Variance spectrum scaling analysis for high-dimensional coordination in human movement

Coordinated movement has been essential to our evolution as a species but its study has been limited by ideas largely developed for one-dimensional data. The field is poised for a change by high-density recording tools and the popularity of data sharing. New ideas are needed to revive classical theoretical questions such as the organization of the highly redundant biomechanical degrees of freedom and the optimal distribution of variability for efficiency and adaptiveness. Methods have been focused on increasing dimensions: making inferences from one or few measured dimensions about the properties of a higher dimensional system. The opposite problem is to record 100+ kinematic degrees of freedom and make inferences about properties of the embedded manifold. We present an approach to quantify the smoothness and degree to which the manifold is distributed among embedding dimensions. The principal components of embedding dimensions are rank-ordered by variance. The power-law scaling exponent of this variance spectrum is a function of the smoothness and dimensionality of the embedded manifold. It defines a threshold value beyond which the manifold becomes non-differentiable. We verified this approach by showing that the Kuramoto model close to global synchronization in the upper critical end of the coupling parameter obeys the threshold. Next, we tested if the scaling exponent was sensitive to participants gait impairment in a full-body motion capture dataset containing short gait trials. Variance scaling was highest in the healthy individuals, followed by osteoarthritis patients after hip-replacement, and lastly, the same patients pre-surgery. Thinking about manifold dimensionality, smoothness, and scaling could inform classic problems in movement science and exploration of the biomechanics of full-body action.