October 2023 Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling
Xavier Erny, Eva Löcherbach, Dasha Loukianova
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Ann. Appl. Probab. 33(5): 3563-3586 (October 2023). DOI: 10.1214/22-AAP1900


We consider the stochastic system of interacting neurons introduced in (J. Stat. Phys. 158 (2015) 866–902) and in (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 1844–1876) and then further studied in (Electron. J. Probab. 26 (2021) 20) in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order 1/N. In between successive spikes, each neuron’s potential follows a deterministic flow. In our previous article (Electron. J. Probab. 26 (2021) 20) we proved the convergence of the system, as N, to a limit nonlinear jumping stochastic differential equation. In the present article we complete this study by establishing a strong convergence result, stated with respect to an appropriate distance, with an explicit rate of convergence. The main technical ingredient of our proof is the coupling introduced in (Z. Wahrsch. Verw. Gebiete 34 (1976) 33–58) of the point process representing the small jumps of the particle system with the limit Brownian motion.

Funding Statement

This work has been conducted as part of the FAPESP project Research, Innovation and Dissemination Center for Neuromathematics (grant 2013/07699-0) and of the ANR project ANR-19-CE40-0024.


The authors thank the anonymous referees for the positive feedback, many useful remarks and careful reading. E.L. thanks Vlad Bally for discussions about Euler schemes and the nonneed of approximating the particle system by its Euler scheme, and Nicolas Fournier for pointing out an error in an early version of this manuscript.


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Xavier Erny. Eva Löcherbach. Dasha Loukianova. "Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling." Ann. Appl. Probab. 33 (5) 3563 - 3586, October 2023. https://doi.org/10.1214/22-AAP1900


Received: 1 November 2021; Revised: 1 May 2022; Published: October 2023
First available in Project Euclid: 3 November 2023

Digital Object Identifier: 10.1214/22-AAP1900

Primary: 60G09 , 60G55 , 60J76 , 60K35
Secondary: 60G09

Keywords: conditional propagation of chaos , exchangeability , KMT coupling , mean field interaction

Rights: Copyright © 2023 Institute of Mathematical Statistics


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Vol.33 • No. 5 • October 2023
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