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Neurocomputing65–66(2005)203–209ModellingavisualdiscriminationtaskB.
Gaillard,J.
FengDepartmentofInformatics,UniversityofSussex,COGS,Falmer,BrightonBN19QH,UKAvailableonline18December2004AbstractWestudytheperformanceofaspikingnetworkmodelbasedonintegrate-and-reneuronswhenperformingabenchmarkdiscriminationtask.
Thetaskconsistsofdeterminingthedirectionofmovingdotsinanoisycontext.
Byvaryingthesynapticparametersoftheintegrate-and-reneurons,weillustratethecounter-intuitiveimportanceofthesecond-orderstatistics(inputnoise)inimprovingthediscriminationaccuracyofthemodel.
Surprisingly,wefoundthatmeasuringtheringrate(FR)ofapopulationofneuronsconsiderablyenhancesthediscriminationaccuracyaswell,incomparisonwiththeringrateofasingleneuron.
r2004ElsevierB.
V.
Allrightsreserved.
Keywords:Discrimination;Firingrate;Inputnoise;Population1.
IntroductionDiscriminatingbetweeninputsisafundamentaltaskforthevisualsystem.
Inmostcases,theaccuracyofthediscriminationisdirectlylinkedtothereactiontime:thisisexpressedastheFittslaw.
Experimentswithrandomdotsstimuliareclassicalwaystostudyit,NewsomeandShadlen[5]haveexperimentedonthisdiscriminationprocessinMacaquemonkeys.
Specically,theyhavestudiedneuronsfromthelateralintraparietal(LIP)areaofthecortex,whosebehaviorARTICLEINPRESSwww.
elsevier.
com/locate/neucom0925-2312/$-seefrontmatterr2004ElsevierB.
V.
Allrightsreserved.
doi:10.
1016/j.
neucom.
2004.
10.
008Correspondingauthor.
E-mailaddresses:bg22@sussex.
ac.
uk(B.
Gaillard),jianfeng@sussex.
ac.
uk(J.
Feng).
dependsbothontheinputcategoryandonthedecisionofthemonkey.
So,thoseneuronsaretypicalofsensorimotordecisionprocesses,neithercompletelydeterminedbythestimulinorcompletelyindependentfromit.
Recently,interestingrelationsbetweenreactiontime(RT)anddiscriminationaccuracyhavebeenshown.
Weimplementedaneuralnetworkmodelforthisdiscriminationtaskusingintegrate-and-re(IF)neurons,sothatwecouldmodelthetimecourseofspikegeneration.
Evenifthemodeltakessimplisticassumptions,thissimplicityrenderstheobviousphenomenonitexhibits.
Wemeasuredtheringrate(FR)bothfromasingleandfromapopulationofneurons,whichenabledustomodeladiscriminationtaskwithinabiologicallyrealistictimescale.
Wecomparedthediscriminativeaccuracyofthepopulationmodeltotheperformanceofthesingleneuron,relativelytothenumberofemittedspikesandtotheprocessingtime.
Inourmodel,theroleofinhibitoryinputsandinputnoisecanaccountfortheFittslaw.
2.
ThediscriminationtaskWehaveimplementedadetailedmodeloftheLIPneuronsthattakepartinthedecisionofthemonkeyduringthetwochoicesdiscriminationtasksetupbyNewsomeetal.
inforexample[5,6].
Inthissetofexperiments,themonkeyshadtowatchadisplayofdots,acertainpercentageofthemmovingconsistentlyinonedirectionoritsopposite,andtherestofthedotsappearingatrandomplacesonthescreenasaperturbingnoise.
Thentheyhadtosignifythedirectionbyaneyemovement.
Thedifcultyofthetaskwascontrolledbymodifyingthepercentageofcoherentlymovingdots.
Weassumethatthediscriminatingneuronsreceivesynapticinputscomposedofanactualsignalperturbedbynoise.
Ifapercentagencofdotsmovescoherentlyinonedirection,thesamepercentageofsynapsesreceivescoherentinput.
Furthermore,weassumethatthespiketrainsarrivingtothosesynapsesarecorrelated.
Therestofthesynapsesreceiverandomlydistributedinputs.
ThesynapticinputsaremodelledasPoissonprocesses.
IthasbeenshownthatthemotiondetectorsofareaMTandMSTthatareinvolvedinthedecisionprocessofthemonkey[1]areconstitutedofcolumnsofneurons,andamodelhasbeenproposedforthisorganization[7].
So,itisprobablethattheneuronsencodingforthesamedirectionareclosetoeachotherandthusresynchronously.
TheoutputsofthediscriminatingneuronsarespiketrainswhoseFRsarerelatedtotheinputofthemovement,sothatwecancrudelymodelthatthisFRbeingbiggerorsmallerthanacriteriameansacommandfortheeyetomoverespectivelyupordown.
SincethereisavariationintheoutputFR,thiscommandcanbeerroneous,e.
g.
theFRisbiggerthanthecriteriumwhenthemovementisdownwards.
Thismimicsanerrormadebythemonkey,andfollowsthebehavioroftherealLIPneuronsthatsuggestthat''thedecisionmightbeembodiedindirecttransforma-tionsbetweentherelevantsensoryandmotorsystems''[5].
Ofcourse,theclearerthestimulus,themorewidelyseparatedtheefferentspiketrains,andthusthelesserrorsthemodelmakes.
ARTICLEINPRESSB.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–2092043.
ModeldescriptionThediscriminatingneuronmodelusedhereistheclassicalIFmodel[4,9].
WesimplisticallyassumedthateachsynapsereceivesaPoissonprocesswhoserateisproportionallylinkedtothedirectionofonemovingdotonthescreen,butindependentonthevelocity.
So,forncdotsthatmovecoherently,thencsynapsesthatreceivecoherentinputsarecorrelatedbyaconstantc,andreectthecorrelationofactivityofdifferentsynapsesasstudiedin[3,11].
Usingthediffusionapproximationasin[8,9],wereachthesimpliedfollowingdescriptionofthedynamicsofourdiscriminatingneuron,withVasthemembranepotential:dVVdtgmdtNsdtp;wheremXNcellsj11rlj;ands2XNcellsj11rljXnci1Xncj1;jaic1rliljp:Theratiobetweeninhibitoryinputsandexcitatoryinputs:risvariable.
Thenumberofincomingsynapses(correspondingtothenumberofdotsintheexperiments):Ncell100:ljisthedirectionofthejthdot.
Thetimedecayparameterg20ms:Thetimestepfortheintegrationdt0:01ms:Thecorrelationcoefcientbetweencoherentmotionc0:1:Thenumberofcoherentinputsncisvariable.
Coherentinputsaredotsthatmoveconsistentlyinonedirection.
Thus,thecoherenceisdenedasnc=Ncell:TherestingmembranepotentialVrest0mV:ThethresholdmembranepotentialVthreshold20mV:Nisanormallydistributedrandomvariable,NdtpistheBrownianmotion.
Insteadofusingonlyoneneuron,wecanmeasuretheFRofawholepopulation.
Onaverage,generating100spikeswith100neuronsonlyrequiresthetimeforoneneurontogenerateonespike;increasingthepopulationenablesustogenerateasmanyspikesaswewantinaveryshorttime.
ThisrehabilitatestheFRmeasure,inavisualsystemthatonlyhastimefor''onespikeperneuron''asarguedin[8].
Alltheneuronsofthepopulation,modelledasabove,receiveindependentinputswiththesamerates.
3.
1.
IncreasingtheinputnoiseWecaninterprettheequationofthedynamicsofthemembranepotentialoftheIFmodel(3)asaleakymembrane(Vdt=g)thatreceivesaninputmmdt;perturbedbyastochasticnoise(sNdtp).
Sincethisstochasticperturbationisproportionalto1randthemeanisproportionalto1r;thestochasticeffectARTICLEINPRESSB.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–209205ofthesynapseincreaseswithr,theratiobetweeninhibitoryandexcitatoryinputs.
Asexplainedin[3],anincreaseinthecoefcientofvariabilityintheinputwillincreasethecoefcientofvariabilityoftheefferentspiketrainoftheneuron.
Thus,intuitively,itshouldbemoredifculttodiscriminatebetweentwoinputsfromtheirefferentFR.
However,Fengandhiscolleagues[2]haveformallyproventhatthisisnotthecasewhenthecoherentinputs(thoseuponwhichwediscriminate)arecorrelated.
Moreprecisely,heobtainedthefollowingconclusion:whenthecorrelationispositive,theaccuracyofthediscriminationincreaseswithr.
Weuseacorrelationcoefcientof0.
1,forsynapsesthatreceivethecoherentinput.
Ithasbeenshown[11]thatinareaV5ofthevisualcortexofthemonkeys,thelevelofcorrelationis0.
1andalthoughbeingweak,hasasignicantimpactontheglobalbehavior.
Thetheoreticallycounter-intuitiveresultsthatthelargerthecoefcientofvariation(CV)oftheinput,thebetterthediscriminationwhichisconrmedbythefollowingsimulationresults.
4.
Simulationresults4.
1.
Aperformancecriterium:totalprobabilityofmisclassication(TPM)Foreachsetofparametervalues,weperform100discriminationtrials,foreachdirection,andmeasuretheFReachtime.
TheFRisthenumberofemittedspikesdividedbythetimewindow.
TheexperimenterusestheFRasdecisiveevidence:iftheFRislargerthana'discriminationboundary',thanthemovementisclassiedupward,iftheFRissmaller,thenthemovementisclassieddownward.
ThisdiscriminationboundarydependsontheFRvalues,thusitisoptimalforeachsetofparametervalues.
4.
2.
Discriminationwitha100spikesExtensivesimulationsovertherangeofr,andovertherangeofinputcoherence(percentageofcoherentlymovingdots),producedthefollowingresults,summarizedinFig.
1:Obviously,theTPMdecreaseswhenthecoherenceincreases:themoreseparatedtheinputsare,theeasierthediscriminationtaskis.
TheTPMdecreaseswhenrincreases.
Thisdecreaseisnotmonotonic.
Forthesingleneuron,thebetterperformanceachievedbyincreasingtheinputnoiseoccursonlyforr40:7:Thepopulationperformsmuchbetter,foralmostoneorderofmagnitude,thanthesingleneuron,anditsTPMdecreasessteadilywithr.
Thebetterperformanceofthepopulationcanbeexplainedasfollows.
Inthepopulationapproach,weusetherst100spikesofa100neuronstomeasuretheFR,whichmeansthatweuseonaverageonespikeperneuron.
Longinterspikeintervals(ISI)areunlikelytobeproduced,becausetherewillbehundredspikesproducedARTICLEINPRESSB.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–209206beforeaspikefollowingalongISIwilleveroccurs.
TheselongerISIsincreasesignicantlythevariabilityoftheefferentFR,thusincreasingtheTPM.
Thisisthereasonforthebetterperformanceofthepopulation.
4.
3.
TimerelatedperformanceFormostbiologicalsystems,theabsoluteperformancemusttakeintoaccountnotonlytheaccuracyatrealizingthetask,butalsothetimespenttoachieveit.
Thetimetogeneratespikesvariesalotwhenrincreases.
Infact,whenr1;theonlypostsynapticinputisnoise,andtheFRisverylow.
WeseeinFig.
2thatgeneratingaARTICLEINPRESS00.
20.
40.
60.
8100.
020.
040.
060.
080.
10.
120.
140.
160.
18RatioTPMSingleNeuron100Neurons5101520253000.
10.
20.
30.
40.
50.
60.
7CoherenceTPM100Neurons,r=0.
98SingleNeuron,r=0.
6SingleNeuron,r=0.
98100Neurons,r=0.
6Fig.
1.
ComparisonoftheTPMofonesingleneuronandofapopulation,forvariousrandcoherences,using100spikes.
Leftpanel,coherence15%:Thetimewindowneededtocollectthese100spikesvariesalotwithparametervalues,especiallyitincreasesdramaticallywithr.
WewillevaluatetheeffectoftimeinFig.
2.
0.
60.
70.
80.
91020004000600080001000012000RATIOTimeto100spikes(ms)1neuron100neurons0.
50.
60.
70.
80.
910100200300400500600RATIOTimetoTPM=0.
1(ms)y=5.
3e+005*x5-1.
9e+006*x4+2.
7e+006*x3-1.
9e+006*x2+6.
6e+005*x-9.
1e+0040200400600800100000.
050.
10.
150.
20.
250.
30.
350.
4Time(ms)TPMr=0.
98cubicinterpolationR=0linearinterpolationFig.
2.
Coherence15%.
Left:timetogetahundredspikesversusr,withapopulationofahundredneuronsandwithasingleneuron.
Middle:Illustrationofthenumericalestimationofthetimetoreachanacceptablediscriminationperformance(TPM0:1).
Right:comparisonoftheevolutionoftheTPMforlongtimewindows,reachingtoonesecond,withr0:98andr0:Whenwewaitforonesecond,theTPMforr0:98is0.
03andtheTPMforr0is0.
09.
B.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–209207numberofspikessufcienttoreliablymeasureanFRincreasesdramaticallytheprocessingtime.
Thepopulationapproachpartlysolvesthisproblem,but,inordertoputtheTPMinperspective,wehavetomeasuretheevolutionofthequantityoferrorswiththesizeofthetimewindowduringwhichwecollectthespikes.
Thosetimeconsiderationsunderminetheadvantagegainedwithincreasingtheinputnoise;asweseeinFig.
2,itismuchquickertoachieveanacceptableperformancewithexclusivelyexcitatoryinputs.
However,theperformanceofthesystemcanbemuchbetter,overalongtimewindow,withbalancedexcitatoryandinhibitoryinputs(r'1).
5.
ConclusionsWehaveshownthatmeasuringtheFRofapopulationofneuronsenablesustoovercomethetimescaleimpossibilitiesoftenassociatedwiththeFRapproach.
Althoughaugmentingr,i.
e.
theinputnoise,increasestheperfor-manceperspike,itincreasesthereactiontimedramatically.
Theprobabilityofmisclassicationdecreasesmuchquickerforsmallerratios.
However,wehaveseenthatonlyratiosclosetoonecanreachacertainlevelofperformanceunreachablebytheFRofapopulationwithexclusivelyexcitatorysynapses.
ThoseverygoodperformancesareachievedatthecostofaverylongRT.
ThisphenomenonofincreasedaccuracywithalongerprocessingtimeinlivingorganismsisknownastheFittslaw.
Furthermore,thefactthatinhibitoryinputsplayacentralroleinadiscriminationtaskisinagreementwithbiologicaldataasreportedin[10,6].
References[1]K.
H.
Britten,W.
T.
Newsome,M.
N.
Shadlen,S.
Celebrini,J.
A.
Movshon,ArelationshipbetweenbehavioralchoiceandthevisualresponsesofneuronsinmacaqueMT,VisualNeurosci.
13(1996)87–100.
[2]Y.
Deng,P.
Williams,F.
Liu,J.
Feng,Neuronaldiscriminationcapacity,J.
Phys.
A:Math.
General36(2003)12379–12398.
[3]J.
Feng,Istheintegrate-and-remodelgoodenough—areview,NeuralNetworks14(2001)955–975.
[4]W.
Gerstner,W.
Kistler,SpikingNeuronModels,SingleNeurons,Populations,Plasticity,CambridgeUniversityPress,Cambridge,2002.
[5]M.
Shadlen,W.
T.
Newsome,Neuralbasisofaperceptualdecisionintheparietalcortex(arealip)oftherhesusmonkey,J.
Neurophysiol.
86(2001)1835–1916.
[6]M.
Shadlen,J.
I.
Gold,Theneurophysiologyofdecisionmakingasawindowoncognition,in:M.
S.
Gazzaniga(Ed.
),TheCognitiveNeuroscience,thirded.
,MITPress,Cambridge,MA,2004.
[7]E.
P.
Simoncelli,D.
J.
Heeger,AmodelofneuronalresponsesinvisualareaMT,VisualRes.
38(1998)743–761.
[8]S.
Thorpe,R.
Vanrullen,Isitabird,isitaplaneUltra-rapidvisualcategorizationofnaturalandartifactualcategories,Perception(2000)539–550.
ARTICLEINPRESSB.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–209208[9]H.
C.
Tuckwell,IntroductiontoTheoreticalNeurobiology(2),CambridgeUniversityPress,Cambridge,1988.
[10]X.
J.
Wang,Probabilisticdecisionmakingbyslowreverberationincorticalcircuits,Neuron36(2002)955–968.
[11]E.
Zohary,M.
Shadlen,W.
Newsome,Correlatedneuronaldischargeanditsimplicationsforpsychologicalperformance,Nature370(1994)140–143.
ARTICLEINPRESSB.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–209209
Gaillard,J.
FengDepartmentofInformatics,UniversityofSussex,COGS,Falmer,BrightonBN19QH,UKAvailableonline18December2004AbstractWestudytheperformanceofaspikingnetworkmodelbasedonintegrate-and-reneuronswhenperformingabenchmarkdiscriminationtask.
Thetaskconsistsofdeterminingthedirectionofmovingdotsinanoisycontext.
Byvaryingthesynapticparametersoftheintegrate-and-reneurons,weillustratethecounter-intuitiveimportanceofthesecond-orderstatistics(inputnoise)inimprovingthediscriminationaccuracyofthemodel.
Surprisingly,wefoundthatmeasuringtheringrate(FR)ofapopulationofneuronsconsiderablyenhancesthediscriminationaccuracyaswell,incomparisonwiththeringrateofasingleneuron.
r2004ElsevierB.
V.
Allrightsreserved.
Keywords:Discrimination;Firingrate;Inputnoise;Population1.
IntroductionDiscriminatingbetweeninputsisafundamentaltaskforthevisualsystem.
Inmostcases,theaccuracyofthediscriminationisdirectlylinkedtothereactiontime:thisisexpressedastheFittslaw.
Experimentswithrandomdotsstimuliareclassicalwaystostudyit,NewsomeandShadlen[5]haveexperimentedonthisdiscriminationprocessinMacaquemonkeys.
Specically,theyhavestudiedneuronsfromthelateralintraparietal(LIP)areaofthecortex,whosebehaviorARTICLEINPRESSwww.
elsevier.
com/locate/neucom0925-2312/$-seefrontmatterr2004ElsevierB.
V.
Allrightsreserved.
doi:10.
1016/j.
neucom.
2004.
10.
008Correspondingauthor.
E-mailaddresses:bg22@sussex.
ac.
uk(B.
Gaillard),jianfeng@sussex.
ac.
uk(J.
Feng).
dependsbothontheinputcategoryandonthedecisionofthemonkey.
So,thoseneuronsaretypicalofsensorimotordecisionprocesses,neithercompletelydeterminedbythestimulinorcompletelyindependentfromit.
Recently,interestingrelationsbetweenreactiontime(RT)anddiscriminationaccuracyhavebeenshown.
Weimplementedaneuralnetworkmodelforthisdiscriminationtaskusingintegrate-and-re(IF)neurons,sothatwecouldmodelthetimecourseofspikegeneration.
Evenifthemodeltakessimplisticassumptions,thissimplicityrenderstheobviousphenomenonitexhibits.
Wemeasuredtheringrate(FR)bothfromasingleandfromapopulationofneurons,whichenabledustomodeladiscriminationtaskwithinabiologicallyrealistictimescale.
Wecomparedthediscriminativeaccuracyofthepopulationmodeltotheperformanceofthesingleneuron,relativelytothenumberofemittedspikesandtotheprocessingtime.
Inourmodel,theroleofinhibitoryinputsandinputnoisecanaccountfortheFittslaw.
2.
ThediscriminationtaskWehaveimplementedadetailedmodeloftheLIPneuronsthattakepartinthedecisionofthemonkeyduringthetwochoicesdiscriminationtasksetupbyNewsomeetal.
inforexample[5,6].
Inthissetofexperiments,themonkeyshadtowatchadisplayofdots,acertainpercentageofthemmovingconsistentlyinonedirectionoritsopposite,andtherestofthedotsappearingatrandomplacesonthescreenasaperturbingnoise.
Thentheyhadtosignifythedirectionbyaneyemovement.
Thedifcultyofthetaskwascontrolledbymodifyingthepercentageofcoherentlymovingdots.
Weassumethatthediscriminatingneuronsreceivesynapticinputscomposedofanactualsignalperturbedbynoise.
Ifapercentagencofdotsmovescoherentlyinonedirection,thesamepercentageofsynapsesreceivescoherentinput.
Furthermore,weassumethatthespiketrainsarrivingtothosesynapsesarecorrelated.
Therestofthesynapsesreceiverandomlydistributedinputs.
ThesynapticinputsaremodelledasPoissonprocesses.
IthasbeenshownthatthemotiondetectorsofareaMTandMSTthatareinvolvedinthedecisionprocessofthemonkey[1]areconstitutedofcolumnsofneurons,andamodelhasbeenproposedforthisorganization[7].
So,itisprobablethattheneuronsencodingforthesamedirectionareclosetoeachotherandthusresynchronously.
TheoutputsofthediscriminatingneuronsarespiketrainswhoseFRsarerelatedtotheinputofthemovement,sothatwecancrudelymodelthatthisFRbeingbiggerorsmallerthanacriteriameansacommandfortheeyetomoverespectivelyupordown.
SincethereisavariationintheoutputFR,thiscommandcanbeerroneous,e.
g.
theFRisbiggerthanthecriteriumwhenthemovementisdownwards.
Thismimicsanerrormadebythemonkey,andfollowsthebehavioroftherealLIPneuronsthatsuggestthat''thedecisionmightbeembodiedindirecttransforma-tionsbetweentherelevantsensoryandmotorsystems''[5].
Ofcourse,theclearerthestimulus,themorewidelyseparatedtheefferentspiketrains,andthusthelesserrorsthemodelmakes.
ARTICLEINPRESSB.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–2092043.
ModeldescriptionThediscriminatingneuronmodelusedhereistheclassicalIFmodel[4,9].
WesimplisticallyassumedthateachsynapsereceivesaPoissonprocesswhoserateisproportionallylinkedtothedirectionofonemovingdotonthescreen,butindependentonthevelocity.
So,forncdotsthatmovecoherently,thencsynapsesthatreceivecoherentinputsarecorrelatedbyaconstantc,andreectthecorrelationofactivityofdifferentsynapsesasstudiedin[3,11].
Usingthediffusionapproximationasin[8,9],wereachthesimpliedfollowingdescriptionofthedynamicsofourdiscriminatingneuron,withVasthemembranepotential:dVVdtgmdtNsdtp;wheremXNcellsj11rlj;ands2XNcellsj11rljXnci1Xncj1;jaic1rliljp:Theratiobetweeninhibitoryinputsandexcitatoryinputs:risvariable.
Thenumberofincomingsynapses(correspondingtothenumberofdotsintheexperiments):Ncell100:ljisthedirectionofthejthdot.
Thetimedecayparameterg20ms:Thetimestepfortheintegrationdt0:01ms:Thecorrelationcoefcientbetweencoherentmotionc0:1:Thenumberofcoherentinputsncisvariable.
Coherentinputsaredotsthatmoveconsistentlyinonedirection.
Thus,thecoherenceisdenedasnc=Ncell:TherestingmembranepotentialVrest0mV:ThethresholdmembranepotentialVthreshold20mV:Nisanormallydistributedrandomvariable,NdtpistheBrownianmotion.
Insteadofusingonlyoneneuron,wecanmeasuretheFRofawholepopulation.
Onaverage,generating100spikeswith100neuronsonlyrequiresthetimeforoneneurontogenerateonespike;increasingthepopulationenablesustogenerateasmanyspikesaswewantinaveryshorttime.
ThisrehabilitatestheFRmeasure,inavisualsystemthatonlyhastimefor''onespikeperneuron''asarguedin[8].
Alltheneuronsofthepopulation,modelledasabove,receiveindependentinputswiththesamerates.
3.
1.
IncreasingtheinputnoiseWecaninterprettheequationofthedynamicsofthemembranepotentialoftheIFmodel(3)asaleakymembrane(Vdt=g)thatreceivesaninputmmdt;perturbedbyastochasticnoise(sNdtp).
Sincethisstochasticperturbationisproportionalto1randthemeanisproportionalto1r;thestochasticeffectARTICLEINPRESSB.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–209205ofthesynapseincreaseswithr,theratiobetweeninhibitoryandexcitatoryinputs.
Asexplainedin[3],anincreaseinthecoefcientofvariabilityintheinputwillincreasethecoefcientofvariabilityoftheefferentspiketrainoftheneuron.
Thus,intuitively,itshouldbemoredifculttodiscriminatebetweentwoinputsfromtheirefferentFR.
However,Fengandhiscolleagues[2]haveformallyproventhatthisisnotthecasewhenthecoherentinputs(thoseuponwhichwediscriminate)arecorrelated.
Moreprecisely,heobtainedthefollowingconclusion:whenthecorrelationispositive,theaccuracyofthediscriminationincreaseswithr.
Weuseacorrelationcoefcientof0.
1,forsynapsesthatreceivethecoherentinput.
Ithasbeenshown[11]thatinareaV5ofthevisualcortexofthemonkeys,thelevelofcorrelationis0.
1andalthoughbeingweak,hasasignicantimpactontheglobalbehavior.
Thetheoreticallycounter-intuitiveresultsthatthelargerthecoefcientofvariation(CV)oftheinput,thebetterthediscriminationwhichisconrmedbythefollowingsimulationresults.
4.
Simulationresults4.
1.
Aperformancecriterium:totalprobabilityofmisclassication(TPM)Foreachsetofparametervalues,weperform100discriminationtrials,foreachdirection,andmeasuretheFReachtime.
TheFRisthenumberofemittedspikesdividedbythetimewindow.
TheexperimenterusestheFRasdecisiveevidence:iftheFRislargerthana'discriminationboundary',thanthemovementisclassiedupward,iftheFRissmaller,thenthemovementisclassieddownward.
ThisdiscriminationboundarydependsontheFRvalues,thusitisoptimalforeachsetofparametervalues.
4.
2.
Discriminationwitha100spikesExtensivesimulationsovertherangeofr,andovertherangeofinputcoherence(percentageofcoherentlymovingdots),producedthefollowingresults,summarizedinFig.
1:Obviously,theTPMdecreaseswhenthecoherenceincreases:themoreseparatedtheinputsare,theeasierthediscriminationtaskis.
TheTPMdecreaseswhenrincreases.
Thisdecreaseisnotmonotonic.
Forthesingleneuron,thebetterperformanceachievedbyincreasingtheinputnoiseoccursonlyforr40:7:Thepopulationperformsmuchbetter,foralmostoneorderofmagnitude,thanthesingleneuron,anditsTPMdecreasessteadilywithr.
Thebetterperformanceofthepopulationcanbeexplainedasfollows.
Inthepopulationapproach,weusetherst100spikesofa100neuronstomeasuretheFR,whichmeansthatweuseonaverageonespikeperneuron.
Longinterspikeintervals(ISI)areunlikelytobeproduced,becausetherewillbehundredspikesproducedARTICLEINPRESSB.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–209206beforeaspikefollowingalongISIwilleveroccurs.
TheselongerISIsincreasesignicantlythevariabilityoftheefferentFR,thusincreasingtheTPM.
Thisisthereasonforthebetterperformanceofthepopulation.
4.
3.
TimerelatedperformanceFormostbiologicalsystems,theabsoluteperformancemusttakeintoaccountnotonlytheaccuracyatrealizingthetask,butalsothetimespenttoachieveit.
Thetimetogeneratespikesvariesalotwhenrincreases.
Infact,whenr1;theonlypostsynapticinputisnoise,andtheFRisverylow.
WeseeinFig.
2thatgeneratingaARTICLEINPRESS00.
20.
40.
60.
8100.
020.
040.
060.
080.
10.
120.
140.
160.
18RatioTPMSingleNeuron100Neurons5101520253000.
10.
20.
30.
40.
50.
60.
7CoherenceTPM100Neurons,r=0.
98SingleNeuron,r=0.
6SingleNeuron,r=0.
98100Neurons,r=0.
6Fig.
1.
ComparisonoftheTPMofonesingleneuronandofapopulation,forvariousrandcoherences,using100spikes.
Leftpanel,coherence15%:Thetimewindowneededtocollectthese100spikesvariesalotwithparametervalues,especiallyitincreasesdramaticallywithr.
WewillevaluatetheeffectoftimeinFig.
2.
0.
60.
70.
80.
91020004000600080001000012000RATIOTimeto100spikes(ms)1neuron100neurons0.
50.
60.
70.
80.
910100200300400500600RATIOTimetoTPM=0.
1(ms)y=5.
3e+005*x5-1.
9e+006*x4+2.
7e+006*x3-1.
9e+006*x2+6.
6e+005*x-9.
1e+0040200400600800100000.
050.
10.
150.
20.
250.
30.
350.
4Time(ms)TPMr=0.
98cubicinterpolationR=0linearinterpolationFig.
2.
Coherence15%.
Left:timetogetahundredspikesversusr,withapopulationofahundredneuronsandwithasingleneuron.
Middle:Illustrationofthenumericalestimationofthetimetoreachanacceptablediscriminationperformance(TPM0:1).
Right:comparisonoftheevolutionoftheTPMforlongtimewindows,reachingtoonesecond,withr0:98andr0:Whenwewaitforonesecond,theTPMforr0:98is0.
03andtheTPMforr0is0.
09.
B.
Gaillard,J.
Feng/Neurocomputing65–66(2005)203–209207numberofspikessufcienttoreliablymeasureanFRincreasesdramaticallytheprocessingtime.
Thepopulationapproachpartlysolvesthisproblem,but,inordertoputtheTPMinperspective,wehavetomeasuretheevolutionofthequantityoferrorswiththesizeofthetimewindowduringwhichwecollectthespikes.
Thosetimeconsiderationsunderminetheadvantagegainedwithincreasingtheinputnoise;asweseeinFig.
2,itismuchquickertoachieveanacceptableperformancewithexclusivelyexcitatoryinputs.
However,theperformanceofthesystemcanbemuchbetter,overalongtimewindow,withbalancedexcitatoryandinhibitoryinputs(r'1).
5.
ConclusionsWehaveshownthatmeasuringtheFRofapopulationofneuronsenablesustoovercomethetimescaleimpossibilitiesoftenassociatedwiththeFRapproach.
Althoughaugmentingr,i.
e.
theinputnoise,increasestheperfor-manceperspike,itincreasesthereactiontimedramatically.
Theprobabilityofmisclassicationdecreasesmuchquickerforsmallerratios.
However,wehaveseenthatonlyratiosclosetoonecanreachacertainlevelofperformanceunreachablebytheFRofapopulationwithexclusivelyexcitatorysynapses.
ThoseverygoodperformancesareachievedatthecostofaverylongRT.
ThisphenomenonofincreasedaccuracywithalongerprocessingtimeinlivingorganismsisknownastheFittslaw.
Furthermore,thefactthatinhibitoryinputsplayacentralroleinadiscriminationtaskisinagreementwithbiologicaldataasreportedin[10,6].
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