Four equations describe changing sleep patterns across 1,106 nights

Turning sleep stages into measurable patterns

A night’s sleep can be represented as a sequence of stages scored in 30-second intervals: wakefulness, light sleep stages N1 and N2, deep sleep N3, and rapid eye movement, or REM, sleep. Gao’s preprint examines how groups of these intervals form cycles and how those cycles change through the night.

The analysis combines 153 nights from Sleep-EDF Expanded with 953 from the Sleep Heart Health Study. It works with existing stage labels, rather than discovering the patterns directly from electrical brain recordings. Its central claim is that four simple equations summarize the evolution of selected sleep statistics across successive cycles.

Here, “governing equations” means empirical descriptions. The paper explicitly distinguishes these fitted relationships from mathematical explanations of the biological mechanisms producing sleep.

How the cycles were defined

The segmentation algorithm places each cycle’s endpoint at the end of a consolidated REM episode. REM episodes separated by no more than five minutes are merged, and consolidated episodes containing less than one minute of REM are discarded. The first cycle begins at the recording’s start; the incomplete segment after the final REM endpoint is excluded.

This procedure produces 4,314 cycles. The median night contains four, although individual nights range from one to nine.

For each cycle, the analysis measures total cycle duration, total REM duration, the longest uninterrupted N3 episode, and the average length of uninterrupted N1 episodes. Those last two measures describe stage persistence—not the total amount of deep or light sleep.

The author fits several candidate equation families to averages at each cycle position, weighting positions by how many nights contribute. Separate analyses fit individual nights and use symbolic regression, a computational search for mathematical expressions, without prescribing a particular equation template.

The four reported relationships

Cycles become shorter. The pooled cycle-length equation has an R² of 0.996. Its dominant feature is a nearly linear decline. Although the reported equation includes a small quadratic term, that term’s confidence interval includes zero, so the evidence does not clearly establish curvature.

REM duration rises and then falls. Total REM per cycle follows an inverted U, with a fitted peak around cycle 2.7 and an R² of 0.873. This is the least closely fitting of the four relationships. It describes REM minutes per extracted cycle, rather than a general claim that REM always decreases toward morning.

Deep-sleep episodes shorten exponentially. The longest N3 run follows an exponential decay with an R² of 0.987 and a time constant of 2.94 cycles. Gao interprets this as consistent with Process S, the declining homeostatic sleep pressure in the two-process model of sleep regulation. Agreement in timescale is not a direct demonstration that the fitted equation measures that mechanism.

N1 episodes show a long–short–long pattern. Average N1 run length follows a U-shaped trajectory, with an R² of 0.956. Positive curvature also appears in 68% of fitted Sleep-EDF nights and 66% of fitted SHHS nights—not every individual night.

These high R² values concern pooled cycle-position averages; they should not be read as equivalent accuracy for predicting a particular person’s sleep.

What the supporting checks establish

The preprint reports successful individual fitting for 142 Sleep-EDF nights and 773 SHHS nights, fewer than the full corpus. Its symbolic search recovers the same broad equation families. A simulator built from the equations also reproduces the four average patterns across 500 generated nights.

These checks support the equations as compact descriptions of the analyzed statistics. The simulator, however, uses fitted means, empirical variability and prescribed stage placement; matching those statistics does not establish an independent biological explanation.

There are also qualifications within the detailed results. Resampling whole nights 1,000 times supports the key REM, N3 and N1 shape terms, but not cycle-length curvature. Changing the REM-merging threshold preserves several broad trends, yet at the strictest threshold the preferred N1 model is a monotonic rise rather than a full U shape. Thus, the detailed findings are narrower than the abstract’s blanket robustness language.

Limits on individual use

An alternative quadratic fits N3 slightly better by the reported model-selection scores. Gao favors the exponential for its physiological interpretation and supporting analyses, rather than because it wins every statistical comparison.

The paper also reports noisy single-night parameters: using one night’s estimates to predict another does not outperform the population equation. Cohort differences in label fragmentation further complicate interpretation. Applicability to pathological sleep, medication effects, naps and circadian disruption remains uncertain.

The practical contribution is therefore a set of testable population descriptions—not a diagnostic tool or a universal nightly template. The source promises release of code and the synthesizer upon publication, limiting immediate reproducibility from the preprint alone.