Persistence in a large network of sparsely interacting neurons

Maximiliano Altamirano*, Roberto Cortez, Matthieu Jonckheere, Lasse Leskelä

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review


This article presents a biological neural network model driven by inhomogeneous Poisson processes accounting for the intrinsic randomness of synapses. The main novelty is the introduction of sparse interactions: each firing neuron triggers an instantaneous increase in electric potential to a fixed number of randomly chosen neurons. We prove that, as the number of neurons approaches infinity, the finite network converges to a nonlinear mean-field process characterised by a jump-type stochastic differential equation. We show that this process displays a phase transition: the activity of a typical neuron in the infinite network either rapidly dies out, or persists forever, depending on the global parameters describing the intensity of interconnection. This provides a way to understand the emergence of persistent activity triggered by weak input signals in large neural networks.

Original languageEnglish
Article number16
Pages (from-to)1-30
Number of pages30
JournalJournal of Mathematical Biology
Issue number1
Early online dateDec 2022
Publication statusPublished - Jan 2023
MoE publication typeA1 Journal article-refereed


  • Biological neural network
  • Interacting particle system
  • Mean-field limit
  • Nonlinear Markov process
  • Phase transition
  • Propagation of chaos


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