsi

dynamis. When the membrane potential V⁽ⁱ⁾(t) ôf ^the ^neuron îndexed ^by i ^reahes îts

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Figure2: A sampletraeofthemembranepotentialfor twoonnetedneuronsindex byi

and j^. ^The ^neuroni îs ^the^rst ^to ^spikeⁱⁿ ^the^network: ît ^has^the ^lowest^rst ^spike^time Xi(0)^. Ât^this^time,^the^neuroniîs^reset^tovrândîts^next^spike^timeîs^resetâording^to

Yi^. Ît^sendsâ^spike^toîts^neighbors,âmong^whihj^. Îf^theînterationîsinstantaneous,the membranepotentialofj îsinstantaneouslyaddedthesynaptiweightwi,j ând^the^time^to

thenextspikefortheneuronj îsînreased^byâ^valueηi,j^. ^This^gure^was^produedⁱⁿ^the

aseofthePerfetIntegrate-and-remodel.

deterministithresholdfuntionθ(t)ât^timet0^,^the^neuronêliitsânâtion^potential. ^Sub-

sequently, its membrane potentialis reset to agiven value Vr⁽ⁱ⁾^, ând ^the ^statesôf âll ^the

postsynaptineuronsj^onneted^to^the^neuroni^is^modied. ^We^denote^byV(i)^the^postsy-

naptineighboorhoodoftheneuroni^,î.e. ^the^setôf^neurons^that^reeive^spikes^from^neuron i^. ^Theêetôfâ^presynapti^spike^reeived^by^neuronj∈ V(i)ân^be^modelledⁱⁿ^dierent

fashions: itanbeonsideredashavinganinstantaneouseetonthemembranepotential

(i.e. V^(j)(t0) =V^(j)(t⁻₀) +wi,j^wherewi,jîs^the^synaptiêienyôf^theônnetioni→j^),

ormoreomplex,inludingforinstaneasynaptiurrent,asynaptipulses, et.... Many

exampleswilltreatedinthetext. Figure2illustratesthedynamisofthenetwork,showing

thestrutureofthenetworkin 1andthedynamisofthemembranepotentialin 2

This type of model wasstudied forinstane byBrunel and Hakim [5℄with the use of

theFokker-Plankequation. Assumingthat thenetworkissparsely onneted,theyfound

that in the limit N → ∞ ^the^network ^exhibited ^a^sharp ^transition ^between^two ^regimes:

a stationnaryregime and a weakly synhronized osillatory regime. In their model, eah

neuron is an integrate-and-re neuron, and is randomly onneted to C neurons of the

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network,andto Cext

externalneurons. Thesparseonnetivityassumptionis ε= _N^C ≪1^.

Interations betweenexternal and internal neuronsare delayed bya onstant delayδ ^(i.e.

whenaspikeisemittedbyaneuronofthenetwork,itdereasesorinreasesthemembrane

potential after a time δ^, ^see ^setion ^4). ^This ^delay ^plays ^a^ruial ^role ⁱⁿ ^the ^generation

of global osillations. We wish to re-express the dynamis from an event-driven point of

view(seefor exampleReutimann,GiuglianoanFusi [20℄),and toonsiderthenoiseinthe

dynamisofeah individualneuron.

Independently,intheeldofstohastinetworksandqueuetheoryandMarkovproesses,

anetwork modelhasbeendevelopedduring thelast10years. It isreferredforinstane as

thehourglass model by Turova[2,7, 28, 27℄. This model hasbeenintrodued fortherst

time by MarieCottrell in [6℄, and the variable taken into aount was initially alled the

inhibitionproess. This nameisveryonfusingin theeld ofneurosiene,andwewill not

usethisexpressionwhendealingwithneuronandpreferthenameofountdownproess.

These modelsaredenedbytworandomparameters:

Therandomvariables(Yi)i=1...N ^whih^desribe^the^interspike^intervaldistributionfor theneuroni^.

(ηi,j)i6=j ^desribing^the^interation ^ofi→j^.

LetthestateofthenetworkbedesribedbyaN-dimensionalvetor(Xt)t≥0=

(X_t⁽ⁱ⁾)i=1...N

t≥0

havingthefollowingdynamis: lett >0^,

(i). if∀i ∈ {1. . . N}, Xi(t)>0 ^then êah ômponentôf X ^dereases^linearly^with^slope

−1ⁱⁿ ^time.

(ii). if∃i∈ {1. . . N}, Xi(t⁻) = 0^,subsequentlywehave:

Xi ^is^reset^to^a^random^variableindependentofallthehistoryoftheproessand withdistributionYi^.

∀j∈ V(i)^,â^positive^random^variableηi,j îsâdded^toXj^:

Xj(t) =Xj(t⁻) +ηi,j

Hene eah nodeof the network is arenewalproess andthe networkstruture makes

theseproessesinteratviapositiverandomvariables.

Inthis paperwebuild abridgebetweenthese twomodels. Wewill see that stohasti

networksofintegrate-and-reneuronsanbedesribedusinganextensionofthehourglass

model,butneedamoregeneralformalismtotakeintoaountthemoreomplexinteration

strutureatthelevelofthemembranepotential.

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2 From Biologial networks to the Hourglass model

Inthissetionwepresenttheequivalenebetweenthebio-inspirednetworkandthehourglass

model. Werstonsiderinhibitory networksforthesakeofsimpliity. Indeed,inthease

ofnon-inhibitorynetworksan appearthe phenomenonof whatweallaspike avalanhe.

Assumethat theinterations areonsideredinstantaneousandexitatory. Inthisasethe

following proessan our: if thesynapti eieniesare big enough,the spikeemission

of a neuron an indue at the very sametime the spike emission of the neurons diretly

onneted to this neuron, whih themselves anindue spikesin their neighboorhood. A

spikeanthereforebetransmittedinthewholenetwork,andtheninduethespikeemission

in therst neuronwhospike, and thereforethis proesswill not stop. Thismehanismis

learlynot biologiallyplausible: rst of allthere are transmissiondelaysin the network,

and hene this avalane, even if it ours, generates a high frequeny ativity, but with

nologialproblems suh astheonewejust desribed. Furthermore,the limitedresoures

in the neuron's environment makes suh a wasteful enegeti proess impossible. From a

omutational and theoretial point of view, suh a phenomenon results in stuking the

dynamis at the time when it ours: this innite loop of simultaneous spikes bloksthe

proessatthistimeandweannotinferwhatwouldhappenafterwards.

We willsee in setion 4that inludingarefratory period and transmissiondelaysbe-

tweenneuronsoveromesthisdiulty.

Thisequivaleneisbuiltupontheintrodutionofanewproessrelatedtothemembrane

potentialproess,theountdownproess,rigorouslydenedasfollows:

Denition 2.1. [Countdown proess℄ For eah neuron i^, ^let ^us ^dene X⁽ⁱ⁾(t) ≥ 0 ^the

duration of time (after timet⁾^til^l^the ^rst ^ring^moment ôf ^this ^neuron,îf ^no înteration

takes plae meanwhile. We will all this stohasti proess the ountdown proess of the

neurons.

This proess is alled ountdown beause of its dynamis, but in fat at any time, its

valuegivesusthetimetowaittillthenextspike,soitanbealsoseenasalok. Itanbe

seenasaountdownset attheinstantofreeptionofthelastspikeorjust after thespike,

tothetimetowaitforthenextspiketoourifnointerationtakesplaemeanwhile. The

dynamisofthisvariableXⁱîs^linearly^dereasing^with^slope−1^during^theîntervalsôf^time

wherenospikeisreeivedorprodued:

dX⁽ⁱ⁾

dt =−1 ^(2.1)

Attimet^,^the^next^spike^will^ourⁱⁿ ^neuroni= Arg Min_j∈1...NX^(j)(t)^at^timet+X⁽ⁱ⁾(t)

(tîs ^theâbsolute ^time). În^mostôf^theâse,^forînstaneⁱⁿ ^theâse^whereâll^the^random

variableshaveadensitywithrespettoLebesgue'smeasure,theprobabilityfortwoneurons

to spikeexatlyat thesame timeis nullwhen the network is inhibitory. Inthat ase, we

negletthis ase and assumethat onlyoneneuron spikesat a given time. At spiketime,

X⁽ⁱ⁾(t)^isinstantaneouslyresetbydrawingthelawofarandomvariablenotedYi^,^whih^has

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the samedistribution asthe rst hitting time of the stohasti proess (V_t⁽ⁱ⁾)t≥0 ^starting

fromVr ^to ^the^boundaryθ(t)^(the distribution oftheinterspikeintervalin termsofneural models). Thestatesofallneuronsjust beforethespikearegivenby: X^(j) (t+X⁽ⁱ⁾)⁻

= X^(j)(t)−X⁽ⁱ⁾(t)^. ^Finally^, ^the ^states ôf âll ^neuronsj ônneted^to ^neuron i âre ^modied

aording to the spike produed by neuron i^. ^Beause ^the înteration îs inhibitory, this amountsto postponingthespikeprodued byneuronj ^byânâmountηi,j ≥0 ^(see^Fig ^3),

beausetheinhibitioninreasesthetimetothenextspike.

Ingeneral,ηi,j îsâ^random^variable^dependingôn^the^membrane^potentialV^(j) ât^time t^. În^mostôf^the^modelsônsideredⁱⁿ^setion^3,ît^dependsⁱⁿ^fatônlyônX^(j)^,^so^that^the

updatereadsX^(j)(t+X⁽ⁱ⁾) =X^(j)(t)−X⁽ⁱ⁾(t) +ηi,j(X^(j)(t)−X⁽ⁱ⁾(t))^,^where ηi,j(x)^is ^a

randomfuntion.

Inallourmathematialstudyweonsider theproess

X(t) := (X⁽ⁱ⁾(t))1≤i≤N ^(2.2)

UptoanadditionalMarkovhain,thismodelwillbeaontinuoustimeMarkovproess,as

wewillshowinsetion3. Theproess(Xt)t^denedîs^pieewiseôntinuous,^so^theânalysis

ofDavisin [11℄anbeapplied here. Ouraseisevenmoresimplesinethedisontinuities

are verysimply related to the valueof theproess. This very partiularpropertyimplies

thatstudyingtheontinuoustimestohastiproessisstritlyequivalenttoonsideringone

ofthetwofollovingdisretetimeMarkovhain(2.3)and(2.4).

Indeed,let(tn)^denote^the^time^sequeneôf^the^spikesêmitted^byôneôfâll^the^neurons, (Zn)^the^sequeneôf^the^states^just^beforeêah^spikeând(Xn)^the^vetorôf^states^justâfter

eah spike.

Zn =X(t⁻_n) ^(2.3)

Xn =X(tn) ^(2.4)

Consider now therandom variable ηi,j ^to âdd ^to ^the ^stateôf â postsynaptineuron j

when reeiving a spike from i ât ^time t^∗^. ^This ^random ^variable îs ^the ^delay âused ^by

theinhibition, i.e. theadditionaltime to wait for j ^to ^spike^beauseôf ^the ^reeptionôf â

presynaptispike.

Alltheworkdoneinthefollowingsetions3and4isaimedtoshowthatmanybiologial

neuronmodelstintotheframeworkdesribedinsetion2andtoidentifytheparametersof

theorrespondingHourglassmodel. Wewillseethatin manyases,theserandomvariables

anberelatedto rsthittingtimesofstohasti proesses.

3 Inhibitory Networks with instantaneous interations

Inthissetionweonsiderdierenttypesofmodelsoflinearintegrate-and-reneuronsand

dierenttypesofinhibitorysynaptiinterations,anduptoatransformationshowthatthe

networkmodelan beonsidered asan hourglassnetwork,and identify theparametersof

themodel.

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Figure 3: A representation of a sample path for the ountdown proess and the related

membranepotentialintheaseoftheperfetintegrate-and-reneuronrepresentedingure

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The rst model we onsider is a noisy integrate-and-re neuron without leak urrent,

whih wereferastheperfetnoisyintegrate-and-reneuron. Wethenaddaleakurrent.

Werststatesomegeneralresultsabouttheserandom variables. Firstof all,itisvery

interestingto note that the reset proess is only linked with the presynapti neuron, and

has the law of the rst hitting time of the membrane potential proess to the threshold

funtion θ(t)^. Îndeed, ^when â ^neuron êliits â ^spike ⁱⁿ ^the integrate-and-re framework withnorefratory period, itsmembranepotentialis resettoaertainvalueVr¹ ^Therefore

theresetrandomvariable,dened asthetime before thenextspikeoftheneuron,hasthe

law of the rst hitting time of the membranepotentialto the threshold. The interation

variableonlydependsonthepostsynaptidynamisofthemembranepotentialandonthe

synapti eieny w^. ^When ^the ^neuron j ^reeives ^a ^spike ^from ^the ^neuron i ^at ^time t^,

thetime to thenext spikeis hanged,and the random variable orrespondingis equal to

thedierenebetweenthe timeto reah thethresholdstartingfrom V^(j)(t) +wij ^and^the

time toreah thethreshold startingfrom V^(j)(t)^. ^Hene ⁱⁿ ^the^general ^ase,^this ^random

variable depends on thevalue of thepotentialat time t^. ^We ^will ^see ^that ⁱⁿ ^the ^simplest

aseswetreatherethisrandomvariableonlydependsonX^(j)^,^the^time^to^the^next^spike^for

thepostsynaptineuronj^. ^This^property^is ^veryinterestingsineitmakesthe ountdown proessanindependentMarkovhain,i.e. thatdependsonnootherproess.

3.1 Perfet integrate-and-re models

3.1.1 Perfet IF neuron with instantaneous synapses

Westartbyonsideringtheperfetintegrate-and-reneuronwithexternalinputsandBrow-

niannoise. Themembranepotentialoftheneuroni^,^denotedV⁽ⁱ⁾(t)^,^is^hene^driven^by^the

followingequationbetweentwospikes:

τidV⁽ⁱ⁾(t) =I_e⁽ⁱ⁾(t)dt+σidW_t⁽ⁱ⁾ ^(3.1)

whereτiîs^the^membrane^potential^timeônstant,Ie⁽ⁱ⁾(t)îs^theînputûrrent,σi^the^standard

deviationofthenoiseand(W⁽ⁱ⁾)1≤i≤N ^areindependentBrownianmotions,whihrepresents externalsynaptistimulations

2

. Theneuronreswhenitsmembranepotentialreahesthe

thresholdθ^: ^the^membrane^potentialîs^reset^toâ^valueVr ândâ^spikeîsêmitted.

V⁽ⁱ⁾(t⁻) =θ⇒V⁽ⁱ⁾(t) =Vr ^(3.2)

1

TheresetvalueVr analsobearandomvariablewithnoadditionalomplexity.Theresultsweobtain foraonstantresetvalueanbereadilyextendedtothismoregeneralmodel.

2

ItouldhavebeenpossibletoreplaetheBrownianmotionsbyinstantaneousspikes(V⁽ⁱ⁾→V⁽ⁱ⁾+δ) triggeredaordingtoaPoissonproess(theequation(3.1 )wouldbethediusionapproximationofthistype

ofexitation). Thiswouldhangeonsiderablythefollowingstudy,sinetheproessisnomoreontinuous

betweentwoonseutivespikes