Mathematical modeling of neuron model through fractal-fractional differentiation based on maxwell electromagnetic induction: application to neurodynamics
Kashif Ali Abro, Abdon Atangana
Abstract
Abstract The electrical activities of the reliable neuron models have different responses within intrinsic biophysical effects and can functionalize for asymmetric coexisting electrical activities under anti-monotonicity phenomenon. This manuscript presents mathematical analysis of neuron model based on Maxwell electromagnetic induction through newly proposed fractal-fractional differential and integral operators. The neuron model based on Maxwell electromagnetic induction changes with time along a fractal dimension that describes the cumulative chaotic phenomenon. The cumulative chaotic phenomenon of neuron model is mathematically modeled via exponential and Mittag–Leffler kernels with variable and fixed fractal and fractional orders. In order to exhibit fractal properties and memory effects, the neuron model is discretized by means of Adams–Bashforth-Moulton method that allows explicitly to compute the approximate solution of neuron model. The comparison of neuron model based on memory effect and fractal dimension have distinguished the evolution of neuron model at (i) variability of fractal order with fixed fractional order, (ii) variability of fractional order with fixed fractal order, and (iii) variability of fractal order as well fractional order.