Induced symmetry breaking in coordination dynamics: Transaction between spatial and temporal asymmetries

Date of Completion

January 2002


Psychology, Experimental|Psychology, Cognitive




The Haken-Kelso-Bunz (HKB) equation of elementary interlimb coordination dynamics possesses reflectional symmetry. Research suggests that when elements are introduced to break the symmetry, the dynamics' fixed points or attractors abide a variant of the Extended Curie Principle. That is, the induced symmetry breaking produces solutions (fixed points) of the HKB dynamics that are related by reflectional symmetry. The present research was motivated by the question of whether the hypothesis of symmetry redistribution could accommodate recent findings of spatially asymmetric HKB dynamics that suggested an origin in the contrasting stabilities of local muscular organizations. In 4 experiments, left and right forearms generated oscillations in a frontoparallel plane about the same (Experiment 1) or different (Experiments 1–4) axes of rotation that created spatial asymmetry. The oscillations were of two pendulum-like manipulanda with eigenfrequencies that were either the same or different. The results of Experiment 1 were favorable to the Extended Curie Principle for both fixed point and fixed-point stability measures and to the ancillary hypothesis of a dissociation of attractor location and attractor strength. In Experiment 2 manipulations of movement speed revealed that the pattern of symmetry redistribution, for both fixed point and fixed-point stability measures, was consistent over speed variations. They also revealed that movement speed interacted with strict spatial asymmetry in the same way that movement speed interacts with strict temporal asymmetry. Experiment 3 examined the influence of degree of perceptual support (vision versus haptic) on symmetry redistribution. Experiment 4 examined the influence on symmetry redistribution of functional asymmetries (i.e., handedness and selective attention) superimposed on the spatial and temporal asymmetries. The last two experiments emphasized the potential generality of the Extended Curie Principle. In particular, Experiment 4 suggested that with the addition of further forms of induced symmetry breaking, a reflectional transform can always be found that eliminates the differences between its different effects. Discussion focused on the implications of the data for the muscle-stability and symmetry-redistribution hypotheses in respect to the consequences for coordination dynamics of induced symmetry breaking. It also raised a challenging question of a generalized imperfection parameter for the HKB equation. ^