A one-dimensional map for the circadian modulation of sleep in a human sleep-wake regulatory network model

The timing of human sleep is strongly modulated by the 24 h circadian rhythm, and desynchronization of sleep-wake cycles from the circadian rhythm can negatively impact health. To investigate the dynamics of circadian modulation of sleep patterns and of re-entrainment of the sleep-wake cycle with the circadian rhythm, we developed a one-dimensional map for a physiologically-based, sleep-wake regulatory network model for human sleep. The map dictates the phase of the circadian cycle at which sleep onset occurs on day $ n+1$ as a function of the circadian phase of sleep onset on day $ n$ . We numerically compute the map for a reduced, though still high-dimensional, version of the sleep-wake network model that incorporates recent measurements of the time constants of the homeostatic sleep drive in humans. Using fast-slow decomposition, we exploit the underlying bifurcation structure of the model to reveal a reduced dimensional manifold, represented by the map, on which the model trajectory travels during re-entrainment of sleep-wake cycles with the circadian rhythm. The map is piecewise continuous with discontinuities caused by circadian modulation of the duration of sleep and wake episodes and the occurrence of REM sleep episodes. Analysis of map structure reveals the changes in sleep patterning, including REM sleep behavior, as sleep occurs over different circadian phases. Thus, the map provides a portrait of the circadian modulation of sleep-wake behavior. We additionally analyze the changes in the structure of the map as model parameters are varied to change the REM sleep patterning that occurs during sleep episodes. Interestingly, discontinuities in the map correspond to changes in the number of REM bouts during sleep episodes