Averaging Principles for Markovian Models of Plasticity

TL;DR

This paper establishes an averaging principle for a stochastic model of synaptic plasticity in neural networks, showing how slow synaptic changes can be approximated by averaged dynamics over fast neuronal processes.

Contribution

It introduces a limit theorem for the slow synaptic process, providing a rigorous mathematical framework for averaging in neural plasticity models.

Findings

Proves an averaging principle for the synaptic strength process

Analyzes unbounded additive functionals of point processes

Develops technical results on shot-noise processes

Abstract

Mathematical models of biological neural networks are associated to a rich and complex class of stochastic processes. In this paper, we consider a simple {\em plastic} neural network whose {\em connectivity/synaptic strength} $(W(t))$ depends on a set of activity-dependent processes to model {\em synaptic plasticity}, a well-studied mechanism from neuroscience. A general class of stochastic models has been introduced in \cite{robert_mathematical_2020} to study the stochastic process $(W(t))$. It has been observed experimentally that its dynamics occur on much slower timescale than that of the main cellular processes. The purpose of this paper is to establish limit theorems for the distribution of $(W(t))$ with respect to the fast timescale of neuronal processes. The central result of the paper is an averaging principle for the stochastic process $(W(t))$. Mathematically, the key variable is the point process whose jumps occur at the instants of neuronal spikes. A thorough analysis of several of its unbounded additive functionals is achieved in the slow-fast limit. Additionally, technical results on interacting shot-noise processes are developed and used in the general proof of the averaging principle.