Bifurcation and Stability Analysis in a Class of Discontinuous FitzHugh–Nagumo Models

This study constructs a discontinuous FitzHugh–Nagumo model incorporating a membrane potential threshold control strategy, and systematically analyzes its dynamical behaviors and bifurcation characteristics. First, the existence conditions and local stability of equilibrium points in the two smooth subsystems are clarified. Furthermore, the types and parameter-dependent properties of admissible equilibria, virtual equilibria, boundary equilibria, and pseudo-equilibria of the full system are characterized. The investigation reveals a wealth of bifurcation phenomena: supercritical Hopf bifurcations and saddle-node bifurcations exist in the subsystems; while the full system exhibits discontinuity-induced bifurcations such as pseudo-saddle-node bifurcations, persistence bifurcations, nonsmooth fold bifurcations, pseudo-Hopf bifurcations induced by tangent point collisions, and grazing bifurcations. In addition, brief-pulse simulations reveal an all-or-none-like stimulus response with a well-defined operational excitation threshold. The Dravet-like parameter modification increases the threshold pulse amplitude by [Formula: see text], such that the selected common stimulus elicits a firing-like response in the normal model but remains subthreshold in the Dravet-like model. These results provide a dynamical explanation for threshold-dependent excitability and stimulus-evoked firing failure in the proposed nonsmooth neuronal model.