The Use of (Symmetry) Group Theory as a Predictive Tool for Studying Bimanual Coordination

Abstract

Symmetry groups-rules that connect different configurations of a given set of components-represent a compact means of coding for effects, a feature that is desirable in both model- and theory-building. The present study was designed to compare the effects of spatial orientation differences with the various other asymmetries (e.g., timing differences, handedness preferences, the direction of attention) that are accommodated by current models of bimanual coordination. The authors used symmetry groups to predict specific patterns of results. In 2 experiments, participants (N = 13, Experiment 1; N = 9, Experiment 2) coordinated the movements of differently oriented (1 downward and 1 upward) pendulum pairs at a low (0.62 Hz) or high (0.82 Hz) movement frequency to establish an in-phase or antiphase pattern. Consistent with previous results (P. G. Amazeen, E. L. Amazeen, & M. T. Turvey, 1998a), the downward-oriented pendulum tended to lead slightly. In contrast to the effects of other bimanual asymmetries, the downward-oriented pendulum lead was amplified at low frequencies. Although the results contradicted the predictions of existing models of bimanual coordination, they were consistent with predictions from symmetry group theory. In the discussion, the authors focus on the application of symmetry groups to both bimanual coordination and other phenomena with more complex symmetric structures.

Disentangling the effects of attentional and amplitude asymmetries on relative phase dynamics.

“…These results, as obtained for our right-handed participants, are in agreement with the effects of the handedness-related parameter d in Equation 3 and are therefore in accordance with the asymmetric relative phase dynamics that encompass a handedness-related asymmetry in coupling strength (de Poel et al, 2007;Peper, Daffertshofer, & Beek, 2004;Treffner & Turvey, 1995). In addition, the observation that the pattern of results is clearly mirror symmetric around the 1:1 amplitude ratio (see Figures 1A and 1B) can be explained with reference to (symmetry) group theory (Mulvey, Amazeen, & Riley, 2005). From this perspective, these results suggest that the production of a 2:1 amplitude pattern is the same as its mirror image, the 1:2 pattern.…”

“…This further confirms the observation that when bimanual coordination patterns are performed with unequal amplitudes, the production is essentially the same as for its mirror image (i.e., when the amplitude assigned to each hand is reversed). Together, this supports the notion of symmetry groups as a general control principle in bimanual coordination (Mulvey et al, 2005), which can be interpreted as a reduction of computational burden (or degrees of freedom) for coordinating bimanual movements (cf. Stewart & Golubitsky, 1992).…”

“…4In addition, for each attention condition separately, the data presented in Figures 3 and 4 showed signs of symmetry groups (Mulvey et al, 2005; see also Discussion section of Experiment 1). That is, the effect of attention on the performed amplitude for each arm separately seems to be mirror symmetric around the 1:1 amplitude ratio, as does the relative phase pattern.…”

Congruency effects in interpersonal coordination.

“…It is known that breaking the typical symmetry (e.g., in-phase patterning) found in coordination through different timing, spatial, or amplitude requirements for each hand often leads to compensation and changes in both hands or effectors (Amazeen, Amazeen, & Turvey, 1998;Franz et al, 1991;Kelso & Jeka, 1992;Mulvey, Amazeen, & Riley, 2005;Schwartz et al, 1995). These findings are analogous to coupled oscillator systems-individual oscillators exhibit a tendency to maintain their own pattern ("maintenance tendency") and accommodate the movements of other oscillators ("magnet effect";von Holst, 1939von Holst, /1973).…”