A mathematical modelling approach to the prediction of circadian phase under conditions of entrainment

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The circadian rhythm in humans is a cycle of physiological and behavioural changes in the body that takes place over the course of the day. Medical interventions may be more effective if they are delivered at particular phases of the circadian cycle but in order to time the interventions correctly we require a convenient method of estimating circadian phase. Dim-light melatonin onset (DLMO) is an accurate physiological marker of circadian phase but the procedure to measure DLMO is costly and impractical for use in a clinical setting. Mathematical models of the circadian rhythm provide a non-invasive method to predict circadian phase but the existing models have only been validated in the laboratory setting and it is not known if the models produce accurate predictions in real world conditions. The thesis will review key properties of the circadian rhythm such as endogenous periodicity and sensitivity to light, and explore how they can be modelled mathematically using one-dimensional or two-dimensional oscillators. The Jewett-Forger-Kronauer (JFK) model is a popular model of the human circadian system based on a two-dimensional limit cycle oscillator, and it was designed to try and reproduce the effects of bright light stimuli on the phase and amplitude of the circadian system in laboratory conditions. The JFK model was not specifically designed to predict the phase of entrainment in subjects entrained in real world conditions. The model will be reviewed in detail with a view to understanding how the oscillator entrains to a periodic light stimulus. The predictions of the model will be tested using light data from subjects wearing monitoring devices in real world conditions. The model's predictions of phase will be compared with measurements of phase in these subjects by DLMO. The JFK model has two principal parts: (i) Process C, which is a mathematical oscillator, and (ii) Process L, whereby a light signal is transduced to a forcing term that acts on the oscillator. These two parts of the model will be considered separately. In many circumstances the angular dynamics of the oscillator part of the JFK model can be well-approximated by a one-dimensional 'phase only' model. Phase only models can facilitate a better understanding of how oscillators entrain to periodic light stimuli under different conditions. A simple phase only clock model can be written down which has analytic solutions. This provides a good starting point to improve our understanding of entrainment. The thesis will investigate the dynamical behaviour of the simple clock model including the effects of perturbations. Process L in the JFK model was developed in the late 1990s but there has been no significant development of this part of the model since its inception. Meanwhile in that time there have been a number of important discoveries about the effect of light on the circadian system, for example we know that it is mediated by intrinsically-photosensitive retinal ganglion cells in the retina which have a different spectral responsivity compared to rods and cones. In particular it is clear that measuring the light intensity in photopic lux (as in the JFK model) is not an appropriate measure of the intensity of the signal received by the circadian system. The thesis will review the latest understanding of the processing and transmission of light information to the circadian system, and try and incorporate this knowledge in a new model of the transduction of a light signal.