{
  "id": "1802.07829",
  "version": "v1",
  "published": "2018-02-21T22:09:07.000Z",
  "updated": "2018-02-21T22:09:07.000Z",
  "title": "Phase transition for infinite systems of spiking neurons",
  "authors": [
    "P. A. Ferrari",
    "A. Galves",
    "I. Grigorescu",
    "E. Löcherbach"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove the existence of a phase transition for a stochastic model of interacting neurons. The spiking activity of each neuron is represented by a point process having rate $1 $ whenever its membrane potential is larger than a threshold value. This membrane potential evolves in time and integrates the spikes of all {\\it presynaptic neurons} since the last spiking time of the neuron. When a neuron spikes, its membrane potential is reset to $0$ and simultaneously, a constant value is added to the membrane potentials of its postsynaptic neurons. Moreover, each neuron is exposed to a leakage effect leading to an abrupt loss of potential occurring at random times driven by an independent Poisson point process of rate $\\gamma > 0 .$ For this process we prove the existence of a value $\\gamma_c$ such that the system has one or two extremal invariant measures according to whether $\\gamma > \\gamma_c $ or not.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-21T22:09:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G55",
      "60K35",
      "92B99"
    ],
    "keywords": [
      "phase transition",
      "infinite systems",
      "spiking neurons",
      "independent poisson point process",
      "extremal invariant measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}