On the Chaotic Pole of Attraction with Nonlocal and Nonsingular Operators in Neurobiology

Abstract

Until the neurologists J.L. Hindmarsh and R.M. Rose improved the Hodgkin–Huxley model to provide a better understanding on the diversity of neural response, features like pole of attraction unfolding complex bifurcation for the membrane potential was still a mystery. This work explores the possible existence of chaotic poles of attraction in the dynamics of Hindmarsh–Rose neurons with external current input. Combining with fractional differentiation, the model is generalized with introduction of an additional parameter, the non-integer order of the derivative \(\sigma \) and solved numerically thanks to the Haar Wavelets. Numerical simulations of the membrane potential dynamic show that in the standard case the control parameter is \(\sigma =1,\) the nerve cell’s behavior seems irregular with a pole of attraction generating a limit cycle. This irregularity accentuates as \(\sigma \) decreases (\(\sigma =0.8\) and \(\sigma =0.5\)) with the pole of attraction becoming chaotic.