Perturbations of Hindmarsh-rose neuron dynamics by fractional operators: Bifurcation, firing and chaotic bursts

Studying & understanding the bursting dynamics of membrane potential in neurobiology is captivating in applied sciences, with many features still to be uncovered. In this study, 2D and 3D neuronal activities given by models of Hindmarsh-Rose (HR) neurons with external current input are analyzed numerically with Haar wavelet method, proven to be convergent through error analysis. Our numerical analysis considers two control parameters: the external current I^ext and the derivative order γ, on top of the other seven usual parameters a, b, c, d, ν1, ν₂ and xrest. Bifurcation scenarios for the model show existence of equilibria, both stable and unstable of type saddle and spiral. They also reveal existence of stable limit cycle toward which the trajectories get closer. Numerical approximations of solutions to the 2D model reveals that equilibria remains the same in all cases, irrespective of control parameters'values, but the observed repeated sequences of impulses increase as γ decreases. This inverse proportionability reveals a system likely to be γ−controlled with γ varying from 1 down to 0. A similar observation is done for the 3D HR neuron model where regular burst is observed and turns into period-adding chaotic bifurcation (burst with uncountable peaks) as γ changes from 1 down to 0.