U.S.-Mexico Collaborative Research: Bursting Electrical Activity of Excitable Cells; Mathematical Models Based on Averaged Slow Dynamics

该奖项将支持John Rinzel博士指导下的美国国立卫生研究院研究人员和Humberto Carrillo博士领导下的墨西哥国立自治大学（UNAM）研究人员之间的合作研究访问。30多年来，数学模型已被制定并用于描述和解释各种神经、肌肉和分泌细胞的电活动（动作电位）。然而，长时间行为往往涉及两个不同时间尺度上的动力学，如动作电位（毫秒时间尺度）和细胞内钙等第二信使的变化（几十秒）。研究人员打算使用包括动作电位短期平均影响但不包括单个事件的详细时间过程的模型来研究这种行为。作为一个模型问题，他们将考虑爆破电振荡现象。该方法将涉及数值和分析方法，特别是平均的数学技术。他们将推导和分析生物物理上详细的“平均”微分方程，用于爆破的缓慢动力学。构想和研究一个具有重要一般性质的理想化的、可分析的模型，将有助于阐明所有这些模型的定性特征。这项工作将在两个机构同时进行，在贝塞斯达进行大量计算，在墨西哥城进行严格的平均论证。该方法将允许生物物理描述和解释。此外，它将深入了解慢变量所起的重要作用，以及与平均方法相关的新的有趣的数学问题。

This award will support collaborative research visits between researchers at the U.S. National Institutes of Health under the direction of Dr. John Rinzel and researchers at the Universidad Nacional Autonoma de Mexico (UNAM) under the leadership of Dr. Humberto Carrillo. For over 30 years, mathematical models have been formulated and used to describe and interpret the electrical activity (action potentials) of various nerve, muscle, and secretory cells. However, long-time behavior often involves dynamics on two different time scales, e.g., action potentials (time scale of msec) and variation of second messengers such as intracellular calcium (tens of seconds). The researchers intend to study such behavior using models that include the short-time averaged influence of action potentials but not the detailed time courses of individual events. As a model problem, they will consider the phenomenon of bursting electrical oscillations. The approach will involve both numerical and analytical methods and, in particular, the mathematical technique of averaging. They will derive and analyze biophysically detailed, "averaged" differential equations for the slow dynamics of bursting. The formulation and study of an idealized, analytically tractable model with important generic properties will be useful in elucidating the qualitative character of all such models. The work will proceed simultaneously at both institutions with extensive computations done in Bethesda and rigorous justification of averaging in Mexico City. The approach will allow a biophysical description and interpretation. Furthermore, it will provide insight into the important role played by slow variables as well as into new and interesting mathematical questions related to the method of averaging.