Emergent dynamics in fractional-order Wilson–Cowan neural network systems

Abstract

The firing dynamics of excitable systems are critical to understand organized responses in cortical networks. In this paper, we examine a fractional-order Wilson–Cowan (W–C) neural network model composed of excitatory and inhibitory neuron populations, utilizing Caputo’s fractional-order derivative formalism to explore the influence of fractional-order dynamics on firing behavior. The significance of extending to the fractional-order domain lies in the model’s theoretical framework, which inherently retains memory and hereditary characteristics. We investigate memory-dependent response functions and average neuronal characteristics, enabling us to formulate a fractional-order model that incorporates past dynamics into the neuronal populations’ features. This generalized model is capable of producing alternations between spiking and bursting phenomena, including mixed-mode oscillations (MMOs). Using stability and bifurcation analyses, we delineate the parameter space within which variations in firing patterns emerge. One notable impact of the model’s memory property is the potential stabilization of neuronal activity. We demonstrate that our numerical findings are in alignment with the analytical predictions and that the memory trace depends on the fractional-order dynamics.