Discharge Characteristics And Phase Synchronization Phenomena Of Neuron Models

As the basic structure and functional unit of the nervous system,neuron has complex nonlinear characteristics.Its discharge activities often exhibit rich dynamical behavior,such as bifurcation and chaos.Therefore,the study of nonlinear dynamics of single neurons and neural circuits is particularly important.Using the theory and method of nonlinear dynamics,this thesis investigates the dynamical properties of three kinds of neuron models by theoretical analysis and numerical simulations.The detailed framework of this dissertation is as follows.Chapter 1 gives a brief survey to the development of nonlinear dynamics as well as the important role of nonlinear dynamics in the study of nervous systems,the mathematical models of neuronal discharge activities and the current research situation,and the main research methods and contents of this thesis.Chapter 2 introduces basic knowledge and concepts of bifurcation theory.First,we introduce the bifurcation and structural stability of the dynamical system,and narrate the normal form of saddle-node bifurcation and Hopf bifurcation.Then we introduce the bifurcation of equilibrium points and limit cycles in the neuron model.Finally,topological normal forms are presented and discussed in detail for the Cusp,Bautin(generalized Hopf)and Bogdanov-Takens bifurcations.We also give the computing method for the normal form coefficients of the Bogdanov-Takens bifurcation.Chapter 3 studies the dynamical properties of the respiratory neuron model.First,we establish the model of respiratory neuron under external electric field.Then,the bi furcation analysis of equilibrium points in the whole system is performed under the direct current electric field.We focus on the topological structure near the Bogdanov-Takens b-ifurcation point,and propose the mathematical expressions for a saddle-node bifurcation curve,a nondegenerate Hopf bifurcation curve,and a saddle homoclinic bifurcation curve.In addition,by adjusting the maximum conductivity of potassium ion,we obtain four distinct types of bursters.Combined with fast-slow dynamics and phase plane analysis,we explain the cause of bursting in the system.At last,the first Lyapunov coefficient at the Hopf bifurcation point is computed to determine the stability of the bifurcated limit cycle.Chapter 4 studies the bifurcation and discharge activities of the pancreatic ? cell.First,we consider the influence of the change of parameters in the model on the discharge rhythm.By numerical calculation of bifurcation diagrams of interspkie intervals with the variation of parameters,we find that interspike interval sequences could exhibit period-adding,period-doubling and inverse period-doubling scenarios.Furthermore,the range of chaotic regions is confirmed by calculating the largest Lyapunov exponents of the system.Then the system is divided into fast and slow subsystems by the fast-slow dynamics analysis method,and we discuss the formation mechanism of tapered and square-wave bursting with the slow variable as the bifurcation parameter.In addition,the dynamical properties of codimension-one bifurcation points of the fast subsystem are studied,and the Hopf bifurcation of equilibrium points is emphatically analyzed.Finally,we discuss the codimension-two bifurcation of the fast subsystem and calculate the parameters and eigenvalues corresponding to these bifurcation points.Chapter 5 studies the phase synchronization of coupled CA1 pyramidal neurons.First,a coupled neuron model is constructed by using electrical synapse to couple two CA1 pyramidal neurons.According to the distribution of phase differences,changing the coupling strength can make the coupled neurons exhibit different synchronization states.It is found that there is a complex synchronous transition behavior between the elec-trically coupled CA1 pyramidal neurons by continuously varying the coupling strength.And the period-doubling phenomenon occurs in the distribution of phase differences.After that,the synchronization state is converted from out-of-phase synchronization to asynchronization.Moreover,changing the membrane capacitance can also induce the transitions of synchronization states.Therefore,we depict the synchronization states of the coupled CA1 pyramidal neurons in a two-dimensional plane of coupling strength and membrane capacitance,and summarize the laws of the transitions of synchronization states.Finally,the ISI-distance method is used to study the level of synchrony.This method can not only distinguish three distinct synchronization states,but also charac-terize the degree of the same synchronization states,complementing the deficiency of phase differences.The results show that when the coupling strength is large enough,the coupled neurons immediately go into in-phase synchronization for any finite values of membrane capacitance.