# Asymptotic behaviour of dynamical systems with plastic self-organising vector fields

## TL;DR

This paper rigorously analyzes the mathematical properties of a dynamical system with plastic self-organising vector fields inspired by brain plasticity, focusing on existence, uniqueness, and attractors to support cognitive modeling.

## Contribution

It provides a rigorous mathematical formulation and analysis of a previously proposed model, establishing foundational properties and paving the way for more complex cognitive system models.

## Key findings

- Proved existence and uniqueness of the vector field and system behavior
- Analyzed pullback attractors and limit sets in the non-autonomous system
- Confirmed the model's mathematical consistency

## Abstract

In [Janson & Marsden 2017] a dynamical system with a plastic self-organising velocity vector field was introduced, which was inspired by the architectural plasticity of the brain and proposed as a possible conceptual model of a cognitive system. Here we provide a more rigorous mathematical formulation of this problem, make several simplifying assumptions about the form of the model and of the applied stimulus, and perform its mathematical analysis. Namely, we explore the existence, uniqueness, continuity and smoothness of both the plastic velocity vector field controlling the observable behaviour of the system, and of the behaviour itself. We also analyse the existence of pullback attractors and of forward limit sets in such a non-autonomous system of a special form. Our results verify the consistency of the problem, which was only assumed in the previous work, and pave the way to constructing models with more sophisticated cognitive functions.