Bifurcation and periodic solutions to neuroscience models with a small parameter

Abstract

The existence of periodic solutions is proven for some neuroscience models with a small parameter. Moreover, the stability of such solutions is investigated, as well. The results are based on a theoretical research dealing with the functional differential equation with parameters $$\dot{x}(t)=L(\tau )x_t+\epsilon f(t,x_t),$$ where $L:{\mathbb{R}}_{+}\to \mathcal{L}(C;\mathbb{R})$ and $f:\mathbb{R}\times C\to \mathbb{R}$ are, respectively, linear and nonlinear operators, and $\epsilon >0$ is a small enough parameter. The theoretical results are applied to a Parkinson's disease model, where the obtained conclusions are illustrated by numerical simulations.