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.