Abstract
In mammals, the circadian pacemaker is a biological clock responsible for the timing of many
of the body's circadian rhythms, including the sleep-wake cycle. Mathematical models of the
circadian pacemaker could fi nd applications in the treatment of sleep disorders or in medical
chronotherapy. Parametric models are phenomenological models that are widely used to model
the circadian systems of diurnal animals. However, current models of the circadian pacemaker
in humans are based on limited data, and they often produce inaccurate predictions when they
are used in novel settings. We consider one- and two-dimensional parametric models, and we
explore how the design of models is informed by experimental data. Our aim is to design models
that accurately reproduce data from various experimental studies.
The circadian pacemaker generates rhythms even in the absence of time cues, or zeitgebers.
However, the pacemaker is responsive to light, which is how it synchronizes, or entrains, to the
day-night cycle on the Earth's surface. In parametric models, the self-sustaining activity of
the pacemaker is modelled by a clock with some intrinsic velocity and the effect of light is to
modulate the velocity of the clock. The velocity response of the clock to light is the product of
a tonic stimulus and a response function, which describes the sensitivity of the clock to light as
a function of the state of the clock.
In one-dimensional, or phase-only, parametric models, we show that it is necessary to make
simplifying assumptions about the response function to design models based on data from studies
in controlled lighting conditions. This motivates us to consider 'simple' clock models in which
the response function is sinusoidal. We obtain two specific results in our study of phase-only
models. Firstly, we show that the shorter period of sighted humans in forced desynchrony
compared to the period of blind people can be explained by a model whose response function
has a larger advance region than delay region. Secondly, we show that simple clock models with
different response functions are required to reproduce the effects of light of different intensity
in studies of humans in controlled lighting conditions. This challenges the prevailing view in
circadian biology that the sensitivity of the pacemaker to light does not vary in different lighting
conditions.
We investigate how two-dimensional parametric models can reproduce the phenomena of type
0 phase resetting and after-effects. We show that the strength of attraction of the unperturbed
limit cycle of the clock is a key feature of the models. We use a clock with a strongly attracting
limit cycle to reproduce Khalsa's type 0 phase transition curve in humans. We use a clock with
a weakly attracting limit cycle to reproduce after-effects in the four-striped grass mouse.