Electrical activities, excitability and multistability transitions of the hybrid neuronal model induced by electromagnetic induction and autapse

Abstract

Electromagnetic induction and autapse play important roles in regulating the electric activities, excitability, and bistable structure of neurons. The firing activities and global bifurcation patterns of a four-dimensional (4D) hybrid neuron model that combines the fast dynamic variables of the Wilson model and the slow feedback variables of the Hindmarsh–Rose (HR) model and magnetic flux are investigated based on the Matcont software and numerical calculation. The effect of electrical autapse on the dynamic evolution of the system is also discussed emphatically. Upon encountering electromagnetic induction, the hybrid neuron model exhibits complex global stability, Hopf bifurcation, and saddle-node bifurcation. Intriguingly, the system presents initial sensitivity and a bistable structure consisting of quiescent and period-1 spiking near the Hopf bifurcation point. It is worth noting that the feedback type of electrical autapse, including positive and negative feedback, has completely different effects on this bistable structure. Notably, the negative feedback autapse can expand and change the bistable region, so that the system generates a new bistable structure consisting of quiescent and periodic bursting states, and its bursting activities are also promoted. Moreover, extensive numerical results show that the system generally maintains a comb-shaped chaotic structure, abundant bifurcation patterns, and multistability. It should be noted that electrical autapse feedback types and time delays do not change the regular bifurcation structures but operate a complex regulatory mechanism for the coexistence of multiple attractors. These results will provide useful insights into the neuron’s dynamics under the atmosphere of electromagnetic induction and also electrical autapse.