Journal of Computational Neuroscience

, Volume 44, Issue 3, pp 297–312 | Cite as

An integrate-and-fire model to generate spike trains with long-range dependence

  • Alexandre RichardEmail author
  • Patricio Orio
  • Etienne Tanré


Long-range dependence (LRD) has been observed in a variety of phenomena in nature, and for several years also in the spiking activity of neurons. Often, this is interpreted as originating from a non-Markovian system. Here we show that a purely Markovian integrate-and-fire (IF) model, with a noisy slow adaptation term, can generate interspike intervals (ISIs) that appear as having LRD. However a proper analysis shows that this is not the case asymptotically. For comparison, we also consider a new model of individual IF neuron with fractional (non-Markovian) noise. The correlations of its spike trains are studied and proven to have LRD, unlike classical IF models. On the other hand, to correctly measure long-range dependence, it is usually necessary to know if the data are stationary. Thus, a methodology to evaluate stationarity of the ISIs is presented and applied to the various IF models. We explain that Markovian IF models may seem to have LRD because of non-stationarities.


Interspike interval statistics Stochastic integrate-and-fire model Long-range dependence Stationarity 



Part of this work was carried out while A.R. was a postdoc at Inria Sophia-Antipolis and at Ecole Polytechnique (the support from ERC 321111 Rofirm is gratefully acknowledged). A.R. and E.T. acknowledge the support from the ECOS-Sud Program Chili-France C15E05 and from the European Union’s Horizon 2020 Framework Program for Research and Innovation under Grant Agreement No. 720270 (Human Brain Project SGA1). P.O acknowledges the support from the Advanced Center for Electrical and Electronic Engineering (Basal Funding FB0008, Conicyt) and the project P09-022-F from the Millennium Scientific Initiative of the Chilean Ministry of Economy, Development, and Tourism.

We thank the reviewers for their remarks which helped to improve significantly the quality of this paper.

Compliance with Ethical Standards

Conflict of Interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.CentraleSupélec, Université Paris-Saclay, Laboratoire MICS et Fédération CNRS - FR3487Gif-sur-YvetteFrance
  2. 2.Instituto de Neurociencia, Facultad de CienciasUniversidad de ValparaísoValparaisoChile
  3. 3.Centro Interdisciplinario de Neurociencia de ValparaísoUniversidad de ValparaísoValparaisoChile
  4. 4.Université Côte d’Azur, InriaSophia-AntipolisFrance

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