Averaging Principles for Markovian Models of Plasticity

Abstract

In this paper we consider a stochastic system with two connected nodes, whose unidirectional connection is variable and depends on point processes associated to each node. The input node is represented by an homogeneous Poisson process, whereas the output node jumps with an intensity that depends on the jumps of the input nodes and the connection intensity. We study a scaling regime when the rate of both point processes is large compared to the dynamics of the connection. In neuroscience, this system corresponds to a neural network composed by two neurons, connected by a single synapse. The strength of this synapse depends on the past activity of both neurons, the notion of synaptic plasticity refers to the associated mechanism. A general class of such stochastic models has been introduced in Robert and Vignoud (Stochastic models of synaptic plasticity in neural networks, 2020, arxiv: 2010.08195) to describe most of the models of long-term synaptic plasticity investigated in the literature. The scaling regime corresponds to a classical assumption in computational neuroscience that cellular processes evolve much more rapidly than the synaptic strength. The central result of the paper is an averaging principle for the time evolution of the connection intensity. Mathematically, the key variable is the point process, associated to the output node, whose intensity depends on the past activity of the system. The proof of the result involves a detailed analysis of several of its unbounded additive functionals in the slow-fast limit, and technical results on interacting shot-noise processes.