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Poisson Hypothesis and large-population limit for networks of spiking neurons

Daniele Avitabile, Michel Davydov

TL;DR

The paper addresses the challenge of deriving rigorous mean-field limits for spatially extended networks of spiking neurons with stochastic time of spikes, focusing on leaky and quadratic integrate-and-fire dynamics within the Galves–Löcherbach framework. It employs a replica-mean-field construction to establish a Poisson Hypothesis where inter-neuron interactions are replaced by independent time-inhomogeneous Poisson processes with rates given by mean firing rates, and proves the hypothesis for both linear and quadratic autonomous dynamics. Under the Poisson Hypothesis, the large-population limit yields a well-posed neural-field model with stochastic resets, providing a tractable description of the macroscopic dynamics. A key contribution is the rigorous demonstration of the Poisson Hypothesis for the quadratic autonomous evolution, which extends prior results beyond the linear case and enables a principled large-population analysis of resetting networks. Overall, the work offers a rigorous pathway from interacting GL networks to a neuroscience-relevant neural-field formulation, with implications for studying spiking networks with spatial structure and resets in the large-population limit.

Abstract

We study mean-field descriptions for spatially-extended networks of linear (leaky) and quadratic integrate-and-fire neurons with stochastic spiking times. We consider large-population limits of continuous-time Galves-Löcherbach (GL) networks with linear and quadratic intrinsic dynamics. We prove that that the Poisson Hypothesis holds for the replica-mean-field limit of these networks, that is, in a suitably-defined limit, neurons are independent with interaction times replaced by independent time-inhomogeneous Poisson processes with intensities depending on the mean firing rates, extending known results to networks with quadratic intrinsic dynamics and resets. Proving that the Poisson Hypothesis holds opens up the possibility of studying the large-population limit in these networks. We prove this limit to be a well-posed neural field model, subject to stochastic resets.

Poisson Hypothesis and large-population limit for networks of spiking neurons

TL;DR

The paper addresses the challenge of deriving rigorous mean-field limits for spatially extended networks of spiking neurons with stochastic time of spikes, focusing on leaky and quadratic integrate-and-fire dynamics within the Galves–Löcherbach framework. It employs a replica-mean-field construction to establish a Poisson Hypothesis where inter-neuron interactions are replaced by independent time-inhomogeneous Poisson processes with rates given by mean firing rates, and proves the hypothesis for both linear and quadratic autonomous dynamics. Under the Poisson Hypothesis, the large-population limit yields a well-posed neural-field model with stochastic resets, providing a tractable description of the macroscopic dynamics. A key contribution is the rigorous demonstration of the Poisson Hypothesis for the quadratic autonomous evolution, which extends prior results beyond the linear case and enables a principled large-population analysis of resetting networks. Overall, the work offers a rigorous pathway from interacting GL networks to a neuroscience-relevant neural-field formulation, with implications for studying spiking networks with spatial structure and resets in the large-population limit.

Abstract

We study mean-field descriptions for spatially-extended networks of linear (leaky) and quadratic integrate-and-fire neurons with stochastic spiking times. We consider large-population limits of continuous-time Galves-Löcherbach (GL) networks with linear and quadratic intrinsic dynamics. We prove that that the Poisson Hypothesis holds for the replica-mean-field limit of these networks, that is, in a suitably-defined limit, neurons are independent with interaction times replaced by independent time-inhomogeneous Poisson processes with intensities depending on the mean firing rates, extending known results to networks with quadratic intrinsic dynamics and resets. Proving that the Poisson Hypothesis holds opens up the possibility of studying the large-population limit in these networks. We prove this limit to be a well-posed neural field model, subject to stochastic resets.

Paper Structure

This paper contains 6 sections, 8 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Formal model definition
  3. Main results
  4. Proofs
  5. Proof of \ref{['th:mf_limit']}
  6. Proof of \ref{['th:RMF_PoC']}

Key Result

Theorem 3.1

\newlabelth:mf_limit0 Let $(x,t)\in D\times [0,T].$ Let $(D_l(x))_{l\geq 1}$ be a sequence of nested subsets of $D$ such that $|D_l(x)|=l$ that densely fill $D,$ that is, for any $l, D_l(x) \subset D_{l+1}(x)$ and $\bigcup_{l=1}^{\infty} D_l(x)=D$ and such that $x \in D_1(x).$ When $K \rightarrow

Figures (1)

  • Figure 1: Relations between interacting particle systems

Theorems & Definitions (16)

  • Definition 2.1: Neural-field model with stochastic resets
  • Definition 2.2: GL dynamics under the Poisson Hypothesis
  • Theorem 3.1
  • Definition 3.2: Replica-mean-field dynamics of GL neurons
  • Theorem 3.3: Propagation of chaos for RMF dynamics
  • Lemma 4.1
  • Proof 1
  • Lemma 4.2
  • Proof 2
  • Lemma 4.3
  • ...and 6 more