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arXiv · 2607.11685

A stochastic Fokker--Planck equation for the mean-field limit of a population of noisy integrate-and-fire neurons

Ben Hambly·Aldaïr Petronilia·Christoph Reisinger·Andreas Søjmark
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Abstract

We study a densely connected system of excitatory integrate-and-fire neurons which are subject to common noise. The system incorporates a gradual transmission of action potentials and captures the re- and hyperpolarization phases through a random refractory period followed by a reset to a randomized level below the rest potential. As the number of neurons tends to infinity, we show that there is weak convergence to a unique membrane potential density governed by a stochastic Fokker--Planck equation with a well-defined spike transmission rate. The latter is driven by the mean cumulative spike count, which is non-differentiable but shown to satisfy a generalized flux condition. We obtain the uniqueness of the Fokker--Planck equation from energy estimates in the dual of the first Sobolev space. Finally, we give a conditional McKean--Vlasov representation of the membrane potential density as the law of a representative neuron given the common noise.

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Ben Hambly, Aldaïr Petronilia, Christoph Reisinger, Andreas Søjmark. 2026-07-13. A stochastic Fokker--Planck equation for the mean-field limit of a population of noisy integrate-and-fire neurons. https://arxiv.org/abs/2607.11685

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