# The speed�?curvature power law of moveme nts: a reappraisal

**URL:** <http://discourse.iapct.org/t/the-speed-curvature-power-law-of-moveme-nts-a-reappraisal/11529>\
**Category:** CSG2017\
**Created:** [November 1, 2017, 9:47am UTC](http://discourse.iapct.org/t/the-speed-curvature-power-law-of-moveme-nts-a-reappraisal/11529 "2017-11-01T09:47:13Z")\
**Posts on this page:** 4\
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**Author:** ![EetuP](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/e/df705f/32.png) [@EetuP](http://discourse.iapct.org/u/EetuP)\
**Post date:** [November 1, 2017, 9:47am UTC](http://discourse.iapct.org/t/the-speed-curvature-power-law-of-moveme-nts-a-reappraisal/11529/1 "2017-11-01T09:47:13Z")

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[Eetu Pikkarainen 2017-11-01]

(at the risk to be ridiculed…)

I should be but as a humanist, I try not to be ashamed because of missing mathematical competence. I practice my understanding  
by composing a non-mathematical description of what is going on in this case of PCT and Power Law.

When a living being is producing a movement trajectory by moving itself (e.g. crawling) or its organs (e.g. hand in a  
drawing task) there seems to be a certain dependence between the curvature of the trajectory (proceeding straight or turning to some direction) and the speed/velocity (understood here as a moved distance along the trajectory during a time unit). The normal  
case is that the speed is stable on the straight line but it slows down in a curve.

Now the first question is why that slowdown in the curve takes place. The second question is why the dependence between  
curve and speed has those certain values, which have been observed in different situations. That latter question will be postponed.

PCT view is generally that the behavior of an organism is that it controls its certain perception(s) by affecting by its  
output some environmental correlates of those perceptions. So when an organism is moving it can be controlling for its distance from some object to be long (if the object is perceived as dangerous) or short (if the object is perceived as desirable). Any regularities  
of behavior are often assumed to be side effects of control. (Not side effects of observation.)

At the moment I have no idea what the larvae are controlling in those experiments but introspectively I can imagine some  
possible perceptions I would control in a drawing task. First, I control seeing me a helpful aid in science making and being an obedient test subject and following to the rules. Then I would control to see the pen moving in a calm and nice way on the paper.  
At the same time, I control that the pen follows the guiding line as strictly as possible. I think that my output function is such that it is easier to follow a line with certain kind of curvature than with another kind. We humans get much practice to draw  
just certain kind of lines when we learn to draw and write. But there can also be some more general reasons why we use just those curves in our letters and typical drawings that we use. Anyway, it feels much more natural to draw a circle or looped circles  
than rounded polygons.

Therefore, the increased difficulty in controlling of the latter perception (following the line) would draw some effort  
away from the control of the first perception (keeping the steady or moderate speed). Or rather the slowing down the speed will make it easier to control the following the line. (A straight line is easier to draw with higher speed than very slowly.) So the  
slowdown could be a side effect of control. Perhaps something similar is going on with larvae? Is it easier for them to go straight forward? Is it an additional challenge to turn – to decide to turn and not to continnue forward? Would the slowdown make it easier  
to control the perception of the new direction?

Is that at all where the PCT view of Power Law could begin?

Eetu

> **···**
>
> **From:** Alex Gomez-Marin [[mailto:agomezmarin@gmail.com](mailto:agomezmarin@gmail.com)]  
> **Sent:** 31. lokakuuta 2017 18:01  
> **To:** Alex Gomez-Marin [agomezmarin@gmail.com](mailto:agomezmarin@gmail.com); csgnet@lists.illinois.edu  
> **Subject:** Re: The speed�curvature power law of movements: a reappraisal
> 
> regardless of your helicopter data and RCT mantras, it would be good if someone from CSGnet took seriously the challenge to PCT that the speed-curvature power law entails.
> 
> any figure panel of our paper proves rick’s mathematical claims wrong: the PL is not a must and when it takes place it is not trivial and can have different exponents.
> 
> now, how can “control of perception” explain that phenomenon? claiming it is an illusion because it does not fit in the dogma is like creationists insisting that dino fossils are bogus.
> 
> so, as adam and myself take this job seriously, and given how many optimal control and nonPCT theories explain the data, I think Bill would really find his edifice crumbling, or at least unable. so, take your best shot at it and really  
> challenge your “revolutionary paradigm changing” theory of behavior.
> 
> On Sun, 29 Oct 2017 at 02:48, Richard Marken [rsmarken@gmail.com](mailto:rsmarken@gmail.com) wrote:
> 
> > [From Rick Marken (2017.10.28.1745)
> 
> > On Sat, Oct 28, 2017 at 3:34 PM, Alex Gomez-Marin [agomezmarin@gmail.com](mailto:agomezmarin@gmail.com) wrote:
> 
> > > attached
> 
> > RM: Finally! Thank you, Alex. I hope the journal gives us an opportunity to respond. But for now I have only one word for you: helicopter movements. Oh, that’s two words But you know how bad I am at math;-)
> 
> > Best
> 
> > Rick
> 
> > –
> 
> > Richard S. Marken
> 
> > "Perfection is achieved not when you have nothing more to add, but when you
> > 
> > have nothing left to take away.�
> > 
> > ```
> > --Antoine de Saint-Exupery
> > 
> > ```
> 
> –
> 
> Alex Gomez-Marin
> 
> [behavior-of-organisms.org](https://urldefense.proofpoint.com/v2/url?u=http-3A__behavior-2Dof-2Dorganisms.org&d=DwMGaQ&c=8hUWFZcy2Z-Za5rBPlktOQ&r=-dJBNItYEMOLt6aj_KjGi2LMO_Q8QB-ZzxIZIF8DGyQ&m=Cdh5wPdfCe0DVDcixav8TBGSeCoj1cL5HIrxyi3qjN0&s=UHYQm6KmXhnAC0fIHvrLxhW1FEPg_z2ISQKuQM48smA&e=)

---

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**Author:** ![EetuP](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/e/df705f/32.png) [@EetuP](http://discourse.iapct.org/u/EetuP)\
**Post date:** [November 2, 2017, 7:43am UTC](http://discourse.iapct.org/t/the-speed-curvature-power-law-of-moveme-nts-a-reappraisal/11529/2 "2017-11-02T07:43:19Z")

</div>

[Eetu Pikkarainen 2017-11-02 9:24]

Good point, Boris!

I should have been stricter with my words. I have used to differentiate between external and internal action. The latter (for example  
thinking or dreaming) is not visible for an external observer and it does not affect any environmental variables outside the organism. Instead, the external action affects things in the external environment and it is especially this case that I have used  
to use the term “behavior� (i.e. observable action). So I think that if there is a case of external action / behavior (as there is if we are studying the movements of the subject) then the subject is controlling its certain perception(s) by affecting by its  
output some environmental correlates of those perceptions.

> **···**
>
> Eetu
> 
> Please, regard all my statements as questions,
> 
> no matter how they are formulated.
> 
> **From:** Boris Hartman [[mailto:boris.hartman@masicom.net](mailto:boris.hartman@masicom.net)]  
> **Sent:** Wednesday, November 1, 2017 8:38 PM  
> **To:** csgnet@lists.illinois.edu  
> **Subject:** RE: The speed�curvature power law of movements: a reappraisal
> 
> Dear Eetu
> 
> **From:** Eetu Pikkarainen [[mailto:eetu.pikkarainen@oulu.fi](mailto:eetu.pikkarainen@oulu.fi)]  
> **Sent:** Wednesday, November 01, 2017 10:47 AM  
> **To:** csgnet@lists.illinois.edu  
> **Subject:** RE: The speed�curvature power law of movements: a reappraisal
> 
> [Eetu Pikkarainen 2017-11-01]
> 
> (at the risk to be ridiculed…)
> 
> I should be but as a humanist, I try not to be ashamed because of missing mathematical competence. I practice my understanding  
> by composing a non-mathematical description of what is going on in this case of PCT and Power Law.
> 
> When a living being is producing a movement trajectory by moving itself (e.g. crawling) or its organs (e.g. hand in a  
> drawing task) there seems to be a certain dependence between the curvature of the trajectory (proceeding straight or turning to some direction) and the speed/velocity (understood here as a moved distance along the trajectory during a time unit). The normal  
> case is that the speed is stable on the straight line but it slows down in a curve.
> 
> Now the first question is why that slowdown in the curve takes place. The second question is why the dependence between  
> curve and speed has those certain values, which have been observed in different situations. That latter question will be postponed.
> 
> EP : PCT view is generally that the behavior of an organism is that it controls its certain perception(s)
> 
> HB : Sorry Eetu to not agree with you. It’s not that behavior “controls perception� because behavior can’t be controlled. It just affects  
> input generally. If you are sleeping how you are “controlling� perception with behavior ?
> 
> EP : ….by affectiing by its output some environmental correlates of those perceptions.
> 
> HB : Output can affect some correlates of perception in environment, but it has nothing to do with control in environment. What correlate  
> you are affecting in environment if you are observing. And what correlate you are affecting in environment if you are sitting and thinking ? You just affect input. That’s what “feedback� is.
> 
> Bill P (LCS III):
> 
> FEED-BACK FUNCTION : The box represents the set of physical laws, properties, arrangements, linkages, by which the action of this system feeds-back to affect its own input,  
> the controlled variable. That’s what feed-back means : it’s an effect of a system’s output on it’s own input.
> 
> HB : You put it very efficiently what is important about PCT in other post :
> 
> Eetu earlier : I think that when I am telling about PCT it is generally best to start from the idea of self-preservation, that any organism - from  
> the most simple to most complex - must control its intrinsic variables to stay alive. From that I can continue to interaction with environment, the must of affecting and stabilizing things in the environment. Only after that it is possible to start to compare  
> organism to thermostat and draw technical diagrams. Thus it goes from soft and warm to hard and technical.
> 
> HB : It’s control in organism that is on first place and behavior just support to control in organism. That’s what Bills’ definition  
> is talking about.
> 
> Bill P (B:CP):
> 
> CONTROL : Achievement and maintenance of a preselected state in the controlling system, through actions on the environment that also cancel the effects of disturbances.
> 
> HB : So you think right that what is important is preselected state in organism which includes »intrinsic variables«. Internal control  
> is of most importance. What happens in outer environment is important but supporting to internal control. Behavior is »fired« from internal environment not external in accordance how internal environment is controlled. This is the function of internal and  
> external effectors.
> 
> Best,
> 
> Boris
> 
> EP : So when an organism is moving it can be controlling for its distance from some object to be long (if the object is  
> perceived as dangerous) or short (if the object is perceived as desirable). Any regularities of behavior are often assumed to be side effects of control. (Not side effects of observation.)
> 
> EP : At the moment I have no idea what the larvae are controlling in those experiments but introspectively I can imagine  
> some possible perceptions I would control in a drawing task. First, I control seeing me a helpful aid in science making and being an obedient test subject and following to the rules. Then I would control to see the pen moving in a calm and nice way on the  
> paper. At the same time, I control that the pen follows the guiding line as strictly as possible. I think that my output function is such that it is easier to follow a line with certain kind of curvature than with another kind. We humans get much practice  
> to draw just certain kind of lines when we learn to draw and write. But there can also be some more general reasons why we use just those curves in our letters and typical drawings that we use. Anyway, it feels much more natural to draw a circle or looped  
> circles than rounded polygons.
> 
> Therefore, the increased difficulty in controlling of the latter perception (following the line) would draw some effort  
> away from the control of the first perception (keeping the steady or moderate speed). Or rather the slowing down the speed will make it easier to control the following the line. (A straight line is easier to draw with higher speed than very slowly.) So the  
> slowdown could be a side effect of control. Perhaps something similar is going on with larvae? Is it easier for them to go straight forward? Is it an additional challenge to turn – to decidde to turn and not to continue forward? Would the slowdown make it easier  
> to control the perception of the new direction?
> 
> Is that at all where the PCT view of Power Law could begin?
> 
> Eetu
> 
> **From:** Alex Gomez-Marin [[mailto:agomezmarin@gmail.com](mailto:agomezmarin@gmail.com)]  
> **Sent:** 31. lokakuuta 2017 18:01  
> **To:** Alex Gomez-Marin [agomezmarin@gmail.com](mailto:agomezmarin@gmail.com);  
> csgnet@lists.illinois.edu  
> **Subject:** Re: The speed�curvature power law of movements: a reappraisal
> 
> regardless of your helicopter data and RCT mantras, it would be good if someone from CSGnet took seriously the challenge to PCT that the speed-curvature power law entails.
> 
> any figure panel of our paper proves rick’s mathematical claims wrong: the PL is not a must and when it takes place it is not trivial and can have different exponents.
> 
> now, how can “control of perception” explain that phenomenon? claiming it is an illusion because it does not fit in the dogma is like creationists insisting that dino fossils are bogus.
> 
> so, as adam and myself take this job seriously, and given how many optimal control and nonPCT theories explain the data, I think Bill would really find his edifice crumbling, or at least unable. so, take your best shot at it and really  
> challenge your “revolutionary paradigm changing” theory of behavior.
> 
> On Sun, 29 Oct 2017 at 02:48, Richard Marken [rsmarken@gmail.com](mailto:rsmarken@gmail.com) wrote:
> 
> > [From Rick Marken (2017.10.28.1745)
> 
> > On Sat, Oct 28, 2017 at 3:34 PM, Alex Gomez-Marin [agomezmarin@gmail.com](mailto:agomezmarin@gmail.com) wrote:
> 
> > > attached
> 
> > RM: Finally! Thank you, Alex. I hope the journal gives us an opportunity to respond. But for now I have only one word for you: helicopter movements. Oh, that’s two words But you know how bad I am at math;-)
> 
> > Best
> 
> > Rick
> 
> > –
> 
> > Richard S. Marken
> 
> > "Perfection is achieved not when you have nothing more to add, but when you
> > 
> > have nothing left to take away.�
> > 
> > ```
> > --Antoine de Saint-Exupery
> > 
> > ```
> 
> –
> 
> Alex Gomez-Marin  
> [behavior-of-organisms.org](https://urldefense.proofpoint.com/v2/url?u=http-3A__behavior-2Dof-2Dorganisms.org&d=DwMGaQ&c=8hUWFZcy2Z-Za5rBPlktOQ&r=-dJBNItYEMOLt6aj_KjGi2LMO_Q8QB-ZzxIZIF8DGyQ&m=Cdh5wPdfCe0DVDcixav8TBGSeCoj1cL5HIrxyi3qjN0&s=UHYQm6KmXhnAC0fIHvrLxhW1FEPg_z2ISQKuQM48smA&e=)

---

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**Author:** ![EetuP](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/e/df705f/32.png) [@EetuP](http://discourse.iapct.org/u/EetuP)\
**Post date:** [November 7, 2017, 8:03am UTC](http://discourse.iapct.org/t/the-speed-curvature-power-law-of-moveme-nts-a-reappraisal/11529/3 "2017-11-07T08:03:18Z")

</div>

[Eetu Pikkarainen 2017-11-07 9:47]

> **···**
>
> [From Adam Matic 2017.11.6]
> 
> AM: Yep, from what I understand, the optimal control approach is more or less exactly that - trying to find what quantity is minimized in the production of curved trajectories.  
> It’s been very fruitful in the sense that observed trajectories fit very nicely with predicted ones in many cases. Huh & Sejnowski (2015) [https://www.ncbi.nlm.nih.gov/pubmed/26150514](https://www.ncbi.nlm.nih.gov/pubmed/26150514) show that  
> human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related variables,  
> but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.
> 
> This fits with my earlier guess that the test subject controls for (her perception of) the steady (or smooth) speed (even if no instruction about speed is given) and (her perception of) staying on the line (which is also  
> instructed). I guess that these could be tested with an experiment where the subject is drawing onto the tablet where the target line is not already visible but drawn by the program. So the task would be similar like a tracking task except (at least) the cursor  
> would also draw its own trajectory. The subject is not instructed to “catch� the cursor but just to follow it as near or well as possible so that her pen or finger would draw similar curved line as the cursor draws. The disturbances would be produced just  
> by altering the speed and curves or the cursor.
> 
> Is here any sense? Done already?
> 
> Eetu
> 
> Please, regard all my statements as questions,
> 
> no matter how they are formulated.

---

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**Author:** ![EetuP](http://discourse.iapct.org/letter_avatar_proxy/v4/letter/e/df705f/32.png) [@EetuP](http://discourse.iapct.org/u/EetuP)\
**Post date:** [November 7, 2017, 9:17am UTC](http://discourse.iapct.org/t/the-speed-curvature-power-law-of-moveme-nts-a-reappraisal/11529/4 "2017-11-07T09:17:39Z")

</div>

Fine, never suspected that 🙂

Eetu

> **···**
>
> Adam is working on it, and has done part of it already. But it is not that easy, nor one can solve the problem by rephrasing Powers 1973: one really has to get  
> wet with the details.
> 
> On Tue, Nov 7, 2017 at 9:03 AM, Eetu Pikkarainen [eetu.pikkarainen@oulu.fi](mailto:eetu.pikkarainen@oulu.fi) wrote:
> 
> > [Eetu Pikkarainen 2017-11-07 9:47]
> 
> > 
> 
> > [From Adam Matic 2017.11.6]
> 
> > 
> 
> > AM: Yep, from what I understand, the optimal control approach is more or less exactly that - trying to find what quantity is minimized in the production of curved trajectories.  
> > It’s been very fruitful in the sense that observed trajectories fit very nicely with predicted ones in many cases. Huh & Sejnowski (2015) [https://www.ncbi.nlm.nih.gov/pubmed/26150514](https://www.ncbi.nlm.nih.gov/pubmed/26150514)  
> > show that human trajectories when tracing specific curves can be very nicely predicted by a system that minimizes jerk (rate of change of acceleration, equivalent of maximal smoothness). There were attempts with minimization of torque and similar effort-related  
> > variables, but the jerk minimization seems most successful. How these trajectories emerge in the end is still an open question.
> 
> > This fits with my earlier guess that the test subject controls for (her perception of) the steady (or smooth) speed (even if no instruction about speed is given)  
> > and (her perception of) staying on the line (which is also instructed). I guess that these could be tested with an experiment where the subject is drawing onto the tablet where the target line is not already visible but drawn by the program. So the task would  
> > be similar like a tracking task except (at least) the cursor would also draw its own trajectory. The subject is not instructed to “catch� the cursor but just to follow it as near or well as possible so that her pen or finger would draw similar curved line  
> > as the cursor draws. The disturbances would be produced just by altering the speed and curves or the cursor.
> 
> > Is here any sense? Done already?
> 
> > 
> 
> > Eetu
> 
> > 
> 
> > Please, regard all my statements as questions,
> 
> > no matter how they are formulated.
> 
> > 
> 
> > 
> 
> >