Dynamical Analysis Of Sherman-Rinzel-Keizer Neuronal Model

Pancreatic ? cells exhibit periodic bursting activity which can be respond by Sherman-Rinzel-Keizer model. The model consists of three nonlinear first-order differential equations, which respectively represent the membrane voltage, activation parameters of voltage gated potassium channel and the dynamical equation of intracellular concentration of calcium. This paper is mainly research on dynamic analysis for Sherman-Rinzel-Keizer model. By using the knowledge of qualitative theory and bifurcation theory of differential equations, we firstly consider the bifurcation of a reduced model of the three-dimensional neuron models. We mainly discuss the equilibrium points of the model, including the number and stability of equilibrium points and present a detailed analysis of the Hopf bifurcation and the Bogdanov-Takens bifurcation of equilibrium points. Thus we get the corresponding bifurcation curves. These curves contain saddle node bifurcation curves, Hopf bifurcation curves and homoclinic bifurcation curves. Secondly, through the application of fast and slow dynamics bifurcation analysis methods, we study the dynamic properties of bursting pattern of a kind of Sherman-Rinzel-Keizer models. This process is mainly based on the number of Hopf bifurcation point, which is different in the upper branch of equilibrium curve formed by the fast subsystem, concerning with the the Hopf bifurcation is stable limit cycle. At the same time, we analyze the bifurcation mechanism produced by different types of bursting.The first chapter introduces the research significance of the Sherman-Rinzel-Keizer model, the present situation and points out the main research direction of this paper. The second chapter introduces basic principles and research methods involved by this paper. The third chapter studies a reduced Sherman-Rinzel-Keizer model. The main work in the third chapter is as follows: First, it is discussed that the condition and stability of the system equilibrium points, then we discuss the bifurcation at the equilibrium point, which include Hopf bifurcation of codimension 1 and Bogdanov-Takens bifurcation of codimension 2; The saddle-node bifurcation, Hopf bifurcation curves and homoclinic bifurcation curves are obtained near the Bogdanov-Takens bifurcation point. The fourth chapter is analysis of the bifurcation of the fast subsystem in a three-dimensional Sherman-Rinzel-Keizer 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 and the Hopf bifurcation number of the fast subsystem, we discuss the bifurcation mechanism produced by different types of bursting.