P-Bifurcation of Stochastic van der Pol Model as a Dynamical System in Neuroscience

doi:10.1007/s42967-021-00176-9

Abstract

Abstract: This study aims to determine the phenomenological bifurcation (P-bifurcation) occurring in the van der Pol (VDP) neuronal model of burst neurons with a random signal. We observe the P-bifurcation under an intense noise stimulus which would become chaotic transitions. Bursting and spiking simulations are used to describe the causes of chaotic transitions between two periodic phases that are the reason for the neuronal activities. Randomness plays a crucial role in detecting the P-bifurcation. To determine the equilibrium points, stability or instability of the stochastic VDP equation, and bifurcation, we use the stochastic averaging method and some related theorems. Apart from theoretical methods, we also examine numerical simulations in the particular case of that stochastic equation that illustrates the outcome of theorems for various noise types. The most striking part of our theoretical findings is that these results are also valid for the Izhikevich-FitzHugh model, Bonhoeffer-van der Pol oscillator in dynamical systems of neuroscience. Finally, we will discuss some applications of the VDP equation in neuronal activity.

Key words: Averaging diffusion system, P-bifurcation, Stability, Neuroscience, VDP stochastic equation

CLC Number:

F. S. Mousavinejad, M. FatehiNia, A. Ebrahimi. P-Bifurcation of Stochastic van der Pol Model as a Dynamical System in Neuroscience[J]. Communications on Applied Mathematics and Computation, 2022, 4(4): 1293-1312.

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