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

I’ve just published a preprint with a model that can produce elliptic trajectories when tracking elliptic targets across different rhytms, from slow to fast, also including non-rhytmic trajectories.

There is a two-level model. On the lower level is a ‘standard’ target tracking model, and on the higher level are two controlled variables - the angle or phase difference (d-phi) and the size difference (ds), as shown on this diagram:


Here is a diagram of the model:

And here is how well it reproduces human tracking:

Figure 6. Direct comparison of one participant and model behavior in the same tasks (model in orange, participant in green). A) In tracking a low frequency target, model and participant behave similarly in measures of position in the x dimension, speed, phase difference and ellipse size. B) In tracking a high-frequency target, the model shows more regularity, while the participant shows more randomness in behavior. C) The model with the same parameters also reproduces participant phase and trajectories from Experiment 3, where both the target phase and target size were smoothed random signals. D) The model ellipse size is closely reproducing participant cursor ellipse size in experiment 3

So, it is a proposal for controlled variables involved in creating elliptic trajectories, including the speed-curvature power law. There is the TCV, there is a model. Not enough participants at the moment to attempt a peer-reviewed publication, but I suppose I’ll come to that.

The number of participants is only important as an expected requirement, right? Modeling individual behavior means different control system parameter values for each individual’s model. No statistical measures across the population. Or will reviewers insist on that too?

Well done Adam!
I’m sure you planning this but the control parameters can be fitted to the individual and then the group statistics applied to the fit indices.

I have three experiments, the first with 3 participants, and the next two experiments with just 1, myself. I think that is fine for a preprint, and the initial model, but maybe not too convincing as model of general tracking behavior in these tasks. Other people could be controlling different variables, for one. And it is not given that changing the parameters will definitelly make the model fit each of the participants’ behavior, I think I should demonstrate that in experiments. If it doesn’t fit well, maybe I would need to change the model.

In fact, there is already an inconsistency between the proposed structure of the model and the parameters I found in this one participant. It looks like the phase control loop, which should be in the higher level according to the diagram, has the delay estimated at the value of the lower level delay. Maybe that is different in different participants, or maybe the phase control loop, or the oscillator, needs to be on a lower level.

Also, it should be interesting to see the medians and ranges of gains and delays across the sample.

It’s nice to see that something I demonstrated on CSGNet in 2016 and described in a publication in 2017 – that the power law is a side effect of control – is now being celebrated as a great discovery by the same people who called my demonstration 'bullshit" and an affront to science.

I think almost everyone agreed that the power law is a side effect of control of something. The question was what do people control when they draw ellipses and similar shapes; what are the exact controlled variables, or best approximations to controlled variables.

You suggested the trajectory as the controlled variable, but did not make any TCVs or models that would control the trajectory. You had a model controlling position, or the cursor-target distance, and you showed one example ellipse of the speed-curvature power law. This model does not produce ellipses like participants do, unless the speed is VERY low.

There were several “bullshit” elements in your paper, in my opinion, the biggest ones being the statistics of the omitted variable bias and the claim of the behavioral illusion.

The lack of TCV is a big problem too, that should have been your starting point.

The current model and the definitions could still be improved, of course. I’m using the angular difference between the cursor and the target, but maybe the arc-length would be better, not sure, I need to test that.

Your memory of this is quite different than mine. But of course you would remember it that way.

Rick, have you references that somebody had claimed that power law or the power relation between the pace and he curvature is (generally) a controlled variable? (Of course ii sometimes can be.)
I remember also that it is usually thought as a side effect. And the research problem has been what mechanism produces that somewhat surprising regularity to the behavior of living beings. And of course PCTers want to study also what is then the CV.
But you claimed that the power law is a statistical artefact produced either by some necessary mathematical dependency or by the statistical method of the researchers - and thus totally a pseudo problem. (That’s the biggest bullshit according to my understanding.)
If power law were a statistical artefact as you have claimed then it of course could NOT be a side effect of the research subject’s controlling!


| rsmarken
July 23 |

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I think almost everyone agreed that the power law is a side effect of control of something.

Your memory of this is quite different than mine. But of course you would remember it that way.

No, and I don’t believe anyone has claimed that it is.

I think it’s very unlikely that the power relationship between speed and curvature of curved movement could be a controlled variable. I think it’s really impossible to perceive the nature of the relationship between speed and curvature, let alone perceive the coefficient of the power relationship. And you have to be able to perceive a variable in order to be able to control it.

I looked for some evidence of that in the reply to my paper with Dennis Shaffer paper on the power law as a behavioral illusion and in the replies to it from Zago et al and Taylor. Of course, it was described as a side effect in our paper but not in the Zago et al reply. Taylor, however, agreed that the power law is a side effect of control but he didn’t think our paper made much of a contribution to understanding that fact. Why am I not going to have a heart attack and die from the surprise?

Once you know that the power law is a side effect of control then you know that it is a result of something other than the control mechanism that produces the movement of which the law is a side effect. A nice illustration of what this means is Bill’s demonstration of having a person write the word “hello” as a side effect of controlling a cursor in two dimensions.

The person writes “hello” as a side effect of compensating for the 2D disturbance to the cursor. Bill designed the disturbance so that, in order to keep the cursor on target, the person controlling the cursor would have to move the mouse in a way that writes out “hello”. The person doing the controlling doesn’t even know that the movements opposing the disturbance are writing “hello”.

The word “hello” is a completely irrelevant side effect of acting to control the cursor; but because “hello” is a familiar word, it is a side effect that captures an observer’s attention. But, in fact, it has nothing to do with what the person is actually “doing”. An observer who didn’t know what was going on and based their research program on trying to understand why people in this situation always wrote “hello” they would be wasting a lot of time and research funding.

I think the power law relationship between curvature and speed of movement is exactly equivalent to the word “hello”; it’s an irrelevant but compelling side effect of controlling the position of one’s arm (or a pen or whatever). It’s particularly compelling because V = kC^-1/3 describes movement that involves slowing down for curves, which is what we do when driving through curves.

In our paper we showed that this particular side effect – the power law – is a result of using linear rather than multiple regression to find the coefficient of the power relationship between curvature and velocity. This is the OVB – omitted variable bias.

OVB analysis shows that simple linear regression of log (C) on log(V) will result in a power coefficient that deviates from -1/3 by an amount proportional to the degree to which log(C) and log (D) (affine velocity, the omitted variable) covary. So the degree to which one finds the 1/3 power law depends on the mathematical properties of the movement produced, not on how it was produced. This is analogous to the fact that seeing “hello” as a side effect controlling the cursor in Bill’s demo depends on the mathematical properties of the disturbance to cursor position, not on how the movements that compensated for this disturbance were produced.

That’s why I called the power law a statistical artifact (the statistical part being the regression analysis). What we still don’t know, however, is why the coefficient of that power “law” is generally close to 1/3 (or -1/3, depending on what you use as the predictor). That is, we don’t know why the correlation between log(C) and log (D) is generally small – and thus the bias added to the actual 1/3 mathematical relationship between C and V is generally small – for the movements that have been tested. I’m going to do some work on this when I get a chance.

I got pretty good results using Cartesian position as the controlled variable in the models we tested.

The problem with calling our conclusions “bullshit” is that it doesn’t help us understand what these critics think is wrong with our analysis.

I hope my comments above have made it clear that this is not the case. The power law is unquestionably an irrelevant side effect of control. Showing that the power law is a statistical artifact of the way researchers have gone about estimating the coefficient of the power law – using linear rather than multiple regression – was aimed at trying to show why researchers have observed this particular side effect – the ~1/3 power relationship – so consistently.


I’m convinced that without the BS word and the occasional arbitrary antagonism attached to it, this pathway from Rick’s initial insight to Adam’s eventual model could be considered as the ‘space-race’ mutual development of a plausible PCT model of how the power law emerges… but I doubt with Rick or Adam would be that generous towards one another!
Is the metaphor Franklin to Watson & Crick,
Or SSSR to NASA?:wink:

Well, you’re right about me not being that generous towards Rick’s work on the power law. Maybe I could go with “he showed the pathways to avoid”, so, you know, not completely useless. I’m giving him zero credit for the insight that the power law is a side effect of control of something other than the relationship between speed and curvature. I’ve found my posts, and Rick found Martin Taylor’s statements from his paper. Everyone basically agreed that the power law was a side effect of controlling something else.

Ironically, Rick suggested that the trajectory is the controlled variable. If that were the case, the power law would be the main effect, and not the side effect, it would be included in the trajectory.

The things I’ve tried to correct are: a) the lack of any formal or informal test of the controlled variable, which should have been the foundation, b) the lack of a model that reproduces human behavior in relevant tasks.

Still, there are many problematic things with my current model, too, and they need to be addressed experimentally, with participants, and with changes in the model. For example, the model can only do ellipses of one eccentricity - they can change size, but not width and height independently. I may be missing an additional contolled variable, the center of the ellipse. Maybe there are alternatives to the size and phase variables, too. Maybe the oscillator needs to be moved to a different place in the hierarchy, and so on.

Looks like you get incoherent when you’re wrong, Rick.

It is a dichotomy: either people agreed that the slowing down in curves was a side effect of controlling something else, or they disagreed and claimed that the slowing down in curves was planned and intended effect. You agree that no one did the latter.

You can reject the dichotomy by sayng that the effect is not real at all, there is no slowing, it is all a statistical artefact.

Criticizing one’s work as “bullshit” is hardly using a word that has “occasional arbitrary antagonism attached to it”. But if you think this then it shouldn’t bother you at all when I non-arbitrarily and non-antagonistically say that your Psych Review paper on consciousness is total, unadulterated bullshit. This will save me the trouble of explaining what’s wrong with it.

My “initial insight” included two control models that fit the data extremely well; one model showed that a power law is an expected side effect of controlling the position of a cursor. It also showed that power law conforming cursor movements could be produced by non-power law conforming mouse movements. So the fact that movements conform to a power law tells you nothing about how those movements were produced.

The other model was the one I used to account for the object interception data collected by my co-author Dennis Shaffer. This model showed that a power law is also found for the movement paths taken by the subjects who were intercepting the objects.

The fact that these two different models – controlling different variables – fit all these data extremely well, in terms of both R^2 and RMS deviation, shows that the controlled variables used in these models are a pretty close approximation to what the subjects were controlling in these situations.

Adam’s model is not a great new achievement; it’s actually a one level, position control model, just like mine, but where position is represented in polar rather than Cartesian coordinates. And his conclusion from his model – that the power law can be explained in terms of the variables controlled – is not the same as my “insight”, which is that the power law is an irrelevant side effect of control and has nothing to do with the variables being controlled.


Always with the ad hominum.

So? Where the incoherence?


Someone’s a bit huffy, eh? :smiley: :smiley: :smiley:

Feisty? Bubbly? Why are you so bubbly, Rick?

In the paper, I’m crediting Gribble and Ostry (1996?) and Schaal and Sternad (2001) for the initial explanation of the power law in terms of low pass filtering. The power law is a side-effect of low pass filtering in the visomotor loop, musculo-skeletal system, and also, of course, of the work of two higher-level loops. It is a side effect of the operation of the whole system, and not an intended effect, not a controlled variable.

Here is my test, Figure 3 from the paper, of the position control system, that you claim fits your data extremely well:

In orange, you see the same model. It’s great for random pursuit tracking, but when tracking targets allong elliptical trajectories, it shows large phase delays, and much larger sizes of the ellipses. It doesn’t explain how people draw ellipses, or track elliptic targets, even though there is a power law in some situations. The power law is there only because of low pass filtering.

  1. The power law is a side effect of control of something other than the relationship between speed and curvature
  2. The power law comes from controlling trajectory
  3. The power law is a statistical artifact

Those three statements are mutually exclusive. You can choose one of them and lay out the arguments. You chose all three in your papers on the power law, and two of them in your post here. I mean, you also switched your position here on CSGnet, but then you got huffy and chanhed your position back to something else. Who the hell knows.

Your argument is incoherent (no ad-hominem) because the statements can’t be true at the same time.

Well, now I’m going to get offenden, Ricky boy. Read the paper before commenting.

But none of them describe what I think is actually going on. What I think is:

  1. The power law is an irrelevant side effect of control.

Could you point out where I did this?

Well that is just #1. You added “irrelevant”, like writing word hello when controlling curosor position. Sure.
And “side effect of control” is meaningless unless you add what is being controlled. The power law is not a side effect of control of temperature or puressure or whatever. We need a specific controlled variable, and also a whole system that draws shapes and produces the power law in the same situations where the participants do, and does not produce it when participants don’t.

Yeah, like two posts up, in your reply to Eetu. And in your paper on the power law. How is your memory today? Do you want me to search for the claim that trajectory is controlled or you remember it?

Yes, it is a two level position control model. I missed that because the lower level system looked like an output function to me; it didn’t seem to be controlling a perception. But now I see that it is controlling a perception of the Euclidean position of the cursor (c.x, c.y). But this makes that lower level system both unnecessary (redundant) and unlikely.

It is unnecessary because what you call the visual target reference can be seen as the error signal, e(t), that drives the output (x = Ke.t-tau - Bx) that moves the cursor (c.x, c.y). It is unlikely because the model has the upper level systems perceiving the difference between target and cursor positions in polar coordinates (d.phi, d.s) while the lower level system perceives cursor position in Euclidean coordinates.

But the model is fine the way it is. I am just not convinced that it shows that the power law is a side effect of controlling the particular variable you used as the controlled variable in the model: the difference between target and cursor perceived in terms of their polar coordinates.

Your target in these tests is moving elliptically and we know that a perfectly executed elliptical movement will result in a perfect -1/3 exponent of a speed-curvature power function. How about trying the model on some different target movements: a circular target and some some random scribbles. And try using Euclidean as well as polar perceptions of x and y target position. I predict that the shape of the movement will have a much bigger effect than the type of perception controlled on the exponent of the power coefficient that is observed.