Dynamic Analysis Of Two Classes Of Four-dimensional Neuron Models

Two classes of neural models are studied in this paper.One is the neocortical neuron model in humans and mammals.The neuron model is governed by the interaction of more than ten ion currents,and the kinetic behavior of the neuron model is very complex.In order to facilitate the study of mathematics,this paper consider a class of simplified four-dimensional neocortical models with four ionic currents.The dynamic behavior of the model is analyzed by using the bifurcation theory of dynamical system,combing the analysis method of slow-fast dynamics and numerical simulation software.In this paper,the one-parameter bifurcation diagrams and the two-parameter bifurcation diagrams of the model are discussed,and the variation of the bifurcation behavior with the change of the parameter is presented,as well as the phenomena of the firing modes such as resting,spiking and bursting are also shown in this article.Finally,a Hopf bifurcation and a Bogdanov-Takens bifurcation are studied respectively.The other is a four-dimensional neuron model describing a sciatic nerve chronic constriction injury model.The different firing patterns of this model are analyzed by using the method of slow-fast dynamics and numerical simulation.The dynamical mechanisms of different firing patterns are obtained by making a one-parameter bifurcation diagrams with the change of the fast subsystem and the stable limit cycle of the corresponding firing state.The full text is divided into four chapters:The first chapter introduce the background of neuron model and the present research situation.The second chapter describe the knowledge points,research ideas and related software in this paper.The third chapter devotes to research the dynamics of the four-dimensional neocortical neuron model by using the method of slow-fast dynamics,and considers the model with a slow variable and two slow variable,respectively.The different types of firing patterns and the bifurcation mechanisms are discussed.At last,we consider Hopf bifurcation and Bogdanov-Takens bifurcation,and get the saddle-node and Hopf bifurcation curves as well as the homoclinic curve near Bogdanov-Takens point.The fourth chapter studies another four-dimensional neural model(a sciatic nerve chronic constriction injury model)by using the slow-fast dynamics.The dynamical mechanism of different firing patterns is obtained by making the one-parameter bifurcation diagrams of the fast subsystem and the stable limit cycle corresponding to the firing state.