# A complete mean-field theory for dynamics of binary recurrent neural networks

## TL;DR

This paper presents a comprehensive mean-field theory for the dynamics of binary recurrent neural networks, capturing finite-size effects, fluctuations, and novel dynamic states beyond traditional deterministic models.

## Contribution

It introduces a unified framework using martingale theory to analyze nonequilibrium fluctuations and inhomogeneous interactions in recurrent networks.

## Key findings

- Discovery of a new dynamic state with collective stochastic fluctuations
- Demonstration of the theory's ability to handle finite-size and inhomogeneous networks
- Extension of mean-field theory to nonequilibrium and stochastic regimes

## Abstract

We develop a unified theory that encompasses the macroscopic dynamics of recurrent interactions of binary units within arbitrary network architectures. Using the martingale theory, our mathematical analysis provides a complete description of nonequilibrium fluctuations in networks with finite size and finite degree of interactions. Our approach allows the investigation of systems for which a deterministic mean-field theory breaks down. To demonstrate this, we uncover a novel dynamic state in which a recurrent network of binary units with statistically inhomogeneous interactions, along with an asynchronous behavior, also exhibits collective nontrivial stochastic fluctuations in the thermodynamical limit.