Department of Mathematics, University of Auckland, Auckland, 1142, New Zealand.
Math Biosci. 2020 Nov;329:108455. doi: 10.1016/j.mbs.2020.108455. Epub 2020 Aug 21.
Crossbridge theory, originally developed by A.F. Huxley more than 60 years ago to explain the behaviour of striated muscle, has since evolved to encompass many different muscle types and behaviours. The governing equations are generally linear hyperbolic partial differential equations, or systems thereof, describing the evolution of probability density functions. Importantly, the macroscopic behaviour is often described not in terms of these distributions themselves, but rather in terms of their first few moments. Motivated by this observation, G.I. Zahalak proposed the distribution-moment approximation to describe the evolution of these moments alone. That work assumed a Gaussian underlying distribution, and was observed to provide reasonable approximation of the moments despite the non-Gaussian character of the underlying distribution. Here we propose two variations on the distribution-moment approximation: (i) a generalized N-moment approximation based on the Gram-Charlier A-series representation, and (ii) perhaps the simplest possible approximation based on a uniform distribution. Study of these variations suggests that Zahalak's original contention may be correct: approximations based on higher order moments may not be worth their complexity. However, the simplified variation shows more promise, with similar accuracy in approximating the moments yet reduced complexity in the derivation of the approximation.
交联桥理论最初是由 A.F.赫胥黎在 60 多年前提出的，用于解释横纹肌的行为，此后已经发展到涵盖许多不同的肌肉类型和行为。控制方程通常是线性双曲偏微分方程或其系统，用于描述概率密度函数的演化。重要的是，宏观行为通常不是根据这些分布本身来描述，而是根据它们的前几个矩来描述。受此观察结果的启发，G.I.扎哈拉克提出了分布矩逼近，仅描述这些矩的演化。该工作假设了一个基本的高斯分布，并观察到即使基本分布具有非高斯特征，它也能对矩提供合理的逼近。在这里，我们提出了两种分布矩逼近的变体：（i）基于 Gram-Charlier A 级数表示的广义 N 矩逼近，以及（ii）基于均匀分布的最简单可能逼近。对这些变体的研究表明，扎哈拉克最初的论点可能是正确的：基于更高阶矩的逼近可能并不值得它们的复杂性。然而，简化的变体显示出更大的潜力，在近似矩方面具有相似的准确性，同时在逼近的推导中复杂性降低。
