Filippov Hindmarsh–Rose Neuronal Model With Threshold Policy Control

Abstract

A Filippov system of Hindmarsh-Rose (HR) neuronal model with threshold policy control is proposed, membrane potential has been taken as the threshold and the corresponding switching function is also established. We first discuss the existence and stability of the equilibria for the two Filippov subsystems based on the 2-D HR model. Subsequently, the sliding dynamics of HR model including the sliding segments, sliding regions, and various equilibria under the Filippov framework are studied. Then, we further consider the equilibria and the sliding bifurcation set of the Filippov system, and find there exist the bistable equilibria and several sliding bifurcation phenomena, such as boundary-node bifurcation, pseudosaddle-node bifurcation, the emergence and disappearance of limit cycles on the sliding line, and so on. Finally, we study the Filippov system of the 3-D HR model, and provide a phase diagram of the system that generates the sliding spiking and the sliding bursting, which lie on the sliding line.