Abstract
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 limit-cycle 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.
Highlights
IntroductionWe first briefly review the underlying theory for dynamic mode decomposition (DMD) called Koopman spectral analysis, and describe the basic and Hankel DMD procedures
Background of dynamic mode decomposition (DMD)Here, we first briefly review the underlying theory for DMD called Koopman spectral analysis, and describe the basic and Hankel DMD procedures
We show an example of segmental angles in human locomotion in Fig. 2a,b, and show representative examples of intersegmental coordination by five decomposition methods in Fig. 3: conventional singular value decomposition (SVD)-based method, basic two DMDs and two Hankel DMDs for each row
Summary
We first briefly review the underlying theory for DMD called Koopman spectral analysis, and describe the basic and Hankel DMD procedures. In Koopman spectral analysis, we consider a nonlinear dynamical system: xt+1 = f(xt), where xt is the state vector in the state space ⊂ p with time index t ∈ T: = N0. The Koopman operator is a linear operator acting on a scalar observable function g: → defined by g = g f , (1). We assume that has only discrete spectra. It generally performs an eigenvalue decomposition: φj(x) = λjφj(x), where λj ∈ is the j-th (Koopman) eigenvalue and φj is the corresponding (Koopman) eigenfunction. We denote the concatenation of scalar functions as g: = [g1, ..., gd]T.
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