Estimating the phase of synchronized oscillators

Abstract

The state of a collection of phase-locked oscillators is determined by a single phase variable or cyclic coordinate. This paper presents a computational method, Phaser, for estimating the phase of phase-locked oscillators from limited amounts of multivariate data in the presence of noise and measurement errors. Measurements are assumed to be a collection of multidimensional time series. Each series consists of several cycles of the same or similar systems. The oscillators within each system are not assumed to be identical. Using measurements of the noise covariance for the multivariate input, data from the individual oscillators in the system are combined to reduce the variance of phase estimates for the whole system. The efficacy of the algorithm is demonstrated on experimental and model data from biomechanics of cockroach running and on simulated oscillators with varying levels of noise.

A single muscle's multifunctional control potential of body dynamics for postural control and running

“…Limb kinematic response. Individual limbs were tracked using high-speed videography (figure 2a) [34] and the resulting limb extensions for the stimulated middle left leg were compared across spike addition conditions. Limb kinematics changed nonlinearly in a manner parallel to vertical impulse response during running (figure 3b).…”

“…Limb phase has been used extensively in the literature [5,7,8,45,47] and is important for determining the relative timing of muscle activity with respect to muscle strain resulting from limb movement [16]. However, limb phase is typically a poor estimate of the actual neuromechanical gait of the animal because trajectories of individual limbs are highly variable [34]. The relative timing of a periodic gait is better captured by extracting a single phase variable from dimensionally reducing the set of all limb and COM positions and velocities in both the fore-aft and lateral directions.…”

“…The relative timing of a periodic gait is better captured by extracting a single phase variable from dimensionally reducing the set of all limb and COM positions and velocities in both the fore-aft and lateral directions. We calculated this phase, termed kinematic phase (f K ), using an instantiation of the phase algorithm from Revzen & Guckenheimer [34]. Kinematic phase was robust to changes in individual limb kinematics.…”

“…We tracked individual limb trajectories using an automated tracking program [34]. We calculated two different phase variables to characterize gait timing (figure 2a,d).…”

“…Rewriting the motor code to individual muscles. We tracked individual limb trajectories (a) using tracking methods adapted from Revzen & Guckenheimer [34]. A typical stride (b) is composed of a stance and a swing period during which the putative control muscle, a ventral femoral extensor, undergoes shortening and lengthening, respectively.…”