  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.
