Abstract:
Many physiological systems have the property that some processes in the
system evolve much faster than others, and mathematical models of these
systems are constructed to have multiple timescales to mimic this property.
This thesis aims to advance the understanding of oscillations in a group
of ordinary differential equation models of calcium dynamics with multiple
timescales. The model was originally developed to explain qualitative features
of the dynamics of three different cell types, and comes in two versions,
called Class I and Hybrid. First we analyse the model using geometric singular
perturbation theory (GSPT) and related ideas. We find that, although
classical GSPT can be used to describe the dynamics in distinct regions
of the phase space, it cannot explain the dynamics globally; we argue that
a non-standard form of GSPT is needed instead. Secondly, we hypothesise
and confirm the existence of a separatrix in the system, and use the separatrix
to predict the response of the model to pulses of one of the variables.
This, in turn, explains the pulse response behaviour seen in experiments on
three different cell types. Third we investigate an intercellular calcium model
constructed by coupling two or more intracellular calcium oscillators, and
analyse the model to see the influence of coupling strength on the dynamics.
We find the coupling produces a type of periodic orbit not observed in single
oscillators and increases the frequency of the oscillators. These results advance
our understanding of the mechanisms underlying calcium oscillations
in real cells, but also show that new mathematical approaches are necessary
in order to gain a full understanding of oscillation of this type.