Existence and stability analysis of homoclinic cycle and periodic solution of quartic integrate-and-fire neuron model

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Journal of Vibration and Shock ›› 2023, Vol. 42 ›› Issue (23) : 209-214.

WU Jianmei,XU Jieqiong,WANG Junjie,XU Qixiang

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Abstract

This paper studies the existence and stability of the periodic solution by analyzing the homoclinic bifurcation of the quartic integrate-and-fire(IF) neuron model with impulse effect (state reset process).We mainly aim at the system has two equilibrium points, qualitatively analyze the dynamic behavior near the saddle point of the system, and investigate the existence of the order-1 homoclinic cycle of the system in different cases. With as the bifurcation parameter, when the system occurs homoclinic bifurcation, the existence and stability of the order-1 periodic solutions near the order-1 homoclinic cycle in different cases are proved by using the theory of impulsive dynamic system and the fixed point theory of the Poincaré map. Finally, the periodic solutions of the system are simulated numerically to verify the theoretical results.The approach employed in this paper provides a strategy for finding periodic solutions to impulse dynamical systems.

Key words

quartic integrate-and-fire neuron model / impulse effort / order-1 homoclinic cycle / homoclinic bifurcation / order-1 periodic solution

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WU Jianmei,XU Jieqiong,WANG Junjie,XU Qixiang. Existence and stability analysis of homoclinic cycle and periodic solution of quartic integrate-and-fire neuron model[J]. Journal of Vibration and Shock, 2023, 42(23): 209-214

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