Exploring dynamic behavior and bifurcations in a Filippov neuronal system with a double-tangency singularity

We investigated the phenomenon of pseudo-Hopf bifurcation in a Filippov Hindmarsh-Rose neuronal system with threshold switching, and the existence of crossing limit cycles was proved by constructing the half-return mapping. Through the threshold control, the firing state of the system could be switched, allowing transitions from a non-periodic state to a periodic state, as well as the evolution from spiking to bursting. Furthermore, through threshold switching, the system exhibited the coexistence of multiple attractors, the system could be in multiple stable states, or have multiple stable sets that could attract system trajectories. This meant that neuronal system could exhibits diverse dynamical behaviors than being limited to a single stable state. The phenomenon of period-doubling bifurcation also indicated that the system will eventually enter a chaotic state. By extending the analysis to nonlinear neuronal systems, this study contributes to a deeper understanding of complex dynamics and provides valuable insights for designing state switching in the application of neural dynamics.