Topological recurrence in EEG dynamics: A complementary, wake-to-REM-graded descriptor of sleep-stage organisation

Background. Sleep-stage EEG analysis conventionally relies on spectral and complexity measures. Whether the geometric organisation of neural dynamics provides complementary information remains untested. Here I test whether a topological recurrence descriptor derived from persistent homology indexes a dimension of sleep-stage EEG organisation that is statistically distinct from spectral and complexity measures. New method. K 0 , a within-subject z-scored summary of persistent homology of time-delay embeddings ( m = 10 , τ = 2 ) of single-channel frontal-midline (Fpz–Cz) EEG, was computed on 197 nights from 100 subjects (Sleep-EDF Expanded, a public polysomnography corpus). Mixed-effects models, binomial generalised linear models (GLMs), and leave-one-subject-out classification benchmarked K 0 against band power, spectral entropy, permutation entropy, and Lempel–Ziv complexity. Results. K 0 , varying systematically across sleep stages, is statistically distinct from spectral content (Pearson r − 0.49 to − 0.25 with log band powers; opposite rapid eye movement (REM)–Wake direction), with construct validity supported by independent-cohort replication, twofold lifespan attenuation, and within-subject benzodiazepine invariance. The REM–Wake separation is large ( + 1.01 standard deviation (SD); p = 2.62 × 10 − 182 ) and preserved across embeddings. Comparison with existing methods. A binomial GLM confirms K 0 carries information statistically independent of band power, especially for REM vs active-ocular wake (ΔAkaike Information Criterion (AIC) = − 744.6 , odds ratio (OR) = 5.04 ). Leave-one-subject-out area under the receiver operating characteristic curve (AUC) values are 0.79 ( K 0 ) and 0.91 (band power); combining adds a small gain (ΔAUC ≈ + 0.004 , paired-bootstrap p < 10 − 3 ). Conclusions. K 0 is a topologically distinct, spectrally non-redundant descriptor of sleep-stage EEG organisation.