# Existence of global attractor for a nonautonomous state-dependent delay differential equation of neuronal type

## TL;DR

This paper establishes the existence and properties of a global attractor for nonautonomous state-dependent delay differential equations of neuronal type, using monotonicity methods and numerical simulations.

## Contribution

It introduces new theoretical results on the existence, shape, and stability of global attractors for nonautonomous SDDEs in neural models.

## Key findings

- Proved existence of global attractor under certain conditions.
- Provided criteria for exponential stability of the attractor.
- Numerical simulations demonstrate the theory's applicability.

## Abstract

The analysis of the long-term behavior of the mathematical model of a neural network constitutes a suitable framework to develop new tools for the dynamical description of nonautonomous state-dependent delay equations (SDDEs). The concept of global attractor is given, and some results which establish properties ensuring its existence and providing a description of its shape, are proved. Conditions for the exponential stability of the global attractor are also studied. Some properties of comparison of solutions constitute a key in the proof of the main results, introducing methods of monotonicity in the dynamical analysis of nonautonomous SDDEs. Numerical simulations of some illustrative models show the applicability of the theory.