Neurobiological transition of magnetized and demagnetized dynamism for fractional Hindmarsh–Rose neuron model via fractal numerical simulations

This manuscript investigates how magnetic and non-magnetic effects influence the firing patterns, oscillations, and synchronization properties of the Hindmarsh–Rose neuron model under different magnetic conditions. The development of a fractal–fractional Hindmarsh–Rose neuron model is proposed for investigating self-similarity across different scales to analyze and understand the complexities when extreme magnetic flux varies and reaches its critical value. The mathematical modeling of the Hindmarsh–Rose neuron model is established under an application of the Caputo–Fabrizio and Atangana–Baleanu fractional differential operators. For the sake of numerical simulations via the Adams–Bashforth–Moulton method, the discretization of spatial and time domains on fractal–fractional derivatives is employed to generate numerically powerful schemes within approximate accuracy. For understanding the brain function and neural oscillations, the magnetized and demagnetized Hindmarsh–Rose neuron model revealed suppressed neuronal activity and the effects of transcranial magnetic stimulation. Our results suggested two aspects: one is trapping of neurons, striking phenomena and firing patterns under demagnetization, while the other is neurological disorders, spiking and bursting in neurons based on neural interfaces under demagnetization.