Non-exchangeable networks of integrate-and-fire neurons: spatially-extended mean-field limit of the empirical measure

The dynamics of exchangeable or spatially-structured networks of $N$ interacting stochastic neurons can be described by deterministic population equations in the mean-field limit $N\to\infty$, when synaptic weights scale as $O(1/N)$. This asymptotic behavior has been proven in several works but a general question has remained unanswered: does the $O(1/N)$ scaling of synaptic weights, by itself, suffice to guarantee the convergence of network dynamics to a deterministic population equation, even when networks are not assumed to be exchangeable or spatially structured? In this work, we consider networks of stochastic integrate-and-fire neurons with arbitrary synaptic weights satisfying only a $O(1/N)$ scaling condition. Borrowing results from the theory of dense graph limits (graphons), we prove that, as $N\to\infty$, and up to the extraction of a subsequence, the empirical measure of the neurons' membrane potentials converges to the solution of a spatially-extended mean-field partial differential equation (PDE). Our proof requires analytical techniques that go beyond standard propagation of chaos methods. In particular, we introduce a weak metric that depends on the dense graph limit kernel and we show how the weak convergence of the initial data can be obtained by propagating the regularity of the limit kernel along the dual-backward equation associated with the spatially-extended mean-field PDE. Overall, this result invites us to re-interpret spatially-extended population equations as universal mean-field limits of networks of neurons with $O(1/N)$ synaptic weight scaling.