The brain functions like a computer composed of subsystems which in­ teract in a hierarchical manner. But it is not a single hierarchy, but a com­ plex system of hierarchies each of which has its very own and unique fea­ ture. One of these concerns the cyclic or rhythmic control of neuronal ac­ tivities which, among others, give rise to alternating states of wakefulness and sleep. The phenomenon of sleep still remains a mystery. The present monograph does not give us any new insights into its meaning and significance. Yet sleep research may not be the same after the appearance of this book be­ cause it gives us a comprehensive mathematical theory which opens our eyes to new insights into the mechanism of the rhythm generation that under­ lies the "wake-sleep" cycle. No one who has worked his way through this book can again look at ex­ perimental data without recognizing features which the "models" developed in its various chapters so strikingly reveal.