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arXiv · 2309.06398

Bifurcation and periodic solutions to neuroscience models with a small parameter

José Oyarce
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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.

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José Oyarce. 2023-09-12. Bifurcation and periodic solutions to neuroscience models with a small parameter. https://arxiv.org/abs/2309.06398

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