Approximation and stability results for the parabolic FitzHugh-Nagumo system with combined rapidly oscillating sources

Abstract

The use of high-frequency currents in neurostimulation has received increased attention in recent years due to its varied effects on tissues and cells. Nonlinear differential equations are commonly used as models for Neurons, and averaging methods are suitable for addressing questions like stability when considering single-frequency sources. A recent strategy called temporal interference stimulation uses electrodes to deliver sinusoidal signals of slightly different frequencies. Thus, classical averaging cannot be directly applied. This paper considers the one-dimensional FitzHugh-Nagumo system under the effects of a source composed of two sinusoidal terms in time and decaying in space. We develop a new averaging strategy to show that the solution of the system can be approximated by an explicit highly-oscillatory term plus the solution of a simpler, non-autonomous system. One of the main novelties is an extension of the contracting rectangles method to the case of parabolic equations with space and time-depending coefficients.