2019/06/24 by Benjamin Aymard, Aymard, Benjamin, Fabien Campillo +3 · 1 citation
Neuroscience · Physics and Astronomy · #FOS: Mathematics #Neural dynamics and brain function #Neuroscience and Neural Engineering #Numerical Analysis (math.NA) #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.1906.10232
openalex publication_date 2019/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work, we propose a nonlinear stochastic model of a network of stochastic spiking neurons. We heuristically derive the mean-field limit of this system. We then design a Monte Carlo method for the simulation of the microscopic system, and a finite volume method (based on an upwind implicit scheme) for the mean-field model. The finite volume method respects numerical versions of the two main properties of the mean-field model, conservation and positivity, leading to existence and uniqueness of a numerical solution. As the size of the network tends to infinity, we numerically observe propagation of chaos and convergence from an individual description to a mean-field description. Numerical evidences for the existence of a Hopf bifurcation (synonym of synchronised activity) for a sufficiently high value of connectivity, are provided.