Stochastic Models of Neural Plasticity: Averaging Principles

Philippe Robert, Gaetan Vignoud

Published 2020-10-17Version 1

Mathematical models of biological neural networks are associated to a rich and complex class of stochastic processes. When the connectivity of the network is fixed, various stochastic limit theorems, such as mean-field approximation, chaos propagation and renormalization have been used successfully to study the qualitative properties of these networks. In this paper, we consider a simple plastic neural network whose connectivity/synaptic strength $(W(t))$ depends on a set of activity-dependent processes to model synaptic plasticity, a well-studied mechanism from neuroscience. In a companion paper, a general class of such stochastic models has been introduced to study the stochastic process $(W(t))$ when, as it has been observed experimentally, its dynamics occur on much slower timescale than that of cellular processes. The purpose of this paper is to establish limit theorems for the distribution of $(W_\varepsilon(t))$ when $\varepsilon$, the scaling parameter of the (fast) timescale of neuronal processes, goes to $0$. The central result of the paper is an averaging principle for the stochastic process $(W_\varepsilon(t))$. Mathematically, the key variable is the scaled point process, whose jumps occur at the instants of neuronal spikes. The main technical difficulty lies in a thorough analysis of several of its unbounded additive functionals of this point process in the slow-fast limit. Additionally, technical lemmas on interacting shot-noise processes are also used to finally establish an averaging principle.