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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.

The calcium (Ca$^{2+}$) signalling system is important for many cellular processes within the human body. Signals are transmitted within the cell by releasing Ca$^{2+}$ from the endoplasmic reticulum (ER) into the cytosol via clusters of Ca$^{2+}$ channels. Mathematical models of Ca$^{2+}$ release via inositol 1,4,5-trisphosphate receptors (IP$_{3}$R) help with understanding underlying Ca$^{2+}$ dynamics but data-driven modelling of stochastic Ca$^{2+}$ release events, known as Ca$^{2+}$ puffs, is a difficult challenge. Parameterising Markov models for representing the IP$_{3}$R with steady-state single channel data obtained at fixed combinations of the ligands Ca$^{2+}$ and inositol-trisphosphate (IP$_{3}$) has previously been demonstrated to be insufficient. However, by extending an IP$_{3}$R model based on steady-state data with an integral term that incorporates the delayed response of the channel to varying Ca$^{2+}$ concentrations we succeed in generating realistic Ca$^{2+}$ puffs. By interpreting the integral term as a weighted average of Ca$^{2+}$ concentrations that extend over a time interval of length $\tau$ into the past we conclude that the IP$_{3}$R requires a certain amount of memory of past ligand concentrations., Comment: 31 pages, 8 figures, 1 table

Phenomena in chemistry, biology and neuroscience are often modelled using ordinary differential equations (ODEs) in which the right-hand-side is comprised of terms which correspond to individual 'processes' or 'fluxes'. Frequently, these ODEs are characterised by multiple time-scale phenomena due to order of magnitude differences between contributing processes and the presence of switching, i.e., dominance or sub-dominance of particular terms as a function of state variables. We outline a heuristic procedure for the identification of small parameters in ODE models of this kind, with a particular emphasis on the identification of small parameters relating to switching behaviours. This procedure is outlined informally in generality, and applied in detail to a model for intracellular calcium dynamics characterised by switching and multiple (more than two) time-scale dynamics. A total of five small parameters are identified, and related to a single perturbation parameter by a polynomial scaling law based on order of magnitude comparisons. The resulting singular perturbation problem has a time-scale separation which depends on the region of state space. We prove the existence and uniqueness of stable relaxation oscillations with three distinct time-scales using a coordinate-independent formulation of GSPT in combination with the blow-up method. We also provide an estimate for the period of the oscillations, and consider a number of possibilities for their onset under parameter variation., Comment: Early sections have been substantially revised in order to emphasise the method as opposed to the application per se