Abstract
Some studies suggest that complex arm movements in humans and monkeys may optimize several objective functions, while others claim that arm movements satisfy geometric constraints and are composed of elementary components. However, the ability to unify different constraints has remained an open question. The criterion for a maximally smooth (minimizing jerk) motion is satisfied for parabolic trajectories having constant equi-affine speed, which thus comply with the geometric constraint known as the two-thirds power law. Here we empirically test the hypothesis that parabolic segments provide a compact representation of spontaneous drawing movements. Monkey scribblings performed during a period of practice were recorded. Practiced hand paths could be approximated well by relatively long parabolic segments. Following practice, the orientations and spatial locations of the fitted parabolic segments could be drawn from only 2–4 clusters, and there was less discrepancy between the fitted parabolic segments and the executed paths. This enabled us to show that well-practiced spontaneous scribbling movements can be represented as sequences (“words”) of a small number of elementary parabolic primitives (“letters”). A movement primitive can be defined as a movement entity that cannot be intentionally stopped before its completion. We found that in a well-trained monkey a movement was usually decelerated after receiving a reward, but it stopped only after the completion of a sequence composed of several parabolic segments. Piece-wise parabolic segments can be generated by applying affine geometric transformations to a single parabolic template. Thus, complex movements might be constructed by applying sequences of suitable geometric transformations to a few templates. Our findings therefore suggest that the motor system aims at achieving more parsimonious internal representations through practice, that parabolas serve as geometric primitives and that non-Euclidean variables are employed in internal movement representations (due to the special role of parabolas in equi-affine geometry).
Highlights
Despite decades of research on the formation of human hand trajectories, the basic mechanisms of neuromotor control underlying the generation of even the simplest drawing movements remain poorly understood [1]
We argue that geometric invariance may provide a more compact representation of complex movements composed of geometric primitives
Smooth and remain parabolic even when undergoing a specific kind of geometric transformations
Summary
Despite decades of research on the formation of human hand trajectories, the basic mechanisms of neuromotor control underlying the generation of even the simplest drawing movements remain poorly understood [1]. Studies in vertebrates have suggested that voluntary movements are composed of basic movement elements combined in parallel or sequentially [10,11,12,13,14,15,16,17]. Invariant properties of drawing movements were formalized by the two-thirds power law [18]. These kinematic constraints were shown to hold both with respect to movement production [19] and perception [20,21]. We argue that geometric invariance may provide a more compact representation of complex movements composed of geometric primitives
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