The Existence and Stability Analysis of Periodic Solution of Izhikevich Model

Abstract

In this paper, based on the Izhikevich model, a more realistic hybrid impulsive neuron model combining model with state-dependent impulsive effects is proposed. By means of the theory of impulsive semidynamic system, the Poincare section and the ordinary differential equation geometry theory, the properties of the equilibrium points and the sufficient conditions for the existence and stability of different order-1 or order-2 periodic solutions of the system are derived near the equilibrium point or limit cycle. In addition, subthreshold bifurcation behavior which are saddle-node bifurcation, Andronov-Hopf bifurcation, Bogdano-Takens bifurcation and bifurcation behavior of the model with state-dependent impulsive effects are also studied. Final, the main results are illustrated by numerical simulations, and the bifurcation diagram shows that the system becomes more complicated due to the existence of impulses, and there exist bifurcations and chaos phenomena, indicating that the system has rich dynamic behavior.