Skip to main content

A Result of Metastability for an Infinite System of Spiking Neurons


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4


  1. 1.

    Ferrari, P.A., Galves, A., Grigorescu, I., Löcherbach, E.: Phase transition for infinite systems of spiking neurons. J. Stat. Phys. 172, 1564–1575 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Galves, A., Löcherbach, E.: Infinite systems of interacting chains with memory of variable length. J. Stat. Phys. 151, 896–921 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Galves, A., Löcherbach, E.: Modeling networks of spiking neurons as interacting processes with memory of variable length. J. de la Société Française de Statistiques 157, 17–32 (2016)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Brillinger, D.: Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybern. 59(3), 189–200 (1988)

    Article  Google Scholar 

  5. 5.

    Hawkes, A.G.: Point spectra of some mutually exciting point processes. J. R. Stat. Soc. Ser. B (Methodological) 33(3), 438–443 (1971)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Harris, T.E.: Additive set-valued Markov processes and graphical methods. Ann. Probab. 6, 355–378 (1978)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bertein, F., Galves, A.: Une classe de systèmes de paticules stable par association. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 41, 73–85 (1977)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Lebowitz, J.L., Penrose, O.: Rigorous treatment of metastable states in the van der Waals-Maxwell theory. J. Stat. Phys. 3(2), 211–236 (1971)

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    Casandro, M., Galves, A., Olivieri, E., Vares, M.E.: Metastable behavior of stochastic dynamics: a pathwise approach. J. Stat. Phys. 35, 603 (1984)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Schonmann, R.H.: Metastability for the contact process. J. Stat. Phys. 41, 445–464 (1985)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Mountford, T.S.: A metastable result for the finite multidimensional contact process. Canad. Math. Bull. 36(2), 216–226 (1993)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Fernandez, R., Manzot, F., Nardi, F.R., Scoppola, E.: Asymptotically exponential hitting times and metastability: a pathwise approach without reversibility. Electron. J. Probab. 20, 1–37 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Olivieri, E., Vares, M.E.: Large Deviations and Metastability. Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  14. 14.

    Harris, T.E.: On a class of set valued Markov processes. Ann. Probab. 4, 175–194 (1976)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ligget, T.M.: Grundlehren der mathematischen Wissenschaften (Interacting Particle Systems), p. 276. Springer, Berlin (1985)

    Google Scholar 

  16. 16.

    Durrett, R.: An Introduction to Infinite Particle Systems. Stoch. Processes Appl. 11, 109–150 (1981)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Holley, R.: Markovian interaction processes with finite range interaction. Ann. Math. Stat. 43, 1961–1967 (1972)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Durrett, R.: Probability: Theory and Examples, 4th edn. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  19. 19.

    Griffeath, D.: The basic contact processes. Stoch. Processes Appl. 11, 151–185 (1980)

    MathSciNet  Article  Google Scholar 

Download references


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).

Author information



Corresponding author

Correspondence to Morgan André.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Eric A. Carlen.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

André, M. A Result of Metastability for an Infinite System of Spiking Neurons. J Stat Phys 177, 984–1008 (2019).

Download citation


  • Systems of spiking neurons
  • Metastability
  • Interacting particle systems

Mathematics Subject Classification

  • 60K35
  • 82C32
  • 82C22