The hidden, period-adding, mixed-mode oscillations and control in a HR neuron under electromagnetic induction

Abstract

Abstract Researching the discharge patterns of neurons under electromagnetic induction is of great practical significance for understanding the complex nervous system. In this paper, the discharge patterns of a HR neuron under electromagnetic induction(here called magnetic flux HR neuron system), including hidden, period-adding, mixed-mode oscillations and their control are researched by combining theoretical analysis and numerical simulations. Firstly, the distribution and stability of the equilibrium point in the magnetic flux HR neuron system is theoretical analyzed based on the Matcont software, and it is found that there are one supercritical and two subcritical Hopf bifurcation points. Through further analysis, the hidden limit cycle attractor and its existence range are shown near the subcritical Hopf bifurcation points. Then, the complex period-adding, mixed-mode and coexistence oscillations are simulated and analyzed in the two-parameter space. In the following, the stability control of subcritical Hopf bifurcation is realized by using the Washout controller, and the undesirable hidden discharge behavior is eliminated. At last, the Hamilton energy feedback controller is designed to control the mixed-mode oscillation of the membrane voltage to desired discharge states effectively, and the energy transform can also be detected in the control process. The results provide useful research for understanding the discharge pattern of the magnetic flux HR neuron and controlling membrane voltage transfer.