The Pathwise Approach to Metastability and its Applications to Galves--L\"ocherbach Models

Abstract

Metastability is the tendency of a system to dwell for a very long time near an apparently stable equilibrium before a rare fluctuation drives it, on a comparatively short time scale, towards another. Among the rigorous frameworks developed to capture this phenomenon, the pathwise approach proceeds by identifying the ``typical'' trajectories of the stochastic dynamics at hand and estimating their probabilities. In this article we review the pathwise approach and its application to the Galves--L\"ocherbach (GL) class of stochastic models of spiking neural networks. After recalling the conceptual and historical roots of the theory -- which goes from chemistry to rigorous probability theory, with fundamental ideas coming mainly from statistical physics -- and illustrating them on two classical examples, we give a general definition encompassing the known variants of the GL model and survey the metastability results already established for some of these variants. As far as we can, we do so in a self-contained fashion, and we sketch the proofs when possible, highlighting their common structure. We close with a discussion on open problems and point to possible further directions.