{
  "id": "2209.03443",
  "version": "v1",
  "published": "2022-09-07T20:04:31.000Z",
  "updated": "2022-09-07T20:04:31.000Z",
  "title": "Topological-numerical analysis of a two-dimensional discrete neuron model",
  "authors": [
    "Paweł Pilarczyk",
    "Justyna Signerska-Rynkowska",
    "Grzegorz Graff"
  ],
  "categories": [
    "math.DS"
  ],
  "abstract": "We conduct computer-assisted analysis of the two-dimensional model of a neuron introduced by Chialvo in 1995 (Chaos, Solitons & Fractals 5, 461-479). We apply the method for rigorous analysis of global dynamics based on a set-oriented topological approach, introduced by Arai et al. in 2009 (SIAM J. Appl. Dyn. Syst. 8, 757-789) and improved and expanded afterwards. Additionally, we introduce a new algorithm to analyze the return times inside a chain recurrent set. Based on this analysis, together with the information on the size of the chain recurrent set, we develop a new method that allows one to determine subsets of parameters for which chaotic dynamics may appear. This approach can be applied to a variety of dynamical systems, and we discuss some of its practical aspects. The data and the software described in the paper are available at http://www.pawelpilarczyk.com/neuron/.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-07T20:04:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37B35",
      "37B30",
      "37M99",
      "37N25",
      "92-08"
    ],
    "keywords": [
      "two-dimensional discrete neuron model",
      "topological-numerical analysis",
      "chain recurrent set",
      "return times inside",
      "global dynamics"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}