Towards a theory of the striate cortex

TowardsaTheoryoftheStriateCortex1PublishedinNeuralComputation,Vol6,number1,January1994,p127-146ZhaopingLiandJosephJ.AtickTheRockefellerUniversity1230YorkAvenueNewYork,NY10021,USAAbstractWeexplorethehypothesisthatlinearcorticalneuronsareconcernedwithbuildingaparticulartypeofrepresentationofthevisualworld—onewhichnotonlypreservestheinformationandtheefficiencyachievedbytheretina,butinadditionpreservesspatialrelationshipsintheinput—bothintheplaneofvisionandinthedepthdi-mension.Focusingonthelinearcorticalcells,weclassifyalltransformshavingtheseproperties.Theyaregivenbyrepresentationsofthescalingandtranslationgroup,andturnouttobelabeledbyrationalnumbers‘(p+q)=p’(p;qintegers).Anygiven(p;q)predictsasetofreceptivefieldswhichcomeatdifferentspatiallocationsandscales(sizes)withabandwidthoflog2[(p+q)=p℄octaves,and,mostinterestingly,withadiversityof‘q’cellvarieties.Thebandwidthaffectsthetrade-offbetweenpreserva-tionofplanaranddepthrelations,and,wethink,shouldbeselectedtomatchstruc-turesinnaturalscenes.Forbandwidthsbetween1and2octaves,whicharetheoneswefeelprovidethebestmatching,wefindforeachscaleaminimumoftwodistinctcelltypesthatresidenexttoeachotherandinphasequadrature,i.e.,differby90ointhephasesoftheirreceptivefields,asarefoundinthecortex,theyresemblethe“even-symmetric”and“odd-symmetric”simplecellsinspecialcases.Aninterest-ingconsequenceoftherepresentationspresentedhereisthatthepatternofactivationinthecellsinresponsetoatranslationorscalingofanobjectremainsthesamebutmerelyshiftsitslocusfromonegroupofcellstoanother.Thisworkalsoprovidesanewunderstandingofcolorcodingchangesfromtheretinatothecortex.1WorksupportedinpartbyagrantfromtheSeaverInstitute.1.IntroductionWhatisthepurposeofthesignalprocessingperformedbyneuronsinthevisualpath-way?Aretherefirstprinciplesthatpredictthecomputationsoftheseneurons?Recentlytherehasbeensomeprogressinansweringthesequestionsforneuronsintheearlystagesofthevisualpathway.InAtickandRedlich(1990,1992)aquantitativetheory,basedontheprincipleofredundancyreduction,wasproposed.Ithypothesizesthatthemaingoalofretinaltransformationsistoeliminateredundancyininputsignals,particularlythatduetopairwisecorrelationsamongpixels—second-orderstatistics.2Thepredictionsofthetheoryagreewellwithexperimentaldataonprocessingofretinalganglioncells(AtickandRedlich1992,Aticketal1992).Giventhesuccessesofthistheory,itisnaturaltoaskwhetherredundancyreductionisacomputationalstrategycontinuedintothestriatecortex.Onepossibilityisthatcor-ticalneuronsareconcernedwitheliminatinghigher-orderredundancy,whichisduetohigher-orderstatistics.Wethinkthisisunlikely.Toseewhy,werecallthefactsthatmakeredundancyreductioncompellingwhenappliedtotheretina,andseethatthesefactsarenotasrelevantforthecortex.First,theretinahasaclearbottleneckproblem:theamountofvisualdatafallingontheretinapersecondisenormous,oftheorderoftensofmegabytes,whiletheretinaloutputhastofitintoanopticnerveofadynamicrangesignificantlysmallerthanthatoftheinput.Thus,theretinamustcompressthesignal,anditcandosowithoutsignificantlossofinformationbyreducingredundancy.Incontrast,afterthesignalispasttheopticnerve,thereisnoidentifiablebottleneckthatrequirescontinuedredundancyreductionbeyondtheretina.Second,eveniftherewerepressuretoreducedata3,eliminatinghigher-orderstatisticsdoesnothelp.Thereasonisthathigher-orderstatisticsdonotcontributesignificantlytotheentropyofimages,andhencenosignificantcompressioncanbeachievedbyeliminat-ingthem(forreviewsofinformationtheoryseeShannonandWeaver1949;Atick1992).Thedominantredundancycomesfrompairwisecorrelations.4Thereisanotherintrinsicdifferencebetweenhigherandsecond-orderstatisticsthatsuggeststheirdifferenttreatmentbythevisualpathway.Fig.1showsimageAandan-otherimageBwhichwasobtainedbyrandomizingthephasesofthefouriercoefficientsofA.Bthushasthesamesecond-orderstatisticsasAbutnohigherorderones.ContrarytoA,Bhasnoclearformsorstructures(cf.Field1989).Thissuggeststhatsecond-orderstatisticsareuseless,whilehigher-orderonesareessential,fordefiningformsandfordis-2Sinceretinalneuronsreceivenoisysignalsitisnecessarytoformulatetheredundancyreductionhy-pothesiscarefullytakingnoiseintoaccount.InAtickandRedlich(1990,1992)ageneralizednotionofredundancywasdefined,whoseminimizationleadstoeliminationofpairwisecorrelationsandtonoisesmoothing.3Forexampletherecouldbeacomputationalbottlenecksuchasanattentionalbottleneckoccuringdeepintothecortex—perhapsinthelinkbetweenV4andIT(VanEssenetal1991).4Thisfactiswellknowninthetelevisionindustry(seee:g:Schreiber1956).Thisiswhypracticalcom-pressionschemesfortelevisionsignalsnevertakeintoaccountmorethanpairwisecorrelations,andeventhen,typicallynearestneighborcorrelations.ThisfactwasalsoverifiedforseveralscannednaturalimagesinourlaboratorybyN.RedlichandbyZ.Li.1criminatingbetweenimages.Actually,eliminatingtheformerhighlightsthehigher-orderstatisticswhichshouldbeusedtoextractformsignalsfrom“noise.”5Sowhatisthecortexthentryingtodo?Ultimately,ofcourse,thecortexisconcernedwithobjectandpatternrecognition.Onepromisingdirectioncouldbetousestatisticalregularitiesof