Process-Oriented Geometric Singular Perturbation Theory and Calcium Dynamics
| Autor: | Samuel Jelbart, Nathan Pages, Vivien Kirk, James Sneyd, Martin Wechselberger |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | SIAM Journal on Applied Dynamical Systems. 21:982-1029 |
| ISSN: | 1536-0040 |
| DOI: | 10.1137/21m1412402 |
| Popis: | 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 |
| Databáze: | OpenAIRE |
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