Space–time behavior of single and bimanual rhythmical movements: Data and limit cycle model.

Abstract

How do space and time relate m rhythmical tasks that reqmre the hmbs to move singly or together m various modes of coordination ? And what kind of minimal theoretical model could account for the observed data9 Ead~er findings for human cychcal movements were consistent w~th a nonhnear, limit cycle oscdlator model (Kelso, Holt, Rubm, & Kugler, 198 l) although no detailed modehng was performed at that Ume In the present study, lonemauc data were sampled at 200 samples/second, and a detmled analysis of movement amphtude, frequency, peak velooty, and relative phase (for the blmanual modes, m phase and anuphase) was performed As frequency was scaled from l to 6 Hz (m steps of l Hz) using a pacing metronome, amphtude dropped reversely and peak veiooty m-creased WRhm a frequency condmon, the movement's amphtude scaled &rectly with lls peak veloc-Ry These &verse lonematlc behaviors were modeled exphotly m terms oflow-&menslonal (nonhn-ear) dlsslpaUve dynamics, wRh hnear stiffness as the only control parameter Data and model are shown to compare favorably The abstract, dynamical model offers a umfied treatment of a number of fundamental aspects of movement coordination and control How do space and time relate m rhythmical tasks that require the hands to move singly or together in various modes of coordi-nation9 And what kind of minimal theoretical model could account for the observed data? The present article addresses these fundamental questions that are of longstanding interest to experimental psychology and movement science (e g, von Hoist, 1937/1973; Scripture, 1899; Stetson & Bouman, 1935) It is well known, for example, that discrete and repetitive movements of different amplitude vary systematically in movement duration (provided accuracy requirements are held constant, e g, Cralk, 1947a, 1947b) This and related facts were later for-mahzed into F~tts's Law (1954), a relation among movement time, movement amplitude, and target accuracy, whose under-pmnmgs have been extensively studied (and debated upon) quite recently (e g.

Deterministic and stochastic features of rhythmic human movement

“…(9) was fitted to the three types of data shown in Fig. 6a; here, x refers to the recorded position and y to the corresponding velocity (11) Figure 6b shows that this model fitting yields phase portraits that are very similar to the reconstructed ones in the case of forearm and wrist cycling (left and middle panels), supporting earlier studies of Kay et al (1987Kay et al ( , 1991 and Beek et al (1996). For the tapping data, however, the results turn out to be quite poor because we ignored most prominent features like asymmetry and anchoring in constructing the model according to Eq.…”

“…Using averaging methods from the theory of nonlinear oscillators, such as the slowly-varying amplitude approximation and harmonic balance analysis, Kay et al (1987Kay et al ( , 1991 derived second-order nonlinear differential equations that mimicked experimentally observed amplitude-frequency relation and the phase response characteristics of rhythmic finger and wrist movements. In particular, these selfsustaining oscillators included weak dissipative nonlinearities that stabilized the limit cycle and caused a drop of amplitude (accounted for by a Rayleigh term) and an increase in peak velocity (accounted for by a van der Pol term) with increasing movement tempo (i.e., frequency).…”

Modeling inter-human movement coordination: synchronization governs joint task dynamics

“…Fundamental research work on fitting nonlinear dynamic models to trajectories of human rhythmic movements is conducted by Kay et al [1987]. Observed functional relationships between the external driving frequency and the amplitudes and peak velocities of the movements are found to be reproduced well by a mixture of van der Pol and Rayleigh oscillators with stable parameter fits.…”

Engineering entrainment and adaptation in limit cycle systems

“…e.g. (Kay, Kelso, Saltzman, & G., 1987;Righetti & Ijspeert, 2006a)) Most of the time oscillators are used for their synchronization properties. Thus, we are interested in how the phase φ behaves over time.…”