An event-driven framework for the simulation of networks of spiking neurons

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

Anevent-drivenframeworkforthesimulationof

networksofspikingneurons

OlivierRochel,DominiqueMartinez

LORIA-Campusscienti queB.P.239

F-54506Vand uvre-les-NancyCedex

E-mail:{rochel,dmartine}@loria.fr

Abstract.Weproposeanevent-drivenframeworkdedicatedtothe

designandthesimulationofnetworksofspikingneurons.Itconsists

ofanabstractmodelofspikingneuronsandane cientevent-driven

simulationenginesoastoachievegoodperformanceinthesimulation

phasewhilemaintainingahighlevelof exibilityandprogrammability

inthemodellingphase.Ourmodelofneuronsencompassesalargeclass

ofspikingneuronsrangingfromusualleakyintegrate-and- reneuronsto

moreabstractneurons,e.g.de nedascomplex nitestatemachines.As

aresult,theproposedframeworkallowsthesimulationoflargenetworks

thatcanbecomposedofuniqueordi erenttypesofneurons.

1Introduction

Inanevent-drivensimulation,thesimulatedtime(oftencalledvirtualtime)isadvancedbycomputingthestateofthesystemateventoccurenceinstantsonly,whereasinatime-drivensimulationitisadvancedusingarbitrarytimesteps[2].Mappingsuchanevent-drivenschemetoapulsedcoupledneuralnetworkisstraightforward:thepulses(orspikes)areinstantaneous,canoccuratanytime,andthereforecanbeseenasthe“events”thatdeterminetheevolutionofthesystem.Inthecontextofaspikingneuralnetworksimulation,abasicevent-drivensimulationenginethusfollowsthisscheme:

1.Findthenexteventtobeprocessed,thatis,thenextneuronthatshould re(orreceive)aspike.

2.Updatethestateoftheneuronconcernedbythisevent

3.Schedulepossibleeventsinducedbythatchange

4.Ifsomeeventsarepending,returntothe rststep.Otherwise,thesim-ulationends.

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

Previousresearchhasproventhatsuchanevent-drivenapproachiswellsuitedtothesimulationoflargenetworksofspikingneurons,sinceitleadstofastsimulationswhilehandlingthedi culttaskofdealingwiththehighpreci-sionrequiredinthecomputationofspiketimes[7].However,theevent-drivensoftwaresimulatorsthathavebeendevelopedsofararespeci ctoparticularmodelsofneuronsornetworks.Forexample,theevent-drivensimulatorsin

[11,4,8,7]areratherdedicatedtointegrate-and- reneurons,theonein[1]isdedicatedtoneuronssimilartoautomatawitha nitenumberofstates.

Incontrast,weproposeinthispaperanevent-drivenframeworkinwhichtheneuronmodelsareonlylimitedbythefactthattheycanbeimplementedinanevent-drivenfashion.Thisencompassesalargeclassofspikingneuronsrang-ingfromusualleakyintegrate-and- reneuronstomoreabstractneurons,e.g.de nedascomplex nitestatemachines.Asaresult,theproposedframeworkfeaturesahighlevelof exibilitythatallowsthesimulationoflargenetworkscomposedofuniqueordi erenttypesofneurons.

2

2.1SpikingneuronmodelsAbstractneuronmodel

We rstneedtode neanabstractmodelofneuronstobeusedwithinourevent-drivenframework.Accordingtothebasicalgorithmdescribedabove,thefollowingrequirementsmustbeful lledbysuchaneuron:wemustknowhowitsinternalstateisa ectedbythereceptionofaspike,howitsinternalstateismodi edwhenemittingaspike,andwhenitsnext ringwilloccur. i},Wethereforede neanabstractmodelofneuronsasaset{xi,ri,si,twith

xi∈XisthestatevariableoftheneuronandXisagivenstatespace.Thisvariablecanchangeonlyatthetimesofsomeeventsoccuringinthesystem.

ri:X×S×R→Xisthefunctionthatdescribesthechangeofthestatevariabledrivenbythereceptionofapulsefromasynapses∈S,whereSisthesetofallsynapses,attimetr∈R.Wewillbemorespeci caboutthesynapsesinsection2.2.

si:X→Xcaracterizesthechangeofstatevariablecausedbythe ringoftheneuron(resetfunction).

i:X→R+∪{+∞}givesthetimeofthenext ring,giventhepresent t

statevariable,withtheadditionalhypothesisthatnoevent-drivenchangeofstatevariablewilloccuruntilthen.Weneedtoprovidethespecialvalue+∞asawaytosignifythatno ringcanoccurwithoutfurtherevents.

i’sto ndthenext ringeventpending.Thesimulationengineusesthet

This,togetherwithamethodtotakecareof(possiblydelayed)receptionevents

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

scheduledinstep3)oftheabove-mentionedalgorithm,permitsthecompletionofstep1).siorriwillthenbeusedtocompletethesecondstep.

2.2Connectivity

Theabstractmodeldescribedinthe

previoussectionisimplicitelybasedontheassumptionthattheconnec-

tivityofthespikingneuralnetworkstobesimulatedisofaveryclassi-

caltype:thereexistsa( xed)setoforientedconnectionsbetweentheneurons,andaneuronhasonlyoneFigure1outputchannel(oneaxon)suchthat

aspikeemittedbyaneuronwillalwaysbetransmittedtoallitssuccessors(alltheneuronslinkedtoitsaxon).Thelatterassumptionexplainswhythefunctionsidoesnotprovideanyexplicitwayoftargetingparticularneurons.

Suchaconnectivitypermitsthatthestep3)oftheevent-drivensimulationalgorithm(receptioneventsscheduling)isperformedinacentralizedwaybythesimulationengine,asitonlyrequirestheknowledgeofthelistofsuccessorsforeachneuron.Indeed,itispossiblethatmorethanoneconnectionexistsbetweenapairofneurons(e.g.withdi erenttimedelays),soasingleneuroncanappearmorethanonceinthesuccessorlist.Then,at ringtime,thesimulationenginewillscheduleexactlyoneeventpersynapseinthesuccessorlist.Moreover,inorderforareceiverneurontoreacttoanincomingspike,itwillhavetoknowwhichsynapseisconcerned,whichrequiresthatallsynapsesbeidenti edinthesuccessorlist.Thesynapseidentitycorrespondstothesparameteroftherifunctiongivenintheprevioussection.

Figure1(left)showsasamplenetworkconnectivity,with3neurons(A,B,C)withthesynapseidentitiesexpressedasa1,..,c2,c3.Ontheright,thecorre-spondingsuccessorlistforeachneuronisrepresented.

2.3Relationtoothermodels

Letusnowconsideraleakyintegrate-and- reneuroni,whosemembranepo-tentialViobeysthefollowingequation

dVi= Vi+Iidt(1)

whereIicorrespondstoaconstantinputcurrent.Theneuronisfurtherde nedbyathresholdmechanism,i.e.itwill rewheneverVi>θianditspotentialwillbesettozero(Vi=0)at ringtimes.Forthesakeofsimplicity,weconsiderthatIi>θiinthefollowing.Wefurtherassumeinthefollowingthatwhenevertheneuronireceivesaspikethroughasynapses,attimetr,themembranepotentialVi(tr)instantaneouslyjumpsofanamplitudews.

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.

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

Usingagooddatastructureforeventordering(priorityqueue)isoftenseenasthecriticalpointwhenimplementinggeneral-purposeevent-drivensimula-tionframeworks[10].Inthecontextofspikingneuronsimulation,thatissuehasbeenaddressedinrecentworksonsimulationalgorithmsforsomespeci cspikingneuronsmodels,suchasintegrate-and- reneuronsin[7]or nitestateautomata-basedneuronsin[1].Itmustbenotedhoweverthatthesimulationenginehastodealwithtworatherdi erenteventtypes:thereceptionevents,oncescheduled,cannotbecancellednorrescheduledatanothertime,whilethe ringeventscanberescheduledorcancelledbyforthcomingreceptionevents.Thatparticularpointmeansthatitisalmostessentialtodesigntwodi erentdatastructuresaimedatproperorderingofeacheventtype.Asanexhaustivestudyofthepossibledatastructuresforimplementinggoodpriorityqueuesineachcaseisbeyondthescopeofthispaper,wewilljustpointoutaworthwhileoptimizationwhichisrelatedtothewaytheneuronsinteract,asexplainedbe-foreinsection2.2.Thesuccessorlistprovidesthebasicstoschedule,at ringtime,thereceptioneventsforeachsuccessorneuron.Whenusingtimedelayedreceptions,itisgenerallyworthwhiletomaintaintheselistsordered,soastoinsertinthependingeventlistonlytheeventassociatedwiththesmallertimedelay,thuse ectivelylimitingthepriorityqueuelength.Assoonasthiseventwillbeprocessed,thenexteventintheorderedlistwillbeexplicitelyscheduleduntilnoremainingconnectionisleft.

Anotherreasonthatcomplicatesthedesignoftheunderlyingdatastruc-turesreliesonthefactthatsomeevents(e.g.pulsereceptions)cansharethesametimestamp(synchrony).Untilnow,weassumedimplicitelythatthesim-ulationenginewasprovidedawayoforderingtheevents,i.e.sortingeventsbytheirtimestamps.Inordertofullyde nethesimulationofanetworkofspikingneurons,wehavethentoprovideanexplicitrulefororderingeventswithequaltimestamps.Multiplerulescanbeused,dependingonthechoiceoftheuser:data-structurebased(FIFO1-like),randomchoice,ormorespeci crulesde nedfromtheavailableparameters(synapseidentity,neuronidentity,typeofevent...).

4Conclusion

Wehavepresentedanevent-drivenframeworkthatconsistsofanabstractmodelofspikingneuronsandane cientevent-drivensimulationengine.Thisframeworkisdedicatedtothedesignandthesimulationofnetworksofspikingneuronsandpresentsahighlevelof exibilityandprogrammability.Thisallowstobuildandsimulatenetworksofclassicalspikingneuronssuchasintegrate-and- reneuronsorofmoreabstractneuronsspeci callydesignedfortheap-plicationathand.Wehaveusedthisevent-drivenframeworkinthesetwosituations:(1)forthesimulationofleakyintegrate-and- reneuronswiththeaimofcontourdetectionbysynchronization[6]and(2)forthesimulationof1Firstin, rstout

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

moreabstractneuronsspeci callydesignedfordetectinganodorindependentofitsconcentration[9].Asimulatorhasbeendeveloppedandthesoftwareshouldbesoonavailableathttp://www.loria.fr/ rochel/.

Besidestheneedofamorein-depthstudyofthedatastructuresusedbythesimulationengine,futureworkswillincludethedesignofahierarchicalabstractmodelthatwillpermiteasiermodellingofcomplexnetworksandmoree cientsimulationsofhomogeneouspopulationofneurons.

References

[1]E.T.Claverol,A.D.Brown,andJ.E.Chad.Discretesimulationoflarge

aggregatesofneurons.Neurocomputing47,277–297,2002.

[2]A.Ferscha.ParallelandDistributedComputingHandbook,chapterParallel

andDistributedSimulationofDiscreteEventSystems.McGrawHill,1996.

[3]W.GerstnerandW.Kistler.SpikingNeuronModels:SingleNeurons,

Populations,Plasticity.CambridgeUniversityPress,2002.

[4]C.GrassmannandJ.K.Anlauf.Fastdigitalsimulationofspikingneu-

ralnetworksandneuromorphicintegrationwithspikelab.InternationalJournalofNeuralSystems,Vol9,No.5473–478,1999.

[5]D.Hansel,G.Mato,C.Meunier,andL.Nelter.Onnumericalsimulations

ofintegrate-and- reneuralnetworks.NeuralComputation10,467-483,1998.

[6]E.Hugues,F.Guilleux,andO.Rochel.Contourdetectionbysynchro-

nizationofintegrate-and- reneurons.2ndIntl.WorkshoponBiologicallyMotivatedComputerVision(BMCV),2002.

[7]G.LeeandN.H.Farhat.Thedoublequeuemethod:anumericalmethod

forintegrate-and- reneuronnetworks.NeuralNetworks14921–932,2001.

[8]M.MattiaandP.DelGiudice.E cientevent-drivensimulationoflarge

networksofspikingneuronsanddynamicalsynapses.NeuralComputation12,2305-2329,2000.

[9]O.Rochel,D.Martinez,E.Hugues,andF.Sarry.Stereo-olfactionwith

asni ngneuromorphicrobotusingspikingneurons.16thEuropeanCon-ferenceonSolid-StateTransducers(Eurosensors),2002.

[10]R.RonngrenandR.Ayani.Acomparativestudyofparallelandsequential

priorityqueuealgorithm.ACMTransactionsonModelingandComputerSimulation,Vol.7,No.2,pp.157–209,1997.

[11]L.Watts.Event-drivensimulationofnetworksofspikingneurons.Pro-

ceedingsoftheSixthNeuralInformationProcessingSystemsConference,pp.927-934.,1993.

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