Visuomotor phase-locked loop reproduces elliptic hand trajectories across different rhythms [preprint]

[rsmarken]

@amatic

@rsmarken

It’s a general statement of one of the most important implications of Bill’s model for understanding the behavior of living organisms. It means that some observations about behavior or the effects thereof are unintended side effects of producing intended results.

No, it is not a general statement, it a meaningless statement.

OK, it’s meaningless to you but it’s quite meaningful to me. To me it means that, unlike all other models of behavior, Bill’s model makes it possible to scientifically distinguish intentional from unintentional behavior.

in the case of the power law, that controlling is the movement behavior itself.

“movement behavior itself”? No, in this paper, the proposed controlled variables are the phase and size difference.

Yes, yours is a model of the elliptical movement behavior itself; this behavior, per your model, is the control of phase and size difference. Your model shows that a -1/3 power relationship is a side effect of controlling those perceptions under the proper circumstances.

Of course, the reason you see a power law side effect of movement is because there is a mathematical power relationship between speed (V) and curvature (C)

Oh, you’re still stuck with that one. No, V and C are generally independent variables. But did you check out the other paper? Angular speed should be avoided when assesing the speed-curvature power law [preprint]

Yes, I read that paper and it is excellent. It was a very nice explanation of why it’s best to use C rather than R as the measure of curvature when doing the regression to evaluate fit to the power law. But a paper that is more relevant to my claim that there is a mathematical relationship between C and V is the one by Maoz, Portugaly, Flash and Weiss (paper presented at the Neural Information Processing Systems conference, 2005).

Maoz et al derive the following relationship between V, which they call v(t), C, which they call kappa (t), and affine velocity, which they call alpha (t) and which we called D in Marken & Shaffer (2017):

.

This is precisely the same mathematical relationship between curvature and velocity that we derived in the 2017 paper, but we used R for the measure of curvature rather than C so the coefficient of R was positive:

Maoz et al noted (under their equation above) that the 1/3 power law will be found when the affine velocity of the movement (alpha for Maoz, D for Marken & Shaffer) is constant, which is true. But it’s also true that the -1/3 power law will be found when affine velocity is variable but uncorrelated with variations in C. This is what we learn from the OVB (omitted variable bias) analysis that we describe in the 2017 paper.

Apparently, Maoz et al were looking at their equations as describing the physical relationship between variables, which is suggested by the fact that they add time indexes to the variables: v(t), kappa (t), alpha (t). But statistical regression analysis is used to determine whether a movement is power law conforming and regression analysis doesn’t care about the time at which the value of each variable was collected; it just cares that the values of each set of variables was collected at the same time.

So what this means is that you will find that curved movements conform to the -1/3 power law when 1) you use simple linear regression which 2) omits inclusion of the the affine velocity variable, D or K, and 3) cov(D, C)/var(C) is close to 0. This is why I call the finding of a -1/3 power law a side effect of control that results from a statistical artifact; it is a side effect that is seen when using a particular form of statistical regression analysis.

RSM