# Introduction to dynamical mean-field theory of randomly connected neural networks with bidirectionally correlated couplings

arXiv: https://arxiv.org/abs/2305.08459 · HTML: https://mlsanity.com/paper/2305.08459/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 48 references that are in the map.
- First posted on arXiv 15 May 2023.

## Builds on (most cited first, 11 of 11)

- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [Self-Consistent Dynamical Field Theory of Kernel Evolution in Wide Neural Networks](https://mlsanity.com/paper/2205.09653/) (2022)
- [The effective noise of Stochastic Gradient Descent](https://mlsanity.com/paper/2112.10852/) (2021)
- [Unified field theoretical approach to deep and recurrent neuronal networks](https://mlsanity.com/paper/2112.05589/) (2021)
- [Theory of gating in recurrent neural networks](https://mlsanity.com/paper/2007.14823/) (2020)
- [Dynamics of random recurrent networks with correlated low-rank structure](https://mlsanity.com/paper/1909.04358/) (2019)
- [Correlations between synapses in pairs of neurons slow down dynamics in randomly connected neural networks](https://mlsanity.com/paper/1707.08337/) (2017)
- [Forgetting leads to chaos in attractor networks](https://mlsanity.com/paper/2112.00119/) (2021)
- [Emergence of hierarchical modes from deep learning](https://mlsanity.com/paper/2208.09859/) (2022)
- [Spectrum of non-Hermitian deep-Hebbian neural networks](https://mlsanity.com/paper/2208.11411/) (2022)
- [Fluctuation-dissipation relations for spiking neurons](https://mlsanity.com/paper/2204.00549/) (2022)

## Built on by (most cited first, 6 of 6)

- [Eight challenges in developing theory of intelligence](https://mlsanity.com/paper/2306.11232/) (2023)
- [Nonequilbrium physics of generative diffusion models](https://mlsanity.com/paper/2405.11932/) (2024)
- [Neural Langevin Machine: a local asymmetric learning rule can be creative](https://mlsanity.com/paper/2506.23546/) (2025)
- [The Effect of Optimal Self-Distillation in Noisy Gaussian Mixture Model](https://mlsanity.com/paper/2501.16226/) (2025)
- [Fokker-Planck to Callan-Symanzik: evolution of weight matrices under training](https://mlsanity.com/paper/2501.09659/) (2025)
- [Neuronal correlations shape the scaling behavior of memory capacity and nonlinear computational capability of reservoir recurrent neural networks](https://mlsanity.com/paper/2504.19657/) (2025)

## Abstract (arXiv)

Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.