Hidden dynamics and control of a Filippov memristive hybrid neuron model

The development of neurocomputing science highlights the necessity and new trend of establishing various feasible functional neuron models. A non-smooth feedback strategy is proposed by employing membrane potential as the threshold to describe the state-dependent effect of electromagnetic induction of neurons. Accordingly, a four-dimensional Filippov hybrid neuron model is established to reveal the firing mechanism of mammalian neocortical neurons. The local stability and bifurcation conditions of the subsystems are analyzed, and then, the bistable structures composed of quiescent state and periodic bursting patterns are identified. Intriguingly, the global dynamic behaviors of the subsystems are exactly consistent with its Hamilton energy, which enables precise localization of the coexisting attractors by analyzing the energy balance. Further, the sliding mode dynamics of the new neuron model, including sliding regions, various equilibrium points, and sliding bifurcations, are investigated based on differential inclusions theory. Importantly, the underlying mechanisms of sliding electrical activities, multistability, and hidden firing modes are studied via the fast–slow variable dissection method. Moreover, a Hamilton energy feedback control strategy is designed to effectively eliminate the hidden sliding electrical activities, which is far more effective than the traditional Washout controller. Finally, the modeling method and the obtained results provide a crucial theoretical basis and guarantee for optimizing the control of abnormal firing behaviors in the nervous system.