Title: Scalable Torus Graphs for Large-Scale Neural Phase Analysis

Abstract:

Oscillatory neural signals, including local field potentials (LFPs) and electroencephalography (EEG), exhibit phase relationships that facilitate communication between different brain regions. While contemporary recording technologies capture hundreds of channels across numerous frequency bins, traditional phase analysis methods are constrained to examining only a limited number of variables. The Torus Graph (TG) model addresses this by employing an exponential-family distribution over phases, where univariate and pairwise potentials extend the von Mises distribution to infer structured relationships among oscillations. However, the standard TG approach is limited to approximately 100 variables due to its $\mathcal{O}(d^{6})$ score matching inference complexity, and it can only model static, undirected dependencies.

To overcome these limitations, we present a stochastic score matching procedure that lowers the computational cost per iteration to $\mathcal{O}(d^{2})$. This enhancement allows for inference on datasets containing thousands of variables. This scalable framework facilitates the analysis of 1,860 frequency-phase features derived from multi-electrode LFPs and permits two significant extensions that were previously unattainable with classical circular statistics or standard TGs: (i) a TG Hidden Markov Model that captures state-dependent variations in phase coupling, such as those associated with sleep spindles, and (ii) an autoregressive TG that infers directional interactions through transfer-entropy estimation. When applied to LFP recordings, these models uncover distinct phase-interaction patterns that differentiate between wakefulness and non-rapid eye movement (NREM) sleep. Collectively, these advances enable a systematic, large-scale mapping of dynamic and directional phase relationships across various brain and cognitive states.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC