Non-smooth Dynamic Analysis And Coupling Synchronization Of McKean Neuron Model

As the structural unit of the neuron dynamic system,neurons are responsible for information transmission and processing,and their discharge activities show abundant dynamic behaviors,such as the bifurcations,existence and stability of periodic solutions.Single neuron is closely connected with other neurons through electrical or chemical synapses to maintain the coordination and stability of the body functions,accompanied by synchronization phenomenon.It is of great significance to study the bifurcation mechanism and its coupling synchronization of neurons.In this paper,the McKean neuron model is used as the research object,which mainly studies two aspects:First,we study the non-smooth bifurcation and the existence of periodic solutions of McKean neuron with synaptic conductance and gated threshold.Second,we study the coupled synchronization behavior of two identical McKean neurons.Firstly,aiming at the McKean neuron model with synaptic conductance and gated threshold.First of all,the parameter conditions for the existence and stability of the equilibrium point of the system are given,the bifurcation of two types of the boundary equilibrium point of the system are theoretically analyzed.And by introducing the generalized Jacobi matrix near the switching manifold,the parameter conditions for the occurrence of discontinuous Hopf bifurcation in the system are theoretically deduced and numerically studied.Then,the solution manifolds of the system in each region are obtained,and the Poincare mapping across the single boundary is constructed,the properties of the Poincare map-ping are theoretically analyzed,which proves that the existence of periodic solutions across the single boundary.Finally,the parameter thresholds of periodic solutions across a single boundary are numerically studied,and the grazing periodic solutions of the system are obtained.Secondly,for the two identical coupled McKean neurons,the synchronous behavior of coupled McKean neurons under electrical synapses and chemical synapses are studied,respectively.First,under the coupling of electrical synapses,the behavior of complete synchronization,approximate synchroniza-tion and peak-irrelevant cluster synchronization of electro-coupled McKean neurons are studied by synchronization difference curve,and the influence of the extra periodic excitation on the discharge pattern of the coupled McKean neurons is analyzed.Then,under the coupling of chemistry synapses,the synchronization difference curve is still made to analyze the synchronization effect of coupling strength and time delay on chemically coupled McKean neurons.In addition,the extra periodic excitation is applied to chemically coupled McKean neurons with time delay to study the synchronization phenomenon of the system.