Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling

Abstract

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 $1/\sqrt{\mathit{N}}$. 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 $\mathit{N}\to \infty$, 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.