Bifurcation and periodic solutions to neuroscience models with a small parameter

2023/09/12 by José Oyarce, Oyarce, José
Medicine · Mathematics · Computer Science · #Mathematical and Theoretical Epidemiology and Ecology Models #Nonlinear Differential Equations Analysis #Neural Networks Stability and Synchronization

paper · pdf · doi:10.48550/arxiv.2309.06398

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 x(t)=L(τ) x_t + ε f(t, x_t), where L: ℝ₊→ L(C; ℝ) and f: ℝ × C → ℝ are, respectively, linear and nonlinear operators, and ε>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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