# Mathematical studies of the dynamics of finite-size binary neural networks: A review of recent progress

## TL;DR

This review discusses recent mathematical advances in understanding the dynamics of small finite-size binary neural networks, moving beyond traditional mean-field approaches applicable only to infinite networks.

## Contribution

It introduces new mathematical techniques for analyzing bifurcations, activity-potential dualism, correlations, and pattern storage in small neural networks.

## Key findings

- New methods for bifurcation analysis in small networks
- Insights into activity and membrane potential relationships
- Understanding of pattern storage mechanisms

## Abstract

Traditional mathematical approaches to studying analytically the dynamics of neural networks rely on the mean-field approximation, which is rigorously applicable only to networks of infinite size. However, all existing real biological networks have finite size, and many of them, such as microscopic circuits in invertebrates, are composed only of a few tens of neurons. Thus, it is important to be able to extend to small-size networks our ability to study analytically neural dynamics. Analytical solutions of the dynamics of finite-size neural networks have remained elusive for many decades, because the powerful methods of statistical analysis, such as the central limit theorem and the law of large numbers, do not apply to small networks. In this article, we critically review recent progress on the study of the dynamics of small networks composed of binary neurons. In particular, we review the mathematical techniques we developed for studying the bifurcations of the network dynamics, the dualism between neural activity and membrane potentials, cross-neuron correlations, and pattern storage in stochastic networks. Finally, we highlight key challenges that remain open, future directions for further progress, and possible implications of our results for neuroscience.