On the chaotic pole of attraction for Hindmarsh-Rose neuron dynamics with external current input

Since the neurologists Hindmarsh and 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 an external current input. Combining with fractional differentiation, the model is generalized with the 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 dynamics show that in the standard case where the control parameter 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 = 0.9 and sigma = 0.85) with the pole of attraction becoming chaotic.