A Compact Representation of Drawing Movements with Sequences of Parabolic Primitives

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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). Although our movements are flexible and versatile, they are nonetheless highly stereotypical. This versatility is similar to that of natural language sentences, which are composed of words which, in turn, are constructed from a small alphabet of elementary phonemes. Parabolic drawings are simple, smooth and remain parabolic even when undergoing a specific kind of geometric transformations. Smoothness, invariance and compactness of representation are important in motion planning and in visual feedback processing. Hence stereotypical parabolic sub-movements may serve as appropriate building blocks of complex movements. Given the similarities between motor organization in monkeys and humans and the greater opportunity to record brain activities in monkeys here we study the spontaneous emergence of stereotypical arm movements in monkeys following practice. We show that practice has indeed led to the emergence of a small alphabet of parabolic elements during spontaneous drawing movements. We further use this alphabet to study sequences of parabolic sub-movements with respect to possible decisions concerning the animal's choice of what elements to concatenate into words and sentences. We also propose that the relative simplicity of movement data compared, for example, to acoustic or semantic data makes their analysis a useful tool in studies of binding and cognitive processing.