{
  "id": "2205.07155",
  "version": "v1",
  "published": "2022-05-15T00:07:12.000Z",
  "updated": "2022-05-15T00:07:12.000Z",
  "title": "Characterization of blowups via time change in a mean-field neural network",
  "authors": [
    "Thibaud Taillefumier",
    "Phillip Whitman"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Idealized networks of integrate-and-fire neurons with impulse-like interactions obey McKean-Vlasov diffusion equations in the mean-field limit. These equations are prone to blowups: for a strong enough interaction coupling, the mean-field rate of interaction diverges in finite time with a finite fraction of neurons spiking simultaneously, thereby marking a macroscopic synchronous event. Characterizing these blowup singularities analytically is the key to understanding the emergence and persistence of spiking synchrony in mean-field neural models. However, such a resolution is hindered by the first-passage nature of the mean-field interaction in classically considered dynamics. Here, we introduce a delayed Poissonian variation of the classical integrate-and-fire dynamics for which blowups are analytically well defined in the mean-field limit. Albeit fundamentally nonlinear, we show that this delayed Poissonian dynamics can be transformed into a noninteracting linear dynamics via a deterministic time change. We specify this time change as the solution of a nonlinear, delayed integral equation via renewal analysis of first-passage problems. This formulation also reveals that the fraction of simultaneously spiking neurons can be determined via a self-consistent, probability-conservation principle about the time-changed linear dynamics. We utilize the proposed framework in a companion paper to show analytically the existence of singular mean-field dynamics with sustained synchrony for large enough interaction coupling.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-15T00:07:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G99",
      "60K15",
      "35Q92",
      "35D30",
      "35K67",
      "45H99"
    ],
    "keywords": [
      "time change",
      "mean-field neural network",
      "interactions obey mckean-vlasov diffusion equations",
      "impulse-like interactions obey mckean-vlasov diffusion",
      "characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}