Bifurcation and periodic solutions to neuroscience models with a small parameter

TL;DR

This paper proves the existence and stability of periodic solutions in neuroscience models with a small parameter, applying the results to a Parkinson's disease model and supporting findings with numerical simulations.

Contribution

It introduces new theoretical results on bifurcation and periodic solutions for functional differential equations with small parameters in neuroscience models.

Findings

Existence of periodic solutions proven for models with small parameters.

Stability analysis of these solutions conducted.

Numerical simulations illustrate the theoretical results.

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 $$ \dot{x}(t)=L(\tau) x_t + \varepsilon f(t, x_t), $$ where $L: \mathbb{R}_+\rightarrow \mathcal{L}(C; \mathbb{R})$ and $f: \mathbb{R} \times C \rightarrow \mathbb{R}$ are, respectively, linear and nonlinear operators, and $\varepsilon>0$ 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.