THE MATHEMATICAL STRUCTURE OF THE HUMAN SLEEP-WAKE CYCLE (CIRCADIAN, RHYTHM, MODELS, DYNAMICS, FREE-RUN)

The influence of the brain's circadian clock on sleep-wake timing has been clarified by experimental studies of subjects living for months in environments shielded from 24-hour time cues. The key finding is "internal desynchronization," in which the circadian rhythms of sleep and body temperature spontaneously adopt different periods. Although several theorists have claimed recently that their oscillator models account remarkably well for the sleep-wake data, disagreement persists about the experimental facts themselves, and about the standards of performance against which the models should be judged. This thesis addresses those two difficulties.

For the first time, much of the world literature on internal desynchronization is compiled and reanalyzed, revealing several patterns with implications for sleep research and circadian biology. The phase of the circadian pacemaker at bedtime determines the durations of sleep, prior wakefulness, and the wake-sleep cycle. The controversial relation of sleep length to prior wake length is clarified, thereby unifying results from synchronized and desynchronized subjects. Circadian phase-dependent rate functions for the sleep-wake transition are estimated, as are the frequency distributions for the timing of sleep onset and awakening. New methods for extracting circadian period and phase from sleep data alone are presented and validated. The most important empirical finding concerns a zone of the circadian cycle occurring 1-3h before habitual bedtime, in which subjects were virtually unable to fall asleep, suggesting that circadian phase dysfunction underlies certain insomnias.

We introduce the first analytically tractable model of sleep-wake timing, based on coupled nonlinear phase-only oscillators. Exact solutions are used to predict various empirical relationships. A second new model, based on beat phenomena, elucidates the influential and more complex models of Daan et al. and Kronauer et al. Phase plane and asymptotic analyses interconnect the models. Computer simulations indicate that the models of Kronauer et al. and Daan et al. perform no better than the two simple models, when tested against internal desynchronization data. However, a modification of the Daan model produces a superior fit to the desynchrony data. The analysis presented here establishes a rational framework for the evaluation of models of the human sleep-wake cycle.