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.

This letter studies frequency-locked rotations in a phase-locked loop (PLL) circuit as FM demodulator. A rotation represents a desynchronized steady state in the PLL circuit and is regarded as another type of self-excited oscillations with natural rotation frequencies. The rotation frequency can be locked at driving frequencies of modulation signals. This letter shows response curves for harmonic amplitude of frequency-locked rotations. They have several different features from response curves of van der Pol oscillator.