On the existence of traveling fronts in the fractional-order Amari neural field model

Abstract

In this work, we establish the existence of traveling fronts in a fractional-order formulation of the Amari neural field model. Fractional-order models act as a memory index of the underlying dynamical system. Therefore, in a fractional-order neural field model, we potentially incorporate the effect of neuronal collective memory. Considering Caputo’s fractional derivative framework and a fractional-order of 0≤α≤1, we establish explicit front solutions that allow us to analyze frontspeed and frontshape features directly. Furthermore, considering an exponential synaptic connectivity kernel, we find a bifurcation on the effect of fractional-order on front features. In particular, we find the existence of a critical synaptic threshold, k⋆, that qualitatively modifies the effect of fractional order on frontspeed. Below this critical threshold, fractional-order increases frontspeed whereas, above this threshold, fractional-order decreases frontspeed. In particular, less fractional-order implies a more substantial impact on frontspeed (either by increasing or decreasing frontspeed). Also, we find that lower fractional orders imply, in general, a slower power-law tendency towards the excited state. Therefore, our results establish the presence of different dynamics in the propagation of spatio-temporal patterns on neural fields due to the incorporation of a fractional-order framework and a potential memory index.