Fractional Calculus-Based Generalization of the FitzHugh-Nagumo Model: Biophysical Justification, Dynamical Analysis and Neurocomputational Implications

Abstract

In this chapter, the dynamical behavior of the incommensurate fractional-order FitzHugh-Nagumo model of neuron is explored in details from local stability analysis. First of all, considering that the FitzHugh-Nagumo model is a mathematical simplification of the Hodgkin-Huxley model, the considered model is derived from the fractional-order Hodgkin-Huxley model obtained taking advantage of the powerfulness of fractional derivatives in modeling certain biophysical phenomena as the dielectrics losses in cell membranes, and the anomalous diffusion of particles in ion channels. Then, it is shown that the fractional-order FitzHugh-Nagumo model can be simulated by a simple electrical circuit where the capacitor and the inductor are replaced by corresponding fractional-order electrical elements. Then, the local stability of the model is studied using the Theorem on the stability of incommensurate fractional-order systems combined with the Cauchy’s argument Principle. At last, the dynamical behavior of the model are investigated, which confirms the results of local stability analysis. It is found that the simple model can exhibit, among others, complex mixed mode oscillations, phasic spiking, first spike latency, and spike timing adaptation. As the dynamical richness of a neuron expands its computational capacity, it is thus obvious that the fractional-order FitzHugh-Nagumo model is more computationally efficient than its integer-order counterpart.