A circle map model of human circadian rhythms.

Peculiar relationships between the sleep-wake cycle and temperature rhythm have been found during free-run situations, i.e. internal synchronization, internal desynchronization and phase trapping. The periods of both rhythms and the phase relationships between them could well characterize the above behavioral states. In this paper, firstly, our mathematical interpretations for them are proposed. According to these interpretations, a new circadian system model is developed. Our model consists of two circle maps, one for the dynamics of sleep-onset phase and the other wake-onset phase in reference to the temperature rhythm. The model dynamics is controlled mainly by the map for sleep-onsets which simultaneously determines behavior of wake-onsets. The model parameters are uniquely estimated based on the experimental results. The diverse behavior of the sleep-wake cycle associated with the synchronized and desynchronized states is suggested to be understood as bifurcation phenomena of the model dynamics. The bifurcation of the model dynamics is controlled by a single parameter, whose quantitative characteristics are investigated in detail. The results shown here could provide a novel framework to understand the human circadian rhythms systematically.