# Spatial localization for nonlinear dynamical stochastic models for excitable media

## TL;DR

This paper develops a mathematical framework to analyze spatial localization in high-dimensional nonlinear stochastic models of excitable media, with applications in data assimilation and covariance sampling.

## Contribution

It introduces a rigorous theory for covariance decay in coupled nonlinear stochastic systems, validated through models like FitzHugh-Nagumo.

## Key findings

- Covariance decay behavior is characterized by local and nonlocal effects.
- Theoretical results enable efficient covariance sampling strategies.
- Validation with FitzHugh-Nagumo model confirms the framework's applicability.

## Abstract

Nonlinear dynamical stochastic models are ubiquitous in different areas. Excitable media models are typical examples with large state dimensions. Their statistical properties are often of great interest but are also very challenging to compute. In this article, a theoretical framework to understand the spatial localization for a large class of stochastically coupled nonlinear systems in high dimensions is developed. Rigorous mathematical theories show the covariance decay behavior due to both local and nonlocal effects, which result from the diffusion and the mean field interaction, respectively. The analysis is based on a comparison with an appropriate linear surrogate model, of which the covariance propagation can be computed explicitly. Two important applications of these theoretical results are discussed. They are the spatial averaging strategy for efficiently sampling the covariance matrix and the localization technique in data assimilation. Test examples of a surrogate linear model and a stochastically coupled FitzHugh-Nagumo model for excitable media are adopted to validate the theoretical results. The latter is also used for a systematical study of the spatial averaging strategy in efficiently sampling the covariance matrix in different dynamical regimes.