{
  "id": "1410.3263",
  "version": "v1",
  "published": "2014-10-13T11:16:16.000Z",
  "updated": "2014-10-13T11:16:16.000Z",
  "title": "On a toy model of interacting neurons",
  "authors": [
    "Nicolas Fournier",
    "Eva Löcherbach"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We continue the study of a stochastic system of interacting neurons introduced in De Masi-Galves-L\\\"ocherbach-Presutti (2014). The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the neuron potential is reset to 0 and all other neurons receive an additional amount 1/N of potential. Moreover, electrical synapses induce a deterministic drift of the system towards its center of mass. We prove propagation of chaos of the system, as N tends to infinity, to a limit nonlinear jumping stochastic differential equation. We consequently improve on the results of De Masi-Galves-L\\\"ocherbach-Presutti (2014), since (i) we remove the compact support condition on the initial datum, (ii) we get a rate of convergence in $1/\\sqrt N$. Finally, we study the limit equation: we describe the shape of its time-marginals, we prove the existence of a unique non-trivial invariant distribution, we show that the trivial invariant distribution is not attractive, and in a special case, we establish the convergence to equilibrium.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-13T11:16:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60G55",
      "60F17"
    ],
    "keywords": [
      "interacting neurons",
      "toy model",
      "nonlinear jumping stochastic differential equation",
      "unique non-trivial invariant distribution",
      "limit nonlinear jumping stochastic differential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.3263F"
    }
  }
}