Optimal band separation of extracellular field potentials

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ContentslistsavailableatSciVerseScienceDirect

Journal

of

Neuroscience

Methods

jo u rn al h om epa g 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

Optimal

band

separation

of

extracellular

ﬁeld

potentials

Cesare

Magri

a,b

_{, Alberto}

_Mazzoni

c

_{, Nikos}

_K.

_Logothetis

a,d

_{, Stefano}

_Panzeri

c,∗

a_Max_Planck_Institute_for_Biological_Cybernetics,₃₈_{Spemannstrasse,}₇₂₀₇₆_Tübingen,_Germany b_Bernstein_Center_for_{Computational}_Neuroscience_Tübingen,_Germany

c_Robotics,_Brain_and_Cognitive_Sciences_Department,_Italian_Institute_of_Technology,_Via_Morego_30,₁₆₁₆₃_Genova,_Italy d_Division_of_Imaging_Science_and_Biomedical_Engineering,_University_of_Manchester,_Manchester_M13_9PT,_United_Kingdom

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received19July2011

Receivedinrevisedform30October2011 Accepted2November2011

Keywords: Neuralcode LocalFieldPotential Visualcortex Mutualinformation Frequencybands

a

b

s

t

r

a

c

t

LocalFieldPotentials(LFPs)exhibitabroadbandspectralstructurethatistraditionallypartitionedinto distinctfrequencybandswhicharethoughttooriginatefromdifferenttypesofneuraleventstriggered bydifferentprocessingpathways.However,theexactfrequencyboundariesoftheseprocessesarenot knownand,asaresult,thefrequencybandsareoftenselectedbasedonintuition,previousliterature orvisualinspectionofthedata.Here,weaddresstheseproblemsbydevelopingarigorousmethodfor definingLFPfrequencybandsandtheirboundaries.Thecriterionintroducedfordeterminingthe bound-ariesdelimitingthebandsistomaximizetheinformationaboutanexternalcorrelatecarriedjointlybyall bandsinthepartition.ThemethodfirstpartitionstheLFPfrequencyrangeintotwobandsandthen succes-sivelyincreasesthenumberofbandsinthepartition.WeappliedthepartitioningmethodtoLFPsrecorded fromprimaryvisualcortexofanaesthetizedmacaques,andwedeterminedtheoptimalbandpartitioning thatdescribestheencodingofnaturalisticvisualstimuli.ThefirstoptimalboundarypartitionedtheLFP responseat60Hzintolowandhighfrequencies,whichhadbeenpreviouslyfoundtoconveyindependent informationaboutthenaturalmoviecorrelate.Thesecondoptimalboundarydividedthehigh-frequency rangeatapproximately100Hzintogammaandhigh-gammafrequencies,consistentwithrecentreports thatthesetwobandsreflectpartlydistinctneuralprocesses.Athirdimportantboundarywasat25Hz anditsplittheLFPrangebelow50Hzintoastimulus-informativeandastimulus-independentband.

1. Introduction

Local FieldPotentials(LFPs) are a massedsignalwhich cap-turesmultipleneuralcontributions,fromdendrosomaticdipoles generatedbysynapticactivitytonon-synapticslowactivitysuch asvoltage-dependentmembrane oscillationsandspike afterpo-tentials(Logothetis,2003).SinceLFPsaresensitivetosupraand subthresholdprocesses,investigatingstimulusencodingbyLFPs canofferadditionalinsightsintosensoryrepresentationsbeyond those offeredby onlymeasuringneuronal spike trains(Belitski etal.,2010;Logothetis,2003;Montemurroetal.,2008;Nicolelis andLebedev,2009;Rayetal.,2008).Theirstimulustuninghasbeen intensivelyinvestigatedinrecentyears(Belitskietal.,2008;Berens etal.,2008;Frienetal.,2000;Kayser andKonig,2004;Liu and Newsome,2006;SiegelandKönig,2003).LFPsexhibitabroadband spectralstructurewhichis traditionallypartitionedintodistinct frequencybands,initiallyintroducedinthehumanEEGliterature, whicharethoughttooriginatefromdifferenttypesofneuralevents

∗ Correspondingauthor.Tel.:+391070781437. E-mailaddress:stefano.panzeri@iit.it(S.Panzeri).

triggeredbydifferentneuralprocessingpathways(Buzsáki,2006). However,theexactfrequencyboundariesoftheseprocessesare notknown.Frequencybandsarethusoftenselectedbasedon intu-ition,previousresultsintheliteratureorvisualinspectionofthe data.Asaresult,thebandboundariesvarygreatlyacrossstudies. Forexample,thegammabandissometimesreferredtoasanarrow (afewHzwide)frequencyrange(LlinasandRibary,1993;Miltner etal.,1999)andothertimesasawideintervalspanningtensofHz (ChrobakandBuzsaki,1998;KruseandEckhorn,1996).

Arigorousapproachtodefiningtheboundariesofa stimulus-tunedLFPbandhasbeenproposedbySiegelandKönig(2003).Their method,basedonmaximizinganindexofselectivitytoastimulus parameter,allowstheidentificationofboththepartsofthe fre-quencyspectrumthatarebesttunedtoaspecificstimulus,and oftheboundariesthatoptimizethetuningofasinglebandtothe stimulus.Thisapproach,whichwassuccessfullyappliedtoidentify theoptimalboundariesfororientationtuningofthegammaband invisualcortex(SiegelandKönig,2003),requiresamodelofa tun-ingcurvedescribingtherelationshipbetweenanexternalcorrelate ofinterestandtheinducedchangesintheLFPbandconsidered. Acomplementaryapproachwasdevelopedbyourgroup(Belitski etal.,2008)forinvestigatingtherelationshipbetweenthepower 0165-0270/$–seefrontmatter © 2011 Elsevier B.V. All rights reserved.

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ateachfrequencyintheFourierdecomposition–thesmallest fre-quencyelementinwhichwecanbreaktheLFPrange–andcomplex visualstimuli.ThisanalysisrevealedwhichregionsintheLFP spec-trumareinformativeaboutnaturalisticmoviestimuliandwhich regionsare,instead,stimulus-unrelated.However,theseprevious methodscannotdeterminehowtopartitiontheentireLFPrange insuchawaythatthebandsinthepartitioncollectivelycarrythe largestpossibleamountofinformationaboutasetofstimuli.In fact,byﬁne-tuningonebandatatimeonemightfailtotakeinto accounttherelationshipsbetweendifferentfrequencyregions.For instance,tuningonebandatatimemayindividuateasetofbands whichareindividuallyhighlyinformativeandthatallcarryvery similarinformation,butmayfailtoindividuatebandsthatexpress complementaryaspectsof stimulustuningandthuscarry more informationcollectively.

Herewe introducea newprocedureforpartitioning theLFP responseintobandsthatcollectivelycarrymaximalmutual infor-mationabout a setof external correlates.This proceduredoes notmake any assumptionon thetype of relationshipbetween thepower in an LFP band and the external correlate of inter-est.Rather,becauseof theproperties of mutual information, it naturallytakesintoaccountlinearandnon-linearcorrelationsat anyorderaswellasanyrelationshipbetweenthepowersin dif-ferentbands. Thearticleis structuredasfollows.Afterdeﬁning theprocedure,weinvestigateandaddresstheproblemsrelated toits computationalimplementation. We then apply the tech-niquetotheelectrophysiologicalsignalrecordedfromtheprimary visualcortexofanesthetizedmacaques,andwedeterminea sta-blepartitionofLFPsintobandsthatcarriesanoptimalamountof informationaboutallvisualfeaturesinthemovie.

2. Materialsandmethods

2.1. Recordingprocedures,sensorystimulianddataextraction We analyzed sensory evoked LFPs and MUA recorded from primary visualcortex (V1) of two anaesthetized monkeys. The recordingprocedures and stimulation paradigms for the differ-entdatasetsaredescribedshortlyinthefollowingandhavebeen detailedpreviously(Belitskietal.,2008;2010;Montemurroetal., 2008). All experiments were approved by the local authorities (Regierungspräsidium)andwereinfullcompliancewiththe guide-linesoftheEuropeanCommunity(EUVD86/609/EEC)forthecare and useof laboratoryanimals. Prior totheexperiments, form-ﬁttingheadpostsandrecordingchamberswereimplantedduring anasepticandsterilesurgicalprocedure.

Recordings from V1 of two adult rhesus monkeys (Macaca Mulatta)were obtainedwhile the animals were anaesthetized (remifentanil, 1␮g/kg/min), muscle-relaxed (mivacurium, 5mg/kg/h) and ventilated (end-tidal CO2 33mmHg, oxygen

saturation>95%).Bodytemperaturewaskeptconstant and lac-tatedRinger’ssolution supplied (10ml/kg/h).Vital signs(SpO2,

ECG,bloodpressure,endtidalCO2)werecontinuouslymonitored.

Neuronal activity was recorded from opercular V1 (foval and para-foveal representations) using microelectrodes (FHC Inc., Bowdoinham, Maine, 300–800kOhms), signals were high-pass filtered(1Hz;digitaltwopoleButterworthfilter),amplifiedusing anAlphaOmegaamplifiersystem(AlphaOmegaEngineering)and digitizedat20.83kHz.Binocularvisualstimuliwerepresentedat aresolution of640×480pixels(fieldof view:30◦×23◦,24bit truecolor,60Hzrefresh)usingafiberopticsystem(Avotec,Silent Vision, Florida). Stimuli consisted of ‘naturalistic’ complex and commercially available movies (30Hz frame rate), from which 3.5–4.3minlongsequenceswerepresentedandrepeated30–40 times.We confirmed that thereceptive fieldsof allmulti-unit

activityfromeachrecordingsitewerewellwithinareaofvisual stimulation(Raschetal.,2008).

3. Informationtheoreticanalysis

3.1. Deﬁnitionofthemaininformationtheoreticquantities

TodeterminehowwellthepowerinagivenLFPbandencodes thesensorystimuluswecomputedShannon’smutualinformation betweenthestimulusandthepoweroftheLFPband(Shannon, 1948),whichisdeﬁnedas I(S;B)=

b,s P(s)P(b|s)log2 P(b|s) P(b) (1)

whereP(s)istheprobabilityofpresentingstimuluss,P(b|s)isthe probabilityofobservingapowervaluebgivenpresentationof stim-uluss,andP(b)istheprobabilityofobservingpowervaluebacross alltrialstoanystimulus.I(S;B)quantiﬁes,inunitsofbits,the reduc-tionofuncertaintyaboutthestimulusthatcanbegainedfromthe observationofasingle-trialpowervalue.Onebitcorrespondstoa reductionofuncertaintyofafactoroftwo.I(S;B)iszeroonlywhen thestimulusandthepowerarestatisticallyindependent,indicating thatnoknowledgeaboutthestimuluscanbegainedbyobserving theresponse.

Eq.(1)canbeextendedtoquantifytheinformationaboutaset ofstimuligainedbythesimultaneousobservationofthepowerin LdifferentLFPbands,asfollows:

I(S;B1...BL)=

b,s P(s)P(b1...bL|s)log2 P(b1... bL|s) P(b1...bL) (2) where in the above P(b1...bL|s) is the probability of the joint

observationofobservingjointlythepowervaluesb1...bLinthe

consideredbandsduringasingle-trialpresentationofstimuluss. 3.2. Deﬁnitionofstimulusset

Computationofinformationrequiresthedefinitionofasetof sensorystimuliSaboutwhichtheneuralresponsecarries informa-tion.Ourdatasetwasmadeofrecordingsinresponsetorepeated presentationofthesamemovie.Toquantifytheamountofvisual informationcarriedbytheLFPpowerinthisexperiment,weused afeature-independentdefinitionofstimulus,describednext(de RuytervanStevenincketal.,1997;Strongetal.,1998).Themovie presentationtimewasdividedintonon-overlappingtimewindows oflengthT=960msandeachwindowwaslabeledwitha stimu-lusidentificationnumber(s=1,2,...). Inthispaperwepresent resultsobtainedwithawindowoflengthT=960ms,butresults werequalitativelysimilarwhenvaryingthelengthoftherunning windowTintherange480–1920ms(notshown).TheLFPpower wasquantifiedseparatelyineachwindow(seebelow)andwe com-putedtheinformationbetweentheLFPpowerandthesectionofthe movie(“stimuluswindow”)atwhichtheresponsewascollected. Itisimportanttonotethattheresponsecollectedinthes-th win-dowispotentiallyelicitedbythewholetemporalpatternofvisual stimulationuptothelasttimepointofstimuluswindows. There-forethisdefinitionofinformationcarriedbytheresponseabout whichstimuluswindowwasbeingpresentedtakesintoaccount thepotentialcontributionsofallstimulusfeaturesatallprevious movieframesaswellasthecontributionofthevisualfeaturesin thecurrentmovieframe.Assuch,theinformationcalculationdoes notassumeaninstantaneousrelationshipbetweenLFPsand stim-uli,andtakesintoaccounttheeffectsofalltimelagsbetweenthe neuralresponseandanyvisualfeaturethatprovokedit.This def-initionofstimulussethastheadvantagethatit doesnot make anyassumptionsaboutwhichstimulusfeaturesorcombinationsof

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them,eitherpresentedatthecurrenttimeorintheperiodpriorto theconsideredneuralresponse,provokestheneuronalresponse. Thisalsomakestheinformationvalueformallyinvarianttoany rigidtimeshiftbetweenstimuluswindowsandneuralresponse correspondingtoanyassumptionofagivenlatencyvaluebetween stimuliandresponses.

3.3. Computationofspectralpowerineachbandforinformation calculation

Computation of information requires the evaluation of the single-trialspectralpowerofeachbandineachofthe“stimulus windows”oflengthTdefinedabove.Herewecomputedthe single-trialpowerineachwindowusingthemultitapermethod(Percival andWalden,2006)withtime-bandwidthparameterNW=2.This methodofspectralestimationwaschosenbecauseitgivesan excel-lenttrade-offbetweenbiasandvarianceandbecauseitwasshown toworkwellonLFPdata(Belitskietal.,2008;Pesaranetal.,2002). Tocomputethepowerinagivenfrequencybandweintegratedthe spectralpowerbetweenthebandboundaries.However,we veri-fiedthattheuseofalternativemethodsforbandpowerestimation withinawindow,suchasaveragingoverthetimepointswithinthe windowtheinstantaneousHilbertpowerofasignalband-passed withinthedefinedboundboundarieswithaKaiserfilter(asdone e.g.in(Belitskietal.,2010;Kayseretal.,2009)),gaveresultswhich wereessentiallyidenticaltothatobtainedwiththesummationof Multitaperpowerusedhere(see(Magri,2009)formoredetails). 3.4. Informationredundancy

TheinformationredundancybetweenthepowerindifferentLFP bandswasdeﬁnedasthedifferencebetweentheinformation pro-videdbythejointsignalsb1,...,bLandthesumoftheindependent

informationvalues(Belitskietal.,2008;Hatsopoulosetal.,1998; Polaetal.,2003;Schneidmanetal.,2003):

Red(B1...BL)=I(S;B1...BL)− L

i=1

I(S;Bi) (3)

Positivevalues of redundancyindicatethat thejoint knowl-edgeofthepowerofallbandscarries lessinformationthanthe sumoftheinformationprovidedbyeachindividualband.Negative valuesofredundancydenotethepresenceofsynergistic interac-tionbetweendifferentbands,whichmakethetotalinformation greaterthanthesumoftheinformationprovidedindividuallyby eachband.Wenormalizedredundancyexpressingitaspercentage ofthetotalinformationcarriedbythejointobservationofallbands inthepartition.Thispercentageredundancywasthusdeﬁnedas Red(B1...BL)=100×

Red(B1...BL)

I(S;B1...BL)

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4. Computationofinformation

The computation of the information with Eqs. (1) and (2) requirestheestimationofstimulus-conditionalresponse probabil-itiesP(b|s)andP(b1,...,bL|s).Theseprobabilitiesarenotknowna

prioriandmustbemeasuredexperimentallyfromaﬁnitenumber oftrials.Inthisworkwecomparedtwocomplementaryprocedures formeasuringtheprobabilitydistributionsandthecorresponding informationvalues,namely,theDirectMethodandtheGaussian

Methoddescribedbelow.Allinformationanalyseswereperformed usingtheInformationBreakdownToolbox(www.ibtb.org;(Magri etal.,2009)).

4.1. TheDirectMethod

Computation ofinformation according totheDirect Method (Strongetal.,1998)wasasfollows.Weﬁrstdiscretizedtheresponse spacebybinningtheresponsepowerintoMequi-populatedbins. TheparameterMwasvariedbetween3and32inordertoevaluate the effect of the discretization on the Direct information esti-mates.Wethenestimatedtheprobabilitiesofthediscreteneural responsesbysimplycomputingthefractionoftrialsinwhicheach responsevalueisobserved,andthenbyinsertingthese response-probabilityestimates intotheinformation Eqs.(1) and (2).The DirectMethod,beingbasedonempiricallycomputingthe prob-abilityhistogramsofdiscreteordiscretizedneuralresponses,does notmakeanyassumptionontheshapeoftheprobability distri-butions.ThismakestheDirectMethodwidelyapplicabletomany differenttypesofdata.

Evenwheneachstimulusisrepeatedmanytimes,aswasthe casehere,theestimatedprobabilitiessufferfromﬁnitesampling errors,whichinduceasystematicerror(bias)ininformation esti-mates(Panzeriet al.,2007).Tocorrect for thebias, weused a quadraticextrapolationprocedure(Strongetal.,1998)toestimate andsubtractoutthebiasofeachinformationquantity.When com-putingtheinformationconveyedbythejointobservationoftheLFP powerinmorethanonebandweadditionallyappliedthe “shuf-ﬂingprocedure”describedin(Montemurroetal.,2007;Panzeri etal.,2007),whichgreatlyreducesthebiasofmultidimensional informationestimatesandtypicallymakestheresidualbias nega-tive.Thisimpliesthatourestimatesofthejointinformationcarried bytwosignals(andthusofredundancy)tendtobeslightlybiased downward.

4.2. TheGaussianMethod

AnalternativeapproachtotheDirectestimation of informa-tionistouseanalyticalmodelsoftheprobabilitydistributions;ﬁt thesedistributionstothedata;andthencomputetheinformation fromtheseprobabilitymodels.TheGaussianMethodfor comput-inginformationandentropiesistheonebasedonﬁttingresponse probabilitiestoGaussianfunctions.

UndertheGaussianhypothesis,mutualinformationisgivenbya simplefunctionofthetrial-to-trialvarianceofLFPresponses(Cover andThomas,2006)

I(S;B)= 1₂(log2[2e|2(B)|]−log2[2e|s2(B)|]) (5)

where|2₍_B)|_and_|

s2(B)|arethedeterminantsofthematricesof

covariancecomputedacrosstrialsandstimuli,andacrosstrialsto stimuluss,respectively.

TheGaussian Methodhastwo mainadvantages: itdoesnot requiredatadiscretizationpriortotheinformationcalculation,and itdependsonlyonafewparametersthatcharacterizetheneural response(i.e.,thevarianceandcovarianceoftheresponses),which makesitmoredata-robustandlesspronetosamplingbiasthanthe Directcalculation(Magrietal.,2009;Yuetal.,2010).The poten-tialdangerwiththisapproachisthattheestimatesprovidedbyEq. (2)maybeinaccurateiftheunderlyingdistributionsarenotclose enoughtoGaussians.

AlthoughlessseverethanintheDirectcase,thebiasofthe infor-mationcalculationduetolimitedsamplingisstillpresentwhen usingtheGaussianMethod.Whentheunderlyingdistributionsare

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GaussiananexactexpressionforthebiasofI(S;B)canbecomputed asfollows(Goodman,1963;Misraetal.,2005;Oymanetal.,2003): bias[I(S;B)]=gbias(Ntr×Nwin)−

s

p(s)gbias(Ntr) (6)

whereNtr isthenumberofavailabletrials perstimulus(inour

experimentthenumberofmovierepetitions)andNwinisthe

num-berofstimuli(inourcasethenumberofwindowsoflengthTinto whichthemoviepresentationtimewassubdivided).Thefunction gbias(·)isdeﬁnedas gbias(N)= 1 2 ln 2

⎡

⎣

Lln

₂ N−1

+ L

j=1

_N₋_j 2

⎤

⎦

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and isthepolygammafunctionandListhenumberofbands considered.

Whenusing theGaussian Methodwetookthecubic rootof power;weﬁttedthedistributionofthisresponsetoeachstimulus toaGaussian;andweﬁnallycomputedtheinformationthrough Eq.(5)subtractingtheanalyticGaussianbiascorrection,Eq.(6). Thereasonforapplyingthecubicroottransformationisthat mul-titaperpowerestimatesareasymptoticallychi-squaredistributed (PercivalandWalden,2006)thustheircubicrootisapproximately Gaussian(Wilsonand Hilferty,1931).The cubicroot operation, beingmonotonic,doesnotaffecttheunderlyinginformationvalues ofthepower,butitmakesresponseprobabilitiesmuchmore Gaus-sianandthusfacilitatesinformationestimationwiththeGaussian Method.

4.3. DescriptionoftheoptimalprocedureforpartitioningtheLFP response

Theprocedurethatweproposeforanoptimalpartitionofthe LFPrangeintoagivennumberofbandsworksasfollows.First,we definethehighandthelow cutoffsfrequenciesthatdelimitthe partofthespectrumoftheextracellularsignalwhichistakento betheLFP.Thehighfrequencycutoff,usuallytakenintherange betweenoneandafewhundredHz,isnotconsistentlydefinedin theliterature.Ourapproachhereistoconsidertheeffectofdifferent valuesofthisparametersuponthebandpartition.Ontheother hand,thelowfrequencycutoffoftheLFPisusuallyconstrainedonly bythehighpassfilterofthepreamplifierintherecordingsetup, whichinourcasewas1Hz(seeSection2).Becausethepowerof theextracellularsignalisnegligiblebelowthishigh-passvalue,in thefollowingforsimplicitywewillreportourresultsusingalow frequencycutoffof0Hz.However,theresultswouldhavebeen identicalhadweusedalowfrequencycutoffof1Hz.

OncetheLFPcutofffrequencieshavebeenspecified,ourmethod seekstopartitionthisrangeintoanumberofbandschosento maxi-mizetheinformationcarriedbytheLFPpartitionaboutanexternal setofcorrelatesofinterest,suchasasetofsensorystimuli.This meansthat,amongallpossiblepartitionsoftheLFPrangeintothe considerednumberofbands,weselecttheonewhichextractsthe largeststimulus-informationasdefinedinEq.(2).Additionally,we proposetoimplementthisprocedureinanincrementalway,i.e. westartbypartitioningtheLFPresponseintotwobandsandwe thenincreasethenumberofbandsconsidered.Thisisusefulfor tworeasons.First, itprovidesa waytoranktheboundariesby importance,undertheassumptionthat theboundariesdefining theoptimalpartitionwithsmallnumberofbandsaremore funda-mentalthanboundariesappearingonlyinoptimalpartitionswith alargernumberofbands.Second,theincrementalprocedure per-mitstoidentifytheboundariesthataremoststabletothenumber ofbandsconsidered,i.e.ifaboundaryidentifiesanoptimal par-titionintoagivennumber ofbands,thesameboundaryshould

alsobelongtotheoptimalpartition withonemore band. Indi-cationsthatthebandsarestabletorecursivereﬁnementofthe partitionareimportanttosupportthenotionthattheidentiﬁed bandboundariesaretrue“separators”ofdifferentneuralprocesses, oratleastarestableboundariesofspectralregionswhichexpress complementaryinformationaboutthestimulus.

5. Results

WerecordedLFPsfrom23sitesin theprimaryvisualcortex (V1)fromtwoanesthetizedmonkeys,usinganarrayofelectrodes, duringthepresentationofa3.5-to4.3-min-longsequencefrom commerciallyavailablemovies.Eachmovieextractwasrepeated 30–40times.Theresponsesofthesamesitetodifferentmovies wereanalyzedseparately.Insomecases,wewereabletomeasure theresponsesofarecordingsitetorepeatedpresentationsofa sec-ondorthirddifferentmovie.Insuchcases,theresponsesofthe samesitetodifferentmovieswereanalyzedseparately.Thisgave usatotalof57cases,whichwecollectedalltogetherinour popu-lationanalysisandweconsideredeachofthem,forsimplicity,asa differentrecordingsite.

ThespectrogramoftheLFP(Fig.1A)wastypicalofprimary cor-tical recordings,in that itshoweda broadbandspectrum with signiﬁcantpoweratfrequenciesinthewholerange0–100Hzand withthepowerdecreasingathigherfrequenciesafterreachingan initialpeakatlowfrequencies(inthiscasethepeakwasat2Hz). Presentationofthemoviestimulusinducedamarkedincreasein spectralpowerbetween40and150Hzoverthepowerobserved duringspontaneousactivity.

5.1. TheinformationconveyedbysingleLFPfrequencies

Thedatasetusedinthisstudyhasbeeninvestigatedinseveral previousworksfromourgroup(Belitskietal.,2008,2010;Magri etal.,2009).Inthesestudiesweinvestigatedtheinformationabout anaturalisticmoviestimuluscarriedjointlybytheLFPpowerat eachfrequencyoftheFourierspectrum(Belitskietal.,2008;Magri etal.,2009), and thecorrelations betweenLFPpowerat differ-entfrequencies(Magrietal.,2009).Here webrieﬂysummarize theseresultsfromthesepreviousworks,withtheaimof point-ingtotheattentionofthereaderthefactsandﬁndingsthatare mostrelevanttotheanalysispresentedinthisstudy.Wealso facil-itatethecomparisonbetweenthepresentstudyandourprevious workbyre-computingourpreviouslypublishedinformation quan-titieswithexactlythesameinformationestimationalgorithm(the GaussianMethod)usedforthenewoptimalbandpartitionresults. WebeginthissummarybyreportingtheinformationthatLFP powercarriesaboutwhichscenewasbeingpresented.Theaverage overallrecordingsisreportedinFig.1B.Wefoundtwo informa-tivebandsintheLFPspectrum:alowfrequencyrangebelow12Hz (correspondingtothedeltaandthetabands)andahighfrequency range(above50Hz)inthegammaandhigh-gammabands.While theinformationatlowfrequencieswashighonlyoveranarrow bandcenteredaround5Hz,theinformationpeakathigh frequen-cieswasbroad and long-tailedtoward frequencies higher than 100Hz.Intermediate frequencies[12–50Hz]didnotcarrymuch information.

Havingestablishedthatbothgamma,high-gammaandlowLFP frequenciesconveyinformation,thenextimportantquestionisto determinewhetherthedifferentinformativefrequenciesranges areredundantornot,i.e.whetherornottheycarrythesameor differentstimulusinformation.Wecomputedboththejoint infor-mationcarriedbythepowerofpairsofLFPfrequencies(Eq.(2))and theirredundancy(Eq.(3)).Theinformationobtainedbythe com-binedknowledgeofthepoweratlowfrequenciesandthepowerat

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Fig.1.TheinformationcarriedbyLFPpoweratsinglefrequencies.Resultsare aver-agedoverasetof57V1recordingsitesfromanaesthetizedmacaques.Information estimateswerecomputedusingtheGaussianMethodcorrectedwiththeGaussian biascorrectionprocedureandusingthecuberootoftheLFPpowerintensities.(A) AveragepowerspectrumoftheLFPsrecordedduringthepresentationofa naturalis-ticmovie(solidline)andduringspontaneousactivity(dashedline).(B)Information aboutanaturalisticmoviestimuluscarriedbyLFPpoweratsinglefrequencies. TheareaindicatestheSEM.(C)Left:informationaboutanaturalisticmovie car-riedjointlybytheLFPpowerattwofrequencies.Right:insetoftheinformation valuesintheregion50–150Hzindicatedbytherectangleintheleftpanel.(D)Left: theredundancybetweentheinformationcarriedbytheLFPpowerattwo frequen-cies.Right:insetoftheredundancyvaluesintheregion50–150Hzindicatedbythe rectangleintheleftpanel.

gammafrequencieswasnearlythesumoftheinformationcarried bythetwofrequenciesseparately(Fig.1C).Thismeansthatthe redundancybetweentheinformationcarriedbythepowerofhigh andlowfrequenciesisnearlyzero(Fig.1D).Thiszeroredundancy reﬂectedthefactthat,asshownin(Belitskietal.,2008;Magrietal., 2009),LFPfrequenciesbelow40Hzweretotallydecoupledfrom gammaandhigh-gammaoscillations(bothintermsofsimilarity ofstimulusselectivity(“signalcorrelations”),andoftrial-to-trial covariations(“noisecorrelations”).

Incontrast,frequenciesinthegammabandandhigh-gamma bandwerehighlyredundantbetweeneachother(Fig.1D).Thiswas because(Belitskietal.,2008;Magrietal.,2009)gamma frequen-cieshadasimilarresponseproﬁleacrossdifferentmoviescenes(in otherwords,theyhadhigh“signalcorrelation”).Theredundancy wasparticularly pronounced betweenfrequencies in the range 50–100Hz,suggestingthatallfrequenciesinthisrangereﬂecttoa largeextentthesamenetworkphenomenon.Wefound consider-ablysmallerredundancybetweenfrequenciesinthegammarange

[50–100Hz]andthoseinthehigh-gammarange,suggestingthat gammaandhigh-gammamayreﬂectpartlydifferentneural phe-nomena.

5.2. Validationofthemethodsusedtoestimateinformation carriedbythepowerintheLFPbands

AproblemlimitingtheuseoftheDirectMethodinourstudy is that it can only beappliedto computethe information car-riedjointlybyoneormaximumtwoLFPbands.Thesmallnumber (30–40)oftrialsavailabledoesnotallowanaccuratesamplingof theprobabilitydistributionsrequiredforcalculatinginformation withtheDirectMethodwhenseveralbandsareconsidered.Thisis because,asreviewedin(Panzerietal.,2007),thebiasofthedirect methodgrowsexponentiallywiththenumberofdimensionsof theresponsespace(whichinthiscasecorrespondstothe num-berLofbandsconsideredintheinformationcalculation).Thisbias problemisgreatlyalleviatedbytheGaussianMethod,becauseit requiresextractingonlyasmallnumberofparametersfromthe data.Theseparameterscanberobustlycomputedusingfewtrials evenwhenseveralbandsareconsideredatthesametime.Before applyingtheGaussianMethod,however,weneedtoensurethat methodisaccurateonthedatasetunderanalysis.Tothisaim,we comparedinformationvaluesobtainedwiththeGaussianandwith theDirectMethodwhenonlyoneortwoLFPbandswere consid-ered.Ifthetwomethodsreturncomparableinformationresultswe canconcludethattheGaussianMethodcanbeusedsafelyfor com-putingtheinformationcarriedbythepowerofseveralLFPbands. TheGaussianMethodwaspreviouslyvalidatedonsinglefrequency bins(Magrietal.,2009),butnotyetonpowerofextendedbands.

Webeganthis checkbycomputingthestimulus-information carriedbytheLFPpowerwithinafrequencybandB=[f–250Hz] (for61valuesoffuniformlydistributedbetween0and250Hz) usingboththeDirectandtheGaussianMethod.Forthe computa-tionwiththeDirectMethodwevariedM,thenumberofbinsused fordiscretizingtheneuralresponse,between3and32.Thelatter isanupperlimittothenumberofbinsforwhichwecouldstill computelargelyunbiasedinformationestimateswiththeDirect Methodgiventheavailablenumberoftrials,see(Panzerietal., 2007).Wefound(Fig.2A)thattherelationbetweentheinformation valuesobtainedwiththetwomethodswasapproximatelylinear. Whenfew(M=3,4,6)binswereusedwiththeDirect computa-tionthelinearregressorlaybelowthequadrantdiagonal,meaning thattheDirectMethodusedwithsmallnumberofbinsprovides consistentlylowerinformationvaluesthantheonesestimatedby theGaussianMethod.Thislikelyhappensbecausetheuseofcoarse responsediscretizationleadstoaninformationloss.However,the slope,b,ofthelinearregressorincreasedwithincreasingnumber ofbinsusedfortheDirectcomputation,suggestingthattheuseof ﬁnerbinsreducedtheinformationlossoftheDirectMethod.For M>13,theintersectaandtheslopeb(theparametersofthelinear regressor)reachedaplateauata=0andb=1(Fig.2B), demonstrat-ingthatforlargenumberofbinsMtheGaussianandtheDirect Methodsprovideessentiallyidenticalresults.

Fig.2BshowswhathappenswhentheGaussiananalysisis per-formedwithoutapplyingthecuberoottransformationpriortothe computationofinformation:independentlyofthenumberofbins usedfortheDirectMethod,theGaussianandtheDirectMethod failedtoconvergetoacommonestimationvalue.Thisprovesthat thecuberoottransformationisanecessarystepforreliably com-putingtheinformationconveyedbyLFPpowerusingtheGaussian Method.

Finally,wecomputedthevaluesof thestimulusinformation I(S;B1B2)carriedjointlybyapartitionoftheLFPresponseinto two-bandsB1=[0Hz–f]andB2=[f–250Hz]forallpossiblevalues of theboundary parameterfbetween 0and 250Hz. Again,we

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Fig.2.ValidationoftheGaussianMethodtoevaluateinformation.Dataweretaken fromarepresentativecase(channel7ofsessionD04nm1).Unlessotherwisestated, GaussianinformationestimateswerecomputedusingtheGaussianMethod cor-rectedwiththeGaussianbiascorrectionprocedureandusingthecuberootofthe powervalues(seetext);theDirectinformationvalueswerecomputedafter dis-cretizingthepowerresponseintoMequi-populatedlevels.(A)Comparisonofthe informationvaluesobtainedusingtheGaussianMethodandtheDirectMethodfor theinformationaboutanaturalisticmoviestimuluscarriedbyLFPpower.Thered scatterplotshowsGaussianandDirectestimatesofthestimulus-information car-riedbytheLFPpowerinafrequencybandofwidth[f–250Hz]fordifferentvaluesof fbetween0and250Hz.Theorangescatterplotillustratesthestimulus-information carriedjointlybytheLFPpowerintwofrequencybandsofwidth[0Hz–f]and [f–250Hz]forthesamevaluesoftheparameterfconsideredforthesingle-band case.Forbothscatterplots,M=3binswereusedfordiscretizingthepowerresponse fortheDirectcomputation.Thewhitesolidlineandthewhitedashedlineillustrate thelinearfittothepointsinthesingle-bandinformationandinthetwo-bands informationscatterplots,respectively.Thelightgrey,thedarkgreyandtheblack linesindicatethelinearfitwhenM=4,6and16levelsareusedforthe discretiza-tion,respectively(thecorrespondingscatterplotsarenotshownforthesecases). Thesameconventionisusedasforthewhiteline,withthesolidandthedashed linesindicatingthefitforthesingle-bandandforthetwo-bandsinformation val-ues,respectively.Notethat,duetothelimitednumberoftrialsavailable,forM=16 binsitwasnotpossibletocomputetheDirectinformationforthetwo-bandscase, thereforeonlythesolidlineisshown.Thegreendashedlineindicatesthequadrant diagonal.(B)Asymptoticbehavioroftheslopeparameter,b,andoftheintercept parameter,a,ofthelinearfitbetweeninformationvaluesobtainedwiththeGaussian andtheDirectMethodsasafunctionofthenumberofbins,M,usedfordiscretizing thepowerresponsefortheDirectcomputation(blueline).Mwasvariedbetween 3and32.Thecyanlinesindicatethevaluesofaandbwhennocuberoot trans-formationisappliedtotheLFPpowerforthecomputationofinformationwiththe GaussianMethod.

found(Fig.2A)alinearrelationshipbetweentheinformation val-uesobtainedusingtheDirectandtheGaussianMethod,theslopes ofthelinearﬁttingcurvesbeingnearlyidenticaltothoseobtained whenonlyonebandwasconsidered.Note,however,thatinthis casewecouldnotcomputeinformationforM>6usingtheDirect Methodbecausethenumberoftrialsavailabledoesnotallowto properlysampletheresponsespaceforlargevaluesofM.

Taken together theseresults demonstrate that the accuracy oftheGaussianapproximationtocomputeinformationfromLFP powers.Whencomputingtheinformationaboutamovie stimu-luscarried bythepowerinoneor moreLFPbands,differences betweenGaussianandDirectinformationvaluesarisemainlyfrom thediscretizationusedfortheDirectprocedureandthatthetwo methodsprovidecomparableresultswhenenoughbinsareused forthediscretization.

Inwhatfollows,unlessotherwisestated,wewillalwaysusethe GaussianMethodwiththeGaussianbiascorrection(seeSection2) forcomputinginformationvalues.

5.3. PartitioningofLFPresponsesintotwobands

Webegan byinvestigatingwhatisthebestwaytopartition theLFPspectrumintotwofrequencybands.AlthoughtheLFPis consistentlydeﬁnedintheliteratureasthelowfrequencyrange ofthespectrumoftheextracellularlyrecordedneuralsignal,there

Fig.3. PartitioningtheLFPintotwobands.Resultsare averagedoverasetof 57V1recordingsitesfromanaesthetizedmacaques.Informationestimateswere computedusingtheGaussianMethodcorrectedwiththeGaussianbiascorrection procedureandusingthecuberootofthepowervalues.(A)Theinformationabout anaturalisticmoviestimuluscarriedjointlybythepowerinthetwoLFPbands B1=[0Hz–f]andB2=[f–250Hz]asafunctionoftheboundaryparameterfvaried between0and250Hz.ThegreyareaindicatestheSEM.Thedashedanddottedlines illustratetheinformationobtainedwhenvaryingtheupperedgeofB2from250to 200andto300Hz,respectively.(B)Thepercentageredundancybetweenthe infor-mationcarriedbythepowerinthetwoLFPbandsB1andB2deﬁnedasinpanelA. ThegreyareaindicatestheSEM.Thedashedanddottedlinesillustratethe redun-dancyobtainedwhenvaryingtheupperedgeofB2from250to200andto300Hz, respectively.(C)Theinformationaboutanaturalisticmoviestimuluscarried indi-viduallybyLFPpowerinthebandB1=[0–60Hz]andbytheLFPpowerintheband B2=[60–250Hz].TheerrorbarsindicatetheSEM.Thetrianglemarkerindicatesthe informationconveyedbytheunpartitioned[0–250Hz]LFPrange.

isnoagreementabouttheprecisefrequencyrangethatshouldbe consideredasLFPactivity.Someauthorsincludefrequenciesonly upto100Hz,whileothersincludeuptoseveralhundredHz.Here westartedbyassumingtheLFPspectrumtorangefrom0to250Hz. Welatercheckedfortheeffectofconsideringotherdeﬁnitionsof theLFPfrequencyrange.

Wecomputedthestimulus-information,I(S;B1B2),carriedby theLFPpowerwhenpartitionedintothetwobandsB1=[0Hz–f] and B2=[f–250Hz], as a function of the boundary parameterf (with0<f<250Hz).Wefound(Fig.3A)thattheamountof infor-mationcarriedjointlybythetwobandswas,onaverageacross thepopulation,maximalwhensplittingthebandsatf=60Hz.The information,washoweververyhighforabroadfrequencyregion, extendingapproximatelybetween50and70Hz.Thedistribution acrossrecordingsites ofthevalue ofthefrequency boundaryf at which information was maximal in each recording site had a median of62Hz and aninterquartile range of 53–79Hz. The informationaboutthemoviestimulusobtainedaftersplittingthe [0–250Hz]LFPrangeintotwobandsatthef=60Hzoptimal bound-arieswas1.03±0.05bits(hereandhereafter,andunlessotherwise stated,informationvaluesarereportedasmean±SEMover pop-ulation). In comparison, the information in the un-partitioned [0–250Hz] LFP power was 0.30±0.02bits. Thus, splitting the LFP response at the optimal boundary f=60Hz allowed the

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extractionofmore thanthree timestheinformation carriedby theun-partitionedLFPpower.Takentogether,theseresultsclearly indicatethatthemostinformativewaytopartitiontheLFPresponse intotwobandsistoseparatelowandhighLFPfrequencieswitha boundaryplacedatapproximately55–65Hz.

Wenotedthattheinformation(Fig.3A)didnotfall monoton-icallyafterthemaximumat60Hz,butratherasecondlocalpeak waspresentatapproximately105Hz. Similarly,theredundancy (Fig.3B)reachedasecondlocalminimuminthesamefrequency region.

Tounderstandbetterthereasonswhyitwasoptimalto par-tition theLFP range by splittingit at f=60Hz, we investigated howtheredundancybetweentheinformationcarriedbythebands dependedupontheboundaryvaluef.Wefound(Fig.3B)thatthe redundancybetweenthetwobandswasingeneralsmallandwas minimalintherange50–60Hz.Theminimumwasreachedat53Hz andwas−0.05±0.004bits.ThissuggeststhatsplittingtheLFPat theoptimalboundaryrange(50–70Hz)partitionstheLFPintotwo regionswithlargelyindependent(evenweaklysynergistic) infor-mationcontentandstimulustuningproperties.Theseobservations aboutLFPbandsareconsistentwiththeresultsobtainedregarding theinformationcontentofthepowerofsingleLFPfrequencies,see Fig.1Cand(Belitskietal.,2008).

Fig. 3C reports that the information carried by the high-frequency[60–250Hz]bandwas0.69±0.04bits.Incontrast,the low frequency [0–60Hz] band carried much less information (0.29±0.02bits).Thisresult,togetherwiththeaboveinformation redundancyﬁndings,indicatethatthelargeamountofinformation gainedbypartitioning theLFPrangeintothehighand low fre-quencycomponentsarisesmostlybecausethispartitioningallows accessingtheinformationinthelow-powerbuthighlyinformative high-frequencyrange.Intheunpartitionedsignal,instead,the rel-ativelylowpowerofthehigh-frequencyrangeismaskedbythe highpowerofthelessinformativebutverylow-frequencyrange.

Itisinterestingtocomparetheamountofinformationcarried bythepowerofanoptimallydeﬁnedLFPbandwiththeamount of information that can beextracted by the power at a single frequency.Weﬁrstcomparedtheinformationintheentirehigh frequencybandwiththeinformation intheindividual frequen-ciesinthehigh-frequencyrange.Consistentwithpreviousresults (SiegelandKönig,2003),weobservedthatintegratingLFPpower over the range [60–250Hz] greatly increased thetotal amount of information (0.69±0.04bits)compared tothehighest infor-mation carried by a single LFP frequency in the gamma range (0.31±0.02bits).Thiscanbeunderstoodbythefactthat frequen-ciesabove60Hzcarrylargelyredundantinformationwithlargely similarstimulusselectivity,high“signalcorrelation”,andalmost independenttrial-to-trialvariability,i.e.relativelylow“noise cor-relation”(Belitskietal.,2008;Magrietal.,2009).Integratingneural responses with similar stimulus selectivity and weakly corre-latednoisehasbeendemonstratedtogreatlyincreaseinformation (AbbottandDayan,1999;Orametal.,1998;Zoharyetal.,1994).We thencomparedtheinformationintheentirelowfrequencyband [0–60Hz]withtheinformationintheindividualfrequenciesinthe low-frequencyrange.Wefound(Fig.3C)thatintegratingtheLFP poweroverthelow[0–60Hz]frequencyrangeprovided informa-tion(0.292±0.015bits)comparabletothemaximuminformation obtainedforsingleLFPfrequenciesinthisrange(0.289±0.017bits; Fig.1B).Thefactthatessentiallynoinformationisgainedby inte-gratingthepowerinthelowfrequenciesislikelyduetothefact thatlowfrequencieshaveasmallredundancy(Fig.1C)duetoa verysmallsimilarityintheirtuningtothemovie(Belitskietal., 2008;Magrietal.,2009)andsummingdifferentlytunedresponses doesnotleadtoinformationgain.

Aswementionedearlier,theupperfrequencycutoffdeﬁning theLFPrangeisanarbitraryparameternotconsistentlyusedin

theliterature.Weinvestigatedwhethertheoptimalbandsplitting dependedontheupperfrequencycutoffdeﬁningtheLFPbyvarying thisparameterintherange200–300Hz.Wefound(Fig.3)thatboth thevalueoftheoptimalfrequencycutoffandtheamountof infor-mationobtainedfromtheoptimallysplitLFPwerehighlystable whenvaryingthisparameter.

5.4. PartitioningofLFPresponsesintothreebands

Next,weconsideredtheproblemofﬁndingboundariesf1,f2 that optimally partition the LFPs into three bands, B1=[0–f1], B2=[f1–f2] and B3=[f2–250Hz]. Fig. 4A reports the informa-tioncarriedjointly bythethreefrequencyranges,computedas a function of the two boundaries parameters f1 and f2 (with 0Hz<f1<f2<250Hz)andaveragedoverthepopulation.The opti-malpartitionwasfoundatf1=54Hzandf2=97Hz.Asforthecase oftwobands,thepositionoftheinformationpeakwasrobusttothe choiceoftheupperboundaryusedfordeﬁningtheLFPrange(not shown).Again,wefoundthattheredundancy(Fig.4B)was negli-gibleforf1andf2intheoptimallyinformativeregion.Acrossthe populationthedistributionofthepositionoftheinformationpeaks wascentered around 52Hz (median value; interquartile range: 43–57Hz)for f2and around98Hz(medianvalue; interquartile range:92–106Hz).

Thefirstoptimalboundary(atapproximately55Hz)tosplitthe LFPrangeintothreebandsisessentiallycoincidentwiththeoptimal boundaryforsplittingtheLFPintotwobands.Thesecond opti-malboundary(at97Hz)forthethreebandpartitioncorresponds to thelocal peak (Fig.3B) observed when partitioning the LFP responseintotwobands.Thissecondoptimalboundaryseparates thehigh-frequencyrange[54–250Hz]intotworanges[54–97Hz] and[97–250Hz].Thefirstrangecorrespondstothefrequencyrange layinginsidewhatistraditionallycalledthegammarange,andis consistentwiththeoptimalgammabanddefinition[64–103Hz] obtainedforthecatvisualcortexusinganalternativeprocedure forfunctionallydefiningthisband(SiegelandKönig,2003).This alsosuggests,consistentwith(SiegelandKönig,2003),thatwidely useddefinitionofthegammabandastheregionaround40Hz(the frequencyatwhichLFPpowerpeaksinthisrange,andneural syn-chronizationisusuallymaximal)maynottobetheoptimalchoice whenconsideringstimuluscoding.Thesecondrange[97–250Hz] correspondstotherangewhichhasbeentermedthe“high-gamma” band,andwhichwasfound(RayandMaunsell,2011)tocorrelate withspikingactivity.

Thefactthatthepositionofthefirstoptimalboundaryis essen-tiallyunchanged whenpartitioning theLFPresponse intothree bandsindicatesthatthisboundaryisstableandthatitdefinesbands whicharelikelytoreflecttruedistinctneuralphenomena.Italso indicatesthatthemostimportantstep,whenpartitioningtheLFP responseaccordingtoourprocedure,istoseparatelowandhigh LFPfrequencies.Thesignificanceofthisoptimalboundaryisalso indicatedbytheextensionofthepeakoftheinformation curve averagedoverthepopulation(Fig.4A),whichisnarroweralong thef1directioncomparedtof2,indicatingasharpertuningofthe informationcurvetothefirstboundary.Additionally,thebroadness ofthef2peakisconsistentwiththeobservationthattheboundary foundfordelimitingthehighgammarangeshouldnotbethoughtof asastrictboundaryandthatthelowerandupperfrequencylimits ofthehigh-gammabandappeartobesomewhatvariable depend-ingonthetask,corticalsite,andsubject(Croneetal.,2006;Rayand Maunsell,2010).

Weobservedthattheﬁrstoptimalboundarywasstableinthat itappearedbothwhenpartitioningtheLFPrangeintotwoorinto threebands.Totestforthestabilityofthesecondoptimalboundary wepartitionedtherange[54–250Hz]intotwobands.The ratio-naleforconsideringthisrangeseparatelyfromthelowfrequencies

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Fig.4.PartitioningofLFPresponseintothreebands.Resultsareaveragedoverasetof57V1recordingsitesfromanaesthetizedmacaques.Informationestimateswere computedusingtheGaussianMethodcorrectedwiththeGaussianbiascorrectionprocedureandusingthecuberootofthepowervalues.(A)Theinformationabouta naturalisticmoviestimuluscarriedjointlybythepowerinthethreeLFPbandsB1=[0Hz–f1],B2=[f1–f2]andB3=[f2–250Hz]asafunctionoftheboundaryparametersf1 andf2variedbetween0and250Hzwithf1<f2.Thedashedrectangleindicatestheinterquartilerangeforthepositionoftheinformationpeaksacrossthepopulation.(B)The percentageredundancybetweentheLFPpowerinthethreeLFPrangesB1,B2andB3.ThedashedrectangleindicatethesameregionboxedinpanelA.(C)Theinformation aboutanaturalisticmoviestimuluscarriedindividuallybytheLFPpowerinthebandB1=[0–54Hz],inthebandB2=[54–97Hz]andinthebandB3=[97–250Hz].Theerror barsindicatetheSEM.(D)Top:theinformationaboutanaturalisticmoviestimuluscarriedjointlybythepowerinthetwoLFPbandsB1=[54Hz–f]andB2=[f–250Hz]asa functionoftheparameterfvariedbetween54and250Hz.Bottom:theredundancybetweentheinformationcarriedbytheLFPpowerinB1andB2.Thegreyareaindicates theSEM.

isthatwehaveobserved(Figs.1Dand3B)thatLFPpowerbelow 54Hzconveysinformationwhichisindependentfromthatofthe higherfrequencies.Wefound(Fig.4D)thattheinformationwas maximalforfagainintherange[90–110Hz].Wealsofoundthat theredundancybetweentheinformationconveyedbythegamma andthehigh-gammabandswaspositivebutverysmall(reaching aminimumof3%±2%at99Hz)consistentwiththelowvalues oftheredundancy(Fig.1C)observedbetweensinglegammaand high-gammafrequencies.

Amongthebandsintheoptimalthree-bandpartition,the opti-malgammaband[54–100Hz]wastheone carryingthelargest single-band stimulusinformation (0.66±0.03bits; Fig.4C). The secondmostinformativerangewas[97–250Hz](0.61_±0.04bits) followedbythelow(<54Hz)LFPfrequencies(0.29±0.02bits). 5.5. Determiningadditionaloptimalboundaries

Anissuewiththemethodproposedfordeterminingthe opti-malboundariesisthattheparameterspaceinwhichtosearchfor theoptimalboundariesincreasesexponentiallywiththenumber ofboundariesconsidered.Forexample,aspectralwindowoflength T=960mshas241Fouriercoefﬁcientsbetween0and250Hzwith oursamplingrate.IdentifyingtheoptimalpartitionoftheLFPinto

threebandsthusrequirescomputinginformationforseveral hun-dredthousandpossiblecombinationsoftheboundaryparameters foreachrecordedchannel.Thisisdifﬁculttoperformsystematically withcurrentcomputationalmeans.Downsamplingthefrequency spaceisalsonotasolutiontothisproblem,asahighfrequency res-olutionbecomesmoreandmorecrucialwhenconsideringmany bands,sincetheoptimalboundariesbecomecloserandthusharder todiscriminate.

InordertoassesstheutilityofafurtherpartitionoftheLFP spec-trumintofourbands,wefollowedaslightlydifferentapproach: wefixedthetwopreviouslyobtainedprincipalbandboundaries and wecomputed theoptimalwaytofurthersplit each ofthe threeprincipalbands,ratherthanperformingabrute-forcesearch acrosstheentirefour-dimensionalfrequencyspace.Thischoice wasjustifiedbyourfindingsthat theoptimalboundaries at54 and97Hz(whichwecalltheprincipalboundaries)arestablewith respectbothtochangesintheparametersoftheanalysisandto arefinementofthepartition,andthatthethreeprincipalbands ([0–54Hz],[54–97Hz]and[97–250Hz])conveylargely indepen-dentinformationand,consequently,theirfurtherrefiningcanbe studiedindependentlyoftheothers.In whatfollows,the infor-mationgainedbyfurthersplittingabandwasquantifiedasthe differencebetweentheinformationcarriedbythepartitionofthe

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Fig.5. PartitioningtheLFPintofourbands.Resultsareaveragedoverasetof 57V1recordingsitesfromanaesthetizedmacaques.Informationestimateswere computedusingtheGaussianMethodcorrectedwiththeGaussianbiascorrection procedureandusingthecuberootofthepowervalues.Thesolidlinesindicatethe averageoverallavailablerecordingsandthegreyareasindicatestheSEM.(A)The informationgainaboutanaturalisticmoviestimulusobtainedwhenpartitioning theLFPrange[0–54Hz]intotwobandsB1=[0Hz–f]andB2=[f–54Hz]asafunction oftheboundaryparameterf.Therighty-axisexpressesthegainasthepercentage oftheinformationconveyedbytheun-partitionedband.(B)Thepercentage redun-dancybetweentheinformationcarriedbythepowerintheLFPbandsB1=[0Hz–f] andB2=[f–54Hz]asafunctionoftheboundaryparameterf.(C)Sameplotasin panelAbutforapartitionoftheband[54–97Hz].(D)SameplotasinpanelBbut forapartitionoftheband[54–97Hz].(E)SameplotasinpanelAbutforapartition oftheband[97–250Hz].(F)SameplotasinpanelBbutforapartitionoftheband [97–250Hz].

selectedbandandtheinformationcarriedbytheun-partitioned band.Thisinformationgainwasexpressedeitherasanabsolute difference(inbits)betweenthesetwoterms,orwasnormalized aspercentagegainwithrespecttotheinformationcarriedbythe un-partitionedband.

Weﬁrstconsideredhowtosplitthe[0–54Hz]low-frequency range. We found(Fig.5A) thatthe information gain wassmall whentheboundarybetweenthetwo sub-bandswassetbelow 15Hz but that itincreased rapidlyreachinga globalmaximum of0.32±0.03bitsat27Hz,correspondingtoaninformationgain of111%±11%withrespecttotheinformationcarriedbythe un-partitionedband.Thisoptimalsplitat27Hzcorrespondsroughly topartitioningthelowfrequencyrangeintoastimulus-informative [0–27Hz]and tothestimulus-unrelatedfrequencies [27–50Hz] band(see thesinglefrequency informationanalysisreportedin Fig.1Bandin(Belitskietal.,2008;Magrietal.,2009)).Theoverall informationgainofthefurtherpartitionofthe[0–54Hz]bandcan beconceptualizedashelpingisolatingthe“noisy”frequenciesfrom thestimulus-evokedones.

We then considered how best to sub-partition the gamma [54–97Hz] band into two sub-bands. Fig. 5C shows that the optimalsplitwasobtainedusingaboundaryat73Hz.The infor-mation gain of this optimal split over the information in the wholebandwas0.35_±0.06bits(correspondingto53%_±8%more information).Thus,afurthersplitofthegammarangewas worth-whilebecauseitincreasedtheinformation,althoughwithaless

importantproportionalgainwithrespecttotheoneobtainedby furtherpartitioningthelowfrequencyrange.Thefactthatthe par-titionofthegammarangegavealowerproportionalincreasecanbe understoodbyobservingthattheoptimallysplitsub-rangeswithin thegammafrequency domainwere stillalmost 18%redundant (Fig.5D), suggesting that the division at 73Hz separates simi-larfrequencyregionswithaslightlydifferenttuningratherthan identifyingregionswhichqualitativelydifferentstimuluscoding properties.

The smallestinformation gain (bothin absoluteand in pro-portionalterms)wasobtainedwhenpartitioningthehigh-gamma [97–250Hz]range,Fig.5E.Theoptimalsplitofthehigh-gamma rangewasachievedwithaboundaryat111Hz,butitledtoonly asmall(0.16±0.05bits,correspondingto26%±8%)information gain, indicating that littleadditionalinformation can begained byadditionallypartitioningthehighfrequencies.Thisﬁndingis furthersupportedbytheveryhighvaluesofredundancy(Fig.5F) obtainedforanychoiceoftheboundary.

Theseresultssuggestthatthemostconvenientwayofadding athirdoptimalboundaryistopartitionthelowfrequencyrange intoa [0–27Hz]bandand a[27–54Hzrange]since considering thesetwobandsseparatelyprovidesaproportionallylargegainin informationanditseparatesregionswithaqualitativelydifferent relationtothevisualstimulus.

5.6. Distributionofoptimalbandboundariesacrosssitesand animals

IntheabovesectionswepartitionedtheLFPsintooptimalbands usinginformationvaluesaveragedoverallavailablesitesand mon-keys.PreviousworkhasshownthattheLFPspectrumcanchange dependingonfactorssuchasthedetailsofthestimulusused(Ray andMaunsell,2010),thetask(Friesetal.,2001)orrecordingdepth (Maieretal.,2011).Inparticular,itwasshownthatgammaactivity canvarysigniﬁcantlyinpowerandpeakfrequencyacrosssubjects andrecordingsite(Limaetal.,2010;Muthukumaraswamyetal., 2010).Hereweinvestigatetheresultsobtainedbyapplyingour partitioningtechniqueindependentlyforeachrecordingchannel andforeachmonkey.

Results of the single-site two-bands optimal partition are reportedinFig.6B.Wefoundthatthemethodwasabletofind optimalboundariesalsoatthesinglechannellevel.Thepositionof theoptimalboundaryforthetwo-bandspartitionhadamedianof 62Hz(range45–100Hz)forthefirstmonkeyandof45Hz(range 33–69Hz)forthesecondmonkey.Thedistributionofthe bound-ariesinthefirstmonkeywassignificantlyhigherthanthatofthe secondmonkey(p<0.05;Wilcoxonranksumtest).Althoughthe limitednumber ofsubjectsusedinthestudydoesnotallowto perform anexhaustive analysisof thedependency of theband boundariesonthespectralpropertiesofeachanimal,itis tempt-ingtosuggestthatthedifferenceinthedistributionofthefirst optimalboundarybetweenthetwomonkeysmaybeattributedto themuchlargergammapowerobservedinspectrumofthefirst monkey(Fig.6A).

Furthermore,whenpartitioningtheLFPrangeintothreebands independently for each siteof thefirst monkey,we foundthat thefirstoptimalboundarywasdistributedbetween32and63Hz withamedianof54Hzandthatthesecondoptimalboundarywas between74and118Hzwitha medianof98Hz.For thesecond monkeythefirstboundarywasbetween32and49Hz(medianof 38Hz),comparabletotherangeobtainedwhenpartitioningthe LFPsrecordedfromthismonkeyintotwobands(Fig.6B),andthe secondboundarywasbetween79and147Hz(medianof127Hz). Itisimportanttonotethatthepartitioningalgorithmdetermined thatthefirstboundarywaspreservedwhenrefiningthepartition fromtwotothreebandsevenatthesinglerecordingsitelevel,and

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Fig.6.Distributionoftheoptimalboundariesacrosssitesandanimals.Resultsareshownforeachofthetwomonkeysusedinthisstudy.Quantitiesplottedingreyreferto monkey1,quantitiesplottedinblacktomonkey2.(A)AveragepowerspectrumoftheLFPsrecordedduringthepresentationofanaturalisticmovie.(B)Distributionofthe optimalboundariesobtainedforeachrecordingsitewhenpartitioningtheLFPsintotwobands.(C)Distributionoftheﬁrstandsecondoptimalboundariesobtainedforeach recordingsitewhenpartitioningtheLFPsintothreebands.

notonlyatthepopulationlevel.Thissuggeststhatouralgorithmis robustenoughtoanalyzeandpartitionsinglecases,andnotonly grandpopulationaverages.

6. Discussion

WedevelopedanewprocedureforpartitioningtheLFPresponse intoaspecifiednumberofbands,whichopensthewayforamore objective,semi-automaticanalysisoftheroleofcortical oscilla-toryactivity in sensory function. The methodselects theband boundariesthatmaximizetheinformationaboutasetof exter-nalcorrelatescarriedjointlybyallbandsinthepartition.Weused thisproceduretodetermineamaximallyinformativepartitioning ofprimaryvisualcorticalLFPsintotwotofourbands,and iden-tifiedbandboundarieswhicharecompatiblewithourknowledge andintuitionaboutthefunctionalLFPspectruminvisualcortex.It isimportanttonotethatourmethod,thoughexemplifiedforLFPs, canbereadilyappliedtoa varietyofbroad-bandneuralsignals, fromintracellularrecordingsofmembranepotentialfluctuationsto populationspikingactivityortonon-invasiveneuroimagingsignals suchasEEG,magnetoencephalography(MEG)andfunctional mag-neticresonanceimaging(fMRI).Ourmethodcanthereforebecome ausefultoolfortheendeavorofextractinginformationfromthese signals(Alendaetal.,2010;FuhrmannAlpertetal.,2007;Ostwald etal.,2010;Panzerietal.,2008;QuirogaandPanzeri,2009;Schyns etal.,2011).

Theimplicationsofboththemethodologicaladvancesandof theneurophysiologicalresultsarediscussedinthefollowing.

6.1. Potentialapplicationsofmethodsforpartitioningneural responsesintomaximallyinformativebands

Asillustrated by our application to visualcortical LFPs, our methodispotentiallyrelevantforsensoryphysiology.The spec-trumofLFPoscillationsandfluctuationsinresponsetoastimulus isoftenbroadandlacksaclearseparationbetweenspectralregions tunedtodifferentstimulusfeatures.TherelationshipbetweenLFPs andstimuliisparticularlydifficulttountanglewhenconsidering complexstimulimadeofmanyfeaturesvaryingatdifferenttime scales,suchasforexamplenaturalisticmoviesornaturalsounds (KayserandKonig,2004;Kayseretal.,2009).Understandinghow topartitiontheLFPintoaspecificnumberofbandsthatare col-lectivelymaximallyinformativeaboutthestimulusshedslighton differencesinstimulustuningofdistinctLFPfrequencyregions,and helpsindividuatingfrequencyregionsthatmustbeconsidered sep-aratelybecausetheycarrycomplementaryinformationaboutthe externalcorrelates.Beingabletoindividuateobjectivelyandalmost automaticallytheoptimaldefinitionsofsuchbandswillalso facil-itatehomogeneityofanalysisacrossfuturestudiesandwillmake itiseasierandmoremeaningfultocomparedifferentstudiesor preparations.

Althoughherewefocusedonmaximizingtheinformationabout anexternalcorrelate,ourmethodcouldbereadilyusedto deter-minetheoptimalpartitionofbroadbandneuralsignalsintobands whicharemaximallyinformativeaboutinternal(ratherthan exter-nal) correlates. For example, our procedure could be used to determineasetofLFPregressorsprovidinganoptimalprediction oftheBOLDfMRI.

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Finally, determiningthe partitions of theLFPspectra which aremaximallyinformativeaboutsensorystimuliiscrucialforthe developmentofbrainmachineinterfaces(Andersenetal.,2004; Donoghueetal.,1998).TheLFPisaparticularlysuitedneural sig-nalforBMIsbecausetheyhavethepotentialtoprolongthelifetime ofelectrodeimplantsandtoimprovedecodingperformancewhen combinedwithspikes(Andersenetal.,2004;Belitskietal.,2008).It iswidelyrecognizedthatthemainbottlenecktotheperformanceof BMIsistherelativelysmallamountofinformationthatweare cur-rentlyabletoextractfromneuralactivity(Donoghueetal.,1998). Ourmethodprovidesatoolformaximizingtheinformationabout externalcorrelates(suchas,forexample,movementdirection)thus potentiallyenlargingbottlenecksinperformance.

6.2. Signiﬁcanceofthemethodologicaladvances

Onemainadvantageofusinginformationasameasureofthe relationshipbetweenthepowerofthebandpartitionandthe exter-nalcorrelateisthatinformationcapturesalltypesofrelationships (linearandnon-linearandofanyorder).Additionally,by consider-ingtheinformationconveyedsimultaneouslybythepowerinall bands,ourmethoddoesnotonlytakeintoaccountthe relation-shipbetweeneachindividualbandandtheexternalcorrelate,but alsotherelationshipbetweenthedifferentbandsinthepartition. Thesefactorscannotbetakenintoaccountbysingle-frequencyor single-bandanalyses.

Wealsoconsideredthepracticalproblemsinvolvedinthe appli-cation of our methodto empirical neurophysiological datasets. To alleviate the bias problem arising from the limited size of thedatasets thatitisnormallypossibletocollect, weproposed a GaussianMethodand Gaussian biascorrectionforcomputing information.WevalidatedthismethodonvisualcorticalLFPsand demonstratedthat,when oneortwobandsareconsidered, dif-ferencesbetweentheinformation estimatesobtainedusingthe GaussianMethodandthoseobtainedwiththeDirectMethodstem fromthediscretizationusedfortheDirectMethod.Thisresult indi-catesthattheGaussianMethod-basedprocedurethatwepropose canbesuccessfullyappliedtocomputetheinformationconveyed bythepowerinthepartitionoftheLFPresponse.Asecond prac-tical problemis that of implementing thebias corrections and estimationprocedurerequiredfortheproperestimationof infor-mationfromthedata.Wedevelopedafasttoolbox(Magrietal., 2009),whichincludedbothDirectMethodandGaussian compu-tation and canthus be usedto implement and tovalidate our procedure,andmadeitpubliclyavailable(www.ibtb.org).Athird practicalproblemis thatof computationaltime.Besides imple-mentingafasttoolbox,weaddressedthisproblembyproposinga procedureforincrementallyrefiningthepartitionbasedonfurther splittingoptimalbands.Thiscansaveagreatdealofcomputing time whenneedingtopartition theLFPintoa largenumber of bands.Thisapproachisjustifiedbytwokeyfindings:the princi-palbandboundariesarestabletorefinementofthepartition,and optimalpartitionsinmostcasescorrespondtosplitsintolargely independentbands.

Thepartitioningmethodproposedinthisworkalsoprovidesa straightforwardwaytoquantitativelyevaluatetherelativegainof addingextrabandstothepartition.Thiscanbedonebycomparing theinformationobtainedforagivennumberofbandswiththat gainedbyintroducingoneextraband.Byevaluatingthestatistical signiﬁcanceoftheinformationincreasewhenaddingextrabands, thismethodcouldbeextendedtodeterminethesmallestnumber ofbandsthatneedtobeusedtodescribeallstimulusinformation availableintheLFP. However,giventherichnessand complex-ityofLFPsrecordedinresponsetocomplexnaturalisticstimuli, wefeelthatitmaynotbepossibletodescribealltheinformation

contentofLFPsusingonlyaverysmallnumberofbandssuchas thatconsideredintheaboveanalysis.

Theaboveconsiderations,aswellasthelargenumberof differ-entbandsandpartitionsproposedintheneurophysiologicalandin theneuroimagingliterature,raisethequestionofhowtoevaluate theinformationcontentofanumberofbandsmuchlargerthanthe handfulconsideredhere.Boththebiasoftheinformation calcu-lationandthedifficultyoftheevaluationofitssignificancegrow rapidlywiththecardinality ofthespaceinwhichthe stimulus-responseprobabilitiesaredefined,inthisstudywiththenumber ofbandsincludedintheanalysis(Inceetal.,inpress).Evaluation oftheexactinformationcontentofalargenumberofbands(ofthe orderoftenormore)requireseithercollectinglargerdatasetsor developingaccuratestimulusresponsemodelsdescribedbyeven less parameters thantheones consideredhere.We note, how-ever,thattheinformationbiasofthemethodconsideredhere(Eqs. (6) and (7))depends upon thenumber of bands but is largely independentoftheexactbandboundaries.Thismeansthat the partitioningmethodconsideredheremaybestillusefulto deter-minetheboundariesforoptimalpartitionintolargernumberof bands,althoughinsuchcaseitmaynotallowtoevaluatetheprecise informationcontentoftheoptimalpartition.

6.3. Theoptimalbandpartitionsdescribingvisualprimary corticalencodingofnaturalisticinformation

Weusedourmethodtocomputetheoptimallyinformativeband partitionsofLFPsrecordedinV1ofanesthetizedduring presenta-tionofnaturalisticmovies.

Thefirst,andmostimportant,optimalboundarythatappeared inourpartitionwasatapproximately60Hz, andseparatedlow [0–60Hz]andahigh[60–250Hz]LFPfrequencies.Thefindingthat this is the chief boundaryis consistentwithprevious analyses (Belitskiet al.,2008; Magrietal., 2009)reportingthatthe fre-quenciesabove50–60Hzarealmostcompletelyuncorrelatedin bothsignalandnoisetofrequenciesbelowthiscutoff.Thelarge amountofinformationgainedover theunpartitioned LFPwhen splittingtheLFPsatthisfrequencylendssupporttothehypothesis thatthelow-andthehigh-frequencyrangesconveyindependent informationaboutthestimulusbecausetheyoriginatefrom dif-ferentneuralpathways.Inparticular,theoreticalandexperimental studiessuggestedthatlowLFPfrequenciesreflectentrainmentof localneuralactivitytothelowfrequencycomponentsintheinput (Chandrasekaranetal.,2010;Lakatosetal.,2008;Mazzonietal., 2008)whereashighLFPfrequenciesinthegammarangeandabove conveyinformationaboutthestrengthoftheinputbecausethe lat-termodulatesthelocallygeneratedrhythms(BrunelandWang, 2003;Henrieand Shapley,2005;Mazzonietal., 2008;Rayand Maunsell,2010).

Wefoundasecondoptimalboundaryatapproximately100Hz separating the high-frequency LFP band into two ranges cor-respondingto thegamma[60–100Hz]and to thehigh-gamma [100–250Hz]band, leadingtoa largeincrease (approx.30%)in theamountofinformationthatwecanextractaboutthemovie stimuluscomparedtothecaseinwhichonlytwobandsare con-sidered. Gammaoscillationsare thoughttooriginatetoa large extentfromtherecurrentinteractionsamonginhibitoryneurons andamongexcitatoryandinhibitoryneurons(Bartosetal.,2007; BrunelandWang,2003).High-gammapowerhasbeenproposed tooriginatepartlyfrombroadbandLFPactivityandpartly from briefburstsofpowerassociatedwithspikesgeneratednearthe electrode (the socalled“spike bleed-through”).In some condi-tions,forexamplewhenvaryingcontrast,gamma,high-gammaand spikingactivityarelargelycorrelatedandcarrysimilar informa-tionaboutthestimulusparameters.However,inotherconditions, for examplewhen varyingthe stimulussize,gammaactivityis

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partiallydecoupledandhasadifferentstimulusselectivitythan thatofhighgammaandspikingactivity,whichremainhighly cor-relatedinmostconditions(GieselmannandThiele,2008;Rayand Maunsell,2011).Ourﬁndingthatthesplitbetweengammaand highgammafrequencies atapproximately100Hz isthesecond mostimportantsplittomaximizeinformationextractionfromthe data,supportsthehypothesisthatgammarhythmandhigh-gamma activityreﬂectpartlydifferentneuralphenomena,andhighlights theimportanceofdistinguishingandstudyingseparatelythesetwo frequencyrangeswheninvestigatingstimulusselectivityofcortical activity.

Wealsofoundthataddingathirdboundaryatapproximately 25Hzledtoaconsiderablegaininthestimulus-informationthat couldbeextractedfromthelowLFPfrequencies.Thissplitdivides thelowfrequencyLFPrangeintotwoqualitativelydifferentparts: a stimulus-modulatedpart and a stimulusindependent part.A highinformation gain wasobtainedalsowhenfurthersplitting thegammarange[50–100Hz]atapproximately70Hz.Inthiscase, however,theredundancy betweenthetwo sub-bands wasstill considerable.Thissuggeststhattheinformation gainedbyﬁner partitionof thegammarangeresultsfromisolating partofthe spectrumoriginatingfrom thesameneuralphenomenon (most likely recurrent inhibitory–inhibitory and excitatory–inhibitory interactions)withslightlydifferenttuningtothestimulus.Thisis compatiblewiththeoreticalmodelsthatpredictthatthepeak fre-quency,thewidthandthestimulusmodulationoftheoscillation spectrumallchangewiththestrengthoftheinputtothenetwork (BrunelandWang,2003;Mazzonietal.,2008).Accordingtothis model,furtherpartitioningthegammarangewouldallow discrim-inatingtheseﬁnerchangesinLFPsinresponsetothestimuli,hence theinformationincreasedespitethepresenceofredundancy. Acknowledgements

ThisworkwassupportedbytheBMIprojectofIIT,bytheMax PlanckSociety,bytheCompagniadiSanPaolo,and waspartof theresearchprogramoftheBernsteinCenterforComputational Neuroscience,Tübingen,fundedbytheGermanFederalMinistryof EducationandResearch(BMBF;FKZ:01GQ1002).

References

AbbottL,DayanP.Theeffectofcorrelatedvariabilityontheaccuracyofapopulation code.NeuralComput1999;11:91–101.

AlendaA,Molano-MazónM,PanzeriS,MaravallM.Sensoryinputdrives mul-tipleintracellularinformationstreamsinsomatosensorycortex.JNeurosci 2010;30:10872–84.

AndersenRA,MusallamS,PesaranB.Selectingthesignalsforabrain-machine inter-face.CurrOpinNeurobiol2004;14:720–6.

BartosM,VidaI,JonasP.Synapticmechanismsofsynchronizedgammaoscillations ininhibitoryinterneuronnetworks.NatRevNeurosci2007;8:45–56. BelitskiA,GrettonA,MagriC,MurayamaY,MontemurroMA,LogothetisNK,etal.

Low-frequencylocalﬁeldpotentialsandspikesinprimaryvisualcortexconvey independentvisualinformation.JNeurosci2008;28:5696–709.

BelitskiA,PanzeriS,MagriC,LogothetisNK,KayserC.Sensoryinformationinlocal ﬁeldpotentialsandspikesfromvisualandauditorycortices:timescalesand frequencybands.JComputNeurosci2010;29:533–45.

BerensP,KelirisGA,EckerAS,LogothetisNK,ToliasAS.Featureselectivityofthe gamma-bandofthelocalﬁeldpotentialinprimateprimaryvisualcortex.Front Neurosci2008;2:199–207.

BrunelN,WangXJ.Whatdeterminesthefrequencyoffastnetworkoscillations withirregularneuraldischarges?I.Synapticdynamicsandexcitation-inhibition balance.JNeurophysiol2003;90:415–30.

BuzsákiG.Rhythmsofthebrain.USA:OxfordUniversityPress;2006.

ChandrasekaranC,TuressonHK,BrownCH,GhazanfarAA.Theinﬂuenceofnatural scenedynamicsonauditorycorticalactivity.JNeurosci2010;30:13919–31. ChrobakJ,BuzsakiG.Gammaoscillationsintheentorhinalcortexofthefreely

behavingrat.JNeurosci1998;18:388–98.

CoverT,ThomasJ.Elementsofinformationtheory.NewYork:Wiley;2006. CroneNE,SinaiA,KorzeniewskaA.High-frequencygammaoscillationsandhuman

brainmappingwithelectrocorticography.ProgBrainRes2006;159:275–95. deRuytervanSteveninckRR,LewenGD,StrongSP,KoberleR,BialekW.

Repro-ducibilityandvariabilityinneuralspiketrains.Science1997;275:1805–8.

DonoghueJP,SanesJN,HatsopoulosNG,GaálG.Neuraldischargeandlocalﬁeld potentialoscillationsinprimatemotorcortexduringvoluntarymovements.J Neurophysiol1998;79:159–73.

FrienA,EckhornR,BauerR,WoelbernT,GabrielA.Fastoscillationsdisplaysharper orientationtuningthanslowercomponentsofthesamerecordingsinstriate cortexoftheawakemonkey.EurJNeurosci2000;12:1453–65.

FriesP,ReynoldsJH,RorieAE,DesimoneR.Modulationofoscillatoryneuronal syn-chronizationbyselectivevisualattention.Science2001;291:1560–3. FuhrmannAlpertG,Sun FT,HandwerkerD, D’EspositoM,KnightRT.

Spatio-temporalinformationanalysisofevent-relatedBOLDresponses.Neuroimage 2007;34:1545–61.

GieselmannMA,ThieleA.Comparisonofspatialintegrationandsurround suppres-sioncharacteristicsinspikingactivityandthelocalﬁeldpotentialinmacaque V1.EurJNeurosci2008;28:447–59.

GoodmanN.ThedistributionofthedeterminantofacomplexWishartdistributed matrix.AnnMathStat1963;34:178–80.

HatsopoulosNG,OjakangasCL,PaninskiL,DonoghueJP.Informationabout move-mentdirectionobtainedfromsynchronousactivityofmotorcorticalneurons.P NatlAcadSciUSA1998;95:15706–11.

HenrieJA,ShapleyR.LFPpowerspectrainV1cortex:thegradedeffectofstimulus contrast.JNeurophysiol2005;94:479–90.

InceA,MazzoniA,BartelsA,LogothetisN,PanzeriS.Anoveltesttodetermine thesigniﬁcanceofneuralselectivitytosingleandmultiplepotentially corre-latedstimulusfeatures.JNeurosciMeth,doi:10.1016/j.jneumeth.2011.11.013, inpress.

KayserC,KonigP.Stimuluslockingandfeatureselectivityprevailincomplementary frequencyrangesofV1localﬁeldpotentials.EurJNeurosci2004;19:485–9. KayserC,MontemurroMA,LogothetisNK,PanzeriS.Spike-phasecodingboostsand

stabilizesinformationcarriedbyspatialandtemporalspikepatterns.Neuron 2009;61:597–608.

KruseW,EckhornR.Inhibitionofsustainedgammaoscillations(35-80Hz)byfast transientresponsesincatvisualcortex.PNatlAcadSciUSA1996;93:6112–7. LakatosP,KarmosG,MehtaAD,UlbertI,SchroederCE.Entrainmentofneuronal oscillationsasamechanismofattentionalselection.Science2008;320:110–3. LimaB,SingerW,ChenNH,NeuenschwanderS.Synchronizationdynamicsin

responsetoplaidstimuliinmonkeyV1.CerebCortex2010;20:1556–73. LiuJ,NewsomeWT.LocalﬁeldpotentialincorticalareaMT:stimulustuningand

behavioralcorrelations.JNeurosci2006;26:7779–90.

LlinasR,RibaryU.Coherent40-Hzoscillationcharacterizesdreamstateinhumans. PNatlAcadSciUSA1993;90:2078–81.

LogothetisNK.TheunderpinningsoftheBOLDfunctionalmagneticresonance imag-ingsignal.JNeurosci2003;23:3963–71.

MagriC.AnInformationtheoreticanalysisofbrainoscillations.Milano:Department ofComputerSciences,UniversityofMilano;2009.

MagriC,WhittingstallK,SinghV,LogothetisNK,PanzeriS.Atoolboxforthefast informationanalysisofmultiple-siteLFP,EEGandspiketrainrecordings.BMC Neurosci2009;10:81.

MaierA,AuraCJ,LeopoldDA.Infragranularsourcesofsustainedlocalﬁeldpotential responsesinmacaqueprimaryvisualcortex.JNeurosci2011;31:1971–80. MazzoniA,PanzeriS,LogothetisNK,BrunelN.Encodingofnaturalisticstimuliby

localﬁeldpotentialspectrainnetworksofexcitatoryandinhibitoryneurons. PLoSComputBiol2008;4:e1000239.

MiltnerWHR,BraunC,ArnoldM,WitteH,TaubE.Coherenceofgamma-bandEEG activityasabasisforassociativelearning.Nature1999;397:434–5.

MisraN,SinghH,DemchukE.Estimationoftheentropyofamultivariatenormal distribution.JMultivariateAnal2005;92:324–42.

MontemurroM,SenatoreR,PanzeriS.Tightdata-robustboundstomutual infor-mationcombiningshufﬂingandmodelselectiontechniques.NeuralComput 2007;19:2913–57.

MontemurroMA,RaschMJ,Murayama Y,LogothetisNK, PanzeriS. Phase-of-ﬁringcodingofnaturalvisualstimuliinprimaryvisualcortex.CurrentBiol 2008;18:375–80.

MuthukumaraswamySD,SinghKD,SwettenhamJB,JonesDK.Visualgamma oscil-lationsandevokedresponses: variability,repeatabilityand structuralMRI correlates.Neuroimage2010;49:3349–57.

NicolelisMA,LebedevMA.Principlesofneuralensemblephysiologyunderlyingthe operationofbrain–machineinterfaces.NatRevNeurosci2009;10:530–40. OramMW,FoldiakP,PerrettDI,SengpielF.The‘IdealHomunculus’:decodingneural

populationsignals.TrendsNeurosci1998;21:259–65.

OstwaldD,PorcaroC,BagshawAP.Aninformationtheoreticapproachto EEG-fMRIintegration of visually evoked responses. Neuroimage 2010;49:498– 516.

OymanO,NabarRU,BolcskeiH,PaulrajAJ.Characterizingthestatistical prop-ertiesofmutualinformationinMIMOchannels.IEEETransSignalProcess 2003;51:2784–95.

PanzeriS,MagriC,LogothetisNK.Ontheuseofinformationtheoryfortheanalysis oftherelationshipbetweenneuralandimagingsignals.MagnResonImaging 2008;26:1015–25.

PanzeriS,SenatoreR,MontemurroMA,PetersenRS.Correctingforthesamplingbias probleminspiketraininformationmeasures.JNeurophysiol2007;98:1064– 72.

PercivalDB,WaldenAT.Waveletmethodsfortimeseriesanalysis.CambridgeUniv Pr;2006.

PesaranB,PezarisJS,SahaniM,MitraPP,AndersenRA.Temporalstructurein neu-ronalactivityduringworkingmemoryinmacaqueparietalcortex.NatNeurosci 2002;5:805–11.

(13)

PolaG,ThieleA,HoffmannK,PanzeriS.Anexactmethodtoquantifytheinformation transmittedbydifferentmechanismsofcorrelationalcoding.Network-Comp Neural2003;14:35–60.

QuirogaRQ,PanzeriS.Extractinginformationfromneuronalpopulations: informa-tiontheoryanddecodingapproaches.NatRevNeurosci2009;10:173–85. RaschMJ,GrettonA,MurayamaY,MaassW,LogothetisNK.Inferringspiketrains

fromlocalﬁeldpotentials.JNeurophysiol2008;99:14611476.

RayS,CroneNE,NieburE,FranaszczukPJ,HsiaoSS.Neuralcorrelatesofhigh-gamma oscillations(60–200Hz)inmacaquelocalﬁeldpotentialsandtheirpotential implicationsinelectrocorticography.JNeurosci2008;28:11526–36. RayS,MaunsellJHR.Differencesingammafrequenciesacrossvisualcortexrestrict

theirpossibleuseincomputation.Neuron2010;67:885–96.

RayS,MaunsellJHR.Differentoriginsofgammarhythmandhigh-gammaactivity inmacaquevisualcortex.PLoSBiol2011;9:e1000610.

SchneidmanE,BialekW,BerryMJ.Synergy,redundancy,andindependencein pop-ulationcodes.JNeurosci2003;23:11539–53.

SchynsPG,ThutG,GrossJ.Crackingthecodeofoscillatoryactivity.PLoSBiol 2011;9:e1001064.

Shannon CE. A mathematical theory of communication. AT&T Tech J 1948;27:379–423.

SiegelM,KönigP.Afunctionalgamma-banddeﬁnedbystimulus-dependent syn-chronizationinarea18ofawakebehavingcats.JNeurosci2003;23:4251–60. StrongSP,KoberleR,deRuytervanSteveninckRR,BialekW.Entropyand

informa-tioninneuralspiketrains.PhysRevLett1998;80:197–200.

WilsonEB,HilfertyMM.Thedistributionofchi-square.PNatlAcadSciUSA 1931;17:684–8.

YuY,CrumillerM,KnightB,KaplanE.Estimatingtheamountofinformationcarried byaneuronalpopulation.FrontComputNeurosci2010;4:10.

ZoharyE,ShadlenMN,NewsomeWT.Correlatedneuronaldischargerateandits implicationsforpsychophysicalperformance.Nature1994;370:140–3.