The movement of large amounts of ions (e.g., potassium, sodium and calcium) in the nervous system triggers time-varying electromagnetic fields that further regulate the firing activity of neurons. Accordingly, the discharge states of a modified Hindmarsh–Rose (HR) neuron model under an electric field are studied by numerical simulation. By using the Matcont software package and its programming, the global basins of attraction for the model are analyzed, and it is found that the model has a coexistence oscillation pattern and hidden discharge behavior caused by subcritical Hopf bifurcation. Furthermore, the model’s unstable branches are effectively controlled based on the Washout controller and eliminating the hidden discharge states. Interestingly, by analyzing the two-parametric bifurcation analysis, we also find that the model generally has a comb-shaped chaotic structure and a periodic-adding bifurcation pattern. Additionally, considering that the electric field is inevitably disturbed periodically, the discharge states of this model are more complex and have abundant coexisting oscillation modes. The research results will provide a useful reference for understanding the complex dynamic characteristics of neurons under an electric field.
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.
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