Data-driven spectral analysis for coordinative structures in periodic human locomotion

www.nature.com/scientificreports OPEN Data-driven spectral analysis for coordinative structures in periodic human locomotion Keisuke Fujii 1,2*, Naoya Takeishi2, Benio Kibushi3, Motoki Kouzaki4 & Yoshinobu Kawahara 2,5 Living organisms dynamically and flexibly operate a great number of components. As one of such redundant control mechanisms, low-dimensional coordinative structures among multiple components have been investigated. However, structures extracted from the conventional statistical dimensionality reduction methods do not reflect dynamical properties in principle. Here we regard coordinative structures in biological periodic systems with unknown and redundant dynamics as a nonlinear limitcycle oscillation, and apply a data-driven operator-theoretic spectral analysis, which obtains dynamical properties of coordinative structures such as frequency and phase from the estimated eigenvalues and eigenfunctions of a composition operator. Using segmental angle series during human walking as an example, we first extracted the coordinative structures based on dynamics; e.g. the speed-independent coordinative structures in the harmonics of gait frequency. Second, we discovered the speed-dependent time-evolving behaviours of the phase by estimating the eigenfunctions via our approach on the conventional low-dimensional structures. We also verified our approach using the double pendulum and walking model simulation data. Our results of locomotion analysis suggest that our approach can be useful to analyse biological periodic phenomena from the perspective of nonlinear dynamical systems. Living organisms dynamically and flexibly operate a great number of components such as neurons, muscle fibers and skeletal joints. These phenomena can be seemingly regarded as dynamical systems based on some specific rules; however, it is often intractable to find optimal solutions due to a number of combinations of controlled elements and environments. In the field of neuroscience or motor control, for example, this problem has been referred to as Bernstein’s degrees of freedom problem or redundant problem1. For this problem, studies in a forward modeling approach have accomplished forward simulations by introducing various bio-inspired systems2–4, and by solving parameter optimisation problems5,6. However, the governing equation of a real organism behaviour has sometimes been complicated or unclear. Also in other fields, there are many examples of periodic systems with unknown and redundant dynamics, such as population dynamics in ecology and epidemiology and collective motion dynamics in animals and artificial agents. A backward or data-driven approach, which is expected to extract essential information from observed data, would be an effective way to understand them. One of the popular research subjects of the unknown and redundant dynamic phenomena is human locomotion, which can be performed at various speeds by solving the redundancy problem of many skeletal joints. This is because it seems to be based on specific rules (i.e. can be regarded as a cyclic movement) but the governing equation is not fully understood; thus, from a long time ago, it has attracted attention in many scientific and engineering fields such as neuroscience7,8, physics9,10, clinical medicine11,12, behavioural science13,14, robotics4,6 and pattern recognition15. For the redundant problem, some researchers have discovered synergistic and global structures among controlled components called coordinative structures (or low-dimensional spatially coherent structure) in joint motion8 and muscle activities7,16. Computing such coordinative structure may contribute to understanding the redundant problem in many fields. In the early days, for locomotion, researchers have found that the three-dimensional trajectory of elevation angles of limb segments lies close to a two-dimensional plane8,17, and have suggested that this planar law (or low-dimensional structure) may reflect intersegmental coordination. The planar law of intersegmental coordination has been observed at different walking speeds18, forward 1 Graduate School of Informatics, Nagoya University, Nagoya, Japan. 2RIKEN Center for Advanced Intelligence Project, Tokyo, Japan. 3School of Sport Sciences, Waseda University, Tokyo, Japan. 4Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, Japan. 5Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan. *email: Scientific Reports | (2019) 9:16755 | https://doi.org/10.1038/s41598-019-53187-1 1 www.nature.com/scientificreports/ www.nature.com/scientificreports or backward direction19 and different levels of body weight unloading20. It has also been observed in cats21, monkeys and human infants22. The idea of the low-dimensional structures was also supported by the successful clinical application12 and the accomplishment of walking model simulation23. However, the conventional methods to extract coordinative structures have used statistical dimensionality reduction methods with assumptions of independence of sampling such as principal component analysis (PCA). The problem is that the extracted structure does not reflect dynamical properties in principle and still, there has been little theoretical progress in this field. Therefore, an extraction method of the coordinative structures based on dynamical properties from data is needed. In other words, our motivation in this study is to understand the global dynamics behind the data obtained from human locomotion, of which governing equations are not fully known (e.g. neural inputs: for details, see Supplementary Text S5). As a method to obtain a global modal description of nonlinear dynamical systems, operator-theoretic approaches have attracted attention such as in applied mathematics, physics and machine learning. One of the approaches is based on the composition operator (usually referred to as the Koopman operator24,25), which defines the time evolution of observation functions in a function space. The advantage to use the operator-theoretic approach is to lift the analysis of nonlinear dynamical systems to a linear (but infinite-dimensional) regime, which is more amenable to the subsequent analysis. For example, spectral analysis of the Koopman operator can obtain dynamical properties which we define as physical properties regarding time-evolving behaviour such as frequencies with delay/growth rate and coordinative (or coherent) structures corresponding to these properties. Among several estimation methods, one of the most popular algorithms for spectral analysis of the Koopman operator is dynamic mode decomposition (DMD)26,27. The benefit of DMD is to extract such a global modal description of a nonlinear dynamical system from data, unlike other unsupervised dimensionality reduction methods such as PCA or singular value decomposition (SVD) for static data. The extracted coordinative structure based on the above dynam (...truncated)