Dense networks of integrate-and-fire neurons: Spatially-extended mean-field limit of the empirical measure

TL;DR

The paper proves that dense networks of integrate-and-fire neurons with $O(1/N)$ synaptic scaling admit a universal spatially-extended mean-field limit, even without prescribed spatial structure, by leveraging graphon limits to a limit kernel $w$ and a kernel-dependent weak metric. The extended empirical measure $\m{\\mu}^N$ converges (along subsequences and up to neuron re-orderings) to a PDE-valued mean-field solution $\\mu(t,\\xi,dx)$ that obeys a spatially-extended transport-reaction equation with firing-rate and postsynaptic-input terms $r$ and $h$ defined via $f$ and $w$. The proof blends kinetic-theory methods with graphon techniques, introducing a kernel-adapted metric $\\|\\cdot\\|_{\\Phi_w^{-1}}$ and a dual-backward analysis to propagate regularity and control convergence, while a carefully designed change of variables isolates the singular input and renders the limit tractable. Overall, the work reframes spatially-extended population equations as universal mean-field limits for dense, non-exchangeable neural networks and clarifies how graphon limits govern emergent spatial structure in the mean-field dynamics.

Abstract

The dynamics of spatially-structured networks of $N$ interacting stochastic neurons can be described by deterministic population equations in the mean-field limit. While this is known, a general question has remained unanswered: does synaptic weight scaling suffice, by itself, to guarantee the convergence of network dynamics to a deterministic population equation, even when networks are not assumed to be homogeneous or spatially structured? In this work, we consider networks of stochastic integrate-and-fire neurons with arbitrary synaptic weights satisfying a $O(1/N)$ scaling condition. Borrowing results from the theory of dense graph limits, or 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 reinterpret spatially-extended population equations as universal mean-field limits of networks of neurons with $O(1/N)$ synaptic weight scaling.

Theorems & Definitions (32)

• Definition
• Theorem 1
• Corollary 1
• Lemma 1
• Lemma 2
• Theorem 2
• proof : Proof of Theorem \ref{['thm:main']} via Theorem \ref{['thm:main_metric']}
• Proposition 1
• Proposition 2
• Proposition 3