Neuronal dynamics and electrophysiology fractional model: A modified wavelet approach

Abstract

The modeling and numerical simulations of fractional-order cable type problems can provide a proper structure of anomalous-diffusion in the measure of ions in complex neuronal dynamics and electrophysiology. In the present work, a novel approach of Gegenbauer wavelets (GWM) based on the operational matrices with their derivatives is proposed. New operational matrices are established for fractional-order derivative and variable-order derivative with the help of piecewise functions. The given problem is converted into a system of nonlinear equations via Gegenbauer wavelets in the suggested method. The fractional cable equation of variable-order is taken to account and successfully solved using a new algorithm. The present study also contains the convergence and error bound analysis of the proposed approach. Solutions obtained via operational matrix-based algorithm are validating the accuracy, efficiency and reliability of the suggested method. The comparative study in tabular form, as well as graphical plots (2D and 3D) for solutions and absolute error, has been reported. Hence, outcomes are validating the suggested method as an accurate and efficient tool and could be adopted for other types of fractional-order nonlinear complex dynamical problems.