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

该奖项将支持美国国立卫生研究院在John Rinzel博士领导下的研究人员与墨西哥国立自治大学(UNAM)在Humberto Carrillo博士领导下的研究人员之间的合作研究访问。30多年来，数学模型被用来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，长时间行为通常涉及两个不同时间尺度上的动态，例如，动作电位(毫秒的时间尺度)和第二信使的变化，如细胞内钙(几十秒)。研究人员打算使用包括动作电位的短期平均影响的模型来研究这种行为，但不包括个别事件的详细时间过程。作为一个模型问题，他们将考虑突发性电振荡现象。该方法将涉及数值和分析方法，特别是平均化的数学技术。他们将推导和分析生物物理上详细的、“平均”的微分方程式来描述缓慢的爆炸动力学。一个理想化的、易于分析的、具有重要类属性质的模型的形成和研究将有助于阐明所有这类模型的定性特征。这项工作将在两个机构同时进行，在贝塞斯达进行广泛的计算，并在墨西哥城严格证明平均数的合理性。这种方法将允许生物物理描述和解释。此外，它还将提供对慢变量所起的重要作用的洞察，以及与平均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.