Torus Graphs for Large Scale Neural Phase Analysis

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.