Coordinate Dependence of Variability Analysis

Analysis of motor performance variability in tasks with redundancy affords insight about synergies underlying central nervous system (CNS) control. Preferential distribution of variability in ways that minimally affect task performance suggests sophisticated neural control. Unfortunately, in the analysis of variability the choice of coordinates used to represent multi-dimensional data may profoundly affect analysis, introducing an arbitrariness which compromises its conclusions. This paper assesses the influence of coordinates. Methods based on analyzing a covariance matrix are fundamentally dependent on an investigator's choices. Two reasons are identified: using anisotropy of a covariance matrix as evidence of preferential distribution of variability; and using orthogonality to quantify relevance of variability to task performance. Both are exquisitely sensitive to coordinates. Unless coordinates are known a priori, these methods do not support unambiguous inferences about CNS control. An alternative method uses a two-level approach where variability in task execution (expressed in one coordinate frame) is mapped by a function to its result (expressed in another coordinate frame). An analysis of variability in execution using this function to quantify performance at the level of results offers substantially less sensitivity to coordinates than analysis of a covariance matrix of execution variables. This is an initial step towards developing coordinate-invariant analysis methods for movement neuroscience.Author Summary: Over the past decade the identification of synergies has become a prominent theme in motor neuroscience. Like other aspects of neural organization (e.g., vision) the control of coordinated movement is almost certainly hierarchical with synergies a key feature of this hypothesis. In pursuit of identifying synergies, whether flexible or hard-wired in biomechanical or physiological structures, many studies have analyzed variability with techniques of dimensionality reduction such as principal component analysis. Results have been interpreted as evidence for controlled variables in motor control. Our paper demonstrates that such analyses and conclusions based on these methods are exquisitely sensitive to the coordinates of the variables that are the basis for this analysis. As these coordinates are often chosen for convenience of measurement or analysis, any conclusions about neural control are therefore ambiguous at best. The development of coordinate-independent analyses was an important step in the development of modern physics. Here we highlight the problems induced by coordinate-dependency in studies of neural control and present initial steps towards coordinate-independent analyses relevant to computational biology. We critically examine an alternative method proposed to analyze variability for identification of structure and show that it is significantly less sensitive to assumed coordinates than conventional analyses.