Dynamical Analysis Of Two Kinds Of Neuronal Models

The article studies two kinds of neuron models in this paper, one kind of them is a three-dimensional Morris-Lecar model.Through two parameter bifurcation diagram and the single parameter bifurcation diagram and the application of fast and slow dynamics bifurcation analysis methods, We analyze the bifurcation mechanism produced by different types of firing.The other kind is a three-dimensioned neuronal model, the model is composed of fast subsystem of Wilson model and slow subsystem of Hindmarsh-Rose model. And we study the theory of the model by using the bifurcation theory of dynamical systems knowledge, discuss equilibrium point of the model, including the number and stability of equilibrium points, analyse the Hopf bifurcation and the Bogdanov-Takens bifurcation near the equilibrium point, the corresponding conditions and conclusions of bifurcation are obtained.This article is divided into four chapters.The first chapter introduces the significance of neuron model and arrangements for this article.The second chapter describes the basic concepts and methods of this paper, and mathematical software used.The third chapter is analysis of the bifurcation of the fast subsystem in a three-dimensional Morris-Lecar model and make the corresponding two parameter bifurcation diagram and the single parameter bifurcation diagram. These work are mainly based on the bifurcation theory of dynamical system. Then according to the fast and slow dynamics knowledge, we discuss the bifurcation mechanism produced by different types of firingIn the chapter 4.We consider a three-dimensioned model by the use of dynamical system bifurcation theory, including the number and stability of equilibrium points, analyse the Hopf bifurcation and the Bogdanov-Takens bifurcation near the equilibrium point, the corresponding conditions of bifurcation are obtained.At last we analyze the bifurcation near the Bogdanov-Takens bifurcation point, and we get that there are saddle-node bifurcation, Andronov-Hopf bifurcation and homoclinic bifurcation.