Over the past three years I have grown accustomed to the puzzled look which appears on people's faces when they hear that I am a mathematician who studies sleep. They wonder, but are usually too polite to ask, what does mathematics have to do with sleep? Instead they ask the questions that fascinate us all: Why do we have to sleep? How much sleep do we really need? Why do we dream? These questions usually spark a lively discussion leading to the exchange of anecdotes, last night's dreams, and other personal information. But they are questions about the func­ tion of sleep and, interesting as they are, I shall have little more to say about them here. The questions that have concerned me deal instead with the timing of sleep. For those of us on a regular schedule, questions of timing may seem vacuous. We go to bed at night and get up in the morning, going through a cycle of sleeping and waking every 24 hours. Yet to a large extent, the cycle is imposed by the world around us.

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1. Introduction.- 1.1 Beyond Time.- 1.2 The Rosetta Stone.- 1.3 Overview.- 2. Experimental Background.- 2.1 Phenomena and Terminology.- 2.2 History of Free-Run Studies.- 3. Data Bank.- 3.1 Subject 1.- 3.2 Subject 2.- 3.3 Subject 3.- 3.4 Subject 4.- 3.5 Subject 5.- 3.6 Subject 6.- 3.7 Subject 7.- 3.8 Subject 8.- 3.9 Subject 9.- 3.10 Subject 10.- 3.11 Subject 11.- 3.12 Subject 12.- 3.13 Subject 13.- 3.14 Subject 14.- 3.15 Subject 15.- 3.16 Subject 16.- 3.17 Subject 17.- 3.18 Subject 18.- 3.19 Subject 19.- 3.20 Subject 20.- 3.21 Subject 21.- 3.22 Subject 22.- 4. Patterns.- 4.1 Durations Vary with Circadian Phase of Sleep Onset.- 4.2 Sleep Length and Prior Wake Length.- 4.3 Timing of Wake-Up.- 4.4 Timing of Sleep Onset.- 4.5 Wake-Maintenance Zones.- 4.6 Estimating Circadian Parameters from Sleep Data Alone.- 4.7 Phase-Trapping.- 4.8 Slow Changes in Sleep-Wake Cycle Length.- 4.9 Miscellany and Missing Patterns.- 4.10 Napping and Split Sleep.- 4.11 Summary: The Basic Patterns of Internal Desynchrony.- 5. Theoretical Background.- 5.1 Conceptual Model of Aschoff and Wever.- 5.2 Wever's Noninteractive Model.- 5.3 Kronauer's XY Model: Coupled Van der Pol Oscillators.- 5.4 Conceptual Model of Borbely.- 5.5 Winfree's Half-Model.- 5.6 Gated Pacemaker of Daan, Beersma, and Borbely.- 5.7 Other Approaches.- 6. Analysis of Models.- 6.1 Introduction.- 6.2 BEATS Model.- 6.3 PHASE Model.- 6.4 XY Model of Kronauer et al..- 6.5 Model of Daan et al..- 7. Simulations.- 7.1 Transition from Synchrony to Desynchrony.- 7.2 Napping and Split Sleep Simulations.- 7.3 A Representative Simulation of Internal Desynchrony.- 7.4 Overall Performance During Desynchrony.- 7.5 Summary and Discussion.- 8. Epilogue.- 8.1 Contributions.- 8.2 Directions for Future Research.- References.- Index of Authors.