A Hessian-based decomposition characterizes how performance in complex motor skills depends on individual strategy and variability

Abstract

In complex real-life motor skills such as unconstrained throwing, performance depends on how accurate is on average the outcome of noisy, high-dimensional, and redundant actions. What characteristics of the action distribution relate to performance and how different individuals select specific action distributions are key questions in motor control. Previous computational approaches have highlighted that variability along the directions of first order derivatives of the action-to-outcome mapping affects performance the most, that different mean actions may be associated to regions of the actions space with different sensitivity to noise, and that action covariation in addition to noise magnitude matters. However, a method to relate individual high-dimensional action distribution and performance is still missing. Here we introduce a decomposition of performance into a small set of indicators that compactly and directly characterize the key performance-related features of the distribution of high-dimensional redundant actions. Central to the method is the observation that, if performance is quantified as a mean score, the Hessian (second order derivatives) of the action-to-score function determines the noise sensitivity of the action distribution. We can then approximate the mean score as the sum of the score of the mean action and a tolerance-variability index which depends on both Hessian and action covariance. Such index can be expressed as the product of three terms capturing overall noise magnitude, overall noise sensitivity, and alignment of the most variable and most noise sensitive directions. We apply this method to the analysis of unconstrained throwing actions by non-expert participants and show that, consistently across four different throwing targets, each participant shows a specific selection of mean action score and tolerance-variability index as well as specific selection of noise magnitude and alignment indicators. Thus, participants with different strategies may display the same performance because they can trade off suboptimal mean action for better tolerance-variability and higher action variability for better alignment with more tolerant directions in action space.

Footnotes

* In the revised manuscript we have extensively re-organized Methods and Results and made an effort to simplify the exposition. In the Methods, we have added an initial paragraph and a new figure (Fig 2) introducing the key variables and functions used in the derivation of the decomposition and providing a graphical illustration of the basic components of the decomposition in a 2D case (i.e. the same case we present in the Methods to illustrate the decomposition and inter-individual differences in a simplified 2D model of throwing). In the Results we now start from the examples of the action distributions of five representative subjects (section 3.1, Fig 5) to illustrate qualitatively some of the individual strategies and to motivate the Hessian-based decomposition. We then present the validation of the key assumption of the method, i.e. the adequateness of the quadratic approximation of the action-to-score function (section 3.2, Fig 6). We continue with the presentation of the distribution across participants of the basic elements of the decomposition (eigenvalues and eigenvectors of the Hessian and covariance matrices, section 3.3, Fig 7A-D) and their relationship (scalar products, Fig 7E), which highlights the features shared by all participants but also the inter-individual variability. The following section then provides a characterization of inter-individual differences, for one target, in terms of the individual selections of the values of the five different indicators derived from the Hessian-based decomposition (section 3.4, Fig 8). We then show that such indicators are consistent across targets (section 3.5, Fig 9) and that they show specific dependencies on throwing speed, but that speed does not fully determines them (section 3.6, Fig 10). We conclude the Results by presenting a compact yet informative representation of the same distributions of actions shown in Fig 5) provided by the projection onto the plane defined by the first two principal sensitivity directions (section 3.7, Fig 11). Finally, we have updated the Appendix to include comparison of the Hessian-based decomposition with the TNC-costs analysis by Cohen and Sternad 2009 in addition to the comparison presented in the original manuscript of the TNC method by Muller and Sternad in 2004.