  Sleep is essential for the maintenance of the brain and the body, 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 mutual inhibition
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 occur at
grazing bifurcations.This provides the theoretical underpinning for numerical
results showing that the sleep patterns of many mammals can be explained by the
mutual inhibition 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 valuable because not only does it allow some features of
the two-process model to be interpreted physiologically but it also means that
knowledge gained from study of the two-process model can be used to inform
understanding of the mutual inhibition model. This is important because the
mutual inhibition model and its extensions are increasingly being used as a
tool to understand a diverse range of sleep-wake phenomena such as the design
of optimal shift-patterns, yet the values it uses for parameters associated
with the circadian and homeostatic processes are very different from those that
have been experimentally measured in the context of the two-process model.
