Field computation in motor control

FieldComputationinMotorControlBruceMacLennanComputerScienceDepartmentUniversityofTennessee,[e-mail]13,1996AbstractFieldcomputationdealswithinformationprocessingintermsofelds,continuousdistributionsofdata.Manyneuralphenomenaareconvenientlydescribedaselds,includingneuronactivityfromlarge(brainarea)tosmall(dendritic)scales.Further,itisoftenusefultodescribemotorcontrolandsensorimotorcoordinationintermsofex-ternaleldssuchasforceeldsandsensoryimages.Wesurveythebasicconceptsofeldcomputation,includingbothfeed-forwardeldoperationsandelddynamicsresultingfromrecurrentconnections.Adaptiveandlearningmechanismsarediscussedbriey.Theappli-cationofeldcomputationtomotorcontrolisillustratedbyseveralexamples:externalforceeldsassociatedwithspinalneurons(Bizzi&Mussa-Ivaldi1995),populationcodingofdirectioninmotorcor-tex(Georgopoulos1995),continuoustransformationofdirectionelds(Droulez&Berthoz1991a),andlineargaineldsandcoordinatetrans-formationsinposteriorparietalcortex(Andersen1995).Nextwesurveysomeeld-basedrepresentationsofmotion,includingdirect,Fourier,Gaborandwaveletormultiresolutionrepresentations.Fi-nallyweconsiderbrieytheapplicationoftheserepresentationstoconstraintsatisfaction,whichhasmanyapplicationsinmotorcontrol.1MotivationMypurposeinthischapteristointroducethegeneralconceptsofeldcom-putationandtodescribesomepossibleapplicationsofittomotorcontrol.ThispaperwillappearinSelf-Organization,ComputationalMapsandMotorControl,ed.byPietroG.MorassoandVittorioSanguineti,Elsevier-NorthHolland,inpress.Fieldcomputationdealswithcontinuousdistributionsofactivitysuchasarefoundinthetopographicmapsandotherfunctionalareasofthebrain(Knudsenetal.1987),butalsowithexternaldistributionsofquantity,suchasforceelds.Ineldcomputationwearegenerallyconcernedwiththetopologyofthespaceoverwhichaquantityisdistributed;thiscontrastswiththecommonapproachinneuralnetworkmodeling,whichtreatsneuralactivityasavector,thatis,asquantitydistributedoveraspacewithnosig-nicanttopology(sincetheaxesareindependentand,ineect,allequallydistantfromeachother).Afterdeningeldsandsurveyingtheiroccurrenceinthebrain,Iwillgiveabriefintroductiontothemathematicsofeldcomputationandthenconsiderseveralproblemsinmotorcontrolfromtheperspectiveofeldcomputation.2Fields2.1DenitionForthepurposesofeldcomputation,aeldisdenedtobeaspatiallycon-tinuousdistributionofquantity.Fieldcomputationisthenacomputationalprocessthatoperatesonanentireeldinparallel.Oftenwetreattheeldasvaryingcontinuouslyintime,althoughthisisnotnecessary.Itissometimesobjectedthatdistributionsofquantityinthebrainarenotinfactcontinuous,sinceneuronsandevensynapsesarediscrete.How-ever,thisobjectionisirrelevant.Forthepurposesofeldcomputation,itisnecessaryonlythatthenumberofunitsbesucientlylargethatitmaybetreatedasacontinuum,specically,thatcontinuousmathematicscanbeapplied.Thereis,ofcourse,nospecicnumberatwhichtheensemblebe-comes\bigenoughtobetreatedasacontinuum;thisisanissuethatmustberesolvedbythemodelerinthecontextoftheusetowhichthemodelwillbeput.However,sincethereare146000neuronspermm2throughoutmostofthecortex(Changeux1985,p.51),itisreasonabletosaythatactivityinaregionofcortexmorethanasquaremillimeterinsizecanbesafelytreatedasaeld.Mathematically,aeldistreatedasacontinuous,usuallyreal-valued,functionoversomecontinuum,itsdomainorextent.Forexample,ifisacirculardiskrepresentingtheretina,thenforanypointp2,(p)mightbethelightintensityatp.Theeld’sdomainhassometopology(relationsofconnectivityandnearness);forexample,thetopologyoftheretinaisatwo-dimensionalcontinuum.2.2RealizationintheBrainThereareseverallevelsofneuralactivitythatcanbeviewedaseldcom-putation.2Themostobviouselds,whicharemeasuredbymultipleelectroderecord-ingorbynoninvasiveimaging,suchasNMR,arethosecomprisingthespik-ingactivityofneurons.Since,aswehaveseen,thereare146thousandneuronspersquaremillimeterofcortex,regionsofcortexofthissizearemorethanbigenoughtobetreatedascontinua(reasonably,atenthofasquaremillimeterismorethanlargeenough).Indeed,Knudsenetal.(1987)observethatcomputationalmapsinthebrainmaybeassmallasasquaremillimeter,andperhapssmaller.Incorticalregionswheretheinformationisrepresentedbyimpulserate,theeldisreal-valued;thus(p;t)orp(t)representstheinstantaneousim-pulserateatlocationpandtimet.RecentlyHopeld(1995)hasarguedthatinformationmayberepresentedbyacombinationofimpulsefrequencyandphase(relativetoaglobal\clockeldortootherneurons);insomecasesatleast,thephaserepresentsananalogvalueandtheamplituderepresentsitsimportance.Insuchcasesit’snaturaltotreattheeldascomplex-valued,withthecomplexnumber’sphaseanglerepresentingtheimpulsephaseanditsmagnituderepresentingtheimpulseamplitude.Thuswewritep(t)=ap(t)eip(t),whereap(t)isthetime-varyingamplitudeandp(t)thetime-varyingphase.Synapto-dendritictransmissionofsuchaeld,whichaectsbothitsamplitudeandphase,canberepresentedasmultiplicationbyaconstantcomplexnumber.Forexample,supposeaeld=zre-sultsfromtransmittingeldthroughsynapseszp=wpeipthatintroduceamplitude