Abstract
Sleep is essential for most living things to function. Many features of sleep are not yet understood however, mathematical models are playing an important role in developing our understanding of many of the physiological properties of sleep. We introduce the most well-known model of sleep regulation, the two process model which proposes that sleep-wake cycles can be modelled by the interaction between two oscillators. This ostensibly simple model is an interesting example of a nonsmooth dynamical system whose rich dynamical structure has been relatively unexplored. A key aim of this work is to further understand how transitions between monophasic (one sleep a day) and polyphasic (many sleeps a day) sleep occur in the two process model. The two process model can be framed as a one-dimensional map of the circle which, for some parameter regimes, has gaps. As is a feature of continuous circle maps the bifurcation set consists of saddle-node Arnold tongues. We show that border collision bifurcations that arise naturally in maps with gaps extend and supplement these tongues. We see how the periodic solutions that are created by saddle-node bifurcations in continuous maps transition to periodic solutions created by period-adding bifurcations as seen in maps with gaps. With this deeper understanding of the dynamics and bifurcation structure of the two process model we use modified versions of the model to explain two experimental data sets. An ultradian rhythm is a recurrent period or cycle which repeats multiple times across the day. We consider the sleep wake patterns of a the common vole, Microtus Arvalis, which has ultradian rest activity and feeding patterns. By deriving parameters for the two process model from EEG data and sleep/ wake onset times we are able to simulate with high accuracy the key features of spontaneous sleep-wake patterns in the voles. However, to explain phenomena seen in sleep deprivation experiments we include a high amplitude ultradian oscillation alongside the circadian, the results allow us to give some physiological insight into the internal mechanisms which drive sleep/wake onset times in the common vole. Across the human lifespan there are many changes in the physiological properties of sleep, sleep timing and sleep duration. In adolescence sleep timing is delayed and there is a reduction in slow wave sleep which continues into old age as sleep timing gradually becomes earlier. Using a modified two process model which incorporates a van der Pol oscillator driven by external light signals into the circadian process we show that changes in sleep timing and duration across the lifespan can be explained by varying parameters. Model simulation show that these changes can be understood by a simultaneous reduction in the amplitude of the circadian oscillator and the upper asymptote of the homeostatic sleep pressure.