The mean-field limit of sparse networks of integrate-and-fire neurons

We study the mean-field limit of a model of biological neuron networks based on the so-called stochastic integrate-and-fire (IF) dynamics. However, we do not assume any structure on the graph of interactions but consider instead any connection weights between neurons that obey a generic mean-field scaling. To address this, we propose a novel notion of observables that naturally extends the marginals laws in studying classical many-particle systems. We prove the stability of the network in terms of observables, by applying a novel commutator estimate in weak norms to a tree-indexed extension of the BBGKY hierarchy. While we require non-vanishing diffusion, this approach notably addresses the challenges of sparse interacting graphs/matrices and singular interactions from Poisson jumps, and requires no additional regularity on the initial distribution.