Evaluation Methods of Chaotic State in Spiking Neural System with State Dependent Jump

Abstract

Izhikevich neuron model, which combines continuous spike-generation mechanisms and discontinuous resetting process after spiking, can reproduce almost all spiking activities including chaotic spiking in actual neural systems. When the chaotic state is evaluated in this model, it is known that conventional Lyapunov exponent where the continuous trajectory is presupposed cannot be applied due to the state dependent jump in the resetting process. To evaluate Lyapunov exponent in the system with the resetting process, the accurate numerical calculation for the trajectory by Newton method and the consideration for saltation matrix are needed. By virtue of this method, several routes to chaos have been found in Izhikevich neuron model. While on the other hand, in this study,we have proposed the method combining Euler method and Lyapunov exponent on Poincaré section and evaluated the chaotic state in Izhikevich neuron model. As the result, it has been confirmed that this method can also judge the chaotic state by tuning the initial perturbation against the trajectory.