Comments on Marken and Shaffer: The power law of movement: an example of a behavioral illusion

Abstract

Many researchers who have studied movements along curved paths, under a variety of conditions, by different organisms, mostly human but a couple with non-human organisms, have found a consistent form of relation between the tangential (along-track) instantaneous velocity V and the local radius of curvature R. The consistent relation is that V ≈ cR^k , where k is a constant less than unity, often near 0.33 but sometimes far from 0.33, and c is a proportionality constant appropriate to the organism and the situation (see Zago, Matic, Flash, et al. (2017) for many examples in which the power law holds with widely varying values of the power, as well as cases of simple systems for which everything can be calculated exactly and in which the power law fails badly). Marken and Shaffer (Exp Brain Res 235:1835–1842; 2017), following a challenge by Gomez-Marin to see whether it is possible to use Perceptual Control Theory (Powers 1973/2005) to explain the power law results (Alex Gomez-Marin posting to CSGnet@lists.illinois.edu 2016.05.03), claim to have found a mathematical argument that proves the true exponent of the power relating velocity and radius of curvature always to be 1/3. They say that deviations from this value occur because researchers have omitted a critical correction "cross-product" factor that the authors label "D". This note questions the logic of the analysis offered by Marken and Shaffer, and argues that even had the analysis been correct, it would not affect future research into the reasons why and when the power law is observed and the circumstances that determine the value of the power found when it is observed.

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