# Bifurcation analysis of Wilson-Cowan model with singular impulses

## TL;DR

This paper investigates the Wilson-Cowan neural model incorporating singular impulses, introducing a novel analytical technique to study solution existence and bifurcations, revealing complex attractor structures like medusas and rings.

## Contribution

It introduces the concept of singular impulses in the Wilson-Cowan model and develops a new analysis method for bifurcation and solution existence.

## Key findings

- Bifurcations caused by impulses and singularity affect neural dynamics.
- Existence of solutions is established under new conditions.
- Complex attractors such as medusas and rings are identified.

## Abstract

The paper concerns with Wilson-Cowan neural model with impulses. The main novelty of the study is that besides the traditional singularity of the model, we consider singular impulses. A new technique of analysis of the phenomenon is suggested. This allows to consider the existence of solutions of the model and bifurcation in ultimate neural behavior is observed through numerical simulations. The bifurcations are reasoned by impulses and singularity in the model and they concern the structure of attractors, which consist of newly introduced sets in the phase space such that medusas and rings.