We are motivated by a study that seeks to better understand the dynamic relationship between muscle activation and paw position during locomotion. For each gait cycle in this experiment, activation in the biceps and triceps is measured continuously and in parallel with paw position as a mouse trotted on a treadmill. We propose an innovative general regression method that draws from both ordinary differential equations and functional data analysis to model the relationship between these functional inputs and responses as a dynamical system that evolves over time. Specifically, our model addresses gaps in both literatures and borrows strength across curves estimating ODE parameters across all curves simultaneously rather than separately modeling each functional observation. Our approach compares favorably to related functional data methods in simulations and in cross-validated predictive accuracy of paw position in the gait data. In the analysis of the gait cycles, we find that paw speed and position are dynamically influenced by inputs from the biceps and triceps muscles, and that the effect of muscle activation persists beyond the activation itself.

翻译：本研究受一项旨在深入理解运动过程中肌肉激活与足爪位置动态关系的研究所启发。在该实验中，小鼠在跑步机上小跑时，每个步态周期中二头肌和三头肌的激活状态与足爪位置被同步连续测量。我们提出了一种创新的通用回归方法，该方法结合常微分方程与功能数据分析，将这些功能型输入与响应之间的关系建模为随时间演化的动态系统。具体而言，我们的模型弥补了现有文献中的空白，通过跨曲线借力——同时估计所有曲线的常微分方程参数，而非对每个功能观测进行单独建模。在模拟实验及步态数据足爪位置的交叉验证预测精度方面，本方法相较于相关功能数据方法表现出优越性。在步态周期分析中，我们发现足爪速度与位置动态地受二头肌和三头肌输入的影响，且肌肉激活的效应在激活行为结束后仍持续存在。