A novel test to determine the significance of neural selectivity to single and multiple potentially correlated stimulus features

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JournalofNeuroscienceMethodsxxx (2011) xxx–xxx

ContentslistsavailableatSciVerseScienceDirect

Journal

of

Neuroscience

Methods

j o ur n a l ho me p ag e :w w w . e l s e v i e r . c o m / l o c a t e / j n e u m e t h

A

novel

test

to

determine

the

signiﬁcance

of

neural

selectivity

to

single

and

multiple

potentially

correlated

stimulus

features

Robin

A.A.

Ince

a,1

_,

_Alberto

_Mazzoni

b,1

_,

_Andreas

_Bartels

a,d

_,

_Nikos

_K.

_Logothetis

a,e

_,

_Stefano

_Panzeri

b,c,f,∗

a_Max_Planck_Institute_for_Biological_Cybernetics,_{Spemannstrasse}_38,₇₂₀₇₆_Tübingen,_Germany

b_Robotics,_Brain_and_Cognitive_Sciences_Department,_Istituto_Italiano_di_Tecnologia,_Via_Morego_30,₁₆₁₆₃_Genova,_Italy c_Bernstein_Center_for_{Computational}_{Neuroscience,}_Tübingen,_Germany

d_Vision_and_Cognition_Lab,_Centre_for_Integrative_{Neuroscience,}_University_of_Tübingen,_72076,_Tübingen,_Germany e_Division_of_Imaging_Science_and_Biomedical_Engineering,_University_of_Manchester,_Manchester_M13_9PT,_United_Kingdom f_Institute_of_Neuroscience_and_Psychology,_University_of_Glasgow,_Glasgow,_United_Kingdom

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received30June2011

Receivedinrevisedform7November2011 Accepted8November2011 Keywords: Vision LFP Neuralcoding Naturalmovies Mutualinformation Oscillations Gammaband

a

b

s

t

r

a

c

t

Mutualinformationisaprinciplednon-linearmeasureofdependencebetweenstochasticvariables, whichiswidelyusedtostudytheselectivityofneuralresponsestoexternalstimuli.Herewedefine anddevelopasetofnovelstatisticalindependencetestsbasedonmutualinformation,which quan-tifythesignificanceofneuralselectivitytoeithersinglefeaturesortomultiple,potentiallycorrelated stimulusfeatureslikethoseoftenpresentinnaturalisticstimuli.Ifthevaluesofdifferentfeaturesare correlatedduringstimuluspresentation,itisdifficulttoestablishifonefeatureisgenuinelyencodedby theresponse,orifitinsteadappearstobeencodedonlyasasideeffectofitscorrelationwithanother genuinelyrepresentedfeature.Ourtestsprovideawaytodisambiguatebetweenthesetwopossibilities. Weuserealisticsimulationsofneuralresponsestunedtooneormorecorrelatedstimulusfeaturesto investigatehowlimitedsamplingbiascorrectionproceduresaffectthestatisticalpowerofsuch inde-pendencetests,andwecharacterizetheregimesinwhichthedistributionofinformationvaluesunder thenullhypothesiscanbeapproximatedbysimpledistributions(Chi-squareorGaussian).Finally,we applytheseteststoexperimentaldatatodeterminethesignificanceoftuningofthebandlimitedpower (BLP)ofthegamma[30–100Hz]frequencyrangeoftheprimaryvisualcorticallocalfieldpotentialto multiplecorrelatedfeaturesduringpresentationofnaturalisticmovies.WeshowthatgammaBLPcarries significant,genuineinformationaboutorientation,spacecontrastandtimecontrast,despitethestrong correlationsbetweenthesefeatures.

1. Introduction

Mutual information (Cover and Thomas, 1991) (hereafter referredtoasinformation)isameasureofdependencebetween stochasticvariables.Informationquantiﬁesthereductionin uncer-taintyaboutonevariable(forexample,asensorystimulus)after observationof another (forexample, a neural response). Infor-mation is a popularmetric toquantify, on a scale that allows meaningful comparisons acrossdifferent systems,how much a giventypeofaneuralresponseistunedtooneormorestimulus parameters(BorstandTheunissen,1999;Inceetal.,2010;Optican

∗ Correspondingauthorat:Robotics,BrainandCognitiveSciencesDepartment, IstitutoItalianodiTecnologia,ViaMorego30,16163Genova,Italy.

Tel.:+393316933490;fax:+390107170817. E-mailaddress:stefano.panzeri@iit.it(S.Panzeri). 1 _These_authors_contributed_equally_to_this_work.

andRichmond,1987;Strongetal.,1998).Itsmainadvantageisthat

itisbasedonthefullprobabilitydistributionsoftheconsidered variablesandcanthereforequantifytheeffectofanynon-linearor arbitrarilycomplicateddependency.

Thecalculationofneuralinformationvaluesandtheevaluation of theirsigniﬁcance is madedifﬁcultbythefact thatin neuro-physiologicalexperimentsthejointprobabilitiesofstimulusand neuralresponseshavetobeestimatedempiricallyfromalimited numberofdatasamples(“trials”).Limitedsamplinginducesboth asystematicerror(bias)anda varianceintheestimationofthe neuralinformation values. While thelimited samplingbiashas beenstudiedextensivelyintheneuralliterature(Paninski,2003;

Panzerietal.,2007;Victor,2006),theevaluationofthe

statisti-calsignificance of aninformation valuecomputedfromlimited experimentaldatahasbeenlessthoroughlyinvestigated.The sta-tisticalsignificance ofaninformationvaluetellsuswhetherwe can, at a particular confidence level or p-value, rejectthe null hypothesisthatthestimulusandresponseareindependent.The 0165-0270/$–seefrontmatter © 2011 Elsevier B.V. All rights reserved.

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2 R.A.A.Inceetal./JournalofNeuroscienceMethodsxxx (2011) xxx–xxx signiﬁcance of information values is traditionally calculated by

employingaformofbootstraptest(Opticanetal.,1991).The stim-uliandresponseareshuffledacrossexperimentaltrialstoremove anydependence,andtheinformationvalueiscalculatedformany repetitionsofthisshufflingproceduretoobtainthedistributionof valuesunderthenullhypothesisofindependence.However, sev-eralproblemsrelated totheestimationofsignificance havenot yetbeenconsidered.The firstproblemis whetherit ispossible toemployparametricmodelsforthenullhypothesisdistribution. Thesecondproblemregardstheeffectofsamplingbias correc-tionsonthestatisticalpowerofsuchtests.Thethirdproblemisto determinewhichofthemanystimulusfeaturescharacterizing nat-uralisticstimuliactuallyinfluencetheconsideredneuralresponse. Thisisdifficultbecauseindynamicandcomplexstimuli,likethe onesfound in a natural environment and mimicked by movie stimuli,differentstimulusfeatures areoftenstronglycorrelated

(FelsenandDan,2005;Geisler,2008;Simoncelli,2003;Simoncelli

andOlshausen, 2001).Anabrupt movementofan observer,for

instance,whichleadstosuddenappearanceordisappearanceof objects,willnecessarilyproduceconcurrent,andhencecorrelated, changesinalargenumberoffeatures(Bartelsetal.,2008).Ifthe neuralresponsecarriessignificantinformationabouteachoftwo differentbut correlated features, it doesnot follow necessarily thattheneuralresponseisgenuinelyinfluencedbybothofthem. Theresponsecouldinsteadbeinfluencedbyonlyoneofthe fea-tures,withthedependenceontheotherfeatureappearingbecause ofthecorrelations betweenthem.Understandinghowto deter-minetrueneuronalselectivityinthesecasesisessentialtoallow meaningfulinvestigationoftheneuralrepresentation ofnatural stimuli.

Hereweinvestigatetheseissuesbyfirstcomparingbootstrap basedsignificancetestresultsforsinglefeatureinformationvalues includingdifferentbiascorrections,andpresentingcomparisons withananalyticexpressionforthenullhypothesisdistributionas wellasparametricfitsofsimpledistributionstosmallbootstrap samples.Thislaysthebasistounderstandhowbesttoperforma statisticaltestofindependencebasedonmutualinformation.We theninvestigatetheextenttowhichdependenciesbetween vari-ablescomplicatetheestimationofgenuinetuningtoeachfeature. Weconstructstatisticaltestsforinferringtrueselectivityinthe presenceofmultipleco-varyingfeatures.Wevalidatethese meth-odsbytestingthemonrealisticallysimulatedneuronalresponses tomultiplecorrelatedfeatures,andfinallyweillustratethemby investigatingtheextenttowhichthegammabandlimitedpower (BLP)oflocalfieldpotentials(LFPs)recordedfromtheV1areaof anesthetizedmacaquespresentedwithnaturalisticmoviesis gen-uinelytunedtooneormoreofseveralvisualfeaturescharacterizing themovies.We foundthat orientation,spacecontrastand time contrastareeach genuinelyencoded bythegammaBLP,inthe sensethattuningtoeach ofthesefeaturescannotbeaccounted forbytuningwithrespecttotheother,correlatedfeatures. More-over,wefoundthatcorrelationbetweenthesedifferentstimulus featuresacrossdifferentframesinthemovieincreasedtheoverall amountofinformationthatthegammaBLPcarriedaboutthevisual stimuli.

2. Testingthesigniﬁcanceofmutualinformationvalues 2.1. Mutualinformationandlimitedsamplingbias

Supposethatasensorystimulusspeciﬁedbymultiplefeatures ispresentedtoasubjectwhilerecordingneuralresponses.Oneway toquantifywhetheraneuralresponse,R,isselectivetoastimulus S,istomeasuretheinformationbetweenthesetofstimuliandthe setofassociatedneuralresponses(deRuytervanStevenincketal.,

1997;Fairhalletal.,2001;Panzerietal.,2003),whichisdeﬁnedas

follows: I(R;S)=

s P(s)

r P(r|s)log₂P(r|s) P(r) (1)

whereP(s)istheprobabilityofpresentationofstimuluss,P(r|s) istheprobabilityofobservingresponserwhenstimulussis pre-sented,andP(r)istheprobabilityofobservingaresponseracross allstimuluspresentations.Inthefollowing weassumethatthe values ofboth theneuralresponseR andthestimuli Sare dis-crete,oralternativelyarediscretizedintoafinitenumberofbins tofacilitatetheexperimentalsamplingofprobabilities. Informa-tionisnon-negativeandquantifiestheaveragereductionofthe uncertaintyaboutthestimulusthatcanbegainedfromobserving asingle-trialneuralresponse.Herewemeasureitinunitsofbits, onebitcorrespondingtoareductionofuncertaintybyafactorof two.Ifinformationislargerthanzero,theninprinciplethe stim-ulusfeatureandneuralresponsearenotindependent,indicating thattheresponsereflectstheparticularstimuluspresented.

DirectevaluationoftheinformationdefinedinEq.(1)fromthe experimentalhistogramsofstimulus-responseoccurrences (usu-ally calledthe “plugin” estimate, see Nemenmanet al. (2004)) resultsinasystematicupwardbiasintheinformationestimate. Thisoccursbecauseoffinitesampling;thevariabilityofthe dis-tributionswillbesystematicallyunder-estimatedbecauseofthe limitedamountofdataavailable.It canbeshownthat in suffi-cientlywellsampledsituations(moreprecisely,inthesocalled asymptoticregimeinwhicheachresponseisobservedmanytimes duringtheexperiment)thebiasoftheplug-ininformation esti-matecanbeapproximatedbythefollowingexpression(Panzeri

andTreves,1996): BIAS[I(R;S)]=₂_N1_ln2

s ( ˆRs−1)−( ˆR −1)

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whereNisthetotalnumberoftrials, ˆRsisthenumberof possi-bleresponsesuponpresentationofstimulussand ˆR isthenumber ofpossibleresponsesacrossallstimuluspresentations.Thesum intheright-handsideisoverallstimulipresented.Eq.(2)is use-fultobuildanintuitionaboutthebiasproblem. Assumingthat ˆ

Rsisapproximatelyequalto ˆR,andconstantacrossstimuli,then thebias(Eq.(2))isproportionalto(ˆS −1)( ˆR−1)/N,where ˆSisthe numberofpossiblestimuli.IfwedeﬁneNs=(N/ˆS)tobethe aver-agenumberoftrialsperstimulusthenwecanseethebiasofthe informationisproportionalto(( ˆR−1)/Ns)−(( ˆR −1)/N)andhence approximatelyproportionalto ˆR/Ns.WethereforeconsiderNs/ˆRas thekeyparameter,whichaffectsthemagnitudeofthebias;ifthis ratioislarge,thebiaswillbesmallandviceversa.

Thereareseveralmethodstoestimatethebias(Montanietal.,

2007;Nemenmanetal.,2004;Paninski,2003;Panzerietal.,2007).

Forexample,thebiascanbeestimatedfromEq.(2)(andthen sub-tractedout).Thisrequiresestimatingparameterssuchas ˆRs,the numberofpossibleresponsestostimuluss.Thiscanbeestimated eitherfromsimplycountingthenumber ofobservedresponses (irrespectiveoftheprobabilityofanyparticularresponse)(Miller, 1955).However,incasesoflimiteddatathissimplecountingis likelytounderestimatethenumberof possibleresponses,since possibleresponseswithlowprobabilitymaynotappear.Toaddress thisissueaBayesiancountingprocedurecanbeused(seePanzeri

andTreves(1996),referredtointhefollowingasthePTmethod).

Alternatively,thequadraticextrapolation(QE)procedure(Strong etal.,1998)calculatesinformationfromthefulldatasetaswell asfromsubsetsofthedataconsistingofN/2andN/4trials,and thenestimates thetrueasymptoticvalueof informationby ﬁt-tingaquadraticcurvetothesedatapoints.Whilethesemethods

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R.A.A.Inceetal./JournalofNeuroscienceMethodsxxx (2011) xxx–xxx 3 havebeenshowntobegreatlyeffectiveinreducingthesystematic

errorduetolimitedsamplingininformationestimation(in par-ticular,theyareveryeffectiveifNs/ˆRislargerthan1,seePanzeri

etal.(2007)forareview),theeffectofthesebiascorrectionson

estimatingthesigniﬁcanceofinformationvalueshasbeenlargely unexploredsofar.

2.2. Thestatisticalsigniﬁcanceofmutualinformation

Toconcludethataneuralresponseisselectiveforaspecific fea-tureitisnotsufficienttomeasurethattheinformationbetween thetwovariablesislargerthanzero.Toruleoutthepossibilitythat thisinformationvalueresultsonlyfromfluctuationsduetolimited samplingratherthanfromtruedependency,onemustalsoassess thestatisticalsignificanceofthismeasure.

Thestatisticalsignificanceofinformationvaluesistraditionally evaluatedthroughthemethodofsurrogatedata,whichisreferred tohereasthebootstrapmethod(OpticanandRichmond,1987). Theavailablestimuliandresponsedataareshuffledtoremoveany dependencebetweenthevariables. Thedistributionof informa-tionvaluescomputedacrossdifferentinstantiationsoftheshuffled datasetfollowsthe distributionof theinformation valueunder thenullhypothesisthatthevariablesarestatisticallyindependent, sinceanydependencehasbeenremovedbytheshuffling proce-dure.Afterselectingtherequiredsignificancelevel(p-value),the correspondingpercentileofthebootstrappednullhypothesis dis-tributioniscalculatedandthevalueobtainedfromtheun-shuffled dataiscomparedtothistodeterminestatisticalsignificance.

Whilethebootstrapprocedureisstraightforwardtoperform, itisproblematicwhensmallp-valuesarerequired,sinceaccurate characterizationoftheextremesofthenulldistributionrequires highnumbersofbootstraprepetitions.Besidesbeing computation-allydemanding,obtainingahighnumberofindependentshuffled combinationsofthesamedatasetrequiresaverylargeamountof data,whichcaninsomecasesbedifficulttocollect.Itistherefore interestingtoconsiderwhetherparametricmodelscanbefitto smallerbootstrapsamplepopulations,withoutcompromisingthe statisticalpowerofthetest.Infact,itisalongestablished,although apparentlylittleknownresultthatforlargenumbersofsamplesthe distributionofplugin(notbiascorrected)informationestimates underthenullhypothesisofindependence,whenmultipliedbya factorof2Nln2,followsachi-squaredistribution(Fanetal.,2000;

Wilks,1938):

2N ln2·I(R;S)∼2_{(( ˆ}_R₋_1)(ˆ_{S −}₁₎₎ ₍₃₎ Ifthestimulipresentedaredrawnfromasetcontaining ˆS dif-ferent stimuli, and theresponse is represented with ˆR discrete symbols,thenthedegreesoffreedomforthischi-squarenull distri-butionare( ˆR−1)(ˆS−1).Notetherelationshipwiththeﬁrstorder biasestimateintheprevioussection.Themeanofthischi-square distribution,whenconverted backintotheinformationscaleby dividingby2Nln2,ispreciselytheﬁrstorderestimateofthebias.

Thisnull hypothesis distribution is thesame as the asymp-toticnulldistributionforPearson’schi-squaretestofindependence

(Kullback, 1968). In fact, the scaled mutual information value

describedaboveisequaltotheteststatisticforalikelihoodratio testofindependence.Thechi-squareteststatisticcanbederivedas aquadraticapproximationtothislikelihoodratiotest,bytakinga Taylorseriesexpansion(Kullback,1968).However,withthe avail-abilityofmoderncomputingresourcescalculatinglogarithmsisno longerasubstantialchallengeandsothereislessofanadvantageto usingtheapproximatequadraticform.Anadditionalpointof inter-estisthatthemutualinformationstatisticisitselfthedominant terminafullBayesiantestwithuniformpriors(Wolf,1995).

However,notmuchisknownyetabouttheextenttowhichthe nullhypothesisdistributioncanbeapproximatedbyparametric

distributionsinconditionsofmoderatesampling,andaboutthe effectofbiascorrectionsupontheeffectivenessofthebootstrap test.Theseissuesareofhighpracticalimportance.Inthefollowing weinvestigatethembymeansofrealisticnumericalsimulationsof neuralresponses.

2.3. Investigationofperformanceofinformationsigniﬁcancetests usingsimulations

Toinvestigatethestatisticsofinformationvaluesandthe effi-ciencyoftestsofinformationsignificance,weconductedaseriesof numericalexperimentswithmodelsofsimulatedneuralresponses, which are bothrealistic and relevant totheneurophysiological analysisthat we report inSection 5. In brief,thesimulation is constructedtomatchthefirstandsecondorderstatisticsofthe gamma-rangeBLPofLFPsrecordedinprimaryvisualcortex(V1)in responsetovariationsofMichelsoncontrastduringthe presenta-tionofanaturalisticmovie.

Asymptotictheoremsofspectralestimationpredictthat spec-tralpowersareapproximatelychi-squaredistributed(Percivaland

Walden,1993;Reichetal.,2000).Hence,thecuberootoftheBLP

isapproximatelyGaussian(WilsonandHilferty,1931),aproperty thatwasconﬁrmedempirically(notshown)forthedatawe ana-lyzedinSection5(seeMagrietal.(2009)foranexplicittestof thisassumption).Therefore, ourmodelgeneratedLFPresponses byassumingthattheircubicrootfolloweda Gaussian distribu-tionwiththesamemean andvarianceas thecubic rootof BLP powerfromanexamplerecordingsite(seeSection5.1)inmacaque V1inresponseto8quantizedlevelsoftheMichelsoncontrastin thereceptiveﬁeldoftheconsideredrecordingsiteduringthe pre-sentationofaHollywoodmovie.Forcomputinginformation,the simulatedBLPgammaresponseswerediscretizedintoR=4 equi-populatedbins(exactlyaswedoforrealdatainSection5).Werefer

totheAppendixAforfulldetailsaboutthesimulation.

Intuitionsuggeststhathowdifficultitistodeterminewhether aninformationvalueissignificantdependsboth onthenumber oftrialsavailable(themoredatathereare,theeasieritisto sam-pletheprobabilities)andalsoontheamountofinformationinthe neuralresponses(thelessinformationthereisintheresponses,the moretrialsmaybeneededtodeterminingsignificance).To investi-gatetheseeffects,weintroducedaparameter˛thatmodulatedthe informationinthesimulatedresponsesbytransformingthemean andstandarddeviationparametersasfollows:

Pmodel

s =Psdata+(1−˛)(Psdata−Psdata) (4)

where Pmodel

s is the stimulus conditional parameter (mean or standarddeviation)ofthemodelforstimuluss,Pdata

s isthe

cor-respondingparametermeasuredfromthedata,Pdata

s istheaverage oftheparameteracrossstimuliand˛controlsthemodulationof theinformationinthesimulatedresponses.If˛=1,thesimulated responseshavethesameproperties(andinformation)asthe mea-sureddata.Decreasing˛decreasestheinformationinthemodel responses,untilfor˛=0themodelresponsesarenotstimulus mod-ulatedanymore,andhencetheycarrynoinformation.Weshow resultsfor˛=1(fullinformation),˛=0.5and˛=0(noinformation). We first investigated the statistical power of the bootstrap testindetectingsignificantinformationincaseswherethemodel responseshadtrueinformationaboutthesimulatedstimuli.We used100randombootstrapstoestimatethenullhypothesis distri-bution,whichcorrespondstorelativelylargesamplingcompared towhatcanbedonefeasiblyinanalysesofrealdatasets,andthus representagoodbenchmark.Fig.1Ashowshowthepercentage of correct detectionsofsignificant information,across different realizationsof thesimulation, depends onthenumber oftrials perstimulus.For ˛=1 (informationin simulationmatching the

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4 R.A.A.Inceetal./JournalofNeuroscienceMethodsxxx (2011) xxx–xxx

Fig.1.Determiningthesignificanceofmutualinformation.ModelLFPgammaBLPresponsestoeightdifferentvaluesofMichelsoncontrastweresimulated(seetextfor details).Thenumberoftrialsgeneratedperstimuluswasvaried;foreachnumberoftrials100realizationsofthesimulationweregeneratedandthecalculationswere performedoneachrealization.Thex-axesshowtheNs/Rratio,thecriticalparameterforinformationestimates(seetext,hereR=4).Panel(A)showsthepercentageof realizationsthatwereclassedassignificantusingavarietyofmethods.Solidlinesshowthefullinformationmodel(˛=1);dottedlinesshowthe˛=0.5model;dashed linesshowthenoinformation(˛=0)model.FortheQE(redlines)andPT(greenlines;partiallyobscuredbyoverlappingbluecurve)biascorrectionmethodsthecorrected informationvaluewascomparedtothebootstrapdistributionofcorrectedvaluescalculatedfrom100shufflingrepetitions.Forthechi-squaremethod(blacklines)the analyticalnullhypothesisdistributionwasusedandnobootstrapshufflingwasperformed.Panel(B)showsthemean(errorbarsshowstandarddeviation)over100realizations ofthep-valuefromtheKolmogorov–Smirnovtestofthegoodnessoffitofthe100bootstrapsamplestotheanalyticchi-squaredistribution.Panel(C)showsthepercentage ofstimulusconditionalresponsebinswithlessthan5observations.

oneinrealresponses),thebootstrapmethodreliablydetected sig-niﬁcancedownto32 trialsper stimulus(which correspondsto Ns/R=8).Thisshowsthatdetectingsigniﬁcancerequiresmoretrials thanwouldbenormallyrequiredforbiasfreeinformation estima-tion(biastechniquesareshowntobeeffectiveifNs/ˆRislargerthan

one,seePanzerietal.(2007)).Whentheinformationcontentwas

reduced(˛=0.5),moresampleswererequiredtoreliablydetect signiﬁcance.Thismatchestheintuitionthatwithenoughtrialsan arbitrarilysmalldependence couldbereliablydetected,while a verystrongdependencecouldbedetectedwithfewsamples.

Theanalysisofthemodelwithnoinformation(˛=0,Fig.1A, dashedlines)showsthefalsepositiverateofthetests.Wefound thatforallmethodsthiswasstableandclosetothespeciﬁed sig-niﬁcancelevel(p=0.05).

Acrossallinformationlevelssimulated,theuseofbias correc-tionswasneverhelpfulinthebootstrapcalculationofsigniﬁcance: itreduced(QE)ordidnotaffect(PT)thestatisticalpowerofthetest. Thereductioninstatisticalpowerislikelytooccurbecauseofthe increasedvarianceofthebiascorrectedestimates;mostmethods tradeoffreducedbiasintheestimateforanincreasedvarianceof theestimator.Inthefollowing,wewillthereforeconcentrateon usinguncorrectedpluginestimatorstodeterminesigniﬁcance.

We then examined the effectiveness of using the para-metric analytical chi-square distribution of values under the

nullhypothesis,rather thantheempiricalnon-parametric boot-strap distribution. Fig. 1B shows that the analytic chi-square approximation,withdegrees offreedomequal to( ˆR−1)(ˆS−1), described thebootstrap distributionof information values well (Kolmogorov–Smirnov(KS)test,p>0.3)providedthattherewere atleast32trialsperstimulus,i.e.Ns/R>8.Inthissamplingregime, since the chi-squaredistribution fits thebootstrap distribution well,thechi-squarecanbeusedtobothreducecomputationaltime andtocomputesignificanceforsmallp-valueswhichcouldbe dif-ficulttotestwiththebootstrapmethod.Weverifiedthat,inthis samplingregime,theanalyticchi-squaretestwasasgoodasthe bootstraptestinbothdeterminingtruepositivesandavoidingfalse negatives.

Withfewertrials(Ns/R<8)theanalyticalchi-squarenolonger modeledtheobtainedbootstrapdistributionsaccurately.Inthese cases,theresponseswereunder-sampled,witha largenumber ofbins havingoccupancyofless than5 (Fig.1C).These results areinagreementwithconventionalstatisticalpracticeregarding theapplicationofchi-squaretests(Cochran,1954;Larntz,1978), whichsuggestsusingthechi-squareapproximationonlywhenthe numberofexpectedobservationsunderthenullhypothesisof inde-pendenceforeachpossiblestimulusconditionalresponseisgreater than1,andisgreaterthan5foratleast80%ofstimulusconditional responses.Insituationswherethiswasnotthecase,thestatistical

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R.A.A.Inceetal./JournalofNeuroscienceMethodsxxx (2011) xxx–xxx 5

Fig.2.Parametricmodelsofnullhypothesisbootstrapdistribution.ModelLFPgammaBLPresponsestoeightdifferentvaluesofMichelsoncontrastweresimulated(seetext fordetails).Thenumberoftrialsgeneratedperstimuluswasvaried;foreachnumberoftrials100realizationsofthesimulationweregeneratedandthecalculationswere performedoneachrealization.Thex-axesshowtheNs/Rratio,thecriticalparameterforinformationestimates(seetext,hereR=4).Resultsareshownforthefullinformation model(˛=1),the˛=0.5modelandthenoinformation(˛=0)model.Foreachrealizationachi-square(greenlines)orGaussian(redlines)distributionwasfittoeither5 (dottedlines),10(dashedlines)or20(solidlines)shuffledbootstrapvalues,andthisdistributionwasusedtoestimatethesignificanceofthemeasuredinformationvalue.

testbasedonthechi-squaredistributionperformedpoorly,witha markedincreasedfalsepositiverate.

Inmanysituationsitisnotpossibletocomputealargenumber ofbootstrapsandsoitisdifficulttocomputesignificanceforsmall p-values. Besides using the analytic chi-square approximation, anotherpossiblewaytocomputesmallp-valuesignificanceunder suchcircumstancesistofitparametricmodelstonullhypothesis distributionsbasedonsmallnumbersofbootstrapsamples.These empiricallydeterminedparametricmodelscan thenbeusedto evaluatethesignificanceoftheobservedinformationvalue.InFig.2

wetestedtheeffectivenessofsuchprocedures,byfittingeither chi-squareorGaussiandistributionstodistributionsof informa-tionvaluesobtainedfromsmallnumbers(5–20)ofindependent bootstrapsamples. Eveninunder-sampled situationswhere, as discussedabove,theanalyticchi-squareapproximationfailsfor Ns/R<23 (shown byincreased false positiverate inFig. 2C and smallKStestp-valuesinFig.1B),goodresultswerestillobtained withtheseparametricfits.Thechi-squarefitperformancewasvery closetothefullnon-parametricbootstrap(95thpercentileof100 bootstrap samples), even whenonly 5 bootstrap sampleswere used.Thisprovidesa largecomputationalsaving.The Gaussian modelalsoperformedwell,but20bootstraptrialswererequiredto maintainaclosefittothenon-parametricbootstrapinthe under-sampledregime,andcontrolofthefalsepositiveratethroughout thesamplingrangesconsidered.Althoughthedegreesoffreedom arehigh,suggestingthechi-squareisclosetoaGaussian,when fit-tingtheGaussianmodeltwoparametersmustbeestimatedfrom thedataratherthanjustoneforthechi-squaredistribution.This

couldexplaintheincreasedperformanceofthesimplermodelfor lownumbersofsamples.

Insummary,ouranalysisofthetestsofindependencegavea numberofpracticallyusefulresults:(i)whileaccurateestimatesof informationvaluesrequirebiascorrection,whentestingfor signif-icanceofinformation,bestresultsareobtainedwithuncorrected plug-inestimates.(ii)Theanalyticchi-squarenullhypothesis dis-tributionallowsaveryrapidandhighlyprecisetestevenwithsmall p-values providedsamplingissufﬁcient(Ns/R>8inour simula-tion).Theregimeinwhichtheanalyticchi-squareisaccuratecan bedeterminedinrealdatabyusingtheKStestbetweenempirical bootstrapandanalyticchi-squaredistributionsforsub-samplesof differentsizes.(iii)Eveninsub-sampledsituationswherethe ana-lyticchi-squareapproximationisnotaccurateanditisnotpossible tocomputemanybootstrapsamples,itisstillpossibletoobtain goodresultsbyempiricallyﬁttingachi-squaredistributiontoa smallnumberofbootstrapsamples.

2.4. Theeffectofnumberofbinsusedinthediscretizationofa continuoussignalupontheevaluationofsigniﬁcanceof information

Animportantstepintheinformationanalysisofdiscretized ana-logsignals,suchastheLFPBLPresponse,isthebinningofdata intoanumberofcategoricalbins.Twoissuesregardingthe bin-ningprocedurerequirefurtherdiscussion.The ﬁrstissueis the binningalgorithm tobechosen.Here,we settheboundariesof thebinssothattheyareapproximatelyequallyoccupiedoverall

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A

B

Fig.3.Effectofnumberofbinsoninformationestimationandstatisticalsigniﬁcance.ModelLFPgammaBLPresponsestoeightdifferentvaluesofMichelsoncontrastwere simulated(seetextfordetails).Panel(A)showsaverageinformationestimatesover100realizations(errorbarsshowstandarddeviation)ofthesimulationwiththesame numberoftrialsastheexperimentaldata(3460trialsperstimulus)asafunctionofnumberofbinsusedtodiscretizetheresponses.Informationisestimatedwithout biascorrection(‘Plugin’,redline),withPanzeri–Trevesbiascorrection(‘PT’,blueline)andwithquadraticextrapolation(‘QE’,greenline).Panel(B)showsthepercentage ofrealizationsfoundtobestatisticallysigniﬁcant(100realizations,p=0.05,bootstrapmethod),asafunctionofthenumberoftrialsperstimulus,fordifferentnumbersof responsebins.Thecorrespondinginformationvaluesforeachcurveareindicatedbycorrespondingcoloredmarksinpanel(A).Theverticalreddottedlineindicatesthe numberoftrialsavailableintheexperimentaldatasetandusedforthesimulationsinpanel(A).

theavailabletrials(theunconditionalormarginaldistributionsare approximatelyuniform).Thisavoidssamplingproblemscausedby binshavingverylowprobabilityofoccupancy,forexampleif equal-widthbinsareusedtorepresentaGaussiansignal,thosesampling thetailsofthedistributionhaveaverylowprobabilityof occur-rence.Thisproceduremakestheinformationessentiallyareduced rankstatistic,sincetherelativeposition(rank)ofaparticulartrial determineswhichbin itisin.Anadvantageofthisprocedureis that,likeotherrankstatistics,itisnon-parametricandrobustto outliers.Thesecondissuetobeconsideredisthat,whateverthe binningmethodused,thenumberofbinsusedfordiscretization isanimportant,yetpartlyarbitrary,parameterthatmaygreatly affecttheinformationcalculation.

Thechoiceofthenumberofbinsisacompromisebetweenthe needtousea fineenoughdiscretization inordertocaptureall possibleinformativefeaturesinthedata,andthenneedtousea dis-cretizationcoarseenoughthattheprobabilityofeachdiscretized responsecanbesampledaccuratelywiththeavailabledata.Forany givendataset,selectingmorebinsmeansrestrictingtheaverage numberoftrialsperbin.Inthefollowingweusecomputer simu-lationtoillustratehowtoempiricallyevaluatewhatistheoptimal trade-offbetweenthesetwoconflictingconstraintsinorderto eval-uatethesignificanceofinformationvalues.

Fig.3Ashowshowestimatesofthemutualinformationdepend onthenumberofresponsebinschosenforthesimulateddataset generatedwiththemodeldescribedinSection2.3.InFig.3Athe simulateddatasethadexactlythesamenumber oftrialsasthe realLFPdataanalyzedinSection5(3460trialsineachofthe8 stimuluscategories).Westudiedthedependenceofinformation valuesuponthenumberofbinsusingbias-correctedinformation estimates,becausethedependenceofbias-uncorrectedestimates onthenumberofbinsismaskedbythelinearincreaseofthebias withthisparameter(seeEq.(2)andthebias-uncorrected informa-tioncurveinFig.3A).Forverylownumbersofbins,increasingthe numberofbinsincreasestheinformation,becausethemoreﬁnely

discretizedresponses bettercapture theunderlyingcontinuous distributions.However,bias-correctedinformationvaluesplateau overalargerangeofbinnumbers,inwhichtheinformationvalues arestableandlargelyindependentofthebiascorrectionprocedure used.Thismeansthat,withintherangeofnumberofbinsatwhich informationplateausthesamplingissufficienttoallowanaccurate estimationoftheinformation,butincreasingthenumberofbins doesnothaveaneffect;therearealreadysufficientbinstocapture therelevantfeaturesofthedistribution.Foranaccurateestimate oftheinformationvalue,anyparameterwithintheplateauregion canbechosen.Whenestimatingthesignificanceofthe informa-tion, it is intuitive that using more bins reduces the available samplesperbin,andhencethestatisticalpowerofthetest,but there must be sufficient bins for information to be detectable.

Fig.3Bshowsthepercentageofrealizationsinwhichsigniﬁcance wasdetected(p=0.05,bootstrapmethod)forthesimulated sys-temdescribedabove, asa functionoftheamountof generated data.Inthis case,usingjust 2binsprovidesthegreatest statis-ticalpower,sincetheinformationwithtwobinsisalreadylarge enoughandisalargefractionoftheplateauvalueofinformation (Fig.3A).

Theseresultssuggestsomepracticalconsiderationsthatmay guidethechoiceof thenumber ofbinsfor significance testing. Wheneverpossible,bias-correctedinformationestimatesofhow informationscaleswiththenumberofbins,orempirical obser-vationabouttheshapeandwidthofthedistributions,shouldbe usedtoevaluatetherangeofnumberofbinsthatgivessufficiently highinformationvalues. Thesmallestnumberof binssufficient topreservealargefractionoftheinformationshouldbeusedfor theinformationsignificancetestinginordertomaximizethe sta-tistical powerofthesignificance tests.Whenever possible,it is alsoimportanttoverifythestabilityoftheresultsovera range ofbinnumbersbecauseinsomecasesusingtoofewbinsmaylead toverylargelossesofinformation withrespecttotheirplateau values.

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R.A.A.Inceetal./JournalofNeuroscienceMethodsxxx (2011) xxx–xxx 7 3. Themultiplefeaturetuningprobleminthecontextof

informationtheory

Ifasetofstimuliaredeﬁnedbymultiplefeatures,Fi,wecan considerthemutualinformationbetweentheneuralresponseand eachindividualfeature:

I(R;Fi)=

fi P(fi)

r P(r|fi)log2P(r|fi ) P(r) (5)

wherefirepresentsvaluesofthefeatureFiandtheprobabilities areasinEq.(1).However,ifthesefeaturesarenotindependently distributedacrossdifferentpresentationsofthestimuli,apositive valueofinformationI(R;Fi)aboutfeatureFidoesnotnecessarily implythattheneuralresponsespeciﬁcallyencodesthatfeature. TheneuralresponsemaynotbeselectivetoFi,butinsteadbe selec-tivetoanotherfeature(sayFj),whichiscorrelatedwithFi.When usingcorrelatedstimuli,animportantproblemishowto distin-guishthetrueselectivitytoafeaturefromthatacquiredonlyfrom itsrelationstootherfeatures.

3.1. Multiplefeaturetuningandinformationcomplementarity

Hereweaddresstheissueofdetermininggenuinefeature selec-tivityinthepresenceofmultiplecorrelatedfeaturesbyconsidering, forsimplicity,thecaseinwhichthestimulusisdeﬁnedbytwo dif-ferentcorrelatedfeatures,F1 andF2.Theamountofdependency betweenthetwo featuresismeasured, ininformationtheoretic terms,bytheinformationI(F1;F2)betweenthem:

I(F1;F2)=

f1,f2 P(f1f2)log2 P(f1f2 ) P(f1)P(f2) (6)

whereP(f1f2)isthejointprobabilityoffeaturevaluesf1andf2being presentedinthesamestimulus.Thetwofeaturesaredependentif I(F1;F2)>0.Iftheneuralresponseisselectivetobothfeatures(i.e. I(R;F1)>0andI(R;F2)>0,withI(R;F1)>I(R;F2),howcanwemake surethatthegivenneuralresponsetrulyencodesbothfeatures, anddoesnotencodeF2onlybecauseitcorrelateswiththetruly encodedfeatureF1?Onewaytoaddressthisproblemistolook attheinformationtheresponseconveysaboutF2 atfixedvalues ofF1.SincethevalueofF1 isfixed,anydependencebetweenthe responseandF2 representsgenuinecomplementaryinformation thatcannotbeexplainedbythedependencybetweenthefeatures. Thisquantityiscalledtheconditionalmutualinformation(CMI) andisdefinedas:

I(R;F2|F1)=

f1 P(f1)

f,r2 P(rf2|f1)log2 P(rf 2|f1) P(r|f1)P(f2|f1) (7)

whereP(rf2|f1)isthejointprobabilityofobservingresponserand featuref2atﬁxedf1.Inasituationwhere

I(R;F2|F1)=0 (8)

itcanbeproved(CoverandThomas,1991)thatthefollowing equa-tionholdsforallvaluesoff2,f1andr

P(f1,f2,r)=P(f2)P(f1|f2)P(r|f1) (9) ThisillustratesthefactthatthevariableF2 affectstheneural responseonlythroughitscorrelationwithF1.IfinsteadI(R;F2|F1)>0 thenF2stillhasaneffectonthevariableratﬁxedF1andsoP(f1,f2,r) cannotbeexpressedbytheproductintheright-handsideofEq.(9). Insummary,thefeatureF2hasagenuineeffectupontheneural response,beyondtheindirecteffectofcorrelationswithF1,ifand onlyifI(R;F2|F1)>0.

Itisimportanttonotethat,byusingthechainrule(Coverand

Thomas,1991),theCMIcanberewrittenas:

I(R;F2|F1)=I(R;F2F1)−I(R;F1) (10) whereI(R;F1F2)istheinformationthattheneuralresponsecarried aboutthejointpresenceofF1andF2,deﬁnedasfollows:

I(R;F1F2)=

f1f2 P(f1f2)

r P(r|f1f2)log2P(r|f1f2 ) P(r) (11)

ThereforetheconditionI(R;F2|F1)>0isequivalenttothe condi-tion

I(R;F2F1)−I(R;F1)>0 (12)

Therequirementthatthejointinformationbehigherthanthe informationcarriedaboutthevariableF1 impliesthattheneural responsecarries“true”informationaboutF2aboveandbeyondthe informationthatitcarriesaboutF1,orinotherwords,thatthe infor-mationthattheneuralresponsecarriesaboutF2iscomplementary tothatitcarriesaboutF1(Panzerietal.,2010).Alternatively,due tothesymmetryofinformation,theaboveconditionmeansa bet-terpredictionoftheresponsecouldbemadebasedonobservation ofbothfeaturesthancouldbemadebasedontheobservationof F1alone.ItisintuitivethatgenuinecodingofF2requiresthatthe responsecarryinformationaboutF2complementarytothatwhich itcarriesaboutF1.Theaboveequationsformalizethisintuition.

Thisframeworkcaninprinciplebeextendedtothecaseinwhich thestimulusisdefinedbyanarbitrarynumbernoffeatures(F1,..., Fn).ThestimulusfeatureFn isgenuinelyencoded bytheneural responseiftheinformationI(R;Fn|F1...Fn−1)betweenresponse RandfeatureFn,conditionaltoallothervariablesissignificantly positive:I(R;Fn|F1...Fn−1)>0.However, whiletheextension to multiplefeaturesisstraightforwardinprinciple,theformalismmay bedifficulttoapplytomanyfeaturesbecauseofthedifficultiesin evaluatingmultivariateresponseprobabilitiesfromfiniteamounts

ofdata(Panzerietal.,2007).

3.2. Effectofcorrelationoffeaturesonestablishinginformation complementarity

Wementionedabovethattheproblemofcorrelationsbetween stimulusfeaturescomplicatestheproblemofdeterminingwhich stimulusfeaturesaregenuinelyencodedbytheneuralresponse. In order todevelop a methodsuited totackle this problemon experimental data, we investigate theoretically how statistical dependenciesbetweenfeaturesinﬂuencetheamountofjoint infor-mationcarriedaboutthefeatures.Forsimplicitywewilllimitour discussiontoastimulusdeﬁnedbyonlytwofeatures,F1andF2.

TherelationshipbetweentheCMI,I(R;F2|F1),andthecorrelation betweenvariablesI(F1;F2)canbedeﬁnedasfollows(Adelmanetal.,

2003;Lüdtkeetal.,2008;Schneidmanetal.,2003):

I(R;F2|F1)=I(R;F2)+I(F1;F2|R)−I(F1;F2) (13) whereI(F1;F2)istheamountofcorrelationbetweenvariables with-outregard totheneuralresponse(seeEq.(6)),and I(F1;F2|R)is theamountof correlationbetweenthefeatures at ﬁxedneural response,deﬁnedasfollows:

I(F1;F2|R)=

r P(r)

f1,f2 P(f1f2|r)log2_P(fP(f1f2|r) 1|r)P(f2|r) (14)

Eq.(13)hasseveralimportantimplications.First,itshowsthatif thefeaturesareindependentlydistributed(i.e.I(F1;F2)=0)thenthe conditionI(R;F2)>0issufﬁcienttoensurethatF2carries informa-tioncomplementarytothatofF1.Thisisbecausetheterm−I(F1;F2) istheonlyoneintheright-handsideofEq.(13),whichcanbe neg-ative.Moreover,andforthesamereason,ifthevalueofI(F1;F2)is

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8 R.A.A.Inceetal./JournalofNeuroscienceMethodsxxx (2011) xxx–xxx quantitativelymuchsmallerthanthatoftheinformationcarried

bytheneuralresponseabouteachoftheconsideredfeatures,itis safetoconcludethattheinformationabouteachfeatureisgenuine. Thisgivesausefulruleofthumbforinterpretationoftheresults. Second,Eq.(13)showsthatthepresenceofcorrelationbetween featuresdoesnotalwaystendtosuppressthecomplementarityof informationbetweenfeatures.Infactitcanbeshownthattheterm I(F1;F2|R)−I(F1;F2),sometimescalledtheinteractioninformation, canbeeitherpositiveornegativedependingonwhether correla-tionsbetweenfeaturesacrossallscenesareweaker orstronger thancorrelationsbetweenfeaturesatﬁxedneuralresponse(Pola

etal.,2003;Schneidmanetal.,2003).Thisinteractioninformation

isequaltothesynergy(Brenneretal.,2000b)usuallydeﬁnedas Syn(F1,F2;R)=I(F1,F2;R)−I(F1;R)−I(F2;R) (15)

Together,usingEqs.(13)and(15)weseethat

I(R;F2|F1)=I(F2;R)+Syn(F1,F2;R) (16) Thisshows that theabsence of complementaryinformation (I(R;F2|F1)=0)impliesthepresenceofredundancy(negative syn-ergy) between F1 and F2. Similarly, if the features are coded synergistically,thecomplementaryconditionalmutual informa-tioncanbegreaterthantheunconditionalinformationconveyed aboutasinglefeature.Insuchcases,thecodingofgroupsoffeatures wouldbedifﬁculttofullycharacterizewithstandarduncorrelated stimulusdesigns.These theoreticalconsiderationsreinforcethe importanceofsuchmethodstoenablequantitativeinvestigation ofthecodingofcorrelatedfeaturesfoundinnaturalstimulation conditions.Indeed,ithasbeenproposedthatresponsestonatural stimuliareoptimizedtocopewithcorrelationsintheenvironment

(Barlow,1989;Danetal.,1996;Geisler,2008;OlshausenandField,

1996);themethodsdiscussedhereallowquantitative investiga-tionsoftheseideas.

4. Testingthesigniﬁcanceofcomplementarytuningto multiplefeaturesusingconditionalmutualinformation

Wenowconsidertheproblemofhowtoextendtheabovetests forsignificance ofmutual information toevaluatingthe signifi-canceofcomplementarityoftuningtomultiplefeaturesusingCMI. Thisproblemisdifficultbecausetheinclusionofmultiplefeatures exponentiallyincreasesthesizeofthestimulusspace,which corre-spondinglyreducesthenumberoftrialsavailableforeachstimulus combinationandcompoundsthesamplingdifficulties.The corre-lationsbetweenthefeatures canalsocausedifficultiesbecause theytendtoconcentratethejointdistributionsofthepresented combinationsof features, and the potentially uneven sampling introducedbycorrelationsamongfeaturesmustbecorrectlytaken intoaccountwhendesigningtheappropriatebootstrapshuffling andalgorithmsthatevaluatesignificanceofcomplementary infor-mationvalues.

4.1. Thestatisticalsigniﬁcanceofconditionalmutualinformation

Theconditionalmutualinformationcanbewrittenas I(R;F2|F1)=

f1

P(f1)I(R;F2|f1) (17)

whereI(R;F2|f1)istheinformationbetweenRandF2,conditioned onaspeciﬁcﬁxedvaluef1offeatureF1.Bysubstitutingintheabove equationP(f1)=Nf1/N,whereNf1 isthenumberoftrialsinwhich featureF1takesthevaluef1,oneobtains

I(R;F2|F1)= 1 2Nln(2)

f1 2Nf1ln(2)I(R;F2|f1) (18)

AsdiscussedinSection2,whenusingpluginuncorrected infor-mationestimates,eachelementofthesumontheright-handside ofEq.(18)followsachi-squaredistributionwith( ˆR−1)( ˆF2−1) degreesoffreedom,underthenull hypothesisthatRandF2 are independentatfixedvalues of F1.Fromtheadditivity property ofthechi-squaredistributionwefind(assumingthenumberof observedresponses ˆRandfeaturevalues ˆF2arethesameateach fixedf1,seebelow)that2Nln(2)I(R;F2|F1)ischi-squaredistributed with ˆ_F1( ˆR −1)( ˆF2−1)degreesoffreedom.

Itisalsopossibletoevaluatethestatisticalsignificanceofthe conditionalmutualinformationusingthebootstrapapproachas describedinSection2.However,caremustbetakenwhen imple-menting theshuffling proceduretoensureit samplesfromthe requirednullhypothesis.Sincethenullhypothesisisthatfeature F2andresponseRareindependent,atfixedvaluesofF1,samples shouldbedrawnbyshufflingthecombinationofF2andRatfixed F1.ThesimplerprocedureofshufflingF2valueswithout consider-ingthecorrespondingF1valuewouldindeedbeincorrect(andwe verifiedthatitisalsolessaccurate,resultsnotshown)becauseit removesalltheinformationinF2,andnotonlythatadditionaltoF1. Whentheresponsefeaturesarehighlycorrelatedorwhenthe neuralresponseisselectiveonlytoaspecificcombinationof fea-tures,thenthenumberofrelevantresponsebinsforbothfeatures maybesmallerthantheproductofthenumberofrelevantbinsfor eachfeature.TherandompermutationofF2withoutconsidering thevalueofF1wouldthenmakethejointfeaturespaceatfixed responselargerthanintherealdata.Asaconsequencethenull hypothesisdistributionwillhavealargerbiasthanthetrueone

(Panzerietal.,2007),andalsoalargervariance.Thiswouldmake

theevaluationoftherequirednullhypothesisproblematic.Indeed, insuchcasestheaboveexpressionforthedegreesoffreedomof thechi-squareapproximationshouldbemodiﬁed.Insteadof tak-ing ˆF1( ˆR−1)( ˆF2−1),thevaluesof( ˆR−1)( ˆF2−1)observedforeach valuef1shouldbesummedtoobtainthetotaldegreesoffreedom. 4.2. Numericalinvestigationoftheperformanceofstatistical testsforinformationcomplementary

Toinvestigatethestatisticalpowerofthesetests,weperformed simulationswithasystemsimilartothatdescribedinSection2, butextendedtoincludetwocorrelatedfeatures.Again,themodel isbasedascloselyaspossibleonthestatisticsoftheexperimental dataconsideredinSection5,anditmatchesthefirstandsecond orderstatisticsofV1 gammaBLPLFPresponsestovariationsin spatialcontrastandtemporalcontrastduringpresentationof Hol-lywoodcolormovies.Thecorrelationsinthesimulatedfeatures alsomatchedthoseobservedinthemoviesusedintheexperiments. AsinSection2,thesimulatedgammaBLPresponsewasbinnedin R=4equi-populatedlevels,andeachsimulatedfeaturewasbinned into ˆFi=8equi-populatedclasses(seeAppendixAforfulldetails). Itisintuitivethatthesensitivityofanytestfor complementar-ityofinformationmustdependontheactualamountofadditional genuinecomplementary informationof thesecond feature(the largeritisit,theeasieritistodetect).Thusinoursimulationthe responsedependenceonF1(Michelsoncontrast)wasfixedinall simulations,butweconsideredtwodifferentcaseswithregardto thesecondcorrelatedfeatureF2 (temporalcontrast):inthefirst simulationitcarried anamountofadditionalinformationtoF1, andinasecondsimulationitdidnotcarryanyadditional informa-tion,althoughitwasinformativeperseduetoitscorrelationtoF1

(seeAppendixA).

We tested this bootstrap procedure in Fig. 4. We applied this procedure to uncorrected plugin estimates of information because,forthereasonsexplainedinSection2,theseuncorrected information estimates are better at determining signiﬁcance.

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Fig.4.Determiningthesignificanceofconditionalmutualinformation.ModelgammaresponsestoeightdifferentvaluesofMichelsoncontrastandeightdifferentvaluesof temporalcontrastweresimulated(seetextfordetails).Thenumberoftrialsgeneratedperstimuluswasvaried;foreachnumberoftrials500realizationsofthesimulation weregeneratedandthecalculationsperformedoneachrealization.Thex-axesshowtheNs/RratiowhereNsistheaveragenumberoftrialsforeachpossiblestimulusfeature pair(R=4).Panel(A)showsthepercentageofrealizationswheretheCMIbetweenresponseandtemporalcontrastgivenMichelsoncontrastwasfoundtobesignificantusing theuncorrectedbootstrap(redlines)andanalyticchi-square(blacklines)methods.Dottedlinesshowthemodelwithindependentrepresentationoftemporalcontrast; solidlinesshowthemodelwithnoindependentrepresentationoftemporalcontrast.Panel(B)showsthemean(errorbarsshowstandarddeviation)over100realizations ofthep-valuefromtheKolmogorov–Smirnovtestofthegoodnessoffitofthe100bootstrapsamplestotheanalyticchi-squaredistribution.

theCMIwascorrectlyfoundtobesigniﬁcant(i.e.percentageoftrue positives)dependedupontheNs/Rratio.NotethatinthiscaseNs istheaveragenumberoftrialsforeachpossiblestimulusfeature pair,andRisthenumberofresponses.Thebootstraptestwasable toreliablydetectthepresenceofsigniﬁcantcomplementary infor-mationforsystemswithNs/R>4(16trialsforeachpairofstimulus featurevalues,128trialsperF1stimulusfeaturevalue).

Again,thechi-squareapproximationwasaccurateprovidedthe samplingwassufﬁcient.Forthesinglefeaturesystemshownin

Fig.1,thechi-squareapproximationstartedtofailatNs/R=23.For thisconditionalsystemNs/R=25isrequired.Thisisshownbythe increasedfalsepositive ratefortheanalyticchi-square approxi-mationwhennogenuinecomplementaryinformationispresent (Fig. 4A),and by thelow p-values for the KS-testbetweenthe chi-squareapproximationandthebootstrapdistributionforboth simulatedsystems(Fig.4B).Thefactthatmoretrialsarerequired herefortheanalyticchi-squaretoholdislikelyduetothestrong correlations betweenfeatures changing thestatistics of theF1 -conditionaldistributions;thef2valuesatﬁxedf1arenotnecessarily uniformlydistributedastheywereinthesinglefeaturemodel.We veriﬁedthisfactbydemonstratingthat insimulationsinwhich thecorrelationbetweenF1andF2wassettozero,thechi-square approximationwaseffectiveevenwithNs/R=23,asinthe individ-ualfeaturecase(resultsnotshown).

Insummary, ouranalysisoftheeffectivenessof testsof sig-niﬁcanceofcomplementarity ofinformationrevealedthat:(i)a

bootstraptestbaseduponcomputinguncorrectedplugin informa-tionestimatesaftershufflingafeaturesconditionaltothevalueof other(s)isextremelyeffectiveatdetectingsignificancewithsmall amountsofbothfalsepositivesandnegatives(ii)Thebootstraptest canbereplacedbyananalyticalchi-squaretestprovidedsampling issufficient(Ns/R>32inoursimulation).Theregimeinwhichthe analyticchi-squareisaccuratecanbedeterminedinrealdataby usingtheKStestbetweenempiricalbootstrapandanalytic chi-squaredistributionsforsub-samplesofdifferentsizes.

5. Applicationoftheformalismtotheencodingofvisual featuresbygammaBLPinV1

Toillustratethefeasibilityoftheaforementionedformalismon neurophysiologicaldata,weusedittoanalyzeLFPsrecordedinV1 inresponsetorepeatedbinocularpresentationofacolormovie. Thisisausefulapplicationofourtestsforatleasttworeasons.

Theﬁrstreasonisthatthepreciseorigin,meaningandstimulus tuningoflocalﬁeldpotentialsis notyetfullyknownand isthe subjectofcontinuingstudies(Belitski etal.,2008;Berensetal.,

2008;Jiaetal.,2011;Katzneretal.,2009;KayserandKonig,2004;

Mazzonietal.,2008).Becauseofthis,thereisapressingneedfor

tools,whichareabletocorrectlyidentifyandquantifythetuning ofthesesignalstocomplexstimuli.

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10 R.A.A.Inceetal./JournalofNeuroscienceMethodsxxx (2011) xxx–xxx Thesecondreasonisthatthesedatacollectresponsesto

com-plexnaturalisticstimuli,whichcontainmanyvisualfeaturesthat co-varyovertime(Simoncelli,2003);forinstancemovingobjects mayenterorleavetheRForbeoccluded byotherfastmoving objects,orlocalilluminationmaychangedynamically,both caus-ingalargenumberoffeaturestochangeinacorrelatedfashion

(Bartelsetal.,2008).Evidentlysuchcorrelatedchangesmaypose

problemstostudiesexaminingtheselectivityofneuronal popula-tionstoindividualfeaturesofcomplexvisualimages.Duringthe presentationofsimpliﬁed,artiﬁcialvisualstimuli,theproblemcan bereadilysolvedbyvaryingdifferentvisualattributesinan uncor-relatedmanner(Brenneretal.,2000a;deRuytervanSteveninckand

Bialek,1988;Petersenetal.,2008;Rustetal.,2005;Touryanetal.,

2005;YamadaandLewis,1999).Insuchdesigns,demonstration

ofneuronalfeature-selectivitysimplyfollowstheselective stim-ulusparameterization.Yet,suchanuncorrelatedstimulusdesign cannotbeappliedwhenusingnaturalisticstimulisuchasmovies orsoundscapturingthecomplexityandcharacteristicsofa natu-ralenvironment.Analysingneuralresponsestonaturalisticstimuli isimportantforatleasttworeasons.First,naturalisticstimuliare likelytoengagecomplexpatternsofactivityincortical microcir-cuitsthatmaynotbeelicited inthepresenceofsimple stimuli optimizedforthestudyofbasicpropertiesofsingleneurons(Felsen

andDan,2005;Reinagel,2001).Second,naturalisticvisual

stim-ulihavebeenshown toevokemore reliableresponses(Hasson etal.,2009),andmorespeciﬁcinter-regionalcorrelations(Bartels

andZeki,2005),suggestingthatcodinginthevisualsystemmay

beoptimizedforprocessingnaturalisticstimuli(Danetal.,1996). Hereweprobetheabilityofourmethodstoquantifythestatistical signiﬁcanceofinformationcarriedbyLFPsaboutagivenvisual fea-tureaboveandbeyondthatcarriedbyother,potentiallyco-varying visualfeatures.

5.1. Experimentalprocedures

Beforeproceedingwiththeanalysis,webrieflysummarizethe experimentalproceduresusedtorecordneuralresponsesto nat-uralisticcolormoviesinV1.Adetaileddescriptioncanbefound inpreviousstudies(Belitskietal.,2008,2010;Montemurroetal., 2008). All procedures were approved by the local authorities (Regierungspräsidium)andwereinfullcompliancewiththe guide-linesoftheEuropeanCommunity(EUVD86/609/EEC)forthecare anduseoflaboratoryanimals.Werecordedneuralactivitywith anarrayofextracellularelectrodesfromtheopercular(fovaland para-fovealfield-representations)portionofV1,of anesthetized macaquespresentedwithcommerciallyavailablecolormovieclips consistingof scenesfromTheLastSamurai(2003,WarnerBros. Pictures)andStarWars:EpisodeI–ThePhantomMenace(1999, Lucasfilm).Weanalyzedatotalof55recordingsitesfrom3different monkeys.Movieclipslasted150–210sandwerepresented binoc-ularlyataresolutionof640×480pixels(fieldofview:30◦×23◦, 24bittruecolor,60Hzrefresh)usingafiber-opticsystem(Avotec, SilentVision,Florida).Eachmoviewaspresented35–50timesper sessioninordertoadequatelysampletheprobabilitydistribution oftheneuralresponsestoeachpartofthemovie.

Theextracellularfield potentialwasfilteredtoextract Multi UnitActivity(MUA)andlocalfieldpotentialsusingstandardsignal conditioningtechniquesasdescribedindetailelsewhere(Belitski etal.,2008).TheMUAwasusedonlytodeterminethereceptive fields,asdescribedbelow.Thegammaband(30–100Hz)oftheLFPs wasextractedbymeansofzero-phaseshift, bidirectionalfilters (Kaiserfilterwithatransitionbandof1Hz,stop-bandattenuation of60dB,andpass-bandrippleof0.01dB).Theinstantaneouspower ofthisband-limitedsignal,shortenedinthefollowingasgamma bandlimitedpower,wascomputedasthesquaredmodulusofthe discrete-timeanalyticsignalobtainedviatheHilberttransform.

WefocushereonthegammaBLPasaneuralresponsesinceithas beenpreviouslyshowntobeinformativeaboutnaturalisticmovie stimulation(Belitskietal.,2008)andisknowntobemodulatedby visualfeaturessuchasspatialcontrast(HenrieandShapley,2005) andorientation(KayserandKonig,2004).

5.2. Extractionofvisualfeatures

Weﬁrstestimated theaggregatereceptive ﬁeld(RF)of each recordingsitebyusingthereversecorrelationtechnique(Ringach

andShapley,2004),which measuresthesensitivity oftheMUA

responsetothetimecontrastofeachpixelofthescreen.When usingreversecorrelationwithspatiallycorrelatedstimuli(suchas thenaturalmoviesusedhere)theobtainedRFislikelybiasedby thestimuluscorrelationandisthereforelikelytobelargerthanthe

trueRF(Theunissenetal.,2001;Touryanetal.,2005).Forthis

rea-son,theRFchosenforthefeatureextractionanalysiswasfixedasa 1◦× 1◦_region_around_the_peak_of_the_neural_sensitivity,_even_when theresultofreversecorrelationresultedinalargermap.Thissize ofRFwaschosenbecauseitistypicalofmultipleunitV1 recep-tivefields(Sceniaketal.,1999).AnexampleofsuchRFestimation isreportedinFig.5A.Reversecorrelationwasusedheretoobtain anestimateofthelocationoftheRFareaandnottoestimateits size.ToverifythattheprecisesizesetfortheRFwasnot criti-calfortheinformationresults,werepeatedtheanalysisselecting RFssizesof0.5◦×0.5◦and1.5◦×1.5◦.Wefoundthatthefeatures extractedfromRFsoftheselargerorsmallersizeswereessentially identicaltothoseobtainedwiththe1◦×1◦RF(correlationbetween thetimecoursesoffeaturesextractedwithdifferentRFsizeswas >0.95forallelectrodesandrecordingsessions,andinformation val-ueswerenotsignificantlydifferent,p>0.5).Asanadditionalcheck thatthelocationoftheRFwascorrectlydetermined,wecompared theRFpositionsobtainedbyreversecorrelationwiththose manu-allymappedbytheexperimenterwithsmallpolarstimuliduring theexperimentalsession,andwefoundexcellentagreement(not shown)forallsites.

We subsequently computed three different types of image features within each RF: spatial contrast (Michelson contrast, calculatedas thedifferencebetweenthemaximaland minimal luminance dividedby theaveragebetweenmaximal and mini-malluminance),orientation(computedasthedirectionofgradient ofcontrast(Kayseretal.,2003)),andameasureoftimecontrast, deﬁnedasthetotalamountofpixel-wiseluminancechangefrom frametoframe,thatis,theabsolutevalueoftheluminance differ-encebetweentwosuccessiveframes,calculatedforeverypixeland thenaveragedovertheRF(Bartelsetal.,2008).

5.3. Calculationofinformationconveyedaboutdifferentfeatures

Foreachrecordingsite,wecomputedtheinformationcarried bythegammaBLPaboutthecurrentvalueofthemoviefeatures inthecorrespondingRF.Forthiscalculation,thestimulussetwas createdasfollows(seealsoReichetal.(2000)).Weﬁrstdivided themoviepresentationintonon-overlappingadjacentwindows each10frameslong.Thiswindowlengthwaschosenbecauseit matchedthetimescaleofthefeatureautocorrelations.We aver-agedthevalueofeachfeatureoverthe10framesineachstimulus window.Unlessotherwisestated,wediscretizedtheresulting fea-turevalues acrossallwindowsinto8equallypopulatedclasses, andeachclasswasconsideredasadifferent“stimulus”valuefor theinformationcalculation.GammaBLPvalueswereaveragedover thesameintervalsandsubsequentlydiscretizedintoR=4equally populatedbins.We chosetheseparametersbecausetheyallow accuratesamplinggiventheamountofdataavailable.Wechose R=4basedontheconsiderationsinSection2.4,whichsuggest cau-tiouschoicesofthenumberofbins,especiallyconsideringthatthe

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Fig.5.Exampleofmoviefeaturesandneuralactivity.(A)Foreachrecordingsite,thereceptivefieldwasdeterminedasthe1◦_square_area_in_the_screen_where_the_covariance betweenthemulti-unitactivityrecordedinthesiteandthetimecontrastwashighest.Thelocalfieldpotential(LFP)recordedinthesitewasthenstudiedasafunctionofthe visualfeaturesofthemoviecomputedwithintheassociatedreceptivefield.(B)Comparisonofthegammaactivityrecordedduringasingletrialinarepresentativerecording site(fromsessionD04nm1)withthemoviefeaturescomputedintheassociatedreceptivefield.Fromtoptobottom:LFPgammaband(30–100Hz),spatialcontrast,sineof theorientation,timecontrast.

samplingdemandsofconditionalmutualinformationaregreater thanforthesinglefeatureinformationconsideredthere.Wechose alargernumberofbinsforthestimulusfeatures,sinceouraimwas tocharacterizetheroleofcorrelationsbetweentheminevaluating tuningofneuronstomultiplefeatures.

Wecomputeddifferentsetsofinformationmeasurements.For eachinformationvalue,weﬁrstcomputedthesigniﬁcanceofthe uncorrectedplug-ininformationestimatewiththeanalytic chi-square approximationdescribed in Section 2.2for the absolute informationandinSection4.1fortheconditionalinformation.For thisanalysiswehad20,000–30,000trialsavailable(25,500±3100, mean±std).ThiscorrespondstoNs/R∼800fortheestimationof informationaboutanindividualfeature,andNs/R∼100forthe esti-mationofCMI.Thesenumbersarewellinsidetheregimeswhere thechi-squareapproximationsperformwell(Sections2and4). Wethencomputedanestimateoftheinformation,usingEq.(5)

fortheinformationbetweengammaBLPandfeatures,Eq.(6)for theinformationbetweenfeatures,andEq.(7)fortheconditional information,correctingforthebiaswiththePTcorrectionmethod

(PanzeriandTreves, 1996)describedby Eq.(2).Allinformation

measuresonexperimentaldatawerecomputedusingthe Infor-mationBreakdownToolbox2₍_Magri_et_al.,₂₀₀₉_).

5.4. Results:complementaritycodingoforientation,spatial contrastandtimecontrastbytheLFPgammapower

Thetimecourseofthreeextractedmoviefeatures(spatial con-trast, time contrast and orientation)in the RF (Fig. 5A) on an examplerecordingsiteisshowninFig.5B,togetherwiththe asso-ciatedgammabandoftheLFPrecordedfromthatsite.Fromthis example,itisapparentthatthegammaBLPistuned,todifferent extents,tothesethreefeatures.Thisisconsistentwithprevious

2_{http://www.ibtb.org}_.

reportsaboutgamma-bandselectivityobtainedwithsimple

stim-uli(Frienetal.,2000;HenrieandShapley,2005).

Toquantifythesedependencies,wecomputedthemutual infor-mationbetweenthegammaBLPandthedifferentfeaturesandwe testedthesignificanceoftuningofeachofthe55LFPrecording sites.Tothisend,weusedtheanalyticalchi-squareinformationtest onthebiasuncorrectedpluginestimates(becausethisisthemost sensitivetestunderthesamplingconditionsanalyzed).Wefound thatallrecordingsitescarriedsignificantinformationatp<0.001 confidencelevelforallthreefeatures(Fig.6A).While responses weresignificantlymodulatedbyallstimulusfeaturesconsidered, we foundthat the populationaverage(computed withPTbias correction)oftheinformationaboutspatialortemporalcontrast wasmuchhigherthantheinformationaboutorientation(Fig.6B), showingthatorientationcausedweaker(althoughstillhighly sig-nificant)responsemodulationthantheotherfeatures.

Weevaluatedtherobustnessofthesignificancetestbyusing onlyasubsetofthemoviepresentations(andhenceforthofthe trials),similarlytowhatwasdoneinFig.1A.Wefound(Fig.6C) thatforthemostinformativefeatures(spatialandtemporal con-trast)significantresponsemodulationwasdetectedinalmostall recordingsites(>95%)evenwhenonly6moviepresentationswere considered(ofthe35–50available),andusingonlytwomovie pre-sentationswassufficienttodetectsignificantinformation about thesetwocontrastfeaturesinmorethan50%ofrecordingsites. Fortheorientationfeature,moredatawasrequiredtofind depen-denceinasizablefractionofrecordingsitesbecauseofthesmaller information.However,evenwithonlytwomoviepresentations, morethan30%ofrecordingsiteswerefoundtobesignificantly modulatedbyallthreefeatures.

AsdescribedinSection3,tounderstandwhetherthegamma BLPwastrulytunedtoallthreestimulusfeaturesandwasnot car-ryinginformationaboutsomeofthemonlybecauseofcorrelation toanothertrulyencodedfeature,wehadtocharacterizewhether ornotthedifferentfeaturesintheRFvaryindependentlyduring themovietimecourse.Wefoundthatthevisualfeaturesdidnot

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Fig.6.Informationcarriedbygammabandlimitedpower(BLP)aboutspatialcontrast,orientationandtimecontrast.(A)Informationaboutthevalueofthedifferentfeatures carriedbygammaBLPforeachrecordingsite,computedwith4classesforgammaBLP,8classesforfeaturesandnobiascorrection.Significancewascomputedwiththe chi-squareapproximation,p<0.001.SincethenumberofclassesforgammaBLPandfeaturesarefixed,theanalyticchi-squaresignificancethreshold(inbits)dependsonly onN,thenumberoftrialsavailable(Section2.2).Datapointsareorganizedintoverticallinessincedifferentnumbersoftrialswereproducedbydifferentexperimental protocols(dependingonthedurationandnumberofpresentationsofthemovie);experimentswiththesameNformeachverticalline.Thresholdp=0.001isindicated bydashedline.NumbersinlegendsindicatethefractionofrecordingsforwhichthegammaBLPconveyedsignificantinformation.(B)Informationaboutthevalueofthe differentfeaturescarriedbygammaBLP(mean±SEM,n=55recordingsites).Blacklineindicatesmeanvalueofsignificancethreshold(p=0.95ofaveragebootstrapover 400permutations).(C)FractionofrecordingsforwhichthegammaBLPconveyedsignificantinformation(chi-squareinformationtest;p<0.001)asafunctionofthenumber ofmoviepresentationsusedfortheanalysis.

varyindependentlyfromoneanotherinthemoviesectionused here:thepercentageofrecordingsiteRFswithsigniﬁcant infor-mationbetweenfeaturesatp<0.001conﬁdencelevelwas71%,62% and73%forspatialcontrastwithorientation,temporalwithspatial

contrast and orientation with temporal contrast respectively (Fig.7A).Furthermore,thebiascorrectedpopulationaverage val-ues of information (Fig. 7B) were higher than those between features and neural response. This, as explained in previous

0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Information (bits) I(SC;O) I(TC;SC) I(O;TC) Threshold F1=SC F2=O F1=O F2=TC F1=TC F2=SC 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 I(F1; F2) (bits) p<0.001 71% 62% 73%

A

B

p=0.001 sig. threshold

Fig.7. Mutualinformationbetweenvisualfeatures.(A)InformationbetweenfeaturescomputedforeachRFandvaluesaveragedover10framewindowsusing8feature classes.Significanceofinformationiscomputedwiththechi-squareapproximation,p<0.001.Thresholdp=0.001isindicatedbydashedline.Numbersinitalicsindicatethe fractionofrecordingsforwhichthemutualinformationbetweenthefeatureswassignificant.(B)Mutualinformationbetweenfeatures(mean±SEM,n=55RFs)computed withsameparametersof(A),andPTbiascorrection.Blacklineindicatesmeanofsignificancethreshold(p=0.95ofpopulationaverageofbootstrapover400permutations). Resultsshowthattheconsideredvisualfeaturesofthemoviesectionusedherearesignificantlycorrelated.

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R.A.A.Inceetal./JournalofNeuroscienceMethodsxxx (2011) xxx–xxx 13 4 6 8 10 12 14 x 10−3 0 0.05 0.1 0.15 0.2 0.25

Conditional Information (bits)

Bootstrap information over 400 permutations

Probability

8 feature classes

4 feature classes 12 feature classes

C

χ2 p<0.001 100% for all

A

0 0.05 0.1 0 0.05 0.1 0 0.05 0.1

B

Fig.8.ConditionalinformationcarriedbygammaBLPaboutvisualfeatures.(A)InformationcarriedateachrecordingsitebygammaBLPaboutthevalueofeachfeature conditionedoneachoftheotherfeatures,computedwithnobiascorrection.Significance(p<0.001)isassessedwiththechi-squareapproximation,withthresholdvalue ofp=0.001indicatedbyblackdashedline.NumbersinlegendreportthepercentageofrecordingsforwhichthegammaBLPconveyedsignificantCMI.(B)Information carriedbygammaBLPaboutthevalueofeachfeatureconditionedoneachoftheotherfeatures(mean±SEM,n=55).InformationiscomputedwithPTbiascorrection.Black lineindicatesmeanofsignificancethreshold(p=0.95ofaveragebootstrap,400permutations).(C)DistributionofbootstrapvaluesofuncorrectedCMIfordifferentpairsof features(coloredlines,400randompermutations)comparedwiththecorrespondingchi-squaredistribution,fordifferentnumbersoffeatureclasses.Resultsareaveraged overallrecordingsfromsessionH06nm6.

sections,impliedthatthesingle-featureinformationanalysiswas notenoughtoestablishifallthreeconsideredfeaturesweretruly encodedbythegammaBLP.

To shed light on this issue, we computed, for each pair of features, the conditional information I(BLP;F2|F1) carried by thegammaBLP aboutfeatureF2 given thevalue of featureF1. Weestimatedthesignificanceofthisvalueusingthechi-square approximation for the CMI described in Section 4. We found that for every pair of features F1 and F2 the CMI was signifi-cant at the p<0.001level for all recording sites (Fig. 8A). We thencomputedpopulationaveragesofPTbiascorrectedCMI val-uesforeachpairoffeatures(Fig.8B).Again,orientation(Fig.8B, middlerow)had weakerconditionalmodulationthantheother features,althoughitremainedstatisticallysignificantwhen con-ditioned on either remaining feature. Interestingly, the values of conditional information were higher in all cases than the correspondingunconditioned single-featureinformation values, indicatingthatcorrelationsbetweenfeaturesinthenaturalmovie actuallyincreased(ratherthandecreased)thejointinformation thatthegammaBLPcarriedaboutthesemultiplefeatures.

Theoriginofthiseffectwasfurtherinvestigatedbycomputing I(F1;F2|BLP=1,2,3,4),i.e.byanalysinghowthemutual informa-tionbetweenthefeaturepairswasrelatedtothedifferentlevelsof LFPgammabandpower.Fig.9showsthatthemutualinformation

betweenfeaturepairsatfixedresponsewassignificantforall lev-elsofgammaBLP(chi-squareapproximationtestonuncorrected information;p<0.001),andwasalsosignificantlyvaryingamong differentlevelsofgammaBLP(ANOVAtest;p<0.001).Thefactthat thedegreeofdependencebetweenfeaturesdiffersacrosssections ofstimulithatevokedifferentresponsesexplainswhycorrelations betweenfeaturesincreaseinformation.Inthiscase,thedegreeof dependencebetweenfeaturescanbeusedtopredicttheneural responseandsoitincreasestheoverallinformationbetween fea-turesandresponses(seePanzerietal.(1999)orPolaetal.(2003)

foraproof).Moreover,theaverageoverallfourlevelsofgamma BLPofI(F1;F2|BLP=1,2,3,4)washigheroverallthanI(F1;F2)for allfeaturepairs,theaveragegainbeing+26%,+26%and+22%for I(SC;O),I(O;TC)and I(SC;TC)respectively.Inthiscase,according toEq.(13)statisticaldependenciesamongfeaturesmustincrease information.

Wethenevaluatedifagivenfeaturecarriedinformationthat wascomplementarytothatcarriedbybothotherfeatures,by cal-culatingI(BLP;F2|F1,F3)using4classesforthefeatures.Wefound thatthiswasalsosigniﬁcant(chi-squareapproximation,p<0.001) forallpermutationsoffeatures.Theresultsoftheabove compu-tationsshowthatthegammaBLPconveyedgenuineinformation abouteachsinglefeature,eventhoughtheywerestrongly corre-lated.

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14 R.A.A.Inceetal./JournalofNeuroscienceMethodsxxx (2011) xxx–xxx 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 0.3 I(SC;O|γ BLP) bits

γ BLP level γ BLP level γ BLP level

0 0.05 0.1 0.15 0.2 0.25 0.3 bits 0 0.05 0.1 0.15 0.2 0.25 0.3 bits 1 2 3 4 1 2 3 4 I(O;TC|_γBLP)

C

I(SC;TC|_{γ BLP)}

B

A

Fig.9.Informationbetweenvisualfeaturesatfixedresponse.MutualinformationbetweenfeaturesconditionedtofourdifferentlevelsofgammaBLP(mean±SEM,n=55). (A)Spatialcontrastandorientation,(B)orientationandtimecontrast,(C)spatialcontrastandtimecontrast.Informationiscomputedforvaluesaveragedover10frame windowswithPTbiascorrectionand8featureclasses.Asterisksindicatethatthemutualinformationissignificantforeachlevelofresponse(chi-squareapproximationtest onuncorrectedinformation,p<0.001).ANOVAtest(p<0.001)showsthattheinformationbetweenfeaturepairschangessignificantlyfordifferentlevelsoftheresponse.

We performed all the significance tests using the analytic chi-squareapproximationontheuncorrectedplugininformation valuesbecausetheyallowfastcomputationofhighlysignificant p-values,andbecausetheconsiderationsofSection3predictedthis testtobehighlyeffectiveunderthesamplingconditionsdescribed here.Toillustratetheaccuracyofthechi-squareapproximation,we comparedthedistributionofthebootstrapvalues(over400 per-mutations)oftheuncorrectedplugininformationestimateswith theanalyticalchi-squaredistributionwiththedegreesoffreedom predictedbytheequationsinSection4.1.AsshowninFig.8Cfor arepresentativesession,withthisdatasettherewasanexcellent agreementbetweenthebootstrapandthechi-squaredistribution. Inparticular,whenchangingthenumberofbinsintowhichthe featureswerediscretized,thebootstrapdistributionfollowedthe changeinthenumberofdegreesoffreedomofthechi-square dis-tributionpredictedinSection3:whenthefeatureswerebinned with4,8and12classestheaveragep-valuesoftheKStestwere 0.49±0.025,0.36±0.025,0.34±0.015respectively(mean±SEM overallrecordingsites).Theseresultsbothhighlightthevalueof theanalyticalapproximationsreportedinourstudyand demon-stratethatthechi-squarecouldbecorrectlyusedinthisdatasetto accuratelycomputesignificancevaluesdowntosmallp-values. 6. Discussion

Wehaveconsideredtheapplicationofmutualinformationas astatisticaltest toestablishdependencybetweenstimulus fea-turesand neuronalresponse variables.While accuratebiasfree informationvaluesarecrucialtoenablemeaningfulquantitative comparisons,inmanyapplicationsitissufficienttoestablishthe presenceofastatisticallysignificantdependence.Wehaveshown that for determining significance with the bootstrap method, uncorrectedplugininformationestimatesshouldbeused,andthat ifsufficient dataare available, the bootstrap procedurecan be bypassedcompletelyandsignificancedeterminedthroughan ana-lyticalformforthedistributionofvaluesunderthenullhypothesis. Thisallowsaccuratecalculationofp-valuesandsignificancewith greatlyreducedcomputationalrequirements.

Thisﬁnding,particularlytheaccuracyoftheanalyticchi-square approximation,hasimplicationsforanalysisofotherneurological signals,forexamplefunctionalmagneticresonanceimaging(fMRI) data.Anapproachoftentakenwithsuchdataisamass-univariate analysistodeterminedependenciesbetweenvariousstimulation conditionsandthetimecourseofthebloodoxygenlevel depen-dent(BOLD)signalateachvoxel.Theabilitytodirectlycalculate asigniﬁcancep-valueforanydependencecheaplyusingthe chi-squareapproximationsuggeststhatthemethodsdiscussedhere

couldbeeasilyappliedtosuchdata,evenwhentheyconsistoftens orevenhundredsofthousandsofvoxels.Theadvantagesofsuch anapproachisthattheyarecompletelynon-parametricanddonot requireanyconvolutionofthestimulustimecoursewitha hemo-dynamicresponsefunction(HRF).Thismeanstheycanbeappliedto situationswheretheHRFisnotwellcharacterizedorisnot guar-anteedtobeuniversalacrossconditionsandareas.Thismayfor instancebethecasewheninvestigatingtherelationshipbetween thefMRIBOLDresponseandthepowerofEEGorLFPresponses indifferentfrequencybands,orwhencollectingfMRIresponsesto complicateddynamictimevaryingstimuliwithmultiplecorrelated featuressuchasthemoviestimuliconsideredhere.Additionally, thegeneralandnon-linearnatureofmutualinformationallowsit tocaptureanytypeofdependency,incontrasttoconventional lin-earanalysis.Similarconsiderationscanbeappliedtotheproblemof estimatingsigniﬁcanceofactivationinothertypesofneuroimaging signals,suchasthesourcesreconstructedfromEEGorMEGdata

(Grossetal.,2001).

Whilewehaveshownthattheuseofbiascorrectionshasno beneficialeffectonestablishingthepresenceofastatistically sig-nificantstimulusresponsemodulation,oncethatsignificancehas beendetermined,bias correctedvalues are of coursecrucialto obtainanaccuratequantificationofthestrengthofthemodulation. Thebiascorrectedvaluesallowdirectquantitativecomparisons oftheamountsoftheinformationcarriedbydifferentsystemsor experimentswithdifferentnumbersoftrials,andoftheamounts ofinformationcarriedbyneuralcodescharacterizedbya differ-entresponsedimensionality(suchasforexample,spiketimesand spikecounts).Ourfindingssuggestthatitmaybeworth imple-mentingahybridapproach,whereuncorrectedinformationvalues areused(accordingtotheproceduresdescribedhere)todetermine thestatisticalsignificanceofanydependenceeffect,whilebias cor-rectedvalues arecomputedtoproduceanaccurateestimateof theeffectsize.Inpractice,sincemostbiascorrectiontechniques involvecalculationof theuncorrectedvalues thisapproach car-rieslittlecomputationaloverhead.However,caremustbetakento avoidstatisticalerrorsdueto‘non-independent’analysis(Vuletal., 2009),whicharisewhenreportingmeanvalueswherethemeanis calculatedonlyfromsamplesexceedingsomethreshold.Forthis reason,in theanalysisofrealdatapresented herewereported biascorrectedmeaninformationvaluesoverthewholepopulation ratherthantheinformationvaluesaveragedoverthe subpopula-tionwithsignificantinformation. Theaccuratedeterminationof thestatisticalsignificancecanofcoursebeusedasaselectiontool forothertypesofindependentanalysis,forexamplecalculationof receptivefieldsofsignificantlytunedneurons.