Spectrum of power laws for curved hand movements

Significance In curved hand movements around ellipses, the speed tends to scale inversely with the curvature with a power law having an exponent of −1/3. We examined whether this well-known regularity in motor planning holds for more general shapes. Using an optimality principle, we identified a set of basis shapes, each with a single characteristic angular frequency, which subjects drew with power laws whose exponents ranged from 0 to −2/3. More general movements exhibited linear mixtures of power laws. The speed of arbitrary doodling movements with a broad spectrum of frequencies could also be predicted from the curvature with high accuracy. In a planar free-hand drawing of an ellipse, the speed of movement is proportional to the −1/3 power of the local curvature, which is widely thought to hold for general curved shapes. We investigated this phenomenon for general curved hand movements by analyzing an optimal control model that maximizes a smoothness cost and exhibits the −1/3 power for ellipses. For the analysis, we introduced a new representation for curved movements based on a moving reference frame and a dimensionless angle coordinate that revealed scale-invariant features of curved movements. The analysis confirmed the power law for drawing ellipses but also predicted a spectrum of power laws with exponents ranging between 0 and −2/3 for simple movements that can be characterized by a single angular frequency. Moreover, it predicted mixtures of power laws for more complex, multifrequency movements that were confirmed with human drawing experiments. The speed profiles of arbitrary doodling movements that exhibit broadband curvature profiles were accurately predicted as well. These findings have implications for motor planning and predict that movements only depend on one radian of angle coordinate in the past and only need to be planned one radian ahead.

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