Travelling Waves in Neural Fields with Continuous and Discontinuous Neuronal Activation

The main object of our study is travelling waves in vast neuronal ensembles modelled using neural field equations. We obtained conditions that guarantee the existence of travelling wave solutions and their continuous dependence under the transition from sigmoidal neuronal activation functions to the Heaviside activation function. We, thus, filled the gap between the continuous and the discontinuous approaches to the formalization of the neuronal activation process in studies of travelling waves. We provided conditions for admissibility to operate with simple closed-form expressions for travelling waves, as well as to significantly simplify their numerical investigation. This opens the possibilities of linking characteristics of cortical travelling waves, e.g., the wave shape and the wave speed, to the physiological parameters of the neural medium, e.g., the lengths and the strengths of neuronal connections and the neuronal activation thresholds, in the framework of the neural field theory.