Metastable Geometric Phase Structure in Human EEG: Riemannian Manifold Analysis of Covariance Dynamics During Sustained Attention

This paper presents what is, to our knowledge, the first systematic application of Riemannian manifold analysis to the study of dynamical transitions in human EEG covariance structure. Using the affine-invariant metric on symmetric positive-definite matrices, we demonstrate that multichannel EEG covariance evolves through metastable geometric phases (discrete quasi-stable states separated by abrupt transitions) during sustained attention tasks. Across 20 subjects and 40 recording sessions from the publicly available COG-BCI dataset, plus an independent ~60-minute driving simulation, we document heavy-tailed speed distributions (session-level Fisher combined p = 7.32 × 10⁻⁸ across 40 sessions; a more conservative subject-level aggregation gives p = 6.0 × 10⁻⁴), a median 4.0-fold enrichment of effective-rank (dimensionality) changes at transition boundaries (18 of 20 subjects, 90%), and a dissociation between stochastic transition timing and deterministic regime identity consistent with Kramers escape theory. Geometric speed increases with working memory load by an ordered (Jonckheere-Terpstra) trend and correlates modestly with reaction times, establishing functional relevance without circularity. These findings connect to recent geometric analyses of transformer neural networks, suggesting that metastable phase structure may be a convergent computational architecture across biological and artificial information-processing systems. All analyses use publicly available datasets and standard open-source tools.

Version 3 (June 2026) corrects citations, statistical reporting, and figure and method descriptions, and removes a circularity in the boundary-enrichment and lane-overlap analyses: the enrichment measure is now the change in effective rank near transitions (rather than the partly circular Riemannian displacement), and the driving-dataset covariance excludes a non-neural vehicle-position channel that earlier versions had inadvertently retained. No qualitative finding changed; full details are in the in-paper Corrigendum.