Perspec'vesonApplica'onsofaStochas'cSpikingNeuronModelto

NeuralNetworkModelingAntonioC.Roque

USP,RibeirãoPreto,SP,BrazilJointworkwithLudmilaBrochini1,AriadneCosta3,

ViníciusCordeiro2,RenanShimoura2,MiguelAbadi1,OsameKinouchi2andJorgeStolfi3

1USP,SãoPaulo;2USP,RibeirãoPreto;3Unicamp,Campinas

PNLD2016,Berlin

Whystochas'cneuronmodels?

•  Invivoandinvitrorecordingsofsingleneuronspiketrainsarecharacterizedbyahighdegreeofvariability

•  ThefollowingexamplesaretakenfromthebookbyGerstner,Kistler,NaudandPaninski,NeuronalDynamics,CUP,2014

Awakemouse,cortex,freelywhisking

Crochetetal.,2011

Spontaneousac'vityinvivo

Trialtotrialvariabilityinvivo15repe''onsofthesamerandomdotmo'onpa\ern

AdaptedfromBairandKoch,1996;DatafromNewsome,1989

Trialtotrialvariabilityinvitro

4repe''onsofthesame'me-dependents'mulus

ModifiedfromNaudandGerstner,2012

Sourcesofnoise:extrinsicandintrinsictoneurons

Lindner,2016

Twotypesofnoisemodelforaneuron

•  Spikegenera'onisdirectlymodeledasastochas'cprocess

•  Spikegenera'onismodeleddeterminis'callyandnoiseentersthedynamicsviaaddi'onalstochas'cterms

Stochas'cmodelforsystemsofinterac'ngneurons

Thestochas'cmodel

•  Vi(t): time dependent membrane potential of neuron i at time t for i = 1, …, N; •  t: discrete time given by integer multiples of constant step Δ small enough to

exclude possibility of a neuron firing more than once during each step; •  Xi(t): number of times neuron i fired between t and t+1, namely 0 or 1; •  If neuron fires between t and t+1, its potential drops to VR by time t+1; •  wij: weight of synapse from neuron j to neuron i; •  µ: decay factor (in the interval [0, 1]) due to leakage during time step Δ; •  Xi(t) = 1 with probability Φ(Vi(t)); •  Φ(V) is assumed to be monotonically increasing and saturating at some

saturation potential VS.

Comment•  IfΦ(V) = Θ(V−Vth),i.e.0 for V<Vth and1for

V>Vth,themodelbecomesthedeterminis'cdiscrete-'meleakyintegrate-and-firemodel(LIF).

•  AnyotherchoiceofΦ(V) givesastochas'cneuron

Vs

Inthefollowing,Iwillshowsomeanaly'calandnumericalresultsof

networkmodelsusingthisstochas'cneuronmodel

Networkwithall-to-allcouplingMeanfieldanalysis

Analy'calandnumericalresults

Macroscopicquan''es•  Poten'aldistribu'on: frac'onofneuronswithpoten'alin therange(V,V+dV)at'met

•  Networkac'vity: frac'onofneuronsthatfiredbetweentand t+1

Shapeofthepoten'aldistribu'onP(V,t)hasacomponentthatisaDiracpulseatV=VRwithamplitude,accoun'ngfortheneuronsthatfiredbetweentandt+1

Mean-fieldanalysis

•  Fullyconnectednetwork:eachneuronreceivesinputsfromallotherN−1neurons;

•  VR=0;•  Uniformconstantexternalinput:Ii(t)=I;•  Allweightsareequal:

Themean-fieldpoten'aldistribu'on•  Onceallneuronshavefiredatleastonce,thedensityP(V,t)becomesacombina'onofdiscreteimpulseswithamplitudesη0(t),η1(t),η2(t),…,atpoten'alsU0(t),U1(t),U2(t),...,suchthat.

•  Thevaluesofηk(t)andUk(t)evolvebytheequa'ons:

•  Theamplitudeisthefrac'onofneuronswith“age”k:neuronsthatfiredbetweent – k – 1andt – k anddidnotfirebetweent – k andt

•  Forthistypeofdistribu'ondenetworkac'vityρ(t)is:

•  Givenvaluesforµ,Wandthefunc'onΦ(V):– Therecurrenceequa'onscanbesolvednumerically

– Insomecasestheycanbesolvedanaly'cally

ExamplesofΦ(V)Satura'ngmonomialfunc'onofdegreer

Ra'onalfunc'on

Γ = 1; VT = 0 NB.:Thedeterminis'cLIFmodelcorrespondstothemonomialfunc'onwith

S

Resultsforthemonomialsatura'ngfunc'onwithμ=0

•  Inthecasewithµ = 0,neurons“forget”allpreviousinputsignals,exceptthosereceivedat t – 1.

•  P(V,t)containsonlytwopeaksatpoten'als:V0(t)=0andV1(t)=I+Wρ(t−1)

•  Takingintoaccountthenormaliza'oncondi'on,thefrac'onsη0(t)andη1(t)evolveas:

•  Assumingthatneuronscannotfireatrest,Φ(0) = 0:

•  In a stationary regime, the recurrence equations

reduce to:

•  Since Φ(V) ≤ 1, any stationary regime must have ρ ≤

1/2

Φ(V) = (ΓV)r, I = 0; r = 1Con'nuousphasetransi'ons

Absorbing State ρ = 0

Fixed point ρ > 0

2-cycle ρ1 = ½ − a ρ2 = ½ + a

a ≤ ½ − Vs/W

WC=1/Γ

Γ=1

WB=2/Γ

Brochinietal.,2016

Φ(V) = (ΓV)r, I = 0; r > 1Discon'nuousphasetransi'ons

r=1.2 r=2

ρ+

ρ−

Nontrivialsolu'onρ+onlyfor1≤r≤2Forr=2thissolu'onisapointatWC=2/ΓThediscon'nuitygoestozeroforr=1

W=WC(r)

Γ = 1 Γ = 1

Brochinietal.,2016

Φ(V) = (ΓV)r, I = 0; r < 1Ceaselessac'vity

Noabsorbingρ=0solu'on

Brochinietal.,2016

Γ = 1

ρ>0foranyW>0

Numerical solutions for µ > 0 Φ(V) = (ΓV)r, I = 0; r = 1

Brochinietal.,2016

Discon'nuousphasetransi'onsforVT > 0

Φ(V) = (Γ(V-VT))r, I = 0, µ = 0 ; r = 1, Γ = 1

VT=0 VT=0.05 VT=0.1

Thediscon'nuityρCgoestozeroforVTà0

Brochinietal.,2016

Neuronalavalanches(simula'onstudiesatthecri'calpointof

thecon'nuousphasetransi'on)

•  Anavalanchethatstartsat'met=aandendsat'met=bhas:– Dura'ond=b−a;– Size

Neuronalavalanchesatthecri'calpointΦ(V) = (ΓV)r, I = 0, µ = 0; r = 1, ΓC = WC = 1

Brochinietal.,2016

Avalanchesizesta's'cs

Avalanchedura'onsta's'cs

Brochinietal.,2016

Self-organiza'onwithdynamicneuronalgainsIdea:fixtheweightsatW=1andallowthegainsΓtovary

u=1,A=1.1,τ=1000ms Brochinietal.,2016

Networkwithrealis'cconnec'vityExcitatoryandinhibitoryneurons

Simula'onresults

ThePotjans-DiesmannModel

105neurons(80%excit.20%inhibit.)109synapses

Availableatwww.opensourcebrain.org

•  Modelforlocalcor'calmicrocircuit

•  Integratesexperimentaldataofmorethan50experimentalpapers

•  Excitatoryandinhibitoryneuronsmodeledbythesamedeterminis'cLIFmodel

•  Asynchronous-irregularspikingofneurons

•  Higherspikerateofinhibitoryneurons

•  Replicateswellthedistribu'onofspikeratesacrosslayers

Potjans&Diesmann,2014

Fitofaveragebehaviorofstochas'cmodel(monomialΦ(V))toIzhikevichmodelneurons

Regularspikingneuron(excitatory) Fastspiking

neuron(inhibitory)

−40 −35 −30 −25

μ=0.9ΓΓ

win<<wex win<wex

win>wex win>>wex

Computa'onalcost

TIzhikevich

Tstochas'c_______

No.ofsynapses

----------------------------------------------------

Timetosimulate5secofnetworkac'vity(reducednetworkwith4000neurons)

Whichmodeltouseforcor'calspikingneurons?Izhikevich,2004

Conclusions

•  Thestochas'cneuronmodelintroducedbyGalvesandLöcherbachisaninteres'ngelementforstudiesofnetworksofspikingneurons

•  Enablesexactanaly'cresults:– Phasetransi'ons– Avalanches,SOC

•  Simpleandefficientnumericalimplementa'on

ResearchTeam

USP,SaoPaulo USP,RibeirãoPreto

Thanks!

Unicamp,Campinas

NeuroMat

L.Brochini M.Abadi

A.Galves

A.CostaO.Kinouchi J.StolfiR.Shimoura V.Cordeiro

E.LöcherbachUniv.Cergy-PontoiseUSP,SaoPaulo