面向过程的几何奇异扰动理论与钙动力学
Process-Oriented Geometric Singular Perturbation Theory and Calcium Dynamics
摘要：
化学、生物学和神经科学中的现象通常使用常微分方程（ODE）来建模，其中右侧由对应于单个“过程”或“通量”的项组成。通常，这些常微分方程的特征在于多时间尺度现象，这是由于贡献过程之间的数量级差异和切换的存在，即，作为状态变量的函数的特定项的显性或次显性。我们概述了一个启发式的程序识别小参数的ODE模型，特别强调识别小参数的切换行为。这个过程中概述了非正式的一般性，并详细应用于细胞内钙动力学的模型，其特征在于开关和多个（两个以上）的时间尺度动态。总共有五个小参数被确定，并与一个单一的扰动参数的多项式标度律的基础上的数量级比较。由此产生的奇异扰动问题具有时间尺度分离，这取决于状态空间的区域。我们证明了稳定的松弛振荡的存在性和唯一性与三个不同的时间尺度使用坐标独立的制定GSPT结合爆破方法。我们还提供了一个估计的周期的振荡，并考虑了一些可能性，他们的发病下参数的变化。
英文摘要：
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.
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