Modeling trajectories using functional linear differential equations

TL;DR

This paper introduces a novel regression method combining differential equations and functional data analysis to model the dynamic relationship between muscle activation and paw position during mouse locomotion, improving predictive accuracy.

Contribution

It develops an innovative approach that estimates ODE parameters across all curves simultaneously, addressing gaps in existing methods for functional data analysis.

Findings

Paw speed and position are dynamically influenced by muscle activation.

Muscle activation effects persist beyond immediate activation periods.

The method outperforms related functional data approaches in simulations and predictions.

Abstract

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.