tm Series FROM NEURON TO NETWORK MEASUREMENT, ANAL

tmSeriesFROMNEURONTONETWORK:MEASUREMENT,ANALYSISANDMODELINGPart5:ModelsandsimulationtoolsofbiologicalneuronsandnetworksKunoWyler,DanielStainhauser,RolandE.Best,HendrikusW.G.M.BoddekeMay22,1996AbstractTherstfourpartsofthisseriesgaveanintroductiontomethodsofmeasuringandanalysisofsignalsinbiologicalneuralnetworks.Theremainingtwopartswillconsideraspectsofmethodsinneuronalmodeling.Part5willreviewdierentlevelsofmodelingandwillgiveasurveyofappropriatesimulationtools.Finally,part6willshowinatutorialsectionanimplementationofamodelinMatlab/Simulinkandatypicalapplicationofit.1IntroductionInthelastdecadecomputermodelingandsimulationhasbecomeamoreandmoreimportanttoolinneuroscience.Thetraditionalexperimentationcycleformedbyaniterationbetweenhypothesis(theory)andexperiment(measurementandanalysis)hasbeenreplacedbyanewcyclethatincludescomputationalmodeling(simulationoftheory).Thisdevelopmenthasledtotheemergenceofanewsubdiscipline‘computationalneuroscience’betweenneuroscienceandcomputerscience(see[1]).Thegoalofcomputationalneuroscienceistodevelopmodelsdescribinghowthenervoussystemorsomepartofitisoperatingondierentsystemlevelsandthereforetoprovideaninterfacebetweenexperimentalandtheoreticalworkinneuroscience.Forfurtherreadingthefollowingbooksarerecommended:MethodsinNeuronalModeling[2],FoundationsofCellularNeurophysiology[3],HandbookofBrainTheoryandNeuralNetworks[4],andNeuralandBrainModeling[5].Onemajorproblemneuralmodelersarefacedwithistochoosetheappropriatecomplexityofthemodel.Howmuchdetailshadtobeincludedintoamodeltosimulateaccuratelyabiologicalexperimenttoanswerthequestionsasked?Thereisawiderangeofchoiceinmodelcomplexity,fromverysimplepointmodelsofaneurontoverycomplexmodelslikecompartmentalmodelsthatincludedetailedanatomicalandphysiologicaldataofanervecell.Whichmodeltochoosedependsonhowmuchinformationisavailableoftheneuronunderconsiderationandwhatquestionsshouldbeaddressedbythesimulations.Ontheonehand,themodelermaybelimitedbytheavailableexperimentaldataandhastomakereasonableguessesinstead,ontheotherhandverycomplexmodelscaneitherrunintocomputationallimitationsorbeartheproblemoftoomanyfreeparameters(degreesoffreedom)whichwilltanygivensetofdata.Usuallyonelooksforthesimplestmodelthatiscapabletoexplainortoreproducethemeasuredexperimentaldata(principleofOccam’sRazor1).1Thesimplestexplanationofanobservedphenomenaismostlikelytobethecorrectone.1Thisarticlereviewsinsection2themostpopularmodelsofdierentcomplexityformodelingsinglenervecells,whichareusedasbasicbuildingblockstoconstructnetworks.Anexampleofsuchanetworktostudyactivitypatternsinthespinalcordofaswimminglamprey(eel-likesh)isdescribedinsection3.Finally,section4beginswithsomeremarksonnumericalintegrationmethodstypicallyusedbysimulationprogramsforneuronalmodelingandendswithasummaryofavailableprogrampackagestosimulatemodelsofarbitrarycomplexity.2ModelsofsinglecellsFromaconceptualpointofviewallmodelsofsinglecellssharetwofeaturesincommon:(1)theyprocessmanyinputs(excitatoryandinhibitory)toproduceasingleoutputand(2)theyhaveatleastoneinternalstatevariableusuallycorrespondingtothemembranepotentialofthecell,whichwillbeincreasedbyexcitatoryinputsanddecreasedbyinhibitoryinputs.2.1Single-pointmodelsThesimplestmodelsofaneuronaresocalledsingle-pointmodels.Theydonottakeintoaccounttheanatomyofthecellbutonlyitstopologicalconnectivitywithothercells.ThephysiologyofthecellisrepresentedbyasinglestatevariableV(membranepotential).Thusmorphologyandphysiologyofthecellisreducedintoasingle’point’(hencethename’single-point’models).Thesetypesofneuronmodelsarealsousedtobuild’articialneuralnetworks’(ANN’s[6])andthedynamicsaretypicallydiscreteintime(synchronousandasynchronousoperationmodes).IftimeisconsideredtobediscretethemembranepotentialVj(t+1)ofacelljattimet+1isgivenbytheweightedsumoftheoutputsignalsxi(t)ofthepre-synapticcellsi,Vj(t+1)=nXi=1wjixi(t):(1)Thepost-synapticpotentialofthecelljevokedbyasynapticreleaseofthecelliiscontrolledbyreal-valuedcoecientswji.Forpositivevaluesofwjitherewillbeexcitatorypost-synapticpoten-tials(EPSP’s)andfornegativevaluestherewillbeinhibitorypost-synapticpotentials(IPSP’s).TheeectiveactivitylevelxjofacelljdependsonitsactualpotentialVjandismodeledbyacharacteristicoutputfunctionf(Vj).2.1.1LinearmodelsThemostsimplest(andprobablyleastrealistic)neuronmodelisachievedwhenfissettosomelinearfunction,e.g.theidentityfunctionf(V)=V:xj(t+1)=f(Vj(t+1))=Vj(t+1):(2)Theoutputofacellisalinearcombinationofallofitssynapticinputs.Theadvantageofsuchlinearmodelsisthatnetworksmadeofthesecellscanbeanalyzedbyusingmethodsoflinearalgebra.Butunliketorealneurons,outputsoflinearneuronscanbecomearbitrarilylarge(positiveandnegativevalues).2.1.2NonlinearmodelsTheproblemofunboundedoutputsofacellcanbehandledbyreplacingthelinearfunctionbysomenonlinearfunctionwithupperandlowerbounds.Ofcoursetheresultingmodelswillthen2benonlinearandtheanalysisofnetworksbecomesmoredicult.Oneofthemostpopularsingle-pointmodelofneuronsistheoneintroducedbyMcCulloch-Pitts[7].TheirmodelisbasedonasteporHeavisidefunctionH(=f)H(V