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ANeuralFieldModeloftheSomatosensoryCortex:Formation,MaintenanceandReorganizationofOrderedTopographicMapsGeorgiosIs.
Detorakis,NicolasP.
Rougier*INRIACNRS:UMR7503UniversiteHenriPoincare-NancyIUniversiteNancyIIInstitutNationalPolytechniquedeLorraine,Nancy,FranceAbstractWeinvestigatetheformationandmaintenanceoforderedtopographicmapsintheprimarysomatosensorycortexaswellasthereorganizationofrepresentationsaftersensorydeprivationorcorticallesion.
Weconsiderboththecriticalperiod(postnatal)whererepresentationsareshapedandthepost-criticalperiodwhererepresentationsaremaintainedandpossiblyreorganized.
Wehypothesizethatfeed-forwardthalamocorticalconnectionsareanadequatesiteofplasticitywhilecortico-corticalconnectionsarebelievedtodriveacompetitivemechanismthatiscriticalforlearning.
Wemodelasmallskinpatchlocatedonthedistalphalangealsurfaceofadigitasasetof256Merkelendingcomplexes(MEC)thatfeedacomputationalmodeloftheprimarysomatosensorycortex(area3b).
Thismodelisatwo-dimensionalneuralfieldwherespatiallylocalizedsolutions(a.
k.
a.
bumps)drivecorticalplasticitythroughaHebbian-likelearningrule.
Simulationsexplaintheinitialformationoforderedrepresentationsfollowingrepetitiveandrandomstimulationsoftheskinpatch.
Skinlesionsaswellascorticallesionsarealsostudiedandresultsconfirmthepossibilitytoreorganizerepresentationsusingthesamelearningruleanddependingonthetypeofthelesion.
Forseverelesions,themodelsuggeststhatcortico-corticalconnectionsmayplayanimportantroleincompleterecovery.
Citation:DetorakisGI,RougierNP(2012)ANeuralFieldModeloftheSomatosensoryCortex:Formation,MaintenanceandReorganizationofOrderedTopographicMaps.
PLoSONE7(7):e40257.
doi:10.
1371/journal.
pone.
0040257Editor:AndreaSerino,UniversityofBologna,ItalyReceivedJanuary28,2012;AcceptedJune4,2012;PublishedJuly12,2012Copyright:2012Detorakis,Rougier.
Thisisanopen-accessarticledistributedunderthetermsoftheCreativeCommonsAttributionLicense,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalauthorandsourcearecredited.
Funding:Theauthorshavenofundingorsupporttoreport.
CompetingInterests:Theauthorshavedeclaredthatnocompetinginterestsexist.
*E-mail:Nicolas.
Rougier@inria.
frIntroductionEarlyobservationsofLeytonandSherrington[1](asreportedbyLemonin[2])ontheadultanthropoidapesdemonstratedtheabilityofthemotorcortextorecoverfromextensivecorticallesions.
Theauthorshypothesizedconsequentlytheexistenceofaneuralsubstrateand/oramechanismforsuchextensiverecovery.
However,aboutfortyyearslater,HubelandWieselpublishedaveryinfluentialpaper[3]thatpromotedtheideaoffixedcorticalrepresentationsfollowingthepost-nataldevelopmentalperiod.
ThishypothesishasprevailedforalongtimeuntilthestudiesofMerzenich,Kaasetal.
[4–6]providedexperimentalevidenceforsomatosensorycortexreorganizationfollowingaperipheralnerveinjuryoramputationintheadultmonkey.
Severalneurophysio-logicalstudies[7–9]havesinceconfirmedthislatterhypothesisandthecortexisnowconsideredasadynamicstructurethatisabletoreorganizeitsrepresentationsduringthewholelife-timeandnotonlyduringthecriticalperiod.
Ithasbeenconfirmedforthecaseoflesion(e.
g.
strokes)[9,10],ablation[11,12](e.
g.
tumorssurgery),bodyinjury(e.
g.
accident)orseveredegeneracy'softhalamocorticalandcortico-spinalprojections[13,14].
EvenenvironmentalfactorsmaydeeplyimpactcorticalrepresentationsasithasbeendemonstratedbyDanieletal.
in[15].
However,thenatureoftheunderlyingmechanismssupportingsuchcorticalplasticityisstilllargelyunknownevenifsomehypotheseshaveemerged.
Inthisregard,FeldmanandBrecht[16]publishedanextensivereviewofsynapticmechanismsthatcouldberesponsibleforplasticityintheneocortexatboththesynapticphysiologicallevel(long-termpotentiation(LTP),long-termdepression(LTD),spiketimingdependentplasticity(STDP),homeostasis,meta-plasticity,GABAergiccellsandcircuits)andthestructurallevel(thalamocor-ticalandhorizontalcross-columnaraxons).
HickmottandMerzenich[17]proposedasimilarstudyaboutthepropertiesoflocalcircuitunderlyingcorticalreorganizationandidentifiedtwogeneralclassesofmechanisms,oneinvolvesarapidchangeintheefficiencyofexistingsynapseswhiletheotherentailsadelayedphasepromotingthesproutingofnewconnections.
Thislatterstudyisinfactquiteconsistentwiththeformertwo-levelsanalysis.
Itistobenotedthatsomeofthesemechanismswerealreadyhypothesizedtobeinvolvedincorticalplasticity.
Forexample,neuronalaxonsproutingwasalsoreportedbyFlorenceetal.
[18]asapotentialcandidate,LTPandLTDin[19–22],formationofnewsynapses[4,23]andinhibitoryandexcitatorymechanismsin[24,25].
Someothercandidateshavebeenidentifiedaswellastheroleoftheinterhemisphericmodulationofsomatosensoryreceptivefields[26].
Today,nodefinitivehypothesishasemergedandmostprobablytheanswerisacombinationofdifferentmechanismsatdifferenttimescaleproportionallytotheconsideredperiodofdevelopment(prenatal/postnatalcriticalperiod/adultperiod).
Onedifficultyinidentifyingsuchamechanismisthatonemustgiveaccountonboththeinitialformationoforderedtopographicmaps(asithasbeenobservedinprimarysensoryareasV1,A1andS1forexample),themaintenanceofsuchmapsduringthewholelifetime,thereorganizationfollowingatraumaoraninjuryandPLoSONE|www.
plosone.
org1July2012|Volume7|Issue7|e40257thepossiblerefinementaccordingtoexperience(e.
g.
expandingrepresentationsinordertoincreaseaccuracy).
Sincetheanatom-icalorganizationofthecortexfollowsaregularandhierarchicalstructure[27,28]whoseelementarycircuitistheminicolumn(eveniftheymaydisplayconsiderabledifferences[29]),thislatterstructureorthemacrocolumn(whichgathersfrom60to80minicolumns[30])maybothrepresentnaturalcandidatestobeinvestigatedfurther.
Moreprecisely,weknowthateachcorticallayerreceivesinputfromfourdistinctsources,onefromextra-corticalareasandthreefromintra-corticalareas.
First,excitatoryneuronsofasinglelayerreceiveinputfromotherneuronsofthesamelayer.
Second,excitatoryneuronsoftheinputlayer,L4,receiverecurrentfeedbackinputfromL2/3.
Thusapositiverecurrentloopemergeswhichseemstoaccountforgainmodulationforactiveselectionandre-combinationoftherelativelysmallafferentsignals[31].
Thirdisthebackgroundnoiseofthecorticalcircuitthathasbeenproposedtocontributestothemodulationofthegainofthecircuitbyenhancingtheresponsivenessofcorticalpyramidalneurons,[32,33].
Last,butnotleast,excitatoryneuronsofL4receivedirectinputfromthethalamusandiftheyaccountapproximatelyforonly15%ofthesynapses,BrunoandSakmann[34]demonstratedinvivotheymaynonethelessdrivethecortexwithouttheneedforintra-corticalamplificationandrevealedthethalamocorticalpathwayasahighlyefficientone.
Inaddition,thalamicneuronsdevelopdirectmono-synapticconnectionsontoL4corticalexcitatoryneuronsindependentlyofthemorphologicalcharacteristicsoftheseneurons.
Furthermore,Khazipovetal.
[35]describedtheontologicaldevelopmentofthecortexandtherespectivecontributionofdifferentmechanisms.
Thematurationofthebrainisdividedintwomajorperiods,pre-andpostnatal.
Atthebeginningoftheprenatalperiod,geneticinformationleadstotheearlyformationoftheneuralnetworksfollowedbyaspontaneouselectricalactivityperiodthatleadstotheformationofthecolumnarorganization[36,37].
Thisprocesscontinuesafterbirthandstopswhenthecriticalperiodstarts.
Then,corticalcircuitsaredrivenmainlybyexperienceandsynapticplasticity(e.
g.
Hebbianlearning)takesplace.
Afterthecriticalperiodcomestoanendtheadultbraincanstilllearnandcorticalcircuitsareabletoreorganizethemselvesandrefinetheirreceptivefields.
Atthispoint,wethinkthatcomputationalneurosciencemayplayakeyrolebyprovidingcomputationalmodelsthatcanbeusedtotestthisorthatfunctionalhypothesis.
Ithasbeenalreadythecasewiththeself-organizingmapsasproposedbyT.
Kohonen[38,39]inthelateeightiesthathelpedtopromotetheideaofacompetitionamongunitsleadingtotheformationoforderedrepresentations(althoughwithoutthepossibilityofre-organizingthem).
Atthesametime,G.
Edelmanwasproposingtothecommunityhistheoryofneuralgroupselection[40]andmorespecifically,hewasproposingacomputationalmodelofplasticityintheorganizationofthecorticalmaps[41]whereneuronalgroupsserveasthebasicunitformaporganization.
However,thismodeldidnotemphasizetheimportanceofthethalamocorticalpathwayasweexplainedearlierandwethinkwemightneedtoreconsideritsroleintheformationofandmaintenanceofthesensoryrepresenta-tions.
Weproposeinthisarticletoinvestigate(computationally)theformationoftopographicmapsinthesomatosensorycortexaswellasthereorganizationofrepresentationsaftersensorydeprivationorcorticallesion.
Weconsiderboththecriticalperiod(postnatal)whererepresentationsareshapedandthepost-criticalperiodwhererepresentationsaremaintainedandpossiblyreorganized.
Wehypothesizethatfeed-forwardthalamocorticalconnectionsareanadequatesiteofplasticitytogiveaccountforboththeformationandthemaintenanceoftopographicrepresentationsandpartlyforthereorganizationofrepresentationsfollowingasensoryorcorticallesion.
Wethereforefocusontheorganizationandreorganizationofthesomatosensorycortex(area3b)innervatedbythemechanoreceptorsofthehand.
Amodelofskinanditsassociatedmechanoreceptors(Merkelendingcomplexes)isfirstintroducedandthecorticalmodel,basedonthedynamicneuralfieldtheory,ispresentedtogetherwithitsdynamicsthatallowtodrivelearningthroughaHebbian-likelearningrule.
Resultsconcerningtheinitialformationandmaintenanceoforderedrepresentationareanalyzedaswellasresultsconcerningthereorganizationofrepresentationsfollowingacorticalorskinlesion.
Inlightoftheseexperiments,wediscussthecriticalroleoffeed-forwardthalamocorticalconnectionsinthereorganizationprocessaswellasthepotentialroleoflateralconnections.
MethodsSkinModelWemodeledtheMerkelendingcomplexes(MEC)thatarededicatedtosustainedtouchsensationandpressure,neglectingothermodalities(e.
g.
temperature,pain).
FollowingdataprovidedbyPareetal.
in[42],weconsideredasmallskinpatchlocatedonthedistalphalangealsurfaceofadigit(seefigure1)thataccountsroughlyforhalfofthedigitsurfaceinarea3b,restofthesurfacebeingsharedamongproximalandmiddlesurfaceofthesamedigit[43].
Theskinpatchisapproximatelyofsize1mm2,usingareceptordensityof250/mm2[44].
Ithasbeenmodeledasaplanarsurface{1,1|{1,1(arbitraryunits)andweconsidered256MEC'sthatarearrangedinaregulargridoverthewholesurfacewithalocationjitterof5%.
Thisresultsinaquasi-uniformdistributionconsistentwithactualdistributionofMECasreportedin[42]andillustratedinfigure1C.
EachreceptorisfullydescribedbyitsCartesiancoordinates,namely(Rxi,Ryi),wherei[f1,256g.
Weassumethatwhenastimulusisappliedatagivenlocation(x,y)oftheskinpatch,themechanicpropertyoftheskinextendsthepressureleveltonearbylocations[45]suchthattheresponsesofanyreceptor(Rx,Ry)isgivenby:s(Rx,Ry)~exp{121s((Rx{x)2z(Ry{y)2)r!
1Inprimates,thissomatosensoryinformationflowsthroughseveralrelays,whichlieinthespinalcordandthethalamus,beforereachingthecortex.
Moreprecisely,dorsalrootganglion(DRG)receivesinformationfromskinandtransmitsittothedorsalcolumnnuclei(DCN).
DCN,inturn,transmitsinformationfromDRGtotheventralposteriorlateral(VPL)nucleusofthethalamus,crossingthemidlineatthemedullaviathemediallemniscus[46].
Theserelaystationsplayakeyroleinstimulicontrastsharpeningbutwedecidedtoignorethemsinceweconsideredthattheexperimentalsetupprovidesenoughcontroloverthestimulusandensurespropersharpness.
Hence,theoutputofallreceptorsaredirectlyfedtothecorticalmodel(seefigure2).
Skinlesionsweremadebysilencingreceptorsoveraspecificareaoftheskinsurface.
Insteadoftransmittingpropervalues,disabledreceptorstransmitanullvaluewhichmorelikelycorrespondstosensorydeprivation.
Therearethreetypesoflesions(namelytypeI,II&III)asillustratedinfigure2Thesethreetypescorrespondtothreedistincttopologicalsituations.
Thefirsttypeleadstoaskinpatchthatistopologicallyequivalenttotheintactone.
ThesecondtypeintroducesaseparationoftheintactANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
plosone.
org2July2012|Volume7|Issue7|e40257skinpatchintotwodistinctareasandbecauseastimuluscannotspanthetwopatchesatonce,thesetwoskinpatchesareindeedindependent.
Thethirdtypeintroducesaholeinthetopologyoftheskinandisthemostchallengingtorecoverfrom.
CorticalModelAsmallvolumeoftheprimarysomatosensorycortex(SI)wasmodeledusingtheneuralfieldtheory[47–50]whichconsidersagivencorticalvolumeVtobeaspatialcontinuumwheremacro-statevariables(suchasthemeanfiringrate)ofapopulationatpositionxisgivenbyanequationoftype:Figure1.
Skinmodeling.
AApalmarschematicofthehand((https://commons.
wikimedia.
org/wiki/File:Hand_left.
svg)HanddrawingbyCy21availableoncommons.
wikimedia.
orgunderaCreativeCommonsAttribution-ShareAlike3.
0Unportedhttps://creativecommons.
org/licenses/by-sa/3.
0/deed.
enlicense.
)BLocationandrelativesizeofthemodeledskinpatch.
CMagnificationofskinpatchindicatingthetopologyofreceptors.
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g001Figure2.
Amodelofthesomatosensorycortex.
Theskinpatchismodeledasasetof256mechanicreceptors(whitediscsinthefigure)withaquasi-uniformdistributionthatfeedthecorticalpatch.
Bluecirclesrepresentanexampleofastimulusappliedontheskinpatchandthebluesquarerepresentsthestimulationarea.
Thecorticalpatchismodeledusinganeuralfieldwithaspatialdiscretizationofsize32632elementsusinggloballateralexcitationandinhibition.
Redcirclesrepresenta(schematic)typicalcorticalresponseafterlearning.
Thethreesquaresundereachpatchrepresentthedifferentcasesoflesionthathavebeenstudiedwherethegraypartrepresentsthelesionedarea.
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g002ANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
plosone.
org3July2012|Volume7|Issue7|e402571tLu(x,t)Lt~{u(x,t)zVw(x,y)f(u(y,t{Dx{yDv))dyzi(x,t)whereu(x,t)representstheactivity(e.
g.
themembranepotential)atpositionxandtimet,i(x,t)representsthesynapticinput,wisaweightfunctionmeasuringthestrengthofconnectionbetweenpositionsxandy,fisthefiringratefunctionofasingleneuron,visthevelocityofanactionpotentialandtisthetemporaldecay.
Giventhesmallsizeofcorticalvolume,weneglectedvelocityeffects(v~z,see[5]forastudy)andweconsideredthefieldtobehomogeneousandisotropic,leadingtothefollowingsimplifiedequation:1tLu(x,t)Lt~{u(x,t)zaVwl(Dx{yD)f(u(y,t))dyzi(x,t)2whereaisascalingfactorandwlistheweightfunctionoflateralconnections.
Moreover,weuseasimplerectificationforthefiringfunctionfsinceitisthesimplestfunctionthatcanprovidestabilityforasuchfield[52,53]:f(x)~x,ifx§00,ifxv0&3Wealsoconsideredtheinputtobeameasureofthedifferenceofagivenstimuluss(t)(thatcorrespondtothen~256outputsoftheskinreceptors)withasetoffeed-forwardweightswf(x)suchthatforanypositionx,wehave:i(x,t)~1{Ds(t){wf(x)DnG(x;mc,sc)4whereG(x;mc,sc)isacorrectiveGaussianfunction.
Itiscorrectiveinthesenseoftoricconnectionssinceourmodeldoesnotenforceanytorictopology(itisnotimplementedonatorus).
Wethereforemultipliedi(x,t)byafixed-shapeGaussianfunctioninordertocorrectanykindofboundaryconditionsside-effect.
ThroughoutsimulationsthevarianceofthecorrectiveGaussianfunctionwassc~2:1,andthemean,mc~0.
Finally,thelateralconnectionweight,wl,reflectstheusualpatternofshort-rangeexcitation(we)andlong-rangeinhibition(wi):wl(x)~we(x){wi(x)~Keexp{x22s2e{Kiexp{x22s2i5withKe,Kibeingtheamplitudes,se,sibeingthevariancesoftheexcitatoryandinhibitoryGaussianfunctionsrespectively,suchthatgenerallyweconsidersi&se.
Corticallesionsweremadebysilencingunitsoveraspecificareaofthecorticalsurfacesuchthatdeadunit'sactivitywasalwayszero.
Likeforskinlesions,weconsideredthreetypesoflesions(namelytypeI,II&III)withsametopologicalproperties(seefigure2).
NeuralPopulationDynamicsInhisseminalworkonneuralfields[47],Amaristudiedtheequilibriumsolutions,stabilityandformationofdynamicpatternsprovidingtheconditionsforsuchbehaviorsintheone-dimensionalcase.
Sincethen,alotofworkhasbeendoneinthedirectionofextendingtheinitialtheorytootherconditions[48],higherdimensions[49,54]anddifferentbehaviors(seethereviewbyCoombes[50]).
Inthepresentwork,wearemainlyinterestedinthetwo-dimensionalcasesinceweaimatmodelingacorticalsheet.
Morespecifically,weareinterestedinspatiallylocalizedsolutions(a.
k.
a.
bumps)thatmaydrivethecorticalplasticityandwewouldliketheactivityofthefieldtoreflecttosomeextentameasureoftheinput,e.
g.
ameasureofthedistancebetweenthefeed-forwardweightsofthemostactivatedunits(i.
e.
unitsfromthebump)andthecurrentstimulation.
UsingaspecificsetofparametersP~fKe,Ki,se,si,aggivenintable1,thefieldcanachievethefollowingproperty:foranyuniforminputi(x,t)~v,v[0,1,themaximumactivityofthefieldisv.
Theone-dimensionalcaseisillustratedinthefigure3.
Furthermorethesamepropertyholdstrueinthetwodimensionalcaseusingthesamesetofparameters,P.
PlasticityRuleAlearningrulefortheclassicalself-organizingmapalgorithm[38]hasbeenproposedbyRougierandBoniface[55],wheretheoriginaltime-dependent(learningrateandneighborhood)learningfunctionhasbeenreplacedbyatime-invariantlearningrule.
Instead,adynamicneighborhoodfunctionhasbeenintroducedthatdependsexplicitlyonthedistanceofthewinnertothepresentedstimulus.
Ontheonehand,ifthedistanceofthewinningunitisveryclosetothepresentedinput,thedynamicneighborhoodisrenderedverystrongbutnarrow,weakeningthelearningofotherunits.
Ontheotherhand,whenthewinningunitisveryfarfromthepresentedinput,thedynamicneighborhoodexhibitsaverybroadbutweakpattern,promotingweaklearningofeveryunitinthenetwork.
Thisalgorithmhasbeenexperimen-tallyprovedtobeabletoachieveself-organizationinasimilarwayasofaregularself-organizingmap.
Usingthisideaofadynamicneighborhoodbutinthecontextofneuralfields(nonotionofawinningunit),wecanusetheaforementionedmatchpropertytoachievesuchbehavior.
Asexplainedintheintroduction,ourmainhypothesisisthatcorticalplasticitycanbeachievedatthelevelofthalamocorticalconnections,correspondingtothefeed-forwardweightsinourmodel.
Thisimpliesthatthenetworkdoesnotneedtolearnthelateralweights.
Therefore,wetrainedthefieldusingamodifiedOjalearningrule[56]withthefollowingequation:Lwf(x,t)Lt~cs{wf(x)|{z}pre-synapticterm|Le(x)|{z}post-synapticterm6wherecisthelearningrateandLe(x)isthetotalexcitationreceivedatthepointxwhichisgivenbythetwodimensionalspatialconvolutionbetweentheexcitatorypartofthelateralweightfunctionandthefieldactivity.
Moreprecisely,wehave:Le(x)~T0Vf(u(y,t))we(Dx{yD)dydt7whereweistheexcitatorypartofthelateralweightfunctionasitisgivenbyequation(5).
Theideaistoexplicitlymodulatelearningaccordingtothesumofexcitationreceivedatapointxwhiletheinhibitiononlyservesduringthecompetitionstage.
Atthisstage,itisimportanttonotethatweunifiedtheinhibitoryandexcitatoryneuralpopulationintoasinglepopulationandusedpositive/ANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
plosone.
org4July2012|Volume7|Issue7|e40257negativeweightstoreflectexcitatory/inhibitoryactionofaneuronontoanother.
Itwouldbeperfectlyequivalenttouseadualpopulationbutusingasinglepopulationmakescomputationfaster.
Inthiscontext,Lereflectsthecontributionofexcitatoryneurons.
Thelearningruleiscomposedofapre-synaptictermandapost-synapticterm.
Thepre-synaptictermreflectstheexplicitcomparisonbetweenthestimuluspatternandthepatternofthefeed-forwardthalamocorticalsynapsethatentertheneuron.
ThisallowstogracefullyenforcebothHebbian(LTP)andanti-Hebbian(LTD)learning,controllingthegrowthofthesynapse.
Thepost-synaptictermmodulatesthepre-synaptictermaccordingtothetotalsumofexcitation(insteadofthepost-synapticactivityasitwouldbeintheoriginalOjarule).
Asillustratedinfigure4,theactivationataspecificsiteandtheamountofexcitationreceivedatthesamesitearehighlycorrelatedbutarenonethelessdifferent:theextentofexcitationgoesbeyondtheextentofactivation.
Fromamoremathematicalpointofview,wecanalsonotethatLeappearstobesmooth(C)whileuisnot(C0).
Wehavestartedtheformalanalysisofthatpropertysincewethinkthatthismakesacriticaldifferencefortheself-organizationprocedure.
Thissignalprovidesthemodelwiththenecessaryinformationonneighbor-hoodtopology.
Furthermore,thesupportofu(xsuchthatu(x,t)isnotnull)isconstantandindependentoftheinput(thisisapropertyoftheneuralfields)anditmaythusnotprovideenoughinformationforproperself-organization(wenumericallytestedalearningruleusinguinsteadofLewithoutsuccess).
Fromabiologicalpointofview,thismeansthataneuronwhosemembranepotentialisbelowfiringthresholdmaynonethelesslearnsifithasreceivedenoughexcitation.
Atearlystageofthetraining,becauseoftherandomnessofthefeed-forwardweights,anystimuluscancauseaweakresponseofthemodelatarandomplacewithinthefield(seefigure5A).
Asthelearningprocessisongoingandthefeed-forwardweightsconvergeaccordingtoequation(6),theresponseofthemodelbecomesstrongerandoccupiesaspecificspatiallocation(seefigure5B).
SimulationDetailsThesetofstimulithatisusedduringtrainingisinitiallygeneratedbyequation(4)overasubset{0:75,0:752oftheskinpatch{1:0,1:02.
Stimulilocationsaresetonaregulargrid(seefigure4)inordertoensurepropercoverageofthepatch.
Duringtraining,astimulusisuniformlydrawnfromwithinthistrainingset.
Unlessstatedotherwise,thesamestimulisetisusedforallsimulations.
Theneuralfieldhasbeendiscretizedinto32|32spatialelementsandtheintegrationofequation(2)isperformedusingtheforwardEuler'smethod(timestepdtisgivenintable1).
Feed-forwardweightswfarerandomlyinitializedintherange0,1.
Duringallsimulationsweused10000epochs.
Ineachepoch,thestimulusispresentedtothemodelandthefieldisintegratedoverafixedtimewindowwhilethelearningruleisappliedtothefeed-forwardweights.
Spatialconvolutioninequations(2)and(7)iscalculatedusingafastFouriertransform(FFT)inordertoacceleratethisoperation.
Thentheactivityofthefieldisresettozero.
Thisrepresentstheremovalofpressurefromtheskinpatch(wecouldwaitforthefieldtogobacktothesteadystatebutitisnumericallyfastertoresetit).
Thefeed-forwardweightsaverageevolutionEiofaneuroniwasmeasuredbyusingthefollowingequation:EEi~DEwifnew{EwifoldD8whereE:representstheexpectedvalue(i.
e.
themeanvalueofthearray)andD:Drepresentstheabsolutevalue.
LesionsoftypeI,IIandIII(skinorcortex)havebeenimplementedusingthreemasksdisplayedinfigure2.
Forskinlesion,inputwasnullifiedatlesionsitesbeforebeingtransmittedtotheneuralfieldwhileforcorticallesions,lesionedunitswerenullifiedateachtimestep.
Table1.
Modelparameters.
KeKisesimcscsdtatc3.
652.
400.
11.
00.
02.
10.
150.
20.
11.
00.
05KeandKiareamplitudeoftheexcitatoryandinhibitoryweightfunctions.
seandsiarethevariancesofexcitatoryandinhibitoryweightfunctions.
ThemeanandthevarianceofthecorrectiveGaussianfunctionaregivenbymandsc,respectively.
Thevarianceofstimulusisgivenbysandthemeanisvariable,althoughweexplaininthetexthowwecomputeit.
dtisEuler'smethodtimestep.
aisaconstantandtisthetimeconstantofequation(2).
cisthelearningrateoflearningrulegivenbyequation(6).
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t001Figure3.
Threesolutionsofthesameone-dimensionalneuralfield.
Fieldresponse(redcurve)forthreedifferentuniforminputs(bluecurve).
Ineachcase,themaximumactivityofthefieldmatchestheinput.
Thespatialdiscretizationofthefieldis100units.
AResponsetoinput0:25.
BResponsetoinput0:50.
CResponsetoinput0:75.
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g003ANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
plosone.
org5July2012|Volume7|Issue7|e40257Thereceptivefieldofeachneuronhasbeencomputedfromasetofn|n(n~64inthiswork)regularlydistributedstimulusrioverthesubset{0:75,0:752thathavebeenpresentedsequen-tiallytothemodel.
Eachindividualneuronactivityhasbeenrecordedandaggregatedintoa(n,n)matrixofactivities.
ThesizeofthereceptivefieldhasbeenidentifiedwiththenormalizedsumofnonnullvalueswhilethecenterhasbeencomputedasthecenterofmassCofthereceptivefieldgivenbythefollowingequation:C~Pn2i~0ViriPn2i~0Vi9whereridenotestherespectivepositionofstimuliusedtocomputeRFandViistheactivityatpositionri.
Usingself-organizationinformationfromtheintactmodel,wetranslatedthosecentersintotheskinreferencesuchthattopographicalinforma-tioncorresponds,thiseasesthelectureofthefigurewithoutchangingtheresults.
SimulationswereperformedonaHPZ800Workstation.
ThesourcecodeofallsimulationsiswritteninPython(Numpy,ScipyandMatplotlib)anditisavailableon-lineathttp://www.
loria.
fr/,rougier/coding/software/DNF-SOM.
tgz.
Duringasimulationof10000trainingepochs(sweeps),simulationprogramconsumes,190MBofphysicalmemoryandrequires,13minutesofCPUtimeuntilreachingfinalepoch.
ResultsForthesakeofsimplicitywehavesplitfiguresinto6panels,startingfromtheevolutionofaRFoftheneuron(25,15)(exceptthecorticallesionoftypeIIwhereweusedtheneuron(15,25))fromtheepochnumber0andreachingepoch10000throughepochs50,1500and3500.
Thenweillustratethepreferredlocationoftheneuronsbycomputingthecenterofmassaccordingtoequation(9)andthesizeofeachRFofeachneuroninordertodepictdiscsontheskingridwhichindicatethepreferredlocationandthesizeofeachRF.
Inthethirdpanelweillustratetheresponseofthemodelto100differentstimuli,whichcoveruniformlythe{0:75,0:752area(seefigure4).
Inthecaseofskinlesionsthestimuliarelargerthanthelesionandthereforetriggertheneighboringreceptors(asinglestimulusspansalargeportionFigure4.
Locationsoftrainingandvalidationstimuliontheskinpatch.
ATrainingisperformedonasetof16616stimulithatareuniformlydistributedoverthe{0:75,0:752area(skinpatchnormalizedareais{1:0,1:02)suchthatanystimulusisentirelylocatedontheskinpatch(seeexamplestimulusonupperleftcorner).
BValidation(asreportedinCpanelsinresultfigures)isperformedonasetof10610stimulithatareuniformlydistributedoverthe{0:75,0:752area.
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g004Figure5.
Responseofthemodelandlateralexcitation.
Theresponseofthemodelandtheamountoflateralexcitationataspecificsite(centerofactivity).
Plotsrepresenttheresponseprofilecorrespondingtothedashedlines.
ABeforelearning.
BAfterlearning.
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g005ANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
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Thesestimuliarepresentedtothemodelandsimultaneouslytheactivityofthefieldisrecorded.
Inthefourthpanel,wepointouttheevolutionofthefeed-forwardweightsoftheaforementionedneuronandinthetwolastpanelsweusetwohistogramsofthesizeofRFsofthewholeneuralfield.
Thisoverallorganizationallowsustoillustrateconsistentlyimportantalter-ationsthattakeplaceduringcorticalandskinlesions.
Thehistogramsweremadeusing100bins.
Likewise,wemeasuredthemeanandthestandarddeviation(SD)ofthesizeofRFsbeforeandaftercorticallesionsandsensorydeprivations.
EmergenceofOrderedTopographicMapsDuringtheearlystageofthelearningprocess,theresponseofthefieldtoastimulusisnotnulleventhoughfeed-forwardweightshavebeensettorandomvalues.
Itdisplaysinsteadalocalizedbutweakactivityasillustratedinfigure5A.
Thisisduetoneuralfieldpropertiesthatguaranteesuchbehaviordependingontheamountoflateralexcitationandinhibition.
Thisweakactivitybumpishighlycorrelatedwiththepresenceoflateralexcitationatthelocationoftheformer.
Thisallowsmostactiveunitstolearnthepresentedstimulusproportionallytotheirlateralexcitation(seeequation(6)).
Oncethefieldhasbeentrained,theresponsetoanystimulusisstronger(seefigure5B)aswellastheamountoflateralexcitation.
Thisresultsinanincreasedlearningrateforastimulusthatisalreadyknowntothemodelandthus,itdoesnotchangedrasticallythefeed-forwardweightsanymore.
Thisisakeypointofthemodelsincewe'llseeinnextsubsectionhowthisactivelearningrulemayhelptorecoverfromlesions.
TheevolutionovertimeoftheRFoftheneuron(number(25,15))isillustratedinfigure6A.
Initially,theneuronismostlysilent,butafter1500,3500and10000presentedstimuli,onecanseethedevelopmentoftheRFthatisfinallypreciselytunedtoaspecificsetofstimuluslocation.
Thisevolutionoccursintwophases.
Inthefirstphase,theRFisextendedandcoversalargepartoftheskin,theninasecondphase,asthetrainingprocessgoeson,theRFshrinksandcoversonlyasmallpartoftheskin.
Eachneuronnowrespondspreferentiallytoaspecificskinregion.
Moreover,figure6Cshowstheresponseofthemodel(aftertraining)to10610differentstimuli.
Itisquiteclearthatatopographicmaphasemerged.
Eachblockinthisfigurerepresentsaresponseofthemodeltoaspecificstimulus(e.
g.
theblockattheupperleftcornerrepresentstheresponseofthemodeltoastimulusattheupperleftcorneroftheskingrid)providingawaytoverifyself-organizationandalsoprovidesaframeofreferenceofthereceptivefieldlocationontheskinpatch.
Similarresultsareillustratedinfigure6Bwhereweusedequation(9)inordertocomputethelocationofeachreceptivefieldontheskinpatch.
TheradiusofeachcirclehasbeencalculatedbyusingthesizeofeachRF.
Inaddition,thedistributionofRFsizescanundergoalterationsduringtraining.
Panel6EshowsthedistributionofRFsizesbeforeanylearningoccursinthemodel.
Asonecanexpected,thereisnoRFsatallsincetheneuronshavenotyetlearnedanything.
However,oncelearningisfinished,onecanseeinfigure6Fthenormal-likedistributionoftheRFsizes.
Thereisahigh-valuecomponentnearzerowhichindicatesalargenumberofverysmallRFsizesthatisduetoside(border)effectsoftheneuralfield.
OtherRFsizesfollowanormal-likedistributionwithmean0:02246(SD~0:01190).
ThisindicatesthatthereisabetteracquisitionofRFsatthecenterofthefieldthanattheperiphery.
Combiningalltheaforementionedresultswecanconcludethatthemodelhasachievedproperself-organization.
Subsequently,theemergenceofsuchanorderedmaptendstoconfirmtheinitialhypothesisthatthalamocorticalconnectionsareanadequatesiteofplasticityforboththeformationandthemaintenanceoftopographicrepresentations.
Inthiscontext,lateralconnectionsmainlyserveassupportforcompetitionatthecorticallevelfortheemergenceofauniquebumpofactivitythatdriveslearning.
Finally,figure7displaystheRFsofalltheneuronsafter10000presentedstimuli.
Onecanclearlyseethatorderedrepresentationshaveemergedoverthewholefield.
ReorganizationafteraSkinLesionWefirststudiedthecaseofsensorydeprivationresulting,forexample,fromdamagedsensorynervesorphysicaldamagestothereceptors.
Thishasbeenmodeledbysilencingaspecificamountofskinreceptors(25%intypeIandtypeIIskinlesionsand9%fortypeIIIskinlesion)suchthatonlyasubpartofpreviouslysensoryinformationismadeavailabletothecortex.
AlesionwasmadeontothreespecificareaswhicharereferredaslesiontypeI,typeII,andtypeIII(seefigure2forpreciseshape).
Weonlyreport,inthissection,resultsfromtypeIIlesionsincewefoundqualitativelyequivalentresultsregardingtypeIandtypeIIIlesions(seefigures8and9,respectively,fordetails).
Followingaskinlesion,themodelhasbeenretrainedover10000epochsusingthesamesetofstimuliasbeforebutwithmissingvaluesfromdisabledreceptors.
Panel10Ashowsthetemporalevolutionofthereceptivefieldofunit(25,15)andfigure10Cshowstheoverallreorganizationofrepresentationsthathasoccurredaccordingtotheresponseofthemodelto10|10differentstimuli.
Thisismostclearlyillustratedinfigure10Bthatdisplaysthepreferredlocationofunitsthatdonotintersectwiththeskinlesionedarea.
ComparingtheRFsillustratedinthisfigurewiththeonesonfigure6B,onecanconcludethatthesizesofRFswhichwerepreviouslyinnervatedbythelesionedskinareaarenowlarger.
Thisisbecauseneuronslosttheirpreferredinputandthereforethebalanceofexcitationandinhibitionisdisrupted.
ThereforeneuronsexpandthesizeoftheirRFsinordertoacquirenewinputs.
Thisresilientbehaviorcanbeeasilyexplainedbecausethalamusprovidesdivergentinputstothecortex.
NeuronsthatwerepreviouslytunedtodeadreceptorswillexpandtheirRFsinordertoreachneighboringreceptors.
Thisexpansiontakesplaceimmediatelyafterthesensorydeprivationasshowninfigure10Band10AwheretheRFofneuron(25,15)underwentanexpansionimmediatelyaftersensorydeprivation(epoch50).
Panels10Eand10Fshowsthehistogramsbeforeandaftersensorydeprivation,respectively.
Theformercorrespondstotheintactmodelwhichwediscussedinprevioussubsectionandthelatercorrespondstothesensorydeprivationcaseafterretrainingofthemodel.
Thereisasmallshiftofthemainpeakofthedistributionfromthevalueof0:02246towards0:02227,butwithanoticeablespreadingoftheRFs(SD=0:02256)sizeindicatinganewdistributionofRFtowardsbothsmallerandlargerreceptivefields(whilethelargecomponentatzerobecauseofbordereffectsremains).
ThisalterationinthedistributiontendstoshowthatevenifmostRFshaveshrunk,asignificantportionhaveexpandedinsize.
Furthermore,fromfigure10Adescribingthetemporalevolu-tionofunit(25,15),wecanseethatreorganizationoccursintwomajorphases.
Atthebeginning,eachneuroninnervatedbydeprivedskinareaundergoesanexpansionofitsRFsimulta-neouslywithaspatialshiftinginordertocaptureanewskinarea(firstphase).
Thislastsalmostduringthewholeretrainingprocess.
NeartheendoftrainingprocesstheaffectedneuronhasshrunkitsRF(secondphase).
Similartothisfinding,Foellerin[21]proposesathree-phasesmodeloftheRFsreorganization.
InthefirstphaseandduetoreductionofinhibitoryconnectionstheRFsexpandtheirsize.
Duringthesecondphase,afurtherincreaseofRFssizeANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
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Finally,inthethirdphase,ashrinkageofRFsaroundtheirnewcentersoccurringasitisdrivenbyre-establishedinhibitoryconnections.
Wecanmergethetwofirstphasesintooneinourmodelsincewedonotinvolveanykindofneurobiologicalmechanismandthereforeasuchdetailedtimescaleisnotnecessary.
However,therefinementoftheRFsisnotsoexquisitebecause,accordingtoourmainhypothesis,lateralconnectionsremainfixedandnon-plasticthroughoutallsimulations.
ThismeansthatneuronsareabletoreceiveproperexcitationandinhibitionFigure6.
Intactmodel.
AEvolutionofthereceptivefieldofneuron(25,15)duringlearning.
Theneuronisinitiallysilent(epoch0)butlearnsquicklytoanswertoalargerangeofstimuli(epoch1500)untilfinallysettlingonanarrowerrangeofstimuli.
BReceptivefieldsofthewholemodel.
Eachbluecirclerepresentsaneuron.
Thecenterofthecircleindicatesthe(converted)receptivefieldcenterandtheradiusexpressesthe(relative)sizeofthereceptivefield.
CResponseofthemodel(afterlearning)toasetof10610regularlyspacedstimuli.
Eachsquarerepresentaresponsetoaspecificstimulus.
DThisrepresentsthemeanevolutionofthalamo-corticalweightsofneuron(25,15)duringlearning(i.
e.
E(25,15)).
E&FHistogramofreceptivefieldsizes(100bins)before(E)andafter(F)learning.
ThefinaldistributionisGaussian-shapedcenteredaroundameanvalueof0:02246.
Isistobenotedthehighnumberofverysmallreceptivefieldsizethatcorrespondtoneuronsontheborderofthefieldthataremostlysilentduringthewholesimulation.
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Nevertheless,abetterrefinementcouldbepossiblebyusingalearningrulealsoforthelateralconnections.
Furthermoreandasithasbeenexplained,theprecisetypeoflesiondoesnotimpactresultinasignificantway.
Neuronsthatwerepreferentiallytunedtoadisabledskinareatendstohavetheirreceptivefieldsshiftingawayfromthesiteofthelesiontoneighboringlocations.
However,fortypeIIandtypeIIIlesions,thereisanadditionaltopologicalconstraintontothoseneuronsbecausetheycanstillbepartofanactivebumpinthefield(andtunetheirreceptivefieldaccordingly).
TheycanbethusattractedeithertotheleftortotherightpartofthelesionsitefortypeIIandtoanyborderofthelesionsitefortypeIII.
Thisexplainsthatsomeneuronsdonotexpressanykindofresilienceandhavetheirpreferredlocationstillonthelesionedareaevenafterextensiveretraining(figures10Band9B).
Thisalsoexplainstheincreasedoscillationsinaverageevolutionoffeed-forwardweights,Ei(figures10Dand9D).
ReorganizationafteraCorticalLesionWealsoaddressedthecaseofreorganizationbycausingacorticallesion,i.
e.
silencingsomeneuronsintheneuralfield.
Inlivingtissue,suchdamagescanbecausedbyastroke,ahematomaorbyasurgeryeitherfortherapeuticorexperimentalpurposes.
Subsequently,wecausedthreedifferenttypesofcorticallesions(i.
e.
typeI,IIandIII)byapplyingamasktotheself-organizedrepresentationalmapaswepreviouslydescribedinmethodssection.
Theselesionswereofanextentof25%ofthetotalamountofneurons.
WeappliedatypeIlesionclosetotheborderofcorticalsheet,atypeIII,localizedablationandatypeIIband-shapedlesioninthemainlandofcorticalsheet(seefigure2forpreciselesionshapes).
Thus,afterretrainingofthenetworkusing10000stimulipatternsforeachoftheselesioncases,anewrepresentationalmaphasemergedasitisdepictedinfigure11fortypeIlesion.
ComparingtheRFssizebeforeandaftercorticallesion,theyhavebeenclearlyaltered.
Afterlesion,RFstendtobecomelargerandconsequentlytorespondtolargerskinareas.
Moreprecisely,RFssizeafterlesionisalmosttwicebiggercomparedtopre-lesionones.
Asitisshowninfigure11Atheevolutionofneuron(25,15)afteracorticallesionoftypeIhasbeenaltered.
Immediatelyfollowingthelesion(epoch50),RFhasexpandeditselfandcovertheskinpatchwhichwaspreviouslyrepresentedbylesionedneurons.
ThetemporalevolutionoftheRFindicatesthatthisneuronhaschangeditspreferredinputinordertopromotetherecovery.
Inadditionandasitisillustratedinfigure11B,RFsofalmostallneuronshavebeenchanged.
Theradiiofthebluediscshavebeenincreasedinsize,especiallyaroundthelesionsite.
Takingalsointoaccounttheresultscomingfromthehistogramsoffigures11Eand11Fconcerningthepre-lesionandthepost-lesioncases,respectively,onecanseetheoveralldistributionofRFssizehaschangedinfavorofalargernumberoflargeRFs.
ThemeanvalueofRFsaftercorticallesionisequalto0:02235andtheSDisequalto0:02179indicatingasignificantspreadofRFsizes.
ThisisquiteconsistentwithSober[12]whoreportedsimilarresultswiththenoticeabledifference,thatweeksafteralesion,cortexisabletocompletelyrecover,havingitsneuronsRFssharpened.
Thisisbecauseofre-establishmentofinhibitoryconnectionsand/orsproutingofneuralaxonsasithasbeenproposedbyFlorence[57].
Consequently,therefinementofRFsariseintwophases.
Duringthefirstphase,thereisanexpansionofRFstowardslostterritoriesfollowedbyashrinkageofthesecondphase.
Inourcomputationalexperimentsthereisnosuchshrinkageduringthesecondphasebecauseofthefixedsetoflateralconnectionsasitisdepictedinfigures11A,12Aand13A.
Thisleadsustoascertainthatthelateralconnectionsarecrucialtothedevelopmentofstablerepresentationalmaps.
NeuronsarenotabletopreciselyrefinetheirRFssincethereisnobalancemechanismbetweenexcitationandinhibitionwithincorticalcircuits.
Sur[58]hasshownthatintralaminarexcitatoryconnec-tionsarethemajorfactorforexpansionofRFs.
Inconsequence,RFsinfigures11Aand11Bhavesuccessfullyexpandedthemselvesleadingtolargerskinarearepresentationbuthavefailedatshrinkingthemselvesbecauseofthenon-plasticlateralconnec-tions.
Furthermore,itisremarkabletoseethatneuronshavemigratedtocoverthewholeskinsurfaceagain(figures11A,11B)andnon-functionalrepresentations(justafterlesion)havebeenFigure7.
Receptivefieldsoftheintactmodel.
AOfthewholecorticalsheet.
BMagnificationofthewhitebox.
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Themodelisabletorespondagaintostimuliappliedonareasinnervatingneuronswithinthelesionedcorticalarea(figureC).
Thisindicatesthatotherneuronstookoverandrecoveredfromlesionbymigratingtheirrepresentationstowardsthelostones,makingthecorticalpatchfunctionalbutdegraded.
RecoveryfromcorticallesionsoftypeIIandIIIisdisplayedinfigures12and13andshowthedegradedresponseofthemodelwithonlyapartialrecoveryoflostterritories.
Furthermore,figures12A,12Band13A,13BclearlyshowthatmostRFshavebeenshiftedandexpandedspatiallywithoutanykindofrefinementexceptforasmallnumberwhichhaveunderwentashrinkageasitispointedoutbyfigures12Fand13F.
ThisFigure8.
SkinlesiontypeI(grayarea).
AEvolutionofthereceptivefieldofneuron(25,15)duringretrainingafteraskinlesionoftypeI.
BReceptivefieldsofthewholemodel.
CResponseofthemodel(afterretraining)toasetof10610regularlyspacedstimuli.
DThisrepresentsthemeanevolutionofthalamo-corticalweightsofneuron(25,15)duringretraining(i.
e.
E(25,15).
E&FHistogramofreceptivefieldsizes(100bins)before(E)andafter(F)skinlesion.
TheinitialdistributionisGaussian-shapedcenteredaroundameanvalueof0:02246.
However,thefinaldistributionisaPoison-likecenteredaroundameanvalueof0:2241withalongtailindicatingthattherearealotofneuronswhoseRFshaveunderwentanexpansion.
AtthesametimeanalmostequivalentamountofneuronshasmovedtowardsmallerRFsizesunderlyingthatashrinkageofRFshastakenplace.
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ThefirsttypeofcorticallesionistopologicallyequivalenttotheintactonewhiletypeIIlesionintroducesaseparationofthecorticalpatchintotwodistinctpatchesandtypeIIIintroducesaholeinthetopology.
Inbothcases,neuronsfromeithersidesofthelesioncannotcooperatebecausetheirinfluenceismostlyinhibitory(duetotheirrespectivedistancefromeachother).
Thismeansthattheactivepopulationresultingfromthecompetitioncannotexistonanybordersofthelesion.
Thisbringssevereconstraintstotheself-organizationprocessthatcanonlybepartiallyovercomewithoutrelearninganewtopologythroughthemodificationsoflateralconnections.
Figure9.
SkinlesiontypeIII(grayarea).
AEvolutionofthereceptivefieldofneuron(25,15)duringretrainingafteraskinlesionoftypeIII.
BReceptivefieldsofthewholemodel.
CResponseofthemodel(afterretraining)toasetof10610regularlyspacedstimuli.
DThisrepresentsthemeanevolutionofthalamo-corticalweightsofneuron(25,15)duringretraining(i.
e.
E(25,15).
E&FHistogramofreceptivefieldsizes(100bins)before(E)andafter(F)skinlesion.
TheinitialdistributionisGaussian-shapedcenteredaroundameanvalueof0:02246.
Although,thefinaldistributionisaPoison-likecenteredaroundameanvalueof0:2248withalongtailindicatingthattherearealotofneuronswhoseRFshaveunderwentanexpansion.
AtthesametimeanalmostequivalentamountofneuronshasmovedtowardsmallerRFsizesunderlyingthatashrinkageofRFshastakenplace.
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g009ANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
plosone.
org11July2012|Volume7|Issue7|e40257DiscussionWehaveintroducedacomputationalmodelofprimarysomatosensorycortexthatisabletodeveloptopographicmaps,maintainandreorganizetheminthefaceoflesions.
Weusedneuralfieldsasamathematicalandcomputationalframeworkandfocusedonarea3binnervatedbyhandmechanoreceptors.
ThecombinationofsuchneuralfieldwithasimpleHebbian/anti-Hebbianlikelearningruleadvocatesforanunsupervised,distributed,robustandbiologicallyplausiblemodelofa(simplified)somatosensorycorticalmodelwherethalamocorticalconnectionsarethemainsitesofplasticity.
ThemajorfindingofourmodelisFigure10.
SkinlesiontypeII(grayarea).
AEvolutionofthereceptivefieldofneuron(25,15)duringretrainingafteraskinlesionoftypeII.
Immediatelyfollowingskinlesion(epoch50),RFtendstoexpand.
Thisphenomenonpersistsuntilthefinalepochisreachedwhereashrinkagetakesplace.
BReceptivefieldsofthewholemodel.
CResponseofthemodel(afterretraining)toasetof10610regularlyspacedstimuli.
DThisrepresentsthemeanevolutionofthalamo-corticalweightsofneuron(25,15)duringretraining(i.
e.
E(25,15)).
E&FHistogramofreceptivefieldsizes(100bins)before(E)andafter(F)skinlesion.
TheinitialdistributionisGaussian-shapedcenteredaroundameanvalueof0:02246.
However,thefinaldistributionisaPoisson-likecenteredaroundameanvalueof0:2241withalongtailindicatingthattherearealotofneuronswhoseRFshaveunderwentanexpansion.
AtthesametimeanalmostequivalentamountofneuronshasmovedtowardsmallerRFsizesunderlyingthatashrinkageofRFshasalsotakenplace.
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g010ANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
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Thesefeed-forwardconnectionsarecapableofcausingthereorganizationofatopographicmapeveninthepresenceofacorticallesionorasensorydeprivation.
Brunoin[34]hasshownthatexcitatorythalamocorticalconnectionscansynchronizethemselvesinordertodrivecorticalneuronswithoutmakinguseofanykindofcorticalamplificationmechanism.
Thisenhancedourhypothesis,whichstatesthatthemaineffortoftheemergenceandreorganizationofatopographicmapcanbepromotedbythalamocorticalconnections.
Thisalsoholdsforallthreeinvestigatedcases.
First,theformationandemergence(one)ofatopographicmap.
Second,sensorydeprivation(two-congenitalandcontracted).
Andintheend,corticallesions(fourFigure11.
CorticallesiontypeI(redarea).
AEvolutionofthereceptivefieldofneuron(25,15)duringretrainingafteracorticallesionoftypeI.
Immediatelyfollowingthelesion(epoch50),RFtendstoexpand.
Thisphenomenonpersistsuntilthefinalepochisreached.
BReceptivefieldsofthewholemodel.
CResponseofthemodel(afterretraining)toasetof10610regularlyspacedstimuli.
Theactivityofthemodelisnowboundtotheunlesionedarea.
DThisrepresentsthemeanevolutionofthalamo-corticalweightsofneuron(25,15)duringretraining(i.
e.
E(25,15)).
E&FHistogramofreceptivefieldsizes(100bins)before(E)andafter(F)skinlesion.
TheinitialdistributionisGaussian-shapedcenteredaroundameanvalueof0:02246.
However,thefinaldistributionisauniform-likecenteredaroundameanvalueof0:02245(0:02235).
Thisuniform-likedistributionindicatestheexistenceofneuronswhoseRFshaveunderwentanexpansion,butnotashrinkage.
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g011ANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
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Thoseresultsarequiteconsistentwiththeexistingliteratureonthecomputationalmodelingofthesomatosensorycortexeventhoughwethinkwebroughtnewinsightsontheinnermechanisms.
OnetheearliestmodeloftheSIhasbeenproposedbyPearsonetal.
[41].
TheydesignedacomputationalmodelofthesomatosensorycortexbasedonadualpopulationofneuronsFigure12.
CorticalLesiontypeII(redarea).
AEvolutionofthereceptivefieldofneuron(15,25)duringretrainingafteracorticallesionoftypeII.
ThisparticularneuronhasnotexpandeditsRFbutithasreplaceditspreferredlocationasitisdepictedatthefinalprofile(epoch10000).
BReceptivefieldsofthewholemodel.
Thecorticallesionisappearedatthepreferredlocationssincethepreviouslycorrespondingneuronsarenowaffectedbythelesion.
TheRFsaroundthelesionhavebeenincreasedinsizecomparingwiththecorrespondingpre-lesionfigure6B.
CResponseofthemodel(afterretraining)toasetof10610regularlyspacedstimuli.
Theactivityofthemodelisnowboundtotheunlesionedarea.
DThisrepresentsthemeanevolutionofthalamo-corticalweightsofneuron(15,25)duringretraining(i.
e.
E(15,25)).
E&FHistogramofreceptivefieldsizes(100bins)before(E)andafter(F)skinlesion.
TheinitialdistributionisGaussian-shapedcenteredaroundameanvalueof0:02246.
However,thefinaldistributionisaUniform-likecenteredaroundameanvalueof0:02233.
Thisuniform-likedistributionindicatestheexistenceofneuronswhoseRFshaveunderwentanexpansion,butnotashrinkageasincorticallesiontypeIcase.
Inthiscaseweillustrateresultsregardingneuron(15,25)becauseneuron(25,15)liesinthelesion.
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Followingtherepetitivetappingofthesensorysurface,themodelisabletoshapeitselfintoseveralsegregatedneuronalgroupsthatarededicatedtoasubpartofthewholesensoryspace.
Authorsusedbothintrinsicandextrinsicconnectionsmodificationsalthoughtheyunderlinedthattheydonotknowifthisisreallythecaseinvivo.
Joublinetal.
[59]introducedacombinationofneurophysiologicalrecordingsfromratswithcomputationalsimulations.
ThemodelisdescribedusingasetofWilson-Cowanequationsandthearchitectureismadeofthreelayers(receptorsgrid,subcorticalgridandcortex).
Authorstriedtokeepthemodelclosetorealdatainordertobeabletocomparetheirsimulationswithneurophysiologicaldata.
However,themodelreliesonapre-corticallevelwhoseroleistotopographicallyorderrepresentationsinordertosimplifythesimulation.
Furthermore,theydidn'ttreatlesioncasesandfocusedinsteadonthelearningrulepointingoutthatdifferentlearningrulesunderlyingdifferentformsofplasticity.
XingandGerstein[60–62]usedaspikingneuralnetworkwithinathreelayermodel(i.
e.
receptors,thalamus,cortex)payingattentiontothelateralconnections.
Authorsshowedthatinhibitoryconnectionsarecrucialforlimitingthenumberofactivatedcorticalneurons,whilethebalancebetweenexcitationandinhibitioniscrucialtothestabilityofthenetwork.
Thisisquiteconsistentwithourownresultssincethecompetitiveprocessoccurringwithinthemodelreliesonaprecisebalancebetweeninhibitionandexcitation.
Inthecorticallesioncases,we'vealsoshownthatthemodelcannotachievefullrecoverywithoutmodificationsofthelateralconnec-tions.
Thisisagainquiteconsistentwiththeimportantroleoflateralconnectionsgivenbypreviousmodelseventhoughthosemodelsdidnotaddressspecificallythecaseofcorticalandcutaneouslesions.
WecanconcludewiththemthatlateralconnectionsplayanimportantroleandtherefinementofRFsfollowingalesionmaybeduetothemodificationoflateralconnections.
However,wemaintainthatthalamocorticalafferentconnectionsarethemainsitesofplasticityforbothprimaryself-organizationandlaterreorganization.
Wewouldlikenowtopointoutafewmoreinterestingresults.
FoellerandFeldman[21]aswellasFlorenceetal.
[63],proposedthatRFsarecapableofrefinementandshrinkageduringalong-termreorganizationprocessofatopographicmapinthepresenceofasensorydeprivation.
Ourmodel,duetonon-labilelateralconnections,isnotabletoachievesuchpreciserefinementduringthereorganizationofthetopographicmap.
ThisleadsustoclaimthatlateralconnectionsisamajormoderatorofRFs,especiallyduringthereorganizationprocessofthecorticalsheet.
Neverthe-less,wehavebeenabletoshowtheexpansionofRFs,whichmeansthatRFsareable,duringthereorganizationprocess,torepresentalargerskinarearatherthantheydidbeforelesion.
Butthisisonlyonepartofthewholepictureasitisonlyoneoutoftwo(ormaybethree)reorganizationphases.
Thissecondphaseismissinginourmodel.
DuringthatreorganizationphaseashrinkageofRFstakesplaceduetoadaptationoflateralconnections,thesproutingofnewintra-corticalconnectionsandtheleft-overunaffectedthalamocorticalconnections.
Inadditionlateralconnectionsmustbeanimportantandvaluablemechanismofthebalancebetweenexcitatoryandinhibitoryneuralpopula-tions,which,inturn,steerstothereorganizationofrobusttopographicmaps.
Nevertheless,ourmodelindicatesthatevenwithoutrelearninglateralconnections,corticalsheetisabletofullyorpartiallyrecoverfromacorticallesiondependingonitstype.
LesionoftypeIdoesnotmodifythetopologyofthefieldandallowforarobustandfullrecoverywhilelesionsoftypeIIandIIIaremoreproblematicandleadstopartialrecoveryonly.
Fromaneurophysiologicalpointofviewsuchcorticallesionsmeansthattheskinpatchwhichprovidesafferentinputtothedeadneuronslosesitscorticalrepresentation.
Hence,neuronsthatareunaffect-edbylesionreceiveinputfromthenon-representativeskinpatchandareorganizationoftheSItopographicmaptakesplace.
TheconsequenceinourmodelisanexpansionoftheRFfortheunaffectedneurons(seefigure13B).
However,onewouldexpectunaffectedneuronstocoveralmostthewholeskinpatchoratleastalargerpart.
Instead,itisobviousthatthereisacoveringoftheskinpatchbutstillthereisapartoftheskinwhichremainsuncovered.
Thismeansthatthereisinputfromsomeareasoftheskinbuttheresponsibleneuronsarenowdead.
Afterthereorganizationprocessanewtopographicmaphasformedandhencetheunaffectedneuronshavetakenoverthepreviouslynon-representativeterritoriesoftheskinpatch.
Thisphenomenonisillustratedinfigure13C,wherethemodelisabletorespondtodifferentstimuli.
Therefore,thetuningofRFsofunaffectedneuronsisnotoptimal.
Webelievethatthisiscloselyrelatedtothelackofrelearningoflateralconnections.
Moreprecisely,thecorticallesiondisruptsthebalanceinthelateralconnectionsandwedonotallowthemodeltofixitbyrelearningtheseconnections.
Thisseemstobeacriticalprocessbecauselateralconnectionsarenotabletoconveypropercompetitionanymore.
Atthispoint,wecanpointouttwomajorcharacteristicsofacorticallesion(orablation)thatcouldberesponsibleforaproperreorganizationandrecoveryofacorticalsheet.
First,isthelocationofthelesion,Wherethelesionislocatedandsecondtheextentofthelesion,WhatistheamountofthedeadneuronsBoth,locationandextentareintertwined,inafashionthattheformerpervadesthelaterandviceversa.
Therefore,wecandiscriminatetwodifferentcases.
First,ifthelesionislocatedaroundtheborderoftwoormorecorticalrepresentationsandprovidedtheextentofthelesionisnottoolarge,thenrecoveryiseasilyachievable.
Thisisbecauseofthelargeamountofleftoverneuronsandafferentconnections.
However,iftheextentofthelesionislargeenough,thentherepresentationscannotrecovercompletelyandtheymayevennotrecoveratall.
Second,ifthelesionislocatedwithinasinglerepresentation,recoveryisonlyamatteroftheextentofthelesionitself.
Thatisbecause,ifasignificantamountofneuronsareaffectedbythelesion,thereisnoenoughneuronstodeploytheirRFsandrevivepreviouslylostterritory.
Yet,inalocalizedlesioncasewenoticedthatinthevicinityofthelesionedarea,therearesomeneuronswhichdonotrespondsointensivelyastheothersandthoseneuronsdriveotherneuronsleadingthemtoreorganizethemselves.
Asitisdepictedinfigure4thoseneuronshaveaspecificshapewhichisinheritedbytheneighboringneuronsthatexpandtheirRFsomnidirectionaly.
SimilarfindingshavebeenpointedoutbySoberin[12],whereadisinhibitoryhaloaroundcorticalablationhasbeenfound.
Proximalneuronstothishaloareabletodrivethereorganizationoftheneighborhoodneuronsviatheirintactlateralconnections.
ProximalneuronshavebeenloosedtheirinhibitoryconnectionsduetoablationandthereforetheyhaveomnidirectionallyexpandedRFs.
FurthermoredistantneuronshavenarrowerRFs.
Likewise,thecaseofcorticallesiontypeIIIofourmodelpresentsasimilarbehaviorasithasbeenillustratedbyfigures6Band13Bconsideringthepre-andpostlesionstate,respectively.
RFsinthelaterfigurearelargerthanthoseintheformerfigure.
InthelaterfigureneuronsaroundlesionhavelargerRFsinsize.
ThisisinaccordancewiththeresultsofSoberandtheso-calleddisinhibitoryhalo.
Although,inourmodelwekeepthelateralconnectionsfixedandthereforeinthecaseofacorticallesionthereisnowaytorecoverthem.
Hence,adisturbanceoflateralconnectionstriggersadisturbanceANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
plosone.
org15July2012|Volume7|Issue7|e40257ofexcitation/inhibitionbalanceandthusneuronsclosetolesionreceivemostlyexcitatoryconnectionsratherthaninhibitory,whichinturncausestheexpansionoftheRFsaroundlesionandtheshrinkageofthedistantones.
Totestfurtherthishypothesis,wecanobservethatlateralcortico-corticalconnectionsareamajorcomponentofthecompetitivemechanismthatallowstohaveauniqueandcompactactivepopulation.
Theshapeofthispopulationiscriticalforlearningsinceitenforcesthetopologywithinthemodel.
Moreprecisely,wecanpredictthatanymodificationonthesizeoftheactivepopulationwouldhaveaFigure13.
CorticallesiontypeIII(redarea).
AEvolutionofthereceptivefieldofneuron(25,15)duringretrainingafteracorticallesionoftypeIII.
ThisparticularneuronhasexpandeditsRFimmediatelyafterlesionandmoreoverithashasreplacedhispreferredlocationasitisdepictedatthefinalprofile(epoch10000).
BReceptivefieldsofthewholemodel.
Thecorticallesionisappearedatthepreferredlocationssincethepreviouslycorrespondingneuronsarenowaffectedbythelesion.
TheRFsaroundthelesionhavebeenincreasedinsizecomparingwiththecorrespondingpre-lesionfigure6B.
CResponseofthemodel(afterretraining)toasetof10610regularlyspacedstimuli.
DThisrepresentsthemeanevolutionofthalamo-corticalweightsofneuron(25,15)duringretraining(i.
e.
E(25,15)).
E&FHistogramofreceptivefieldsizes(100bins)before(E)andafter(F)skinlesion.
TheinitialdistributionisGaussian-shapedcenteredaroundameanvalueof0:02246.
However,thefinaldistributionisaUniform-likecenteredaroundameanvalueof0:02227.
Thisuniform-likedistributionindicatestheexistenceofneuronswhoseRFshaveunderwentanexpansion,butnotashrinkageasincorticallesiontypeIcase.
doi:10.
1371/journal.
pone.
0040257.
g013ANeuralFieldModeloftheSomatosensoryCortexPLoSONE|www.
plosone.
org16July2012|Volume7|Issue7|e40257directimpactonthereceptivefields.
Forexample,ifweweretodecreasetheinhibitionlevelatthecorticallevelwhileblockinglearning,thesizeoftheactivepopulationwouldgrowandthiswouldresultinlargerreceptivefields.
Thiswouldmeanalossinsensoryrepresentation:thetwopointdiscriminationdistancewouldbeincreased.
Ontheopposite,ifweweretoincreasetheinhibitionlevel,thesizeoftheactivepopulationwouldbecomesmallerandleadtosmallerreceptivefields(andhigherprecision).
Themodelhasbeenkeptdeliberatelysimpleanditcomesasnosurprisethatanumberofknownmechanismshavenotbeentakenintoaccountlikeforexamplehomeostaticmechanismsand/ormetaplasticitywhichhavebeenproposedbyTurrigianoandNelson[64]asmoderatorfactorsoflateralconnections.
Theformerconservesandregulatestheaverageactivityofbraincircuitsbyscalingneuralsynapsesandthelaterpreventsthemfromsaturationeffects[65].
Asfutureworkwelefttheexaminationofhomeostaticmechanismsandmetaplasticityaswebelievethatthismodelisofferedforfurtherinvestigationthroughitsabilitytoadjustitsactivitydependingontheintensityofstimulus.
This,inturn,canpreventnetworksfromsaturationeffectsinthesamewaymetaplasticitymayaffectneuralcircuitsofthebrain.
Anotheraspectwedidnottreatisthephenomenonofspontaneousactivity.
Itisrationaltodiscussaboutthisbecauseitseemstoplayakeyroleinthedevelopmentofatopographicmapwithincortex.
Forinstance,KatzandShatz[66]andKhazipovandLuhmann[35]havefoundamechanismwhichcouldexplaintheearlyformationofL4inbarrelcortexandV1inrats,respectively.
Inbothcasesthetopographicmaphasbeenformedalmostcompletelybeforebirth.
AddingtothatfindingsfromKhazipovetal.
[36],wecanconcludethatafetusinuteromaytakeadvantageofspontaneousmovementsinordertoestablishanearlyformedtopographicmapinL4withinasensorimotorloop.
Wealsoneglectedtop-downmechanismssuchasattentionalmodulatorysignals.
Knightetal.
in[67]haveproposedthattheprefrontalcortexactsasamodulatorofbalanceofexcitationandinhibitionofthebrain.
Thisprovidesastraight-forwardattentionalmechanismsincethisregulationofbalancecanaffectthereceptivefieldsofneurons.
Furthermore,accordingtotheresultsofSchaefer[68],prefrontalcortexseemstoprovidetosomatosensorycorticalareasagatingmechanismwhichisabletorefinereceptivefieldsthroughinhibition/excitationregulationregardingtoattention.
Weneglectedsuchmechanismsinthisworkbecausewebelievethattheyareoutofthescopeduetothelackofaclosedloop(e.
g.
sensorimotorloop).
Wethusleftasfutureworktheinvestigationoftheroleoftop-downmechanismsintopographicmapsformationandreorganization.
Inconclusion,eventhoughthismodeldoesnotconsiderallneurophysiologicalaspectswhichmightplayanimportantroleintheoverallorganizationprocess,webelievethatitcanhelptoinvestigatefurthertheemergenceofsomatotopicmapsduringtheearlymonthsoflife.
Themodelissimpleenoughfromamathematical/computationalpointofviewtoallowforfurtherrefinementthatcouldpotentiallygiveaccountonmoreexperi-mentaldata.
AuthorContributionsConceivedanddesignedtheexperiments:GIDNPR.
Performedtheexperiments:GID.
Analyzedthedata:GIDNPR.
Contributedreagents/materials/analysistools:GIDNPR.
Wrotethepaper:GIDNPR.
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