MULTI-MODE OSCILLATIONS AND HAMILTON ENERGY FEEDBACK CONTROL OF A CLASS OF MEMRISTOR NEURON

According to Faraday's electromagnetic induction law, the electrophysiological environment in and out the cell will produce electromagnetic induction effects in the case of ions penetrating the cell membrane or in the case of being exposed to external electromagnetic radiation, which will affect the electrical activity behavior of neural systems. Based on this principle, this paper studies the mixed-mode oscillation discharge characteristics of the Hindmarsh-Rose(HR) neuron model (here we call it as magnetic flux HR neuron model) with the influence of electromagnetic induction, and designs a Hamilton energy feedback controller to manage the mode to different periodic cluster discharge states. First, through theoretic analysis, it is found that the stability of equilibrium point in the magnetic flux HR neuron model is changed by the occurrence of Hopf bifurcation in the magnetic flux HR neuron model and a limit cycle is generated. Besides, some discharge characteristics of the membrane voltage near the Hopf bifurcation point are also discussed in detail. Then, it is also displayed there are abundant bifurcation structures in the magnetic flux HR neuron model based on the two-parameter numerical simulations, which includes multi-period bifurcation, period-adding bifurcation with chaos, period-adding bifurcation without chaos and co-existing mixed-mode oscillations in different initial conditions. At last, with the purpose of controlling the mixed-mode oscillation of membrane voltage, the Hamilton energy function is calculated by utilizing the Helmholtz theorem, and a Hamilton energy feedback controller is designed further. Additionally, it can be seen that the controller can effectively control the membrane voltage in different periodic clustering discharge modes, with the analysis of the discharge states of membrane voltage under different feedback gains in view of the numerical simulation. The results of this paper provide a useful theoretical support for the study of bifurcation structure in artificial neuron system under electromagnetic induction and the field of energy control related to the neurons.