Dynamic Analysis Of Several Kinds Of Neuron Models

The biological nervous system is interconnected by myriad neurons to form a large nonlinear dynamical system.When the internal and external environment of the organism changes,the biological nervous system can encode and transmit information through different discharge patterns of neurons.Several types of neuron models are studied in the thesis.With the aid of nonlinear dynamics theory,dynamics theory of biological neural system and computer simulation methods,the dynamic characteristics of several neuron models are analyzed.By controlling the parameters,the discharge region of the nervous system that meets actual needs can be found to control the stable region of the system.The main results of the thesis are listed as follows:The first part is devoted to modeling the dynamics of a three-dimensional HR neuron system.From the theoretical point of view,the influence of parameter changes on the number and stability of the equilibrium points of the HR neuron model is analyzed.The Shilnikov theorem is used to obtain the critical value of the external electric field excitation of the chaotic attractor in the HR neuron model.The Hopf bifurcation theory is used to verify the existence of Hopf bifurcation and derive the critical values of Hopf bifurcation in HR neuron model.From the perspective of numerical simulation,the single-parameter bifurcation diagram,two-parameter bifurcation diagram,Lyapunov exponent diagram,time response diagram and phase diagram of HR neuron system are simulated by computer software,which verifies the correctness of the above theoretical results and describes the complex information coding process of the HR neuron model.The second part discusses the eHR neuron model.The influence of system parameters on the number and stability of the equilibrium points in the eHR neuron model is analyzed theoretically.The Hopf bifurcation theory is used to calculate the Hopf bifurcation conditions in the eHR neuron system.The single-parameter bifurcation diagram,two-parameter bifurcation diagram,phase diagram,time response diagram,and Lyapunov exponent diagram of the eHR neuron model are simulated by computer software,which describes the existence of the chaotic attractor and Hopf bifurcation more accurately and provides help for the selection of parameters in the eHR neuron model.The coupled model of eHR neuron and the influence on complex process of coupled eHR neuron as the coupling strength changes are given in part three.The sufficient conditions for the complete synchronization of the system are deduced by using the mode decomposition.The theoretical results above are verified by means of numerical simulation.It is concluded that if the coupling strength is greater than a certain critical value,the coupled eHR neuron will always maintain the full synchronization state.Therefore,we can control the coupled neuron model by controlling the magnitude of the coupling strength,which will provide an important reference for the study of the cluster movement of neurons.