# Torus Graphs for Large Scale Neural Phase Analysis

arXiv: https://arxiv.org/abs/2606.00496 · HTML: https://mlsanity.com/paper/2606.00496/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 33 references that are in the map.
- First posted on arXiv 30 May 2026.

## Builds on (most cited first, 4 of 4)

- [Self-Supervised Graph Neural Networks for Improved Electroencephalographic Seizure Analysis](https://mlsanity.com/paper/2104.08336/) (2021)
- [EEG2Rep: Enhancing Self-supervised EEG Representation Through Informative Masked Inputs](https://mlsanity.com/paper/2402.17772/) (2024)
- [EvoBrain: Dynamic Multi-Channel EEG Graph Modeling for Time-Evolving Brain Networks](https://mlsanity.com/paper/2509.15857/) (2025)
- [Torus Graphs for Multivariate Phase Coupling Analysis](https://mlsanity.com/paper/1910.11044/) (2019)

## Abstract (arXiv)

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 ~ ! 100 variables because its score matching inference scales as O(d6). We introduce a stochastic score matching procedure that reduces the per-iteration cost to O(d2), 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.