  Oscillations of free intracellular calcium concentration are thought to be
important in the control of a wide variety of physiological phenomena, and
there is long-standing interest in understanding these oscillations via the
investigation of suitable mathematical models. Many of these models have the
feature that different variables or terms in the model evolve on very different
time-scales, which often results in the accompanying oscillations being
temporally complex. Cloete et al [5] constructed an ordinary differential
equation model of calcium oscillations in hepatocytes in an attempt to
understand the origin of two distinct types of oscillation observed in
experiments: narrow-spike oscillations in which rapid spikes of calcium
concentration alternate with relatively long periods of quiescence, and
broad-spike oscillations in which there is a fast rise in calcium levels
followed by a slower decline then a period of quiescence. These two types of
oscillation can be observed in the model if a single system parameter is varied
but the mathematical mechanisms underlying the different types of oscillations
were not explored in detail in [5]. We use ideas from geometric singular
perturbation theory to investigate the origin of broad-spike solutions in this
model. We find that the analysis is intractable in the full model, but are able
to uncover structure in particular singular limits of a related model that
point to the origin of the broad-spike solutions.
