Spatiotemporal dynamics of a non-smooth reaction–diffusion Wilson neuron model with time delay

To complement the vacuum in traditional functional neural models that cannot quantify neurodynamic behaviors in heterogeneous and non-smooth physiological environments, we consider the effects of ion diffusion, electrical autapse, and state-dependent induced electric field on actual bioelectrical activities. Accordingly, a Filippov-type Wilson neuron model with delay and diffusion is established under homogeneous Neumann boundary conditions, and the spatiotemporal properties of the model are qualitatively analyzed to reveal the intricate operation mechanism of cortical neurons. The local dynamics of delayed subsystems, including the existence of equilibrium points and diffusion-driven Turing instability, are discussed emphatically. Further, the analytical conditions of delay-induced Hopf bifurcation, Hopf–Hopf bifurcation, and Turing–Hopf bifurcation are provided. Subsequently, the discriminant formulas for the direction of the local Hopf bifurcation and the stability of its periodic solution are derived. Quantitative numerical verifications manifest that the delay can not only trigger the periodic spatiotemporal firing modes but also stimulate the marvelous multiple stability switching and bistable phenomena under proper feedback. Importantly, the properties of sliding segments, various equilibrium points, and local and global sliding bifurcations under the effect of non-smooth thresholds are clarified. Besides, the sliding spatiotemporal response features of the model are determined analytically, and the evolution mechanism of hidden sliding limit cycles is elucidated. The obtained results confirm the validity of the proposed threshold strategy, which plays a substantial role in the treatment of neurological diseases and the design of artificial intelligence sensors.