Self-organization of trajectory formation

Abstract.

 Most studies examining the stability and change of patterns in biological coordination have focused on identifying generic bifurcation mechanisms in an already active set of components (see Kelso 1994). A less well understood phenomenon is the process by which previously quiescent degrees of freedom (df ) are spontaneously recruited and active df suppressed. To examine such behavior, in part I we study a single limb system composed of three joints (wrist, elbow, and shoulder) performing the kinematically redundant task of tracing a sequence of two-dimensional arcs of monotonically varying curvature, κ. Arcs were displayed on a computer screen in a decreasing and increasing κ sequence, and subjects rhythmically traced the arcs with the right hand in the sagittal plane at a fixed frequency (1.0 Hz), with motion restricted to flexion-extension of the wrist, elbow, and shoulder. Only a few coordinative patterns among the three joints were stably produced, e.g., in-phase (flexion-extension of one joint coordinated with flexion-extension of another joint) and antiphase (flexion-extension coordinated with extension-flexion). As κ was systematically increased and decreased, switching between relative phase patterns was observed around critical curvature values, κ_c. A serendipitous finding was a strong 2:1 frequency ratio between the shoulder and elbow that occurred across all curvature values for some subjects, regardless of the wrist-elbow relative phase pattern. Transitions from 1:1 to 2:1 frequency entrainment and vice versa were also observed. The results indicate that both amplitude modulation and relative phase change are utilized to stabilize the end-effector trajectory. In part II, a theoretical model is derived from three coupled nonlinear oscillators, in which the relative phases (φ) between the components and the relative joint amplitudes (ρ) are treated as collective variables with arc curvature as a control parameter.