Visuomotor phase-locked loop reproduces elliptic hand trajectories across different rhythms

A well-known phenomenon in human hand movement is the correlation between speed and curvature, also known as the speed-curvature power law (V{approx}kC{beta}). In drawing elliptic shapes, the exponent is often found to be {beta} {approx} -1/3, however it is not clear why the power law appears and why the exponent is near -1/3. More fundamentally, it is not clear how do people track elliptic targets. In answering these questions, Ive analyzed trajectories of participants cursors while they tracked visual targets moving along elliptical paths, across different target speed profiles and cycle frequencies. The speed-curvature power law emerged when drawing ellipses at about 1 Hz or faster, regardless of the target speed profile, and it did not emerge for lower frequency movements. Analysis of the position frequency spectrum shows that the target-cursor trajectory transformation may be seen as a low-pass filter. Comparison of different hypothetical salient features of the visual field shows that phase difference (angular difference between the cursor and the target) and size difference (difference in the sizes of the elliptic paths) are the features most likely used in the task. The next experiment confirmed that phase and size difference could be controlled variables because participants kept them stable even under direct pseudorandom disturbances. A numerical model simulating the sensorimotor processes of the participant, similar to a phase-locked loop, using the visual features of phase and size difference as controlled variables, performed the same target tracking tasks as the participants. When fitted, the model closely replicated position and speed profiles of the participants across all trials, as well as the emergence of the power law at high frequencies. The model also reproduced the trajectories of participants in the experiment with direct pseudorandom disturbances. In conclusion (1) the speed-curvature power law emerges as a side effect of movement system properties, namely low-pass filtering in the sensorimotor loop; (2) people could be tracking elliptical targets by varying the frequency and amplitude of an internal pattern generator until the produced phase and shape size match the targets phase and shape size. The model generates new hypotheses about the neural mechanisms of rhythmic movement control.