Governing Equations of Human Sleep Macro-Structure:Symbolic Discovery from 1,106 Nights
Junhui Gao
Abstract
Human sleep is organized into a small number of cycles per night: deep sleep dominates early cycles, REM lengthens toward morning, and light sleep clusters at cycle boundaries. These regularities have long been described qualitatively but never distilled into closed-form laws from large-scale data. A total of 1,106 nights (Sleep-EDF 153 + SHHS 953) are segmented into REM-anchored cycles, and four cycle-indexed statistics follow simple closed-form equations: cycle length declines nearly linearly (R^2=0.996), REM duration follows an inverted U (R^2=0.873), deep-sleep (N3) run length decays exponentially (R^2=0.987) with a time constant of 2.94 cycles, consistent with the homeostatic Process S of the two-process model, and light-sleep (N1) run length follows a U shape (R^2=0.956), confirming a "long-short-long" pattern across the night. The laws are reproduced by per-night differentiable fitting (142/142 and 773/773 nights converge), by unconstrained symbolic regression (PySR), and by a 500-night generative synthesizer built from them. Their robustness is established against the two main methodological threats: bootstrap CIs computed by resampling whole nights (1,000 replications) exclude zero for every shape term, and a five-level scan of the REM-merge segmentation threshold preserves every qualitative form. These appear to be the first closed-form laws governing the macro-structure of sleep, distilled and tested across an age-mixed sample (healthy young adults and community-based older adults).