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nirtoudtcoinembmarenptoneitlasaaj(mu)piduisnorpoecssoPsiosnpskiertanisniofmrtaoinrtasnmviewthemembranepotentialbetweensuccessivespikesasastochasticprocessof(jump)diffusiontype:siisnonialgreystsmesfosynapses,dendrites,soma:additivityandexponentialdecayoneneuronhasO(104)synapses,90%excitatory,10%inhibitorycontributionofincomingspikestothemembranepotential:left:singleexcitingsynapsis;middle:singleinhibitorysynapsis;right:2excitingand1inhibitorysynapsescombinedenrunoststasiitaclniefercn,eomedlevirctaoinerefercnse
nirtoudtcoinembmarenptoneitlasaaj(mu)piduisnorpoecssoPsiosnpskiertanisniofmrtaoinrtasnimssoinnialgreystsmesfoenrunoexample3:spiketrainsrecordedinthevisualcortex210iidexperiments-identicalvisualstimulus(Shadlen-Newsome98)hence:viewthespiketrainµemittedbyoneneuronasarandompointmeasureon[0,)withstochasticintensitysuchthatmeanvalueofintensityattimetcorrespondstostimulusattimetststasiitaclniefercn,eomedlevircationerefercnse
nirtoudtcoinembmarenptoneitlasaaj(mu)piduisnorpoecssoPsiosnpskiejumpdiffusionprocessmodelizationrtanisniofmrtaoinrtasnimssoinnialgrforthemembranepotentialbetweensuccessivespikeseystsmesfoenrunostsamanymodelsaretimehomogeneous,e.g.mean-revertingOrnstein-Uhlenbeck(Lansky-Lanska87,Tuckwell89,Lansky-Sato99,Lansky-Sacerdote01,Ditlevsen-Lansky05,...)orCox-Ingersoll-Ross(Lansky-Lanska87,Giorny-Lansky-Nobile-Ricciardi88,Lansky-Sacerdote-Tomassetti95,Ditlevsen-Lansky06,Brodda-Ho¨pfner06,...)stage1(timehomogeneousandstationary):CIRtypemodel(Vt)t0foraneuronbelongingtoanactiveneuronalnetworkpdVt=([KR+f]Vt)τdt+σ(VtK0)+τdWtwithconstantsσ,τ>0,referencelevelsK0<KR<KEK0:lowerboundforpossiblevaluesofthemembranepotentialKR:meanvalueofmembranepotentialforneuron’atrest’KE:excitationthresholdandsomequantitymeasuringthedegreeofactivityofthenetworkf0:cnotsnaterrpseneitgntsergnhtfoxeetnrlatsmilusuittscilaniefercn,eomedlevirctaoinerefercnse
nirtoudtcoinewmllmerbnaeptoneitlasaaj(mu)piduisnorpoecssoPsiosnpskiertanisniofmrtaoinrtasnimssoinnialgreystsmesfoenruknown:shiftingmembranepotentialVbyK0,process(VtK0)t0´`isergodicwithinvariantlawΓσ22(KRK0+f),σ22on[0,)2has(stationary)meanKRK0+fandvarianceσ2(KRK0+f)notdependingonthetimeconstantτ(Cox-Ingersoll-Ross85,Ikeda-Watanabe89,...)timehomogeneousCIRmodelgivesconvincingfitforthemembranepotentialdataofexample1(newelectronicstabilizationdevicewasusedbyKilb)(Jahn09)noreasonablefitforsomeofthemembranepotentialdatainexample2(atleastinlevels8,9,10whereneuronisabletogeneratespikes)(Ho¨pfner07)butinmanydatasetswhichseemCIRcompatibleevidencefortimedependenceconcerningtermf0inthedriftssomeindicationforpresenceofjumpsopenquestion;PRO:biologicalreasons;CONTRA:sophisticatedsemimartingaletools(Jacod09,AitSahalia-Jacod09)donotworkasastheyshouldforsure,neuronscanbehavedifferently:OU,othertypesofdiffusions,nodiffusionatall,...,but:CIRseemssuitableforslowlyspikingneuronsbelongingtoanactivenetworktstasiitaclniefercn,eomedlevirctaoinerefercnse
nirtoudtcoinembmarenptoneitlasaaj(mu)pidusionrpoecssoPsiosnpskiertmorerealistic:betweensuccessivespikesanisniofmrtaoinrtasnimssoinnialgreystsmesfoenrunousedeterministicfcttf(t)oftime(strengthofexternalstimulus)introducePoissonjumps,positiveandsummable:PRMµ(dt,dy)on(0,M)×(0,)withintensityτfe(t)dtν(dy)sindependentofBMW,forsomedeterministicfunRctiontfe(t)andsomeσ-finitemeasureν(dy)on(0,M)suchthat(0,M)yν(dy)<stage2:timeinhomogeneousmodelwithjumps:ZpdVt=([KR+f(t)]Vt)τdt+y|µ(d{t,zdy})+σ(VtK0)+τdWt-fe(t)tsapathwiseuniqueness,uniquestrongsolution(Yamada-Watanabe71,Dawson-Li06,Fu-Li08,...)explicitLaplacetransformsfortransitionprobabilities(Kawazu-Watanabe71,...)ittscilaniefercn,eomedlevirctaoinerefercnse
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