PACS. 02.50.Ey -Stochastic processes. PACS. 05.40.Fb -Random walks and Levy flights. PACS. 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion.Abstract. -We study the dynamics of the awakening during the night for healthy subjects and find that the wake and the sleep periods exhibit completely different behavior: the durations of wake periods are characterized by a scale-free power-law distribution, while the durations of sleep periods have an exponential distribution with a characteristic time scale. We find that the characteristic time scale of sleep periods changes throughout the night. In contrast, there is no measurable variation in the power-law behavior for the durations of wake periods. We develop a stochastic model which agrees with the data and suggests that the difference in the dynamics of sleep and wake states arises from the constraints on the number of microstates in the sleep-wake system.

Sleep is not just the absence of wakefulness but a regulated process with an important restorative function. Based on electroencephalographic recordings and characteristic patterns and waveforms we can distinguish wakefulness and five sleep stages grouped into light sleep, deep sleep, and rapid-eye-movement (REM) sleep. In order to explore the functions of sleep and sleep stages, we investigated the dynamics of sleep stages over the night and of heart-rate variability during the different sleep stages. Sleep stages and intermediate wake states have different distri butions of their duration and this allows us to create a model for the temporal sequence of sleep stages and wake states. Heart rate is easily accessed with a high precision by the recording and analysis of the electrocardiogram (ECG). Heart-rate regulation is part of the autonomous nervous system and sympathetic tone is strongly influenced by the sleep stages.

Sleep disorders have a high prevalence. Sleep disorders are recognized first by the complaint of nonrestorative sleep. A quantification of the disorder is done by the investigation in a sleep laboratory. The investigation in the sleep laboratory examines the EEG, EOG and EMG to derive sleep stages. This is a labor intensive sleep scoring after the polysomnography investigation. Usually the time course of the sleep stages is quantified in terms of percentages of stages related to total sleep time and the latencies for the individual stages. The additional feature of transitions between sleep stages and the disruption of sleep, which corresponds to periods of wakefulness during sleep are not evaluated systematically. We have evaluated these transitions using a statistical approach. We have detected systematic differences in the distributions o sleep stages and wake states during sleep. This differences were investigates in normal subjects and patients with sleep apnea. Then these differences were investigated in different species. The difference in the distributions can be explained only by fundamentally different regulation of sleep and wakefulness.