The Mathematical Structure of the Human Sleep-Wake Cycle

In 1972, Michel Siffre spent six months alone in an underground cave. His account (Siffre, 1975) of that harrowing experience begins dramatically:

Overcome with lethargy and bitterness, I sit on a rock and stare at my campsite in the bowels of Midnight Cave, near Del Rio, Texas. Behind me lie a hundred days of solitude; ahead loom two and a half more lonely months. But I — a wildly displaced Frenchman — know none of this, for I am living “beyond time,” divorced from calendars and clocks, and from sun and moon, to help determine, among other things, the natural rhythms of human life.

This chapter contains both raw data and analyses of many sleep-wake records. The emphasis is on internally desynchronized subjects, with a few examples of nappers and split sleepers. The subjects are discussed one at a time, whereas in the next chapter their records are pooled and general patterns are extracted.

Chapter 3 may have given the impression that sleep-wake records are highly variable. Indeed they are. Yet there are certain regularities or “patterns” which transcend the idiosyncrasies of individual subjects. Regarded in this way, each record becomes a variation on common themes.

Internal desynchronization has stimulated a great deal of theorizing. Loosely speaking, most of the proposed models postulate one of two time-keeping mechanisms: a pair of self-sustained oscillators with continuous coupling (Sections 5.1–5.3) or a fatigue variable building up and decaying between circadian-modulated thresholds; (Sections 5.4–5.6). Other approaches are discussed in Section 5.7.

This chapter explores the mathematical structure of models of the sleep-wake cycle. As in other chapters, we focus on the empirical phenomenon of spontaneous internal desynchron-ization between the sleep-wake and temperature cycles during free-run. Other important phenomena such as those occurring during sleep deprivation, continuous bedrest or entrainment to 24h or other schedules, will not be discussed. As we shall see, understanding and accounting for internal desynchronization alone is a difficult task.

The previous chapters have set the stage for a confrontation of experiment and theory. Chapter 3 presented many records of internally desynchronized subjects, regarded individually and in all their particulars. Chapter 4 extracted from those records certain generalizations, empirical patterns that underlie the timing of the sleep-wake cycle under free-running conditions. Turning from experiment to theory, Chapter 5 reviewed previous models of the sleep-wake cycle, and Chapter 6 explored the analytical structure of the two leading contenders: the XY model of Kronauer et al. (1982, 1983) and the gated pacemaker model of Daan et al. (1984). Chapter 6 also introduced two simpler alternative models, BEATS and PHASE, with a dual purpose. First, they were used in Chapter 6 to illuminate the structure of the more sophisticated models. In this chapter, they act as “controls” during tests of the models. That is, the BEATS and PHASE models offer a standard of performance, against which the sophisticated models are measured. The models of Daan et al. and Kronauer et al. are to be judged successful, not when they reproduce some empirical pattern, but when they reproduce it more faithfully than the control models do.

This final chapter reviews the main contributions of the monograph, and suggests some possible directions for future research.