Abstract
Several types of curvilinear movements obey approximately the so called 2/3 power law, according to which the angular speed varies proportionally to the 2/3 power of the curvature. The origin of the law is debated but it is generally thought to depend on physiological mechanisms. However, a recent paper (Marken and Shaffer, Exp Brain Res 88:685–690, 2017) claims that this power law is simply a statistical artifact, being a mathematical consequence of the way speed and curvature are calculated. Here we reject this hypothesis by showing that the speed-curvature power law of biological movements is non-trivial. First, we confirm that the power exponent varies with the shape of human drawing movements and with environmental factors. Second, we report experimental data from Drosophila larvae demonstrating that the power law does not depend on how curvature is calculated. Third, we prove that the law can be violated by means of several mathematical and physical examples. Finally, we discuss biological constraints that may underlie speed-curvature power laws discovered in empirical studies.
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Notes
Bold characters denote vector quantities throughout.
Although the terms velocity and speed are often used interchangeably in the literature (including M/S), the former denotes the vector with a magnitude and direction while the latter denotes the magnitude only.
Notice, however, that orthogonal harmonic oscillations at a different frequency generate Lissajous motions that do not comply necessarily with the 2/3 power law (Lebedev et al. 2001).
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Acknowledgements
The authors declare no competing financial interests. The work was supported by the Italian Space Agency (grant n. I/006/06/0 to F.L. and grant n. 2014-008-R.0 to M.Z.), the Italian University Ministry (PRIN grant 2015HFWRYY_002 to F.L.), the Spanish Ministry of Economy and the Severo Ochoa Center of Excellence programs (SEV-2013-0317 start-up funds to A.G.-M., grant BFU-2015-74241-JIN to A.G.-M., and pre-doctoral contract BES-2016-077608 to A.M.). Tamar Flash is an incumbent of Dr. Haim Moross professorial chair. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. We thank the anonymous reviewers for helpful suggestions.
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Zago, M., Matic, A., Flash, T. et al. The speed-curvature power law of movements: a reappraisal. Exp Brain Res 236, 69–82 (2018). https://doi.org/10.1007/s00221-017-5108-z
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DOI: https://doi.org/10.1007/s00221-017-5108-z