School of Mathematical and Statistical Sciences Faculty Publications and Presentations
Document Type
Article
Publication Date
3-19-2025
Abstract
This article studies the dynamics of the mean-field approximation of continuous random networks. These networks are stochastic integrodifferential equations driven by Gaussian noise. The kernels in the integral operators are realizations of generalized Gaussian random variables. The equation controls the time evolution of a macroscopic state interpreted as neural activity, which depends on position and time. Such a network corresponds to a statistical field theory (SFT) given by a momenta-generating functional. Discrete versions of the mentioned networks appeared in spin glasses and as models of artificial neural networks. Each of these discrete networks corresponds to a lattice SFT, where the action contains a finite number of neurons and two scalar fields for each neuron. Recently, it has been proposed that these networks can be used as models for deep learning. In this type of applications, the number of neurons is astronomical; consequently, continuous models are required. In this article, we develop mathematically rigorous, continuous versions of the mean-field theory (MFT) approximation and the double-copy system that allow us to derive a condition for the criticality of continuous stochastic networks via the largest Lyapunov exponent. It is essential to mention that the classical methods for MFT approximation and the double-copy based on the stationary phase approximation cannot be used here because we are dealing with oscillatory integrals on infinite dimensional spaces. To our knowledge, the approach presented here is completely new. We use two basic architectures; in the first one, the space of neurons is the real line, and then the neurons are organized in one layer; in the second one, the space of neurons is the p-adic line, and then the neurons are organized in an infinite, fractal, tree-like structure. We also studied a toy model of a continuous Gaussian network with a continuous phase transition. This behavior matches the critical brain hypothesis, which states that certain biological neuronal networks work near phase transitions.
Recommended Citation
Zúñiga-Galindo, W. A. "Dynamic mean-field theory for continuous random networks." Journal of Physics A: Mathematical and Theoretical 58, no. 12 (2025): 125201. https://doi.org/10.1088/1751-8121/adbc50
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication Title
Journal of Physics A: Mathematical and Theoretical
DOI
10.1088/1751-8121/adbc50
Comments
© 2025 The Author(s)
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