Global multistability and mechanisms of a memristive autapse-based Filippov Hindmash-Rose neuron model

Electromagnetic induction in the nervous system can be emulated by memristive autapses, which plays a critical role in regulating physiological functions. A discontinuous control strategy has been proposed by taking membrane potential as the threshold. Accordingly, a four-dimensional Filippov Hindmash-Rose (HR) neuron model is established by improving the memristive autapse with the threshold control strategy. The existence, stability, and bifurcation conditions of the two subsystems are discussed, and the bistable regions and their internal mechanism are further revealed with the help of two-parameter bifurcation analysis and global basins of attraction. Subsequently, the complex sliding mode dynamics of the model including sliding segments, various equilibrium points, and sliding bifurcations are analyzed via differential inclusions theory. Then, extensive numerical results exhibit that the proposed threshold strategy leads to the occurrence of sliding bursting, sliding limit cycle and coexisting attractors, and so on. Moreover, the mechanism of sliding electric activities, mode transition, and multistability under the threshold strategy feedback is revealed based on the fast-slow variable dissection method. Finally, the internal mechanism of multistable evolutionary patterns and stochastic bifurcations induced by Gaussian white noise is confirmed. The obtained results will contribute to the further design of functional neural networks and provide potential guidance for the treatment of patients with intellectual disabilities. •A 4-D Filippov HR neuron model is established by introducing the threshold control strategy of membrane potential.•The global stability, bifurcation patterns, and initial sensitivity of the subsystems are investigated.•The complex sliding mode dynamics of the model are analyzed via differential inclusions theory.•The mechanism of multistability of the Filippov system is revealed by the fast-slow variable dissection method.•The evolutionary laws of multistability and stochastic P-bifurcation induced by Gaussian white noise are clarified.