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.
Systems of spiking neurons Metastability Interacting particle systems
Mathematics Subject Classification
60K35 82C32 82C22
This is a preview of subscription content, log in to check access.
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).
Ferrari, P.A., Galves, A., Grigorescu, I., Löcherbach, E.: Phase transition for infinite systems of spiking neurons. J. Stat. Phys. 172, 1564–1575 (2018)ADSMathSciNetCrossRefGoogle Scholar
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)MathSciNetzbMATHGoogle Scholar
Brillinger, D.: Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybern. 59(3), 189–200 (1988)CrossRefGoogle Scholar
Hawkes, A.G.: Point spectra of some mutually exciting point processes. J. R. Stat. Soc. Ser. B (Methodological) 33(3), 438–443 (1971)MathSciNetzbMATHGoogle Scholar
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)MathSciNetCrossRefGoogle Scholar
Olivieri, E., Vares, M.E.: Large Deviations and Metastability. Encyclopedia of Mathematics and its Applications, vol. 100. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar