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一类含时滞的扩散单神经元模型的分岔分析（英文）.docx

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一类含时滞的扩散单神经元模型的分岔分析（英文） Title:BifurcationAnalysisofaDelayedDiffusiveSingleNeuronModel Abstract: Neuronaldynamicsplayacrucialroleinunderstandingvariousphysiologicalandpathologicalprocessesoccurringinthebrain.Thestudyofsingleneuronmodelshelpsustocomprehendthecomplexbehaviorofneuralnetworks.Inthispaper,wefocusonaparticularclassofsingleneuronmodelsthatincorporatetimedelaysanddiffusion.Weconductabifurcationanalysistoinvestigatetheimpactoftimedelaysonthedynamicsofthesystem.Ourfindingsshedlightontheemergenceofdifferenttypesofbifurcationsandtheirimplicationsinneuronalactivity. 1.Introduction Thebrain'sabilitytoprocessandtransmitinformationreliesheavilyonthecollectivebehaviorofindividualneurons.Singleneuronmodelsofferasimplifiedyeteffectivewaytostudythedynamicsofneuralactivity.Inthisstudy,weextendtheclassicalsingleneuronmodelbyincorporatingtimedelaysanddiffusion,aimedatcapturingmorerealisticneuronbehavior.Weaimtoexploretheimpactofthesefeaturesonthesystem'sbifurcationbehavior. 2.MathematicalModel Webeginbypresentingthemathematicalformulationofourdelayeddiffusivesingleneuronmodel.Themodelincorporatesbothtemporaldelays,whichaccountforthefinitetimetakenforsignalpropagationalongneuronalaxons,anddiffusion,whichrepresentsthespatialspreadofactivityintheneuron.Wedescribethemodelusingasetofpartialdifferentialequationsandappropriateboundaryandinitialconditions. 3.StabilityAnalysis Tounderstandthestabilitypropertiesofourmodel,weperformalinearstabilityanalysis.Bylinearizingthemodelequationsaroundtherestingstate,weobtainacharacteristicequationwhoserootsdictatethestabilitybehaviorofthesystem.Weinvestigatetheeffectsofvariousparameterssuchasdelayanddiffusioncoefficientsonstability.TheanalysisrevealstheexistenceofcriticalpointsandHopfbifurcations. 4.HopfBifurcationAnalysis TheoccurrenceofHopfbifurcationssignifiestheemergenceofoscillatorybehaviorinthesystem.WeanalyzetheconditionsforHopfbifurcationstooccurinourdelayeddiffusivesingleneuronmodel.Byemployingacombinationofnumericalsimulationsandana