Dynamical Analysis Of Two Kinds Of Neuronal Models

This article is composed of two parts: the first part consider a two-dimensional model ofcerebellar Purkinje cells is. By using bifurcation methods and numerical simulations, weexamine the global structure of bifurcations of the model. Results are summarized in varioustwo-parameter bifurcation diagrams with the stimulating current as the abscissa and the otherparameter as the ordinate. We also give the one-parameter bifurcation diagrams and pay muchattention to the emergence of periodic solutions and multistability. Different classes ofmembrane excitability are obtained by bifurcation analyses and frequency-current curves. Wealso show that the neural models possess bistability and tristability. The second part weconsider the bifurcation of a three-dimensioned Hindmarsh-Rose model by the use ofdynamical system.The conditions of Hopf and Bogdanov-Takens bifurcations are discussed.This article is divided into four chapters.The first chapter introduces the significance of neuron model and arrangements for thisarticle.The second chapter describes the basic concepts and methods of this paper,lists the toolsXPP,Matlab and matcont for drawing the bifurcation diagrams,phase portraits andwaveforms.In the chapter3.By using bifurcation methods and numerical simulations, we examinethe global structure of bifurcations of the model. Results are summarized in varioustwo-parameter bifurcation diagrams with the stimulating current as the abscissa and the otherparameter as the ordinate. We also give the one-parameter bifurcation diagrams and pay muchattention to the emergence of periodic solutions and multistability. Different classes ofmembrane excitability are obtained by bifurcation analyses and frequency-current curves. Wealso show that the neural models possess bistability and tristability,and we also get transitionsamong three attractors.In the chapter4. we consider the bifurcation of a three-dimensioned Hindmarsh-Rosemodel by the use of dynamical system bifurcation theory.At last we do numerical simulationnear the Bogdanov-Takens bifurcation point,and we get that there are saddle-nodebifurcation,Andronov-Hopf bifurcation and homoclinic bifurcation.