A peculiar application of Atangana-Baleanu fractional derivative in neuroscience: Chaotic burst dynamics

Recent discussions on the non validity of index law in fractional calculus have shown the amazing filtering feature of Mittag-Leffler function foreseing Atangana-Baleanu derivative as one of reliable mathematical tools for describing some complex world problems, like problems of neuronal activities. In this paper, neuronal dynamics described by a three dimensional model of Hindmarsh-Rose nerve cells with external current are analyzed analytically and numerically. We make use of the Atangana-Baleanu fractional derivative in Caputo sense (ABC derivative) and asses its impact on the dynamic, especially the role played by its derivative order in combination with another control parameter, the intensity of the applied external current. Our analysis proves existence of equilibria whose some are unstable of type saddle point, paving the ways for possible bifurcations in the process. Numerical approximations of solutions reveal a system with initially regular bursts that evolve into period-adding chaotic bifurcations as the control parameters change, with precisely the Atangana-Baleanu fractional derivative's order decreasing from 1 down to 0.5.