Torus Graphs for Large Scale Neural Phase Analysis

2026-05-29

AI Summary

Traditional neural phase analysis methods struggle to handle the hundreds to thousands of frequency–phase variables present in high-throughput electrophysiological data and are unable to model dynamic or directed coupling. This work addresses these limitations by extending torus graphs—a statistical framework for circular data—to scale to thousands of variables, introducing hidden Markov and autoregressive variants that respectively enable state-dependent switching and directed phase interactions. By integrating stochastic score matching with exponential family distributions, the approach achieves computationally efficient inference. Applied to 1,860-dimensional local field potential (LFP) recordings, the method successfully uncovers dynamic reconfiguration of phase coupling between wakefulness and non-rapid eye movement sleep, yielding a large-scale, systematic map of neural phase relationships.

Abstract

Oscillatory neural signals such as electroencephalography (EEG) and local field potentials (LFPs) show phase relationships that coordinate communication across brain regions. Modern recordings capture hundreds of channels across many frequency bins, yet standard phase analyses are restricted to only a few variables. The Torus Graph (TG) model, an exponential-family distribution over phases whose univariate and pairwise potentials generalize von Mises distributions, infers principled structure among oscillations but models only static, undirected dependencies and is limited to $\sim \! 100$ variables because its score matching inference scales as $\mathcal{O}(d^{6})$. We introduce a stochastic score matching procedure that reduces the per-iteration cost to $\mathcal{O}(d^{2})$, enabling inference on datasets with thousands of variables. This scalable foundation supports analyses of 1,860 frequency-phase features from multi-electrode LFPs and enables two extensions previously inaccessible to TGs or classical circular statistics: (i) a TG Hidden Markov Model capturing state-dependent phase-coupling changes (e.g., spindle-related states during sleep) and (ii) an autoregressive TG inferring directional interactions via transfer-entropy estimation. Applied to LFP recordings, these models reveal state-dependent phase-interaction patterns between wakefulness and NREM sleep. Together, they enable systematic, large-scale mapping of dynamic and directional phase relationships across brain and cognitive states.

Problem

• phase coupling
• large-scale neural data
• oscillatory signals
• dynamic interactions
• directional connectivity

Innovation

• Torus Graph
• stochastic score matching
• phase coupling
• directed interactions
• scalable inference