Gell-Mann-Low criticality in neural networks

TL;DR

This paper develops a renormalized theoretical framework for neural criticality, revealing how scale interactions enable complex computation and information storage in neural systems.

Contribution

It introduces a Gell-Mann-Low type renormalized theory for neural dynamics, analyzing scale-dependent interactions in a neural field model.

Findings

Interactions flow towards Gaussian fixed point logarithmically slowly

Critical structure balances linearity and nonlinearity for computation

Theory extends understanding of neural criticality beyond mean-field approximations

Abstract

Criticality is deeply related to optimal computational capacity. The lack of a renormalized theory of critical brain dynamics, however, so far limits insights into this form of biological information processing to mean-field results. These methods neglect a key feature of critical systems: the interaction between degrees of freedom across all length scales, which allows for complex nonlinear computation. We present a renormalized theory of a prototypical neural field theory, the stochastic Wilson-Cowan equation. We compute the flow of couplings, which parameterize interactions on increasing length scales. Despite similarities with the Kardar-Parisi-Zhang model, the theory is of a Gell-Mann-Low type, the archetypal form of a renormalizable quantum field theory. Here, nonlinear couplings vanish, flowing towards the Gaussian fixed point, but logarithmically slowly, thus remaining effective on most scales. We show this critical structure of interactions to implement a desirable trade-off between linearity, optimal for information storage, and nonlinearity, required for computation.