Mathematical modeling of the neuron autocoling in the cell membrane using the fractional model of FitzHugh-Nagumo with the function of irritant intensity

The article studies the process of temporary propagation of a nerve impulse in a cell membrane. For this purpose, a new mathematical model based on the fractional FitzHugh-Nagumo oscillator with a stimulus intensity function was proposed. A feature of the fractional oscillator is that the model equation contains derivatives of fractional variables of the Gerasimov-Caputo type. The proposed mathematical model is a Cauchy problem. Due to the nonlinearity of the model equation, the solution to the Cauchy problem was sought using a numerical method of a nonlocal explicit finite-difference scheme of the first order of accuracy. The numerical method was implemented in the Maple 2022 language. Using a numerical algorithm, the simulation results were visualized, oscillograms and phase trajectories were constructed for various values of the model parameters. It is shown that the solution to the new mathematical model can have relaxation oscillations. In addition, an example is given in which the limit cycle is stable. It is also shown that the proposed FitzHugh-Nagumo fractional oscillator with stimulus intensity function has rich dynamics: various regular and chaotic modes.