# The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states

arXiv: https://arxiv.org/abs/1710.05596 · HTML: https://mlsanity.com/paper/1710.05596/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Oct 2017.

## Built on by (most cited first, 1 of 1)

- [Stabilizing Direct Training of Spiking Neural Networks: Membrane Potential Initialization and Threshold-robust Surrogate Gradient](https://mlsanity.com/paper/2511.08708/) (2025)

## Abstract (arXiv)

Neural network dynamics emerge from the interaction of spiking cells. One way to formulate the problem is through a theoretical framework inspired by ideas coming from statistical physics, the so-called mean-field theory. In this document, we investigate different issues related to the mean-field description of an excitatory network made up of leaky integrate-and-fire neurons. The description is written in the form a nonlinear partial differential equation which is known to blow up in finite time when the network is strongly connected. We prove that in a moderate coupling regime the equation is globally well-posed in the space of measures, and that there exist stationary solutions. In the case of weak connectivity we also demonstrate the uniqueness of the steady state and its global exponential stability. The method to show those mathematical results relies on a contraction argument of Doeblin's type in the linear case, which corresponds to a population of non-interacting units.