Switching dynamics of a Filippov memristive Hindmash-Rose neuron model with time delay

Considering the existence of magnetic induction effect with different intensities in the process of subthreshold and suprathreshold oscillations of bioelectrical activities, a non-smooth feedback strategy for memristive current with time delay is proposed, and then a four-dimensional Filippov Hindmarsh–Rose (HR) neuron model is established. The local stability and bifurcation patterns of delayed subsystems are qualitatively analyzed. Accordingly, the discriminant formula for the direction and stability of periodic solutions generated by Hopf bifurcation is obtained on the center manifold. Importantly, the stability of subsystems has switching behavior, which is accompanied by abundant hidden electrical activities under the effect of time delay. The theoretical analysis clarifies that the proposed feedback strategy leads to complex sliding mode dynamics, including sliding segments, various equilibrium points and sliding bifurcations. Meanwhile, the analytical conditions for motions of grazing, sliding, and crossing are developed and verified based on the flow switching theory. Moreover, the mechanism and evolutive rule of the self-excited and hidden sliding electrical activities are revealed by the fast-slow variable dissection method. Finally, it is verified that the time delay can not only induce bistable structures composed of the quiescent state and periodic bursting, but also eliminate the hidden sliding dynamics.