323
Torre is with the Sensorimotor Neuroscience Laboratory, MacMaster University, Canada, and Motor
Eficiency and Deiciency, University Montpellier I, France. Balasubramaniam is with the Sensorimotor
Neuroscience Laboratory, MacMaster University, Canada. Delignières is with Motor Eficiency and
Deiciency, University Montpellier I, France.
Motor Control, 2010, 14, 323-343
© 2010 Human Kinetics, Inc.
Oscillating in Synchrony
with a Metronome: Serial Dependence,
Limit Cycle Dynamics, and Modeling
Kjerstin Torre, Ramesh Balasubramaniam,
and Didier Delignières
We analyzed serial dependencies in periods and asynchronies collected during
oscillations performed in synchrony with a metronome. Results showed that asyn-
chronies contain 1/f luctuations, and the series of periods contain antipersistent
dependence. The analysis of the phase portrait revealed a speciic asymmetry
induced by synchronization. We propose a hybrid limit cycle model including
a cycle-dependent stiffness parameter provided with fractal properties, and a
parametric driving function based on velocity. This model accounts for most
experimentally evidenced statistical features, including serial dependence and
limit cycle dynamics. We discuss the results and modeling choices within the
framework of event-based and emergent timing.
Keywords: Oscillation, asynchrony, 1/f noise, limit cycle model, parametric driving.
So far, studies of single-limb oscillations have essentially focused on oscil-
lators within-cycle dynamics, especially through the analysis of the phase-plane
representation of motion. Serial dependence (i.e., cycle-to-cycle dynamics) has
been largely disregarded, except in a few studies focusing on self-paced oscillations
(Daffertshofer, 1998; Delignières et al., 2004, 2008; Schöner, 1994). Especially,
Delignières et al. (2008) revealed the presence of fractal serial dependencies in
series of oscillation periods, which could not be accounted for by classical limit
cycle models. We focus in the present paper on oscillations performed in synchrony
with a metronome. Our irst aim was to combine the analyses of within-cycle and
cycle-to-cycle oscillation dynamics, to provide a complete characterization of the
impact of synchronization on the limb dynamics. In a second step, we propose to
assess the capability of some candidate models to account for the empirical results.
ORIGINAL RESEARCH