{
  "id": "1211.0299",
  "version": "v5",
  "published": "2012-11-01T20:25:20.000Z",
  "updated": "2015-06-19T07:45:36.000Z",
  "title": "Global solvability of a networked integrate-and-fire model of McKean-Vlasov type",
  "authors": [
    "François Delarue",
    "James Inglis",
    "Sylvain Rubenthaler",
    "Etienne Tanré"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/14-AAP1044 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Applied Probability 2015, Vol. 25, No. 4, 2096-2133",
  "doi": "10.1214/14-AAP1044",
  "categories": [
    "math.PR"
  ],
  "abstract": "We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by $\\alpha$, is of great importance as the resulting system is known to blow-up for large values of $\\alpha$. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when $\\alpha$ is small enough.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-06-02T18:48:26.000Z",
      "abstract": "We here investigate the well-posedness of a networked integrate-and-fire model describing an infinite population of neurons which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the potential of a neuron increases when some of the others fire: precisely, the kick it receives is proportional to the instantaneous proportion of firing neurons at the same time. From a mathematical point of view, the coefficient of proportionality is of great importance as the resulting system is known to blow-up as this becomes large. In the current paper, we focus on the complementary regime and prove that existence and uniqueness hold for all time when the coefficient of proportionality is small enough.",
      "comment": "Version 4: shortened version",
      "journal": "33 pages (2012)",
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2015-06-19T07:45:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "networked integrate-and-fire model",
      "global solvability",
      "mckean-vlasov type",
      "infinite population",
      "uniqueness hold"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.0299D"
    }
  }
}