An event-driven framework for the simulation of networks of(4)

Abstract. We propose an event-driven framework dedicated to the design and the simulation of networks of spiking neurons. It consists of an abstract model of spiking neurons and an efficient event-driven simulation engine so as to achieve good performance

Torewritetheaboveintegrate-and- reneuronaccordingtoourmodelweconsiderthatthestatevariablexiisde nedasthevector(Vi0,t0i)inwhich0Vi0=Vi(t0i)representsthepotentialatthetimetiofthelatestevent(emissionorreceptionofaspike).Theintegrate-and- reneuronisnowfullydescribedby:

Vi(tr)+wsri(xi,s,tr)=tr 0si(xi)= ti(xi) 0t i(xi)=itI V00ti+logii(2)(3)ifVi0≥θi,otherwise.(4)

. twhereVi(tr)isfoundbyintegratingEq.(1):Vi(tr)=Ii+(Vi0 Ii)expt0riMoregenerally,torephraseathreshold-basedspikingneuronwithinthe

event-drivenframeworkdescribedabove,the ringtimehastobecomputedexplicitely.Thisispossiblefortheleakyintegrate-and- re(seeabove).How-ever,itiscommonthatnoanalyticalsolutionisavailableifoneconsidersmorebiologicallyplausiblesynapticinteractions,suchastheuseofpostsynapticcurrentsgivenbyalphafunctions.Insuchacase,wecanstilluseanumericalschemetoestimatethenext ringtimewithagivenprecision,asstatedin

[7].AsimilartechniquehasbeenusedbyHanselandal.in[5].Notehoweverthatwhenthefocusofastudyismainlyonmodellingpreciselytheshapeofapostsynapticpotential,aframeworksuchastheSpikeResponseModel[3]isprobablybettersuitedthanours.

Obviously,ourabstractmodeldoesnotrequirethatthestatevariableoftheneuronshouldbederivedfromthetimecourseofanunderlyingmembranepotential.Otherchoicesarepossible:forexample,theneuronmodelsbasedon nitestateautomatain[1]couldbeusedaswell.Anotherpossiblechoiceistotakethetimeofthenext ring(assumingitalwaysexists)asthestatevariable.Interactionsfromotherneuronswillleadtoadvance(excitatoryconnections)ordelay(inhibitoryconnections)thattime.

3Simulationengine

Sofar,wehavedescribedanabstractmodelofanetworkofspikingneurons.Throughoutthedescription,thebasicinteractionswithanappropriatesim-ulationenginehavebeenunderlinedaswell.Wenowneedtocompletethedescriptionoftheengine,morepreciselytoexplainhowitcanhandletheevent owinadeterministicway.Thebasicalgorithmforasimulationenginehasalreadybeendescribedinsection1,andcorrespondstoaclassicalevent-drivensimulationalgorithm.Wenowfocusonsomekeypointsthatstronglydeter-minethee ciencyandcorrectnessofeventordering.