Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Replica Mean Field limits for neural networks with excitatory and inhibitory activity

Ioannis Papageorgiou

TL;DR

The paper develops replica mean-field limits for an infinite neural network with excitatory and inhibitory interactions under the Galves-Löcherbach model, extending analysis beyond the Uniform Summability Principle. It builds a finite replica framework with cross-replica interactions and uses the Poisson Hypothesis to derive an ordinary differential equation for the RMF limit of the moment generating function, yielding analytic expressions for stationary means. The study treats both linear and nonlinear intensity and interaction regimes, providing explicit RMF equations and illustrating that regeneration arguments ensure ergodicity in the RMF limit. The results offer analytic access to the invariant state and balance properties of large sparse networks, enabling study of finite-size effects and excitatory-inhibitory balance in neural dynamics.

Abstract

We study Replica Mean Field limits for a neural system of infinitely many neurons with both inhibitory and excitatory interactions. As a result we obtain an analytical characterisation of the invariant state. In particular we focus on the Galves-Löcherbach model with interactions beyond the Uniform Summability Principle.

Replica Mean Field limits for neural networks with excitatory and inhibitory activity

TL;DR

The paper develops replica mean-field limits for an infinite neural network with excitatory and inhibitory interactions under the Galves-Löcherbach model, extending analysis beyond the Uniform Summability Principle. It builds a finite replica framework with cross-replica interactions and uses the Poisson Hypothesis to derive an ordinary differential equation for the RMF limit of the moment generating function, yielding analytic expressions for stationary means. The study treats both linear and nonlinear intensity and interaction regimes, providing explicit RMF equations and illustrating that regeneration arguments ensure ergodicity in the RMF limit. The results offer analytic access to the invariant state and balance properties of large sparse networks, enabling study of finite-size effects and excitatory-inhibitory balance in neural dynamics.

Abstract

We study Replica Mean Field limits for a neural system of infinitely many neurons with both inhibitory and excitatory interactions. As a result we obtain an analytical characterisation of the invariant state. In particular we focus on the Galves-Löcherbach model with interactions beyond the Uniform Summability Principle.
Paper Structure (4 sections, 8 theorems, 57 equations)

This paper contains 4 sections, 8 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Replica Mean Field Limits
  3. Linear intensity and linear interaction.
  4. Non-linear intensity and non linear interaction.

Key Result

Proposition 3.1

Assume that $\phi$ is an increasing function of polynomial order. For every finite set of neurons $K=\{1,2,...,s\}$ for some $s\in {\mathbb N}$, such that for $K^{i}\subset \{1,...,i-1,i+1\}$ for any $i\in K$ and $u>0,$ there exist $c>0$ and $d>0$ such that

Theorems & Definitions (15)

  • Proposition 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • Corollary 3.6
  • Remark 3.7
  • ...and 5 more