Mathematical Analysis of Phase Resetting Control Mechanism during Rhythmic Movements

Kazuki NAKADA  Keiji MIURA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E103-A   No.2   pp.398-406
Publication Date: 2020/02/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.2019MAI0002
Type of Manuscript: INVITED PAPER (Special Section on Mathematical Systems Science and its Applications)
rhythmic movements,  locomotion,  central pattern generator (CPG) model,  phase resetting control,  phase response curve (PRC),  

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Possible functional roles of the phase resetting control during rhythmic movements have been attracting much attention in the field of robotics. The phase resetting control is a control mechanism in which the phase shift of periodic motion is induced depending on the timing of a given perturbation, leading to dynamical stability such as a rapid transition from an unstable state to a stable state in rhythmic movements. A phase response curve (PRC) is used to quantitatively evaluate the phase shift in the phase resetting control. It has been demonstrated that an optimal PRC for bipedal walking becomes bimodal. The PRCs acquired by reinforcement learning in simulated biped walking are qualitatively consistent with measured results obtained from experiments. In this study, we considered how such characteristics are obtained from a mathematical point of view. First, we assumed a symmetric Bonhoeffer-Van der Pol oscillator and phase excitable element known as an active rotator as a model of the central pattern generator for controlling rhythmic movements. Second, we constructed feedback control systems by combining them with manipulators. Next, we numerically computed the PRCs of such systems and compared the resulting PRCs. Furthermore, we approximately calculated analytical solutions of the PRCs. Based on the results, we systematically investigated the parameter dependence of the analytical PRCs. Finally, we investigated the requirements for realizing an optimal PRC for the phase resetting control during rhythmic movements.