One-Dimensional Map for the Circadian Modulation of Sleep in a Sleep-Wake Regulatory Network Model for Human Sleep

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 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. The map is piecewise continuous with discontinuities caused by circadian modulation of the duration of sleep and wake episodes and the occurrence of rapid eye movement (REM) sleep episodes. We analyze the map as model parameters are varied to affect the occurrence of REM sleep within the sleep period, and we determine that the structure of the map changes with different REM sleep patterning. Specifically, variations in the discontinuities in the map correspond to changes in the number of REM bouts during sleep episodes. 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 entrainment of sleep-wake cycles with the circadian rhythm. Analysis of map structure reveals 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 and its effects on REM sleep patterning.