A Criterion for the Convergence of Learning with Spike Timing Dependent Plasticity

8 Pages·0.185 MB·English

by  Legenstein R., Maass W.

A Criterion for the Convergence of Learning
with Spike Timing Dependent Plasticity
RobertLegensteinandWolfgangMaass
InstituteforTheoreticalComputerScience
TechnischeUniversitaetGraz
A-8010Graz,Austria
flegi,[email protected]
Abstract
Weinvestigateunderwhatconditionsaneuroncanlearnbyexperimen-
tallysupportedrulesforspiketimingdependentplasticity(STDP)topre-
dict the arrival times of strong “teacher inputs” to the same neuron. It
turns out that in contrast to the famous Perceptron Convergence Theo-
rem, which predicts convergence of the perceptron learning rule for a
strongly simplified neuron model whenever a stable solution exists, no
equally strong convergence guarantee can be given for spiking neurons
withSTDP.Butwederiveacriteriononthestatisticaldependencystruc-
tureofinputspiketrainswhichcharacterizesexactlywhenlearningwith
STDP will converge on average for a simple model of a spiking neu-
ron. This criterion is reminiscent of the linear separability criterion of
thePerceptronConvergenceTheorem,butitappliesheretotherowsof
a correlation matrix related to the spike inputs. In addition we show
throughcomputersimulationsformorerealisticneuronmodelsthatthe
resultinganalyticallypredictedpositivelearningresultsnotonlyholdfor
the common interpretation of STDP where STDP changes the weights
ofsynapses,butalsoforamorerealisticinterpretationsuggestedbyex-
perimentaldatawhereSTDPmodulatestheinitialreleaseprobabilityof
dynamicsynapses.
1 Introduction
NumerousexperimentaldatashowthatSTDPchangesthevaluew ofasynapticweight
old
afterpairingofthefiringofthepresynapticneuronattimetpre withafiringofthepostsy-
napticneuronattimetpost =tpre+(cid:1)ttow =w +(cid:1)waccordingtotherule
new old
w = minfwmax; wold+W+(cid:1)e(cid:0)(cid:1)t=(cid:28)+g ; if(cid:1)t>0 (1)
new (cid:26) maxf0; wold(cid:0)W(cid:0)(cid:1)e(cid:1)t=(cid:28)(cid:0)g ; if(cid:1)t(cid:20)0;
withsomeparametersW+;W(cid:0);(cid:28)+;(cid:28)(cid:0) > 0(see[1]). Ifduringtrainingateacherinduces
firingofthepostsynapticneuron,thisrulebecomessomewhatanalogoustothewell-known
perceptron learning rule for McCulloch-Pitts neurons (= “perceptrons”). The Perceptron
ConvergenceTheoremstatesthatthisruleenablesaperceptrontolearn,startingfromany
initial weights, after finitely many errors any transformation that it could possibly imple-
ment. However, we have constructed examples of input spike trains and teacher spike
trains (omitted in this abstract) such that although a weight vector exists which produces
thedesiredfiringandwhichisstableunderSTDP,learningwithSTDPdoesnotconverge
toastablesolution. Ontheotherhandexperimentsinvivohaveshownthatneuronscanbe
teachedbysuitableteacherinputtoadoptagivenfiringresponse[2,3].Weshowinsection
2thatsuchconvergenceoflearningforspikingneuronscanbeexplainedbySTDPinthe
averagecase,providedthatacertaincriterionismetforthestatisticaldependenceamong
Poissonspikeinputs. Thevalidityoftheproposedcriterionistestedinsection3formore
realisticmodelsforneuronsandsynapses.
2 Ananalytical criterionfortheconvergenceofSTDP
TheaveragecaseanalysisinthissectionisbasedonthelinearPoissonneuronmodel(see
[4,5]). ThisneuronmodeloutputsaspiketrainSpost(t)whichisarealizationofaPoisson
processwiththeunderlyinginstantaneousfiringrateRpost(t). Werepresentaspiketrain
S(t)asasumofDirac-(cid:14)functionsS(t)= (cid:14)(t(cid:0)t ),wheret isthekthspiketimeof
thespiketrain. TheeffectofaninputspikePatkinputiakttimet0 iskmodeledbyanincrease
intheinstantaneousfiringrateofanamountw (t0)(cid:15)(t(cid:0)t0), where(cid:15)isaresponsekernel
i
andw (t0)isthesynapticefficacyofsynapseiattimet0. Weassume(cid:15)(s) = 0fors < 0
i
1
(causality), ds(cid:15)(s) = 1(normalizationoftheresponsekernel),and(cid:15)(s) (cid:21) 0foralls
0
aswellaswRi (cid:21) 0foralli(excitatoryinputs). Inthelinearmodel,thecontributionsofall
inputsaresummeduplinearly:
n 1
Rpost(t)= dsw (t(cid:0)s)(cid:15)(s)S (t(cid:0)s), (2)
j j
Xj=1Z0
where S1;:::;Sn are the n presynaptic spike trains. Note that in this spike generation
process,thegenerationofanoutputspikeisindependentofpreviousoutputspikes.
The STDP-rule (1) avoids the growth of weights beyond bounds 0 and w by simple
max
clipping. Alternatively one can make the weight update dependent on the actual weight
value. In [5] a general rule is suggested where the weight dependence has the form of a
powerlawwithanon-negativeexponent(cid:22). Thisweightupdateruleisdefinedby
(cid:1)w = W+(cid:1)(1(cid:0)w)(cid:22)(cid:1)e(cid:0)(cid:1)t=(cid:28)+ ; if(cid:1)t>0 (3)
(cid:26) (cid:0)W(cid:0)(cid:1)w(cid:22)(cid:1)e(cid:1)t=(cid:28)(cid:0) ; if(cid:1)t(cid:20)0 ,
where we assumed for simplicity that w = 1. Instead of looking at specific input
max
spiketrains,weconsidertheaveragebehavioroftheweightvectorfor(possiblycorrelated)
homogeneousPoissoninputspiketrains.Hence,thechange(cid:1)w isarandomvariablewith
i
a mean drift and fluctuations around it. We will in the following focus on the drift by
assumingthatindividualweightchangesareverysmallandonlyaveragedquantitiesenter
thelearningdynamics,see[6]. LetS bethespiketrainofinputiandletS(cid:3) betheoutput
i
spiketrainoftheneuron. Themeandriftofsynapseiattimetcanbeapproximatedas
1 0
w_i(t)=W+(1(cid:0)wi)(cid:22)Z0 dse(cid:0)s=(cid:28)Ci(s;t)(cid:0)W(cid:0)wi(cid:22)Z(cid:0)1dses=(cid:28)Ci(s;t) , (4)
whereC (s;t)=hS (t)S(cid:3)(t+s)i istheensembleaveragedcorrelationfunctionbetween
i i E
inputiandtheoutputoftheneuron(see[5,6]).
ForthelinearPoissonneuronmodel,input-outputcorrelationscanbedescribedbymeans
of correlations in the inputs. We define the normalized cross correlation between input
spiketrainsS andS withacommonrater >0as
i j
hS (t)S (t+s)i
C0(s)= i j E (cid:0)1, (5)
ij r2
whichassumesvalue0foruncorrelatedPoissonspiketrains. Weassumeinthisarticlethat
C0 isconstantovertime. Inoursetup,theoutputoftheneuronduringlearningisclamped
ij
totheteacherspiketrainS(cid:3) whichistheoutputofaneuronwiththetargetweightvector
w(cid:3). Therefore, the input-output correlations C (s;t) are also constant over time and we
i
denotethembyC (s)inthefollowing. Inourneuronmodel,correlationsareshapedbythe
i
response kernel (cid:15)(s) and they enter the learning equation (4) with respect to the learning
window. Thismotivatesthedefinitionofwindowcorrelationsc+ andc(cid:0) forthepositive
ij ij
andnegativelearningwindowrespectively:
1 1
1
c(cid:6) =1+ dse(cid:0)s=(cid:28) ds0(cid:15)(s0)C0((cid:6)s(cid:0)s0) . (6)
ij (cid:28) Z0 Z0 ij
We call the matrices C(cid:6) = fc(cid:6)ijgi;j=1;:::;n the window correlation matrices. Note that
windowcorrelationsarenon-negativeandthatforhomogeneousPoissoninputspiketrains
and for a non-negative response kernel, they are positive. For soft weight bounds and
(cid:22) > 0,asynapticweightcanconvergetoavaluearbitrarilycloseto0or1,butnottoone
ofthesevaluesdirectly. Thismotivatesthefollowingdefinitionoflearnability.
Definition2.1 We say that a target weight vector w(cid:3) 2 f0;1gn can approximately be
learnedinasupervisedparadigmbySTDPwithsoftweightboundsonhomogeneousPois-
soninputspiketrains(short:“w(cid:3)canbelearned”)ifandonlyifthereexistW+;W(cid:0) >0,
suchthatfor(cid:22)!0theensembleaveragedweightvectorhw(t)i withlearningdynamics
E
givenbyEquation4convergestow(cid:3)foranyinitialweightvectorw(0)2[0;1]n.
Wearenowreadytoformulateananalyticalcriterionforlearnability:
Theorem2.1 A weight vector w(cid:3) can be learned for homogeneous Poisson input spike
trainswithwindowcorrelationmatricesC+ andC(cid:0) toalinearPoissonneuronwithnon-
negativeresponsekernelifandonlyifw(cid:3) 6=0and
nk=1wk(cid:3)c+ik > nk=1wk(cid:3)c+jk
Pn w(cid:3)c(cid:0) Pn w(cid:3)c(cid:0)
k=1 k ik k=1 k jk
P P
forallpairshi;ji2f1;:::;ng2 withw(cid:3) =1andw(cid:3) =0.
i j
Proofidea: WeobtainthecorrelationsbetweenaninputandtheoutputbyusingEq. 2:
n 1
(cid:3) (cid:3) 0 0 0
C (s)=hS (t)S (t+s)i = w ds (cid:15)(s)hS (t)S (t+s(cid:0)s)i .
i i E Xj=1 j Z0 i j E
SubstitutionofthisequationintoEq. 4yieldsthesynapticdrift
n n
w_i =(cid:28)r22W+f+(cid:22)(wi) wj(cid:3)c+ij (cid:0)W(cid:0)f(cid:0)(wi) wj(cid:3)c(cid:0)ij3 . (7)
Xj=1 Xj=1
4 5
We find the equilibrium points w of synapse i by setting w_ = 0 in Eq. 7. This yields
(cid:22)i i
(cid:0)1
w = 1+ 1 , where (cid:3) denotes W+ nj=1wj(cid:3)c+ij. Note that the drift is zero if
(cid:22)i (cid:18) (cid:3)1=(cid:22)(cid:19) i W(cid:0)Pn w(cid:3)c(cid:0)
w(cid:3) = 0 whichiimplies that w(cid:3) = 0 cannot bePleaj=rn1edj. iFjor w(cid:3) 6= 0, one can show that
w(cid:22) = (w(cid:22)1;:::;w(cid:22)n) is the only equilibrium point of the system and that it is stable.
Furthermore,oneseesthatlim(cid:22)!0w(cid:22)i = 1ifandonlyif(cid:3)i > 1,andlim(cid:22)!0w(cid:22)i = 0if
andonlyif(cid:3)i <1. Therefore,lim(cid:22)!0w(cid:22) =w(cid:3) holdsifandonlyif(cid:3)i >1foralliwith
w(cid:3) =1and(cid:3) <1foralliwithw(cid:3) =0. Thetheoremfollowsfromthedefinitionof(cid:3) .
i i i i
For a wide class of cross-correlation functions, one can establish a relationship between
learnabilityby STDP and thewell-known conceptof linearseparabilityfromlinearalge-
bra.1 Because of synaptic delays, the response of a spiking neuron to an input spike is
delayedbysometimet0. Onecanmodelsuchadelayintheresponsekernelbytherestric-
tion(cid:15)(s) = 0foralls (cid:20) t0. InthefollowingCorollaryweconsiderthecasewhereinput
correlationsC0(s)appearonlyinatimewindowsmallerthanthedelay:
ij
Corollary2.1 Ifthereexistsat0 (cid:21)0suchthat(cid:15)(s)=0foralls(cid:20)t0andCi0j(s)=0for
all s < (cid:0)t0; i;j 2 f1;:::;ng, then the following holds for the case of homogeneous
PoissoninputspiketrainstoalinearPoissonneuronwithpositiveresponsekernel(cid:15):
Aweightvectorw(cid:3)canbelearnedifandonlyifw(cid:3) 6=0andw(cid:3)linearlyseparatesthelist
L=hhc+;w(cid:3)i;:::;hc+;w(cid:3)ii,wherec+;:::;c+aretherowsofC+.
1 1 n n 1 n
(cid:0)
Proofidea: Fromtheassumptionsofthecorollaryitfollowsthatc =1. Inthiscase,the
ij
conditioninTheorem2.1isequivalenttothestatementthatw(cid:3) linearlyseparatesthelist
L=hhc+;w(cid:3)i;:::;hc+;w(cid:3)ii.
1 1 n n
Corollary2.1canbeviewedasananalogonofthePerceptronConvergenceTheoremforthe
averagecaseanalysisofSTDP.Itsformulationistightinthesensethatlinearseparabilityof
thelistLalone(asopposedtolinearseparabilitybythetargetvectorw(cid:3))isnotsufficient
to imply learnability. For uncorrelated input spike trains of rate r > 0, the normalized
cross correlation functions are given by C0(s) = (cid:14)ij(cid:14)(s), where (cid:14) is the Kronecker
ij r ij
deltafunction. ThepositivewindowcorrelationmatrixC+isthereforeessentiallyascaled
versionoftheidentitymatrix. ThefollowingcorollarythenfollowsfromCorollary2.1:
Corollary2.2 Atargetweightvectorw(cid:3) 2f0;1gncanbelearnedinthecaseofuncorre-
latedPoissoninputspiketrainstoalinearPoissonneuronwithpositiveresponsekernel(cid:15)
suchthat(cid:15)(s)=0foralls(cid:20)0ifandonlyif w(cid:3) 6=0.
3 ComputersimulationsofsupervisedlearningwithSTDP
In order to make a theoretical analysis feasible, we needed to make in section 2 a num-
ber of simplifying assumptions on the neuron model and the synapse model. In addition
a number of approximations had to be used in order to simplify the estimates. We con-
sider in this section the more realistic integrate-and-fire model2 for neurons and a model
forsynapseswhicharesubjecttopaired-pulsedepressionandpaired-pulsefacilitation,in
additiontothelongtermplasticityinducedbySTDP[7]. Thismodeldescribessynapses
withparametersU (initialreleaseprobability),D(depressiontimeconstant),andF (facil-
itation time constant) in addition to the synaptic weight w. The parametersU, D, and F
1Letc1;:::;cm 2Rnandy1;:::;ym 2f0;1g.Wesaythatavectorw2Rnlinearlyseparates
the list hhc1;y1i;:::;hcm;ymii if there exists a threshold (cid:2) such that yi = sign(ci (cid:1)w (cid:0)(cid:2))
fori=1;:::;m.Wedefinesign(z)=1ifz(cid:21)0andsign(z)=0otherwise.
2ThemembranepotentialV oftheneuronisgivenby(cid:28) dVm = (cid:0)(V (cid:0)V )+R (cid:1)
m m dt m resting m
(I (t)+I +I (t))where(cid:28) =C (cid:1)R =30msisthemembranetimeconstant,
syn background inject m m m
R =1M(cid:10)isthemembraneresistance,I (t)isthecurrentsuppliedbythesynapses,I
m syn background
isaconstantbackgroundcurrent,andI (t)representscurrentsinducedbya“teacher”. IfV
inject m
exceedsthethresholdvoltageV itisresettoV = 14:2mV andheldthereforthelength
thresh reset
T =3msoftheabsoluterefractoryperiod.Neuronparameters:V =0V,I
refract resting background
randomlychosenforeachtrialfromtheinterval[13:5nA;14:5nA]. V wassetsuchthateach
thresh
neuronspikedatarateofabout25Hz. Thisresultedinathresholdvoltageslightlyabove15mV.
Synapticparameters:Synapticcurrentsweremodeledasexponentiallydecayingcurrentswithdecay
timeconstants(cid:28) =3ms((cid:28) =6ms)forexcitatory(inhibitory)synapses.
S S
were randomly chosen from Gaussian distributions that were based on empirically found
dataforsuchconnections. Wealsoshowthatinsomecasesalessrestrictiveteacherforc-
ing suffices, that tolerates undesired firing of the neuron during training. The results of
section 2 predict that the temporal structure of correlations has a strong influence on the
outcomeofalearningexperiment. Weusedinputspiketrainswithcrosscorrelationsthat
decay exponentially with a correlation decay constant (cid:28) .3 In experiment 1 we consider
cc
relativelybroadtemporalcorrelations((cid:28) =10ms)andinexperiment2weconsidersharper
cc
correlations((cid:28) =6ms).
cc
Experiment 1 (correlated input with (cid:28) =10ms): In this experiment, a leaky integrate-
cc
and-fireneuronreceivedinputsfrom100dynamicsynapses. 90%ofthesesynapseswere
excitatory and 10% were inhibitory. For each excitatory synapse, the maximal efficacy
w was chosen from a Gaussian distribution with mean 54 and SD 10:8, bounded by
max
54(cid:6)3SD. The90excitatoryinputsweredividedinto9groupsof10synapsespergroup.
Spiketrainswerecorrelatedwithingroupswithcorrelationcoefficientsbetween0and0:8,
whereastherewerevirtuallynocorrelationsbetweenspiketrainsofdifferentgroups.4 Tar-
get weight vectors w(cid:3) were chosen in the most adverse way: half of the weights of w(cid:3)
withineachgroupwassetto0,theotherhalftoitsmaximalvaluew (seeFig.1C).
max
A
target
trained
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
time [sec]
B 1 C 60 target
40
0.8 w
20
0.6 0
0 20 40 60 80
angular error [rad] Synapse
0.4 spike correlation [s=5ms] D 60 trained
40
0.2 w
20
0 0
0 500 1000 1500 2000 2500 0 20 40 60 80
time [sec] Synapse
Figure1: Learningatargetweightvectorw(cid:3)oncorrelatedPoissoninputs.A)Outputspiketrainon
testdataafteronehouroftraining(trained)comparedtothetargetoutput(target). B)Evolutionof
theanglebetweenweightvectorw(t)andthevectorw(cid:3)thatimplementsF inradiant(angularerror,
solidline),andspikecorrelation(dashedline). C)Targetweightvectorw(cid:3) consistingofelements
withvalue0orthevaluew assignedtothatsynapse. D)Correspondingweightsofthelearned
max
vectorw(t)after40minutesoftraining.
Beforetraining,theweightsofallexcitatorysynapseswereinitializedbyrandomlychosen
small values. Weights of inhibitory synapses remained fixed throughout the experiment.
Informationaboutthetargetweightvectorw(cid:3) wasgiventotheneurononlyintheformof
shortcurrentinjections(1(cid:22)Afor0.2ms)atthosetimeswhentheneuronwiththeweight
vector w(cid:3) would have produced a spike. Learning was implemented as standard STDP
(seerule1)withparameters(cid:28)+ =(cid:28)(cid:0) =20ms,W+ =0:45,W(cid:0)=W+ =1:05. Additional
inhibitoryinputwasgiventotheneuronduringtrainingthatreducedtheoccurrenceofnon-
3Weconstructedinputspiketrainswithnormalizedcrosscorrelations(seeEquation5)approxi-
matelygivenbyC0(s)= ccij e(cid:0)jsj=(cid:28)cc betweeninputsiandjforameaninputrateofr=20Hz,
ij 2(cid:28)ccr
acorrelationcoefficientc ,andacorrelationdecayconstantof(cid:28) =10ms.
ij cc
4Thecorrelationcoefficientc forspiketrainswithinwithingroupkconsistingof10spiketrains
ij
wassettoc =cc =0:1(cid:3)(k(cid:0)1)fork=1;:::;9.
ij k
teacher-inducedfiringoftheneuron(seetextbelow).5 Twodifferentperformancemeasures
were used for analyzing the learning progress. The “spike correlation” measures for test
inputs that were not used for training (but had been generated by the same process) the
deviationbetweentheoutputspiketrainproducedbythetargetweightvectorw(cid:3) forthis
input,andtheoutputspiketrainproducedforthesameinputbytheneuronwiththecurrent
weight vector w(t)6. The angular error measures the angle between the current weight
vector w(t) and the target weight vector w(cid:3). The results are shown in Fig. 1. One can
seethatthedeviationofthelearnedweightvectorshowninpanelDfromthetargetweight
vectorw(cid:3) (panelC)isverysmall,evenforhighlycorrelatedgroupsofsynapseswithhet-
erogeneoustargetweights. Nosignificantchangesintheresultswereobservedforlonger
simulations(4hourssimulatedbiologicaltime),showingstabilityoflearning. On20trials
(each with a new random distribution of maximal weights w , different initializations
max
w(0)oftheweightvectorbeforelearning,andnewPoissonspiketrains),aspikecorrela-
tionof0:83(cid:6)0:06wasachieved(angularerror6:8(cid:6)4:7degrees).Notethatlearningisnot
only based on teacher spikes but also on non teacher-induced firing. Therefore, strongly
correlated groups of inputs tend to cause autonomous (i.e., not teacher-induced) firing of
theneuronwhichresultsinweightincreasesforallweightswithinthecorrespondinggroup
ofsynapsesaccordingtowell-knownresultsforSTDP[8,5]. Obviouslythiseffectmakes
it quite hard to learn a target weight vector w(cid:3) where half of the weights for each corre-
lated group have value 0. The effect is reduced by the additional inhibitory input during
trainingwhichreducesundesiredfiring. However,withoutthisinputaspikecorrelationof
0:79(cid:6)0:09couldstillbeachieved(angularerror14:1(cid:6)10degrees).
A 1 B 0.35
predicted to be learnable
0.9 0.3 predicted to be not learnable
n0.8 on0.25
spike correlatio000...567 1−spike correlati0.001..512
0.4 0.05
0.3 0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 5 10 15
input correlation cc weight error [°]
Figure 2: A)Spikecorrelationachievedforcorrelatedinputs(solidline). Someinputswerecor-
relatedwithccplottedonthex-axis. Also,asacontrolthespikecorrelationachievedbyrandomly
drawnweightvectorsisshown(dashedline,wherehalfoftheweightsweresettow andtheother
max
weightsweresetto0). B)Comparisonbetweentheoryandsimulationresultsforaleakyintegrate-
and-fireneuronandinputcorrelationsbetween0:1and0:5((cid:28) = 6ms). Eachcross(opencircle)
cc
marksatrialwherethetargetvectorwaslearnable(notlearnable)accordingtoTheorem2.1. The
actuallearningperformanceofSTDPisplottedforeachtrialintermsoftheweighterror(x-axis)and
1minusthespikecorrelation(y-axis).
Experiment2(testingthetheoreticalpredictionsfor(cid:28) =6ms): Inordertoevaluatethe
cc
dependence of correlation among inputs we proceeded in a setup similar to experiment
1. 4inputgroupsconsistingeachof10inputspiketrainswereconstructedforwhichthe
correlations within each group had the same value cc while the input spike train to the
other 50 excitatory synapses were uncorrelated. Again, half of the weights of w(cid:3) within
5Weadded30inhibitorysynapseswithweightsdrawnfromagammadistributionwithmean25
andstandarddeviation7:5,thatreceivedadditional30uncorrelatedPoissonspiketrainsat20Hz.
6ForthatpurposeeachspikeinthesetwooutputspiketrainswasreplacedbyaGaussianfunction
with an SD of 5ms. The spike correlation between both output spike trains was defined as the
correlationbetweentheresultingsmoothfunctionsoftime(forsegmentsoflength100s).
eachcorrelatedgroup(andwithintheuncorrelatedgroup)wassetto0,theotherhalftoa
randomlychosenmaximalvalue. Thelearningperformanceafter1houroftrainingfor20
trials is plotted in Fig. 2A for 7 different values of the correlation cc ((cid:28) = 6ms) that is
cc
appliedin4oftheinputgroups(solidline).
In order to test the approximate validity of Theorem 2.1 for leaky integrate-and-fire neu-
rons and dynamic synapses, we repeated the above experiment for input correlations
cc=0:1;0:2;0:3;0:4,and0:5. Foreachcorrelationvalue,20learningtrials(withdifferent
targetvectors)weresimulated. Foreachtrialwefirstcheckedwhetherthe(randomlycho-
sen)targetvectorw(cid:3)waslearnableaccordingtotheconditiongiveninTheorem2:1(65%
of the 100 learning trials were classified as being learnable).7 The actual performance of
learningwithSTDPwasevaluatedafter50minutesoftraining.8 TheresultisshowninFig.
2B. Itshows that the theoreticalprediction of learnability or non-learnability forthe case
ofsimplerneuronmodelsandsynapsesfromTheorem2.1translatesinabiologicallymore
realisticscenariointoaquantitativegradingofthelearningperformancethatcanultimately
beachievedwithSTDP.
A 1 B target
0.2
0.8 U
angular error [rad] 0.1
0.6 wspeiikgeh tc odrerveilaattiioonn 0 5 10 15 20
Synapse
0.4 C trained
0.2
0.2 U
0.1
0 0
0 500 1000 1500 2000 2500 5 10 15 20
time [sec] Synapse
Figure3: ResultsofmodulationofinitialreleaseprobabilitiesU.A)PerformanceofU-learningfor
agenericlearningtask(seetext).B)TwentyvaluesofthetargetU vector(eachcomponentassumes
itsmaximalpossiblevalueorthevalue0).C)CorrespondingU valuesafter42minutesoftraining.
Experiment 3 (Modulation of initial release probabilities U by STDP): Experimental
data from [9] suggest that synaptic plasticity does not change the uniform scaling of the
amplitudes of EPSPs resulting from a presynaptic spike train (i.e., the parameter w), but
ratherredistributesthesumoftheiramplitudes. IfoneassumesthatSTDPchangesthepa-
rameterU thatdeterminesthesynapticreleaseprobabilityforthefirstspikeinaspiketrain,
whereastheweightwremainsunchanged,thenthesameexperimentaldatathatsupportthe
classicalruleforSTDP,supportthefollowingruleforchangingU:
U = minfUmax;Uold+U+(cid:1)e(cid:0)(cid:1)t=(cid:28)+g ; if(cid:1)t>0 (8)
new (cid:26) maxf0;Uold(cid:0)U(cid:0)(cid:1)e(cid:1)t=(cid:28)(cid:0)g ; if(cid:1)t(cid:20)0 ;
withsuitablenonnegativeparametersUmax;U+;U(cid:0);(cid:28)+;(cid:28)(cid:0).
Fig. 3 shows results of an experiment where U was modulated with rule (8) (similar to
experiment1,butwithuncorrelatedinputs). 20repetitionsofthisexperimentyieldedafter
42minutesoftrainingthefollowingresults: spikecorrelation0:88(cid:6)0:036,angularerror
27:9(cid:6)3:7degrees,forU+ =0:0012,U(cid:0)=U+ =1:055. Apparentlytheoutputspiketrain
is less sensitive to changes in the values of U than to changes in w. Consequently, since
7Wehadchosenaresponsekerneloftheform(cid:15)(s)= (cid:28)1(cid:0)1(cid:28)2(e(cid:0)s=(cid:28)1 (cid:0)e(cid:0)s=(cid:28)2)with(cid:28)1 =2ms
and(cid:28)2 =1ms(Leastmeansquaresfitofthedoubleexponentialtotheperi-stimulus-timehistogram
(PSTH)oftheneuron,whichreflectstheprobabilityofspikingasafunctionoftimessinceaninput
spike),andcalculatedthewindowcorrelationsc+ andc(cid:0) numerically.
ij ij
8Toguaranteethebestpossibleperformanceforeachlearningtrial,trainingwasperformedon27
differentvaluesforW(cid:0)=W+between1:02and1:15.
only the behavior of a neuron with vector U(cid:3) but not the vector U(cid:3) is made available to
theneuronduringtraining,theresultingcorrelationbetweentarget-andactualoutputspike
trains is quite high, whereas angular error between U(cid:3) and U(t), as well as the average
deviationinU,remainratherlarge.
We also repeated experiment 1 (correlated Poisson inputs) with rule (8) for U-learning.
20repetitionswithdifferenttargetweightsanddifferentinitialconditionsyieldedafter35
minutes of training: spike correlation 0:75(cid:6)0:08, angular error 39:3(cid:6)4:8 degrees, for
U+ =8(cid:1)10(cid:0)4,U(cid:0)=U+ =1:09.
4 Discussion
Themainconclusionofthisarticleisthatformanycommondistributionsofinputspikes
a spiking neuron can learn with STDP and teacher-induced input currents any map from
inputspiketrainstooutputspiketrainsthatitcouldpossiblyimplementinastablemanner.
We have shown in section 2 that a mathematical average case analysis can be carried out
for supervised learning with STDP. This theoretical analysis produces the first criterion
that allows us to predict whether supervised learning with STDP will succeed in spite of
correlationsamongPoissoninputspiketrains. Forthespecialcaseof“sharpcorrelations”
(i.e. whenthecrosscorrelationsvanishfortimeshiftslargerthanthesynapticdelay)this
criterioncanbeformulatedintermsoflinearseparabilityoftherowsofacorrelationmatrix
related to the spike input, and itsmathematicalform istherefore reminiscent of the well-
knownconditionforlearnabilityinthecaseofperceptronlearning. InthissenseCorollary
2.1 can be viewed as an analogon of the Perceptron Convergence Theorem for spiking
neuronswithSTDP.
FurthermorewehaveshownthatanalternativeinterpretationofSTDPwhereoneassumes
thatitmodulatestheinitialreleaseprobabilitiesU ofdynamicsynapses, ratherthantheir
scalingfactorsw,givesrisetoverysatisfactoryconvergenceresultsforlearning.
Acknowledgment: We would like to thank Yves Fregnac, Wulfram Gerstner, and espe-
ciallyHenryMarkramforinspiringdiscussions.
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