Abstract
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 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 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 behaviour 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.
Highlights
Reduced or mis-timed sleep is increasingly recognized as presenting a significant health risk and has been correlated with increases in a diverse range of medical problems including allcause mortality, cardio-vascular disease, diabetes and impaired vigilance and cognition [1,2,3,4,5]
Phillips and Robinson model (PR model) At the core of the PR model are two groups of neurons: monoaminergic (MA) neurons in the ascending arousal system that promote wake and neurons based in the ventro-lateral pre-optic (VLPO) area of the hypothalamus that promote sleep
Where QS is the mean maximum firing rate of the neuronal population and hS is the value at which the switch occurs, we show in the Methods section that the parameters for the slow dynamics of the PR model with a switch can be exactly mapped to parameter values in the two-process model, where the drive to the VLPO, Dv and to the MA, Dm are given by Dv ~nvh H {nvc C {Av, H0z~
Summary
Reduced or mis-timed sleep is increasingly recognized as presenting a significant health risk and has been correlated with increases in a diverse range of medical problems including allcause mortality, cardio-vascular disease, diabetes and impaired vigilance and cognition [1,2,3,4,5]. Switching from wake to sleep and from sleep to wake occurs at upper and lower threshold values of the sleep pressure respectively, where the thresholds are modulated by an approximately sinusoidal circadian oscillator. This model has proved compelling for both its physiological grounding and its graphical simplicity and has been used extensively (there are over 1500 citations to [9] and 600 citations to [10] to-date). It remains difficult to relate the threshold values in the two-process model and its extensions to physiological processes
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