Journal of Statistical Physics

, Volume 165, Issue 3, pp 545–584 | Cite as

On the Dynamics of Random Neuronal Networks

  • Philippe Robert
  • Jonathan TouboulEmail author


We study the mean-field limit and stationary distributions of a pulse-coupled network modeling the dynamics of a large neuronal assemblies. Our model takes into account explicitly the intrinsic randomness of firing times, contrasting with the classical integrate-and-fire model. The ergodicity properties of the Markov process associated to finite networks are investigated. We derive the large network size limit of the distribution of the state of a neuron, and characterize their invariant distributions as well as their stability properties. We show that the system undergoes transitions as a function of the averaged connectivity parameter, and can support trivial states (where the network activity dies out, which is also the unique stationary state of finite networks in some cases) and self-sustained activity when connectivity level is sufficiently large, both being possibly stable.


Mean-field limit Jump process Spiking neural network Stationary solutions Stability 



The authors acknowledge N. Fournier for his comments on a preliminary version.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Inria Paris (RAP and Mycenae Teams)ParisFrance
  2. 2.Mathematical Neuroscience Team, Center for Interdisciplinary Research in Biology (CIRB)College de France, CNRS, INSERM, PSL Research UniversityParisFrance

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