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A Result of Metastability for an Infinite System of Spiking Neurons

Abstract

In 2018, Ferrari et al. wrote a paper called “Phase Transition for Infinite Systems of Spiking Neurons” in which they introduced a continuous time stochastic model of interacting neurons. This model consists in a countable number of neurons, each of them having an integer-valued membrane potential, which value determine the rate at which the neuron spikes. This model has also a parameter \(\gamma \), corresponding to the rate of the leak times of the neurons, that is, the times at which the membrane potential of a given neuron is spontaneously reset to its resting value (which is 0 by convention). As its title says, it was proven in this previous article that this model presents a phase transition phenomenon with respect to \(\gamma \). Here we prove that this model also exhibits a metastable behavior. By this we mean that if \(\gamma \) is small enough, then the re-normalized time of extinction of a finite version of this system converges toward an exponential random variable of mean 1 as the number of neurons goes to infinity.

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Acknowledgements

This work is part of my PhD thesis. I thank my PhD adviser Antonio Galves for introducing me to the subject of metastability and for fruitful discussions. Many thanks also to Christophe Pouzat who introduced me to the field of neuromathematics. This article was produced as part of the activities of FAPESP Research, Innovation and Dissemination Center for Neuromathematics (Grant Number 2013/07699-0 , S.Paulo Research Foundation), and the author was supported by a FAPESP scholarship (Grant Number 2017/02035-7).

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Correspondence to Morgan André.

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Communicated by Eric A. Carlen.

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André, M. A Result of Metastability for an Infinite System of Spiking Neurons. J Stat Phys 177, 984–1008 (2019). https://doi.org/10.1007/s10955-019-02402-4

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Keywords

  • Systems of spiking neurons
  • Metastability
  • Interacting particle systems

Mathematics Subject Classification

  • 60K35
  • 82C32
  • 82C22