Mathematical models have been foundational to the development of the sleep research field. Models have provided valuable insight into both the regulation of the timing of the sleep/wake cycle and the regulation of the cycling between the two main sleep states: rapid eye movement sleep (REMS) and non-rapid eye movement sleep (NREMS). There are some aspects of human sleep that, to date, have not yet been systematically understood from a modelling perspective. These include (i) a wide inter-individual variation in phase relationships, i.e. in the timing of sleep relative to the body's internal circadian `clock' and the time of day, and (ii) the progressive lengthening of REMS episodes across the sleep period. In this thesis, we adopt an `anatomical' approach and study the fundamental structure of existing mathematical models of the sleep/wake cycle, and in so doing, evaluate their ability to capture and explain these two features of human sleep.

Firstly, we show that a hierarchy of Arnold tongues in the homeostatic-circadian-light model (Skeldon et al. 2023 PLoS Comput Biol), grounded in the Arnold tongue structure of the two-process model (Daan et al. 1984 Am J Physiol), allows model solutions to display a wide variation in phase relationships. In particular, we demonstrate for the first time, to our knowledge, that altering the model parameters can generate a range of variation comparable to that documented in the sleep literature. We describe how the gating of light by sleep affects model solutions, providing novel insights into the physiological mechanisms that underlie sleep timing.

Secondly, we perform a novel three-timescale decomposition of the Booth-Diniz Behn wake-NREMS-REMS cycling model (Booth et al. 2017 SIAM J Appl Dyn Syst) to reveal that two critical manifolds organise the model dynamics. Our analysis clarifies aspects of REMS regulation in the model and provides insight into non-intuitive features of the circle map representation of the model. Solutions of the Booth-Diniz Behn model do not display a progressive lengthening of REMS. However, from our deeper geometric understanding of the manifolds in the model, we propose a modification that allows solutions to capture this feature of sleep.

Our work sits within a broader context that seeks to develop personalised mathematical models of sleep, with a view towards designing bespoke interventions for poor sleep. Sleep disturbances and disorders are common, are associated with negative health outcomes and come at a large economic cost. The results contained in this thesis therefore provide valuable groundwork for addressing a topic of major concern to society.