Mathematical models for sleep-wake dynamics: a comparison

Sleep is essential for the maintenance of human life, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the reciprocal interaction of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes can be classified as grazing bifurcations. This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the reciprocal interaction model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is important because it brings a deeper insight to sleep-wake dynamics and allows some features of the two-process model to be interpreted physiologically