Peter J. Beek

Dynamical models of movement coordination

This article examines the status of dynamical models of movement coordination qua phenomenological models. After a brief outline of the aims, methods and strategic assumptions of the dynamical systems approach, a survey is provided of the theoretical and empirical progress it has made in identifying general principles of coordination. Although dynamical models are constructed for phenomena at a particular level of analysis for which they provide descriptive explanations, their dynamics can sometimes be linked to or associated with the dynamics of processes at other levels of analysis. The article concludes with a tentative scheme to clarify the position of the dynamical approach relative to other extant approaches in movement science.

Dynamical substructure of coordinated rhythmic movements

A coordinated rhythmic movement pattern is a dynamical activity involving many hidden layers of rhythmic subtasks. To investigate this dynamical substructure, spectroscopic concepts and methods were applied to an interlimb rhythmic movement task requiring 1:1 frequency locking of two hand-held pendulums in 180 degrees phase relation. The pendulums could be of identical or very different dimensions, thereby providing different values of the ratio omega of uncoupled frequencies. Analyses focused on the power spectrum of continuous relative phase as a function of variation in omega. Predictions were derived from the theories of mode locking and fractal time. Experimental results were in agreement with theoretical expectations and were discussed in terms of the possible recruiting of rhythmic subtasks in the assembling of interlimb absolute coordination, the interdependence of these subtasks, and the general dynamical principles that relate coordinative processes occurring at different length and time scales.

Frequency-induced phase transitions in bimanual tapping

The stability of bimanual performance of the frequency ratios 3∶8 and 5∶8 was examined from the perspective of the sine circle map and the associated Farey mode-locking hierarchy. By gradually increasing movement frequency, abrupt transitions from the initial frequency ratios to other frequency ratios were induced. In general, transitions occurred to frequency ratios that were near the initial frequency ratio but lower in the Farey ordering and, hence, of higher stability in the sine circle map. A fair percentage of these transitions were to unimodularly related ratios. The transition routes from 3∶8 and 5∶8 remained largely unaffected by extensive practice of the lower-order ratios 2∶5 and 3∶5. Collectively, these results suggest that (i) bimanual tapping occurs in a domain in which frequency-locked states either overlap or are located sufficiently close to each other to make stochastic switching possible (coupling parameter K > 1 or close to 1); (ii) the overall stability of these frequency-locked states decreases as movement frequency increases (due to a decrease in K) and, consequently, (iii) the probability of transitions to nearby frequency ratios increases as movement frequency increases, due to the differential stability of the frequency locks.

Linear and nonlinear stiffness and friction in biological rhythmic movements

AbstractBiological rhythmic movements can be viewed as instances of self-sustained oscillators. Auto-oscillatory phenomena must involve a nonlinear friction function, and usually involve a nonlinear elastic function. With respect to rhythmic movements, the question is: What kinds of nonlinear friction and elastic functions are involved? The nonlinear friction functions of the kind identified by Rayleigh (involving terms such as n

Coupling strength in tapping a 2:3 polyrhythm

dot theta ^3

Spatiotemporal variability in cascade juggling

n) and van der Pol (involving terms such as n

Perspectives on the relation between information and dynamics: An epilogue

theta ^2 dot theta