Torus Graphs for Large Scale Neural Phase Analysis

Current electroencephalography (EEG)-based dream detection relies on power spectral density (PSD) and statistical moment features, achieving a state-of-the-art area under the receiver operating characteristic curve (AUC) of approximately 0.70 on the DREAM database (Wong et al., 2025, Nature Communications). We introduce PHINN-EEG (Persistent Homology Inspired Neural Network for EEG), the first topological time-series framework for dream mentation analysis. Using sliding-window Takens delay embeddings and Vietoris-Rips filtrations on multichannel pre-awakening EEG epochs, we extract Dynamic Betti Curves that characterize the geometric architecture of neural activity, not merely its energy. These topological invariants, combined with topology-conditioned flow matching, are analytically projected to outperform existing PSD and catch22 benchmarks, targeting AUC = 0.82-0.90 on the 1,462-awakening open-access subset of the DREAM database (drawn from a full registry of 3,191 total awakenings from 263 participants across 20 independent laboratories). We further introduce a topology-conditioned rectified flow model for dream-state EEG synthesis-with a spectral-conditioned flow model of comparable feature dimensionality as an additional ablation baseline to isolate the value of topological conditioning specifically-and propose a set of candidate Betti transition archetypes linking topology to phenomenological dream report categories, presented as an exploratory hypothesis space pending empirical validation. If validated, this work represents a paradigm shift from spectral energy to phase-space geometry in neural rare-event detection, with potential future implications for wearable BCI dream monitoring.

Transformer models have redefined sequence learning, yet dot-product self-attention introduces a quadratic token-mixing bottleneck for long-context time-series. We introduce the Phasor Transformer block, a phase-native alternative representing sequence states on the unit-circle manifold $S^1$. Each block combines lightweight trainable phase-shifts with parameter-free Discrete Fourier Transform (DFT) token coupling, achieving global $\mathcal{O}(N\log N)$ mixing without explicit attention maps. Stacking these blocks defines the Large Phasor Model (LPM). We validate LPM on autoregressive time-series prediction over synthetic multi-frequency benchmarks against honest baselines: it beats a zero-parameter persistence baseline and, with the corrected gradient path, improves monotonically with depth before saturating, while remaining competitive-but-not-superior to self-attention at a fraction of the parameter count. Our results establish an explicit efficiency--accuracy frontier, showing that scalable temporal modeling in oscillatory domains can emerge from geometry-constrained phase computation with deterministic global coupling.

Electroencephalographic (EEG) abnormalities arise from dynamic changes in neural synchrony across spatial and temporal scales, yet many computational approaches reduce these dynamics to static features. We present a Spatial Multi-Expert Graph Transformer that models each EEG recording as a sequence of dynamic functional connectivity graphs. Time-resolved connectivity is estimated using the weighted Phase Lag Index (wPLI), and hierarchical graph encoding aggregates information from electrode to regional and global levels. A multi-expert transformer architecture enables subtype-aware reasoning, with a gating mechanism adaptively fusing expert outputs for global abnormality prediction. Experiments on the TUAB dataset show competitive abnormal EEG detection performance and demonstrate the potential of dynamic graph modeling with adaptive expert fusion for interpretable, subtype-aware spatial--temporal analysis.