{
  "id": "2010.08790",
  "version": "v1",
  "published": "2020-10-17T13:52:41.000Z",
  "updated": "2020-10-17T13:52:41.000Z",
  "title": "Stochastic Models of Neural Plasticity: Averaging Principles",
  "authors": [
    "Philippe Robert",
    "Gaetan Vignoud"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-17T13:52:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "averaging principle",
      "stochastic models",
      "neural plasticity",
      "stochastic process",
      "simple plastic neural network"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}