Process-Oriented Geometric Singular Perturbation Theory and Calcium Dynamics

Phenomena in chemistry, biology, and neuroscience are often modeled 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 characterized by multiple time-scale phenomena due to order of magnitude differences between contributing processes and the presence of switching, i.e., dominance or subdominance 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 behaviors. This procedure is outlined informally in generality, and applied in detail to a model for intracellular calcium dynamics characterized 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 geometric singular perturbation theory in combination with the blowup method. We also provide an estimate for the period of the oscillations, and consider a number of possibilities for their onset under parameter variation.

Automated summary

This article discusses ordinary differential equations (ODEs) used to model phenomena in chemistry, biology, and neuroscience, and presents a heuristic procedure for identifying small parameters in these ODE models. The procedure is applied to a model of intracellular calcium dynamics characterized by switching and multiple time-scale dynamics. Using geometric singular perturbation theory, the existence and uniqueness of stable relaxation oscillations with three distinct time scales are proven, and an estimate for the period of the oscillations is provided.