Diffusion-Guided Renormalization of Neural Systems via Tensor Networks

TL;DR

This work develops diffusion-guided renormalization as a scalable framework for modeling non-equilibrium neural dynamics across scales by marrying stochastic thermodynamics with geometric representations and tensor networks. It introduces Latent Graph Diffusion (LGD) to infer directed temporal graphs from subsampled data and Diffusion Tensor Network Renormalization (DTNR) to generate multiscale renormalization group flows over joint probability distributions. The methods are demonstrated on leaky integrate-and-fire networks and clustered neural architectures, revealing diffusion modes and community structures that drive information processing and anti-correlated activity. Collectively, the approach enables coarse-to-fine prediction, decoding, and control of dissipative neural trajectories with interpretable, scale-aware models that bridge neural computation and quantum-inspired computational techniques.

Abstract

Far from equilibrium, neural systems self-organize across multiple scales. Exploiting multiscale self-organization in neuroscience and artificial intelligence requires a computational framework for modeling the effective non-equilibrium dynamics of stochastic neural trajectories. Non-equilibrium thermodynamics and representational geometry offer theoretical foundations, but we need scalable data-driven techniques for modeling collective properties of high-dimensional neural networks from partial subsampled observations. Renormalization is a coarse-graining technique central to studying emergent scaling properties of many-body and nonlinear dynamical systems. While widely applied in physics and machine learning, coarse-graining complex dynamical networks remains unsolved, affecting many computational sciences. Recent diffusion-based renormalization, inspired by quantum statistical mechanics, coarse-grains networks near entropy transitions marked by maximal changes in specific heat or information transmission. Here I explore diffusion-based renormalization of neural systems by generating symmetry-breaking representations across scales and offering scalable algorithms using tensor networks. Diffusion-guided renormalization bridges microscale and mesoscale dynamics of dissipative neural systems. For microscales, I developed a scalable graph inference algorithm for discovering community structure from subsampled neural activity. Using community-based node orderings, diffusion-guided renormalization generates renormalization group flow through metagraphs and joint probability functions. Towards mesoscales, diffusion-guided renormalization targets learning the effective non-equilibrium dynamics of dissipative neural trajectories occupying lower-dimensional subspaces, enabling coarse-to-fine control in systems neuroscience and artificial intelligence.

Figures (28)

• Figure 1: Subsampled neural networks may be coarse-grained through diffusion-guided renormalization, which identifies higher-order structures in latent graph communities.
• Figure 2:
• Figure 3: Open neural systems exchange energy, matter, and heat with the external environment. Recurrent neural networks give rise to stochastic diffusion modes across space and time. Diffusion modes have fast and slow components as revealed by spectral analysis of time-series and graphs.
• Figure 4: Representational geometry offers a unified approach to symmetry-breaking neural networks across multiple spatiotemporal scales. Translational symmetry of neuronal dynamics is used to build temporal node features. Permutation symmetry of unordered sets is constrained by the equivariance of directed temporal graphs. Time-reversal symmetry of effective non-equilibrium dynamics is broken by chaotic mixing and current flows. Isometry and gauge symmetries of neural manifolds are used to compress high-dimensional attractors into kinetic networks for neural codewords.
• Figure 5: Relation of subsampled neural systems to the dissipative, non-equilibrium thermodynamics of open neural systems. (Left) Schematic representation of a neural circuit with sparse, signed asymmetric coupling. (Right) Dissipation from the neural system to the external environment drives multiscale fluctuations.