the speed-curvature power law in humans and flies

[Alex_Gomez-Marin]

Rupert, velocity is of course the velocity at which the organism moves (there is not a line “to move along to”; the organism simply locomotes and “traces” a line “behind” of a certain local curvature and at a certain instantaneous speed). Angle alpha is where the path is locally pointing to in space, so as alpha changes in time, it means that the animal is walking that path and changing direction.

Warren et al: Many arguments for it:

It is essential to realize that the constraint between speed and
curvature that we (and others) have empirically found complying with the formal
dependency of a power function (or power law) is actually not deducible from
mathematics or physics alone. Therefore, it must an emergent biological
phenomenon (whose origin still remains debated, and whose explanation might be
considered “trivialâ€? but only after the fact, namely, once it has been
understood).

Â

First, intuitively, one can be aware that the shape of a path in
principle only restricts the direction of movement but not the speed at with
that path was traced. In other words, path curvature imposes “towards whereâ€? to
move (let’s say, a pen while scribbling or the path of a fly maggot) but is
agnostic so as to “how fastâ€? to move.

Â

Second, mathematically, it is true that one can formally express
the curvature of a path as a function of velocity (or angular velocity) but such
formal expression precisely reflects how path curvature (whose natural argument
is arc length) can be parametrized as a function of time only when one adds a
specification of the speed profile. In other words, one could write the
expression of curvature in a coordinate system that only defines the shape of a
movement path and that is totally independent on the speed temporal profile
(see Huh and Sejnowski PNAS 2015 for an in-depth mathematical treatment).

Â

Third, mathematically, there is another way to see the
non-trivial relation between kinematics and geometry. In the expression of
curvature we see its explicit dependence on speed but also a term in the
denominator that contains a mixture of first and second derivatives, thus acceleration,
which cannot be specified by curvature, but are a property of the temporal
profile of speed, which can vary in different ways.

Â

Fourth, by simulations, one can actually synthetically create an
arbitrary path with a given local curvature and then change the time rate at
which that path is virtually drawn, and actually see how the power law does not
hold there.

Â

Fifth, more generally, curvature is by definition a geometric
quantity (only in space, that’s why we refer to it s “localâ€? curvature) whereas
speed is obviously a kinematic quantity (in time, that is why we refer to it as
“instantaneousâ€? speed) and so by basic notions of geometry and mechanics we can
be sure that geometry cannot dictate kinematics: purely spatial forms have no
way of acquiring temporal determination.

Â

As we briefly mention in the paper: L57: “The law is not
dictated by physical constraints: even when the path is imposed, as in
hand-drawing, movement speed could in principle vary in infinite ways.
Therefore, the law must result from physiological constraints�

Â

Sixth, physically, the laws of motion by Newton
describe/prescribe the relation between force and acceleration but do not
describe/prescribe that for any given force, the acceleration shall be so that
speed is so that curvature will be so.

 Â

Seventh, biologically and empirically, the law is not “a mustâ€?
for each and every fly larva we studied. As we actually report in the paper: “A
few individuals did not comply with the law (…) corroborating that it is not
an obligatory outcome of our analyses� (L142).

Â

Eight, biologically and empirically, we have explored the
speed-curvature relation from data from other organisms (walking adult flies
and running mice, as well as swimming fish) and we have found huge deviations
from a power-law (data not shown in the paper).

Â

If none of the many arguments above is convincing due to its abstract
nature, the empirical demonstration that some flies do not show the power law
and all fish and mice test do not show the power law either should convince
anyone that: (i) the power law is not a trivial given result of the
mathematical relation between speed and curvature, and that therefore (ii) to
find it in human scribbling on paper and in the maggot, but not in other
organisms so far, is a relevant and interesting empirical result that opens new
perspectives to investigate the origin of such relation.

···

On Wed, Aug 31, 2016 at 1:53 PM, rupert@perceptualrobots.com wrote:

So the velocity is that at which the organism is moving along the line? And it is found that organisms slow down on curves, is that right?

What is the angle alpha between?

On 26 August 2016 21:33:32 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alphaÂ

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.Â

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

[rupert]

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,

Donations welcome!

Regards,

Rupert

···

https://www.justgiving.com/crowdfunding/cycle-for-humanity

[Alex_Gomez-Marin]

C is curvature indeed.

R = 1/C

A is rate of change of alpha

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

···

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

[wmansell]

Hi Alex,

AGM: …one that physics alone does not impose and one that is found in biology.

WM: can you point me to the evidence for this? For example has anyone plotted the motion of ball from an aerial perspective l rolling on a landscape of randomly varying height, or on a platform subject to randomly placed vertical forces? Does this not obey the power law? I wonder what law it does obey within this pure physics example?

···

On 26 Aug 2016, at 21:33, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alpha

A = d alpha / dt
also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

[rupert]

So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship between the slower speed and the speed along the straight path for the experimental occasion (given that the organism could began at a different speed).

Am I understanding correctly?

And you are saying that relationship is a property of the organism rather than the environment, is that right?

Rupert

···

On Wed, Aug 31, 2016 at 1:53 PM, rupert@perceptualrobots.com wrote:

So the velocity is that at which the organism is moving along the line? And it is found that organisms slow down on curves, is that right?

What is the angle alpha between?

On 26 August 2016 21:33:32 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alphaÂ

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.Â

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

[Alex_Gomez-Marin]

Thanks for your questions Rupert, which allow to bring some clarity by answering to the point without “playing start on the PCT radio speech”:

 So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship

Yes, the EMPIRICAL finding is that flies slow down the sharper the turn in a way that there is a power law relation between these speed and curvature (which, I hope it is clear by now, is not a trivial mathematical nor physical constrain, but something observed in some living organisms).

 And you are saying that relationship is a property of the organism rather than the environment, is that right?

No, no, no. I am not saying anything of that (yet). Otherwise I would be immediately excomulgated from the purist PCTers. I am just saying (and that is what I have been saying since the very first email!) that I find this EMPIRICAL relation, which is not a trivial one, but something evolution may have pushed as a biological constraint, AND I wonder then what is the PCT interpretation of it, or better, what is the perceived quantity. For some reason curvature and speed are mutually constrained. I want to find out the origin of that.

Plus, the question does not need to be EITHER OR, namely, the power law relation (as a FACT) does not have to be either 100% illusion (what PCTers seem to call environment property) nor 100% true (what PCTers seem to call organism property). It can be, and most likely is, a MIXTURE of both! Why, because for sure the animal is trying to control for someting AND at the same time the properties of the substrate (friction, viscossity, etc) have an influence. So, again, not being a FUNDAMENTALIST or a CONTROL OF PERCEPTION CHAUVINIST, I am interested, as a biologist, to understand both the organism, the world in which it lives and the interaction between both.

···

On Fri, Sep 2, 2016 at 8:49 PM, rupert@perceptualrobots.com wrote:

So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship between the slower speed and the speed along the straight path for the experimental occasion (given that the organism could began at a different speed).

Am I understanding correctly?

And you are saying that relationship is a property of the organism rather than the environment, is that right?

Rupert

On 31 August 2016 13:55:11 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Rupert, velocity is of course the velocity at which the organism moves (there is not a line “to move along to”; the organism simply locomotes and “traces” a line “behind” of a certain local curvature and at a certain instantaneous speed). Angle alpha is where the path is locally pointing to in space, so as alpha changes in time, it means that the animal is walking that path and changing direction.

Warren et al: Many arguments for it:

It is essential to realize that the constraint between speed and
curvature that we (and others) have empirically found complying with the formal
dependency of a power function (or power law) is actually not deducible from
mathematics or physics alone. Therefore, it must an emergent biological
phenomenon (whose origin still remains debated, and whose explanation might be
considered “trivialâ€? but only after the fact, namely, once it has been
understood).

Â

First, intuitively, one can be aware that the shape of a path in
principle only restricts the direction of movement but not the speed at with
that path was traced. In other words, path curvature imposes “towards whereâ€? to
move (let’s say, a pen while scribbling or the path of a fly maggot) but is
agnostic so as to “how fastâ€? to move.

Â

Second, mathematically, it is true that one can formally express
the curvature of a path as a function of velocity (or angular velocity) but such
formal expression precisely reflects how path curvature (whose natural argument
is arc length) can be parametrized as a function of time only when one adds a
specification of the speed profile. In other words, one could write the
expression of curvature in a coordinate system that only defines the shape of a
movement path and that is totally independent on the speed temporal profile
(see Huh and Sejnowski PNAS 2015 for an in-depth mathematical treatment).

Â

Third, mathematically, there is another way to see the
non-trivial relation between kinematics and geometry. In the expression of
curvature we see its explicit dependence on speed but also a term in the
denominator that contains a mixture of first and second derivatives, thus acceleration,
which cannot be specified by curvature, but are a property of the temporal
profile of speed, which can vary in different ways.

Â

Fourth, by simulations, one can actually synthetically create an
arbitrary path with a given local curvature and then change the time rate at
which that path is virtually drawn, and actually see how the power law does not
hold there.

Â

Fifth, more generally, curvature is by definition a geometric
quantity (only in space, that’s why we refer to it s “localâ€? curvature) whereas
speed is obviously a kinematic quantity (in time, that is why we refer to it as
“instantaneousâ€? speed) and so by basic notions of geometry and mechanics we can
be sure that geometry cannot dictate kinematics: purely spatial forms have no
way of acquiring temporal determination.

Â

As we briefly mention in the paper: L57: “The law is not
dictated by physical constraints: even when the path is imposed, as in
hand-drawing, movement speed could in principle vary in infinite ways.
Therefore, the law must result from physiological constraints�

Â

Sixth, physically, the laws of motion by Newton
describe/prescribe the relation between force and acceleration but do not
describe/prescribe that for any given force, the acceleration shall be so that
speed is so that curvature will be so.

 Â

Seventh, biologically and empirically, the law is not “a mustâ€?
for each and every fly larva we studied. As we actually report in the paper: “A
few individuals did not comply with the law (…) corroborating that it is not
an obligatory outcome of our analyses� (L142).

Â

Eight, biologically and empirically, we have explored the
speed-curvature relation from data from other organisms (walking adult flies
and running mice, as well as swimming fish) and we have found huge deviations
from a power-law (data not shown in the paper).

Â

If none of the many arguments above is convincing due to its abstract
nature, the empirical demonstration that some flies do not show the power law
and all fish and mice test do not show the power law either should convince
anyone that: (i) the power law is not a trivial given result of the
mathematical relation between speed and curvature, and that therefore (ii) to
find it in human scribbling on paper and in the maggot, but not in other
organisms so far, is a relevant and interesting empirical result that opens new
perspectives to investigate the origin of such relation.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Wed, Aug 31, 2016 at 1:53 PM, rupert@perceptualrobots.com wrote:

So the velocity is that at which the organism is moving along the line? And it is found that organisms slow down on curves, is that right?

What is the angle alpha between?

On 26 August 2016 21:33:32 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alphaÂ

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.Â

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

[rupert]

Thanks.

A perhaps analogous situation comes to mind which might help me understand better where you’re coming from. Suppose you are driving a car and are doing so by controlling the angle of the accelerator pedal at a constant reference value; i. e. keeping it, and foot, in same position (let’s say mid-range). And you are driving over (straight) undulating terrain. What would an external person observe about the speed of the car? Would it be possible to determine a relationship between the changes in speed and the terrain? Would the changes in speed, or the relationship, be a property of the organism or the environment, or both?

···

On Fri, Sep 2, 2016 at 8:49 PM, rupert@perceptualrobots.com wrote:

So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship between the slower speed and the speed along the straight path for the experimental occasion (given that the organism could began at a different speed).

Am I understanding correctly?

And you are saying that relationship is a property of the organism rather than the environment, is that right?

Rupert

On 31 August 2016 13:55:11 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Rupert, velocity is of course the velocity at which the organism moves (there is not a line “to move along to”; the organism simply locomotes and “traces” a line “behind” of a certain local curvature and at a certain instantaneous speed). Angle alpha is where the path is locally pointing to in space, so as alpha changes in time, it means that the animal is walking that path and changing direction.

Warren et al: Many arguments for it:

It is essential to realize that the constraint between speed and
curvature that we (and others) have empirically found complying with the formal
dependency of a power function (or power law) is actually not deducible from
mathematics or physics alone. Therefore, it must an emergent biological
phenomenon (whose origin still remains debated, and whose explanation might be
considered “trivialâ€? but only after the fact, namely, once it has been
understood).

Â

First, intuitively, one can be aware that the shape of a path in
principle only restricts the direction of movement but not the speed at with
that path was traced. In other words, path curvature imposes “towards whereâ€? to
move (let’s say, a pen while scribbling or the path of a fly maggot) but is
agnostic so as to “how fastâ€? to move.

Â

Second, mathematically, it is true that one can formally express
the curvature of a path as a function of velocity (or angular velocity) but such
formal expression precisely reflects how path curvature (whose natural argument
is arc length) can be parametrized as a function of time only when one adds a
specification of the speed profile. In other words, one could write the
expression of curvature in a coordinate system that only defines the shape of a
movement path and that is totally independent on the speed temporal profile
(see Huh and Sejnowski PNAS 2015 for an in-depth mathematical treatment).

Â

Third, mathematically, there is another way to see the
non-trivial relation between kinematics and geometry. In the expression of
curvature we see its explicit dependence on speed but also a term in the
denominator that contains a mixture of first and second derivatives, thus acceleration,
which cannot be specified by curvature, but are a property of the temporal
profile of speed, which can vary in different ways.

Â

Fourth, by simulations, one can actually synthetically create an
arbitrary path with a given local curvature and then change the time rate at
which that path is virtually drawn, and actually see how the power law does not
hold there.

Â

Fifth, more generally, curvature is by definition a geometric
quantity (only in space, that’s why we refer to it s “localâ€? curvature) whereas
speed is obviously a kinematic quantity (in time, that is why we refer to it as
“instantaneousâ€? speed) and so by basic notions of geometry and mechanics we can
be sure that geometry cannot dictate kinematics: purely spatial forms have no
way of acquiring temporal determination.

Â

As we briefly mention in the paper: L57: “The law is not
dictated by physical constraints: even when the path is imposed, as in
hand-drawing, movement speed could in principle vary in infinite ways.
Therefore, the law must result from physiological constraints�

Â

Sixth, physically, the laws of motion by Newton
describe/prescribe the relation between force and acceleration but do not
describe/prescribe that for any given force, the acceleration shall be so that
speed is so that curvature will be so.

 Â

Seventh, biologically and empirically, the law is not “a mustâ€?
for each and every fly larva we studied. As we actually report in the paper: “A
few individuals did not comply with the law (…) corroborating that it is not
an obligatory outcome of our analyses� (L142).

Â

Eight, biologically and empirically, we have explored the
speed-curvature relation from data from other organisms (walking adult flies
and running mice, as well as swimming fish) and we have found huge deviations
from a power-law (data not shown in the paper).

Â

If none of the many arguments above is convincing due to its abstract
nature, the empirical demonstration that some flies do not show the power law
and all fish and mice test do not show the power law either should convince
anyone that: (i) the power law is not a trivial given result of the
mathematical relation between speed and curvature, and that therefore (ii) to
find it in human scribbling on paper and in the maggot, but not in other
organisms so far, is a relevant and interesting empirical result that opens new
perspectives to investigate the origin of such relation.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Wed, Aug 31, 2016 at 1:53 PM, rupert@perceptualrobots.com wrote:

So the velocity is that at which the organism is moving along the line? And it is found that organisms slow down on curves, is that right?

What is the angle alpha between?

On 26 August 2016 21:33:32 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alphaÂ

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.Â

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

[Alex_Gomez-Marin]

what if my way of keeping the pedal was shape beyond proximate circular causation so that the undulation of my eco-system had cristalized in evolutionary or developmental time into my way of controlling my perceptions? pct needs a wider Tinbergen-like perspective. the car has been driven for millions of years…

···

On Fri, Sep 2, 2016 at 8:49 PM, rupert@perceptualrobots.com wrote:

So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship between the slower speed and the speed along the straight path for the experimental occasion (given that the organism could began at a different speed).

Am I understanding correctly?

And you are saying that relationship is a property of the organism rather than the environment, is that right?

Rupert

On 31 August 2016 13:55:11 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Rupert, velocity is of course the velocity at which the organism moves (there is not a line “to move along to”; the organism simply locomotes and “traces” a line “behind” of a certain local curvature and at a certain instantaneous speed). Angle alpha is where the path is locally pointing to in space, so as alpha changes in time, it means that the animal is walking that path and changing direction.

Warren et al: Many arguments for it:

It is essential to realize that the constraint between speed and
curvature that we (and others) have empirically found complying with the formal
dependency of a power function (or power law) is actually not deducible from
mathematics or physics alone. Therefore, it must an emergent biological
phenomenon (whose origin still remains debated, and whose explanation might be
considered “trivial� but only after the fact, namely, once it has been
understood).

Â

First, intuitively, one can be aware that the shape of a path in
principle only restricts the direction of movement but not the speed at with
that path was traced. In other words, path curvature imposes “towards where� to
move (let’s say, a pen while scribbling or the path of a fly maggot) but is
agnostic so as to “how fast� to move.

Â

Second, mathematically, it is true that one can formally express
the curvature of a path as a function of velocity (or angular velocity) but such
formal expression precisely reflects how path curvature (whose natural argument
is arc length) can be parametrized as a function of time only when one adds a
specification of the speed profile. In other words, one could write the
expression of curvature in a coordinate system that only defines the shape of a
movement path and that is totally independent on the speed temporal profile
(see Huh and Sejnowski PNAS 2015 for an in-depth mathematical treatment).

Â

Third, mathematically, there is another way to see the
non-trivial relation between kinematics and geometry. In the expression of
curvature we see its explicit dependence on speed but also a term in the
denominator that contains a mixture of first and second derivatives, thus acceleration,
which cannot be specified by curvature, but are a property of the temporal
profile of speed, which can vary in different ways.

Â

Fourth, by simulations, one can actually synthetically create an
arbitrary path with a given local curvature and then change the time rate at
which that path is virtually drawn, and actually see how the power law does not
hold there.

Â

Fifth, more generally, curvature is by definition a geometric
quantity (only in space, that’s why we refer to it s “local� curvature) whereas
speed is obviously a kinematic quantity (in time, that is why we refer to it as
“instantaneous� speed) and so by basic notions of geometry and mechanics we can
be sure that geometry cannot dictate kinematics: purely spatial forms have no
way of acquiring temporal determination.

Â

As we briefly mention in the paper: L57: “The law is not
dictated by physical constraints: even when the path is imposed, as in
hand-drawing, movement speed could in principle vary in infinite ways.
Therefore, the law must result from physiological constraints�

Â

Sixth, physically, the laws of motion by Newton
describe/prescribe the relation between force and acceleration but do not
describe/prescribe that for any given force, the acceleration shall be so that
speed is so that curvature will be so.

 Â

Seventh, biologically and empirically, the law is not “a must�
for each and every fly larva we studied. As we actually report in the paper: “A
few individuals did not comply with the law (…) corroborating that it is not
an obligatory outcome of our analyses� (L142).

Â

Eight, biologically and empirically, we have explored the
speed-curvature relation from data from other organisms (walking adult flies
and running mice, as well as swimming fish) and we have found huge deviations
from a power-law (data not shown in the paper).

Â

If none of the many arguments above is convincing due to its abstract
nature, the empirical demonstration that some flies do not show the power law
and all fish and mice test do not show the power law either should convince
anyone that: (i) the power law is not a trivial given result of the
mathematical relation between speed and curvature, and that therefore (ii) to
find it in human scribbling on paper and in the maggot, but not in other
organisms so far, is a relevant and interesting empirical result that opens new
perspectives to investigate the origin of such relation.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Wed, Aug 31, 2016 at 1:53 PM, rupert@perceptualrobots.com wrote:

So the velocity is that at which the organism is moving along the line? And it is found that organisms slow down on curves, is that right?

What is the angle alpha between?

On 26 August 2016 21:33:32 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alphaÂ

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.Â

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

[rupert]

Alex, I’m afraid I didn’t understand your response, perhaps you could rephrase. Was the situation query I raised understandable? Rupert

···

On Fri, Sep 2, 2016 at 8:49 PM, rupert@perceptualrobots.com wrote:

So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship between the slower speed and the speed along the straight path for the experimental occasion (given that the organism could began at a different speed).

Am I understanding correctly?

And you are saying that relationship is a property of the organism rather than the environment, is that right?

Rupert

On 31 August 2016 13:55:11 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Rupert, velocity is of course the velocity at which the organism moves (there is not a line “to move along to”; the organism simply locomotes and “traces” a line “behind” of a certain local curvature and at a certain instantaneous speed). Angle alpha is where the path is locally pointing to in space, so as alpha changes in time, it means that the animal is walking that path and changing direction.

Warren et al: Many arguments for it:

It is essential to realize that the constraint between speed and
curvature that we (and others) have empirically found complying with the formal
dependency of a power function (or power law) is actually not deducible from
mathematics or physics alone. Therefore, it must an emergent biological
phenomenon (whose origin still remains debated, and whose explanation might be
considered “trivial� but only after the fact, namely, once it has been
understood).

Â

First, intuitively, one can be aware that the shape of a path in
principle only restricts the direction of movement but not the speed at with
that path was traced. In other words, path curvature imposes “towards where� to
move (let’s say, a pen while scribbling or the path of a fly maggot) but is
agnostic so as to “how fast� to move.

Â

Second, mathematically, it is true that one can formally express
the curvature of a path as a function of velocity (or angular velocity) but such
formal expression precisely reflects how path curvature (whose natural argument
is arc length) can be parametrized as a function of time only when one adds a
specification of the speed profile. In other words, one could write the
expression of curvature in a coordinate system that only defines the shape of a
movement path and that is totally independent on the speed temporal profile
(see Huh and Sejnowski PNAS 2015 for an in-depth mathematical treatment).

Â

Third, mathematically, there is another way to see the
non-trivial relation between kinematics and geometry. In the expression of
curvature we see its explicit dependence on speed but also a term in the
denominator that contains a mixture of first and second derivatives, thus acceleration,
which cannot be specified by curvature, but are a property of the temporal
profile of speed, which can vary in different ways.

Â

Fourth, by simulations, one can actually synthetically create an
arbitrary path with a given local curvature and then change the time rate at
which that path is virtually drawn, and actually see how the power law does not
hold there.

Â

Fifth, more generally, curvature is by definition a geometric
quantity (only in space, that’s why we refer to it s “local� curvature) whereas
speed is obviously a kinematic quantity (in time, that is why we refer to it as
“instantaneous� speed) and so by basic notions of geometry and mechanics we can
be sure that geometry cannot dictate kinematics: purely spatial forms have no
way of acquiring temporal determination.

Â

As we briefly mention in the paper: L57: “The law is not
dictated by physical constraints: even when the path is imposed, as in
hand-drawing, movement speed could in principle vary in infinite ways.
Therefore, the law must result from physiological constraints�

Â

Sixth, physically, the laws of motion by Newton
describe/prescribe the relation between force and acceleration but do not
describe/prescribe that for any given force, the acceleration shall be so that
speed is so that curvature will be so.

 Â

Seventh, biologically and empirically, the law is not “a must�
for each and every fly larva we studied. As we actually report in the paper: “A
few individuals did not comply with the law (…) corroborating that it is not
an obligatory outcome of our analyses� (L142).

Â

Eight, biologically and empirically, we have explored the
speed-curvature relation from data from other organisms (walking adult flies
and running mice, as well as swimming fish) and we have found huge deviations
from a power-law (data not shown in the paper).

Â

If none of the many arguments above is convincing due to its abstract
nature, the empirical demonstration that some flies do not show the power law
and all fish and mice test do not show the power law either should convince
anyone that: (i) the power law is not a trivial given result of the
mathematical relation between speed and curvature, and that therefore (ii) to
find it in human scribbling on paper and in the maggot, but not in other
organisms so far, is a relevant and interesting empirical result that opens new
perspectives to investigate the origin of such relation.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Wed, Aug 31, 2016 at 1:53 PM, rupert@perceptualrobots.com wrote:

So the velocity is that at which the organism is moving along the line? And it is found that organisms slow down on curves, is that right?

What is the angle alpha between?

On 26 August 2016 21:33:32 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alphaÂ

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.Â

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

[Alex_Gomez-Marin]

All you say (and I mean, all of you) is understandable, but incomplete because it places a lot of emphasis on the organism the organism and the organism while ignoring that the organism is shaped by the world in many ways and at many timescales. So, evolution and development have enough time to make the brain-body-world confluence inseparable so that the animal integrates in its “behavior” aspects of the environment, and so it is not “only about discovering what is controlled and that’s it”.

···

On Sat, Sep 3, 2016 at 7:42 PM, rupert@perceptualrobots.com wrote:

Alex, I’m afraid I didn’t understand your response, perhaps you could rephrase. Was the situation query I raised understandable? Rupert

On 3 September 2016 13:41:46 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

what if my way of keeping the pedal was shape beyond proximate circular causation so that the undulation of my eco-system had cristalized in evolutionary or developmental time into my way of controlling my perceptions? pct needs a wider Tinbergen-like perspective. the car has been driven for millions of years…

On Friday, 2 September 2016, rupert@perceptualrobots.com wrote:

Thanks.

A perhaps analogous situation comes to mind which might help me understand better where you’re coming from. Suppose you are driving a car and are doing so by controlling the angle of the accelerator pedal at a constant reference value; i. e. keeping it, and foot, in same position (let’s say mid-range). And you are driving over (straight) undulating terrain. What would an external person observe about the speed of the car? Would it be possible to determine a relationship between the changes in speed and the terrain? Would the changes in speed, or the relationship, be a property of the organism or the environment, or both?

On 2 September 2016 20:45:02 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Thanks for your questions Rupert, which allow to bring some clarity by answering to the point without “playing start on the PCT radio speech”:

 So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship

Yes, the EMPIRICAL finding is that flies slow down the sharper the turn in a way that there is a power law relation between these speed and curvature (which, I hope it is clear by now, is not a trivial mathematical nor physical constrain, but something observed in some living organisms).

 And you are saying that relationship is a property of the organism rather than the environment, is that right?

No, no, no. I am not saying anything of that (yet). Otherwise I would be immediately excomulgated from the purist PCTers. I am just saying (and that is what I have been saying since the very first email!) that I find this EMPIRICAL relation, which is not a trivial one, but something evolution may have pushed as a biological constraint, AND I wonder then what is the PCT interpretation of it, or better, what is the perceived quantity. For some reason curvature and speed are mutually constrained. I want to find out the origin of that.

Plus, the question does not need to be EITHER OR, namely, the power law relation (as a FACT) does not have to be either 100% illusion (what PCTers seem to call environment property) nor 100% true (what PCTers seem to call organism property). It can be, and most likely is, a MIXTURE of both! Why, because for sure the animal is trying to control for someting AND at the same time the properties of the substrate (friction, viscossity, etc) have an influence. So, again, not being a FUNDAMENTALIST or a CONTROL OF PERCEPTION CHAUVINIST, I am interested, as a biologist, to understand both the organism, the world in which it lives and the interaction between both.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Fri, Sep 2, 2016 at 8:49 PM, rupert@perceptualrobots.com wrote:

So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship between the slower speed and the speed along the straight path for the experimental occasion (given that the organism could began at a different speed).

Am I understanding correctly?

And you are saying that relationship is a property of the organism rather than the environment, is that right?

Rupert

On 31 August 2016 13:55:11 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Rupert, velocity is of course the velocity at which the organism moves (there is not a line “to move along to”; the organism simply locomotes and “traces” a line “behind” of a certain local curvature and at a certain instantaneous speed). Angle alpha is where the path is locally pointing to in space, so as alpha changes in time, it means that the animal is walking that path and changing direction.

Warren et al: Many arguments for it:

It is essential to realize that the constraint between speed and
curvature that we (and others) have empirically found complying with the formal
dependency of a power function (or power law) is actually not deducible from
mathematics or physics alone. Therefore, it must an emergent biological
phenomenon (whose origin still remains debated, and whose explanation might be
considered “trivial� but only after the fact, namely, once it has been
understood).

Â

First, intuitively, one can be aware that the shape of a path in
principle only restricts the direction of movement but not the speed at with
that path was traced. In other words, path curvature imposes “towards where� to
move (let’s say, a pen while scribbling or the path of a fly maggot) but is
agnostic so as to “how fast� to move.

Â

Second, mathematically, it is true that one can formally express
the curvature of a path as a function of velocity (or angular velocity) but such
formal expression precisely reflects how path curvature (whose natural argument
is arc length) can be parametrized as a function of time only when one adds a
specification of the speed profile. In other words, one could write the
expression of curvature in a coordinate system that only defines the shape of a
movement path and that is totally independent on the speed temporal profile
(see Huh and Sejnowski PNAS 2015 for an in-depth mathematical treatment).

Â

Third, mathematically, there is another way to see the
non-trivial relation between kinematics and geometry. In the expression of
curvature we see its explicit dependence on speed but also a term in the
denominator that contains a mixture of first and second derivatives, thus acceleration,
which cannot be specified by curvature, but are a property of the temporal
profile of speed, which can vary in different ways.

Â

Fourth, by simulations, one can actually synthetically create an
arbitrary path with a given local curvature and then change the time rate at
which that path is virtually drawn, and actually see how the power law does not
hold there.

Â

Fifth, more generally, curvature is by definition a geometric
quantity (only in space, that’s why we refer to it s “local� curvature) whereas
speed is obviously a kinematic quantity (in time, that is why we refer to it as
“instantaneous� speed) and so by basic notions of geometry and mechanics we can
be sure that geometry cannot dictate kinematics: purely spatial forms have no
way of acquiring temporal determination.

Â

As we briefly mention in the paper: L57: “The law is not
dictated by physical constraints: even when the path is imposed, as in
hand-drawing, movement speed could in principle vary in infinite ways.
Therefore, the law must result from physiological constraints�

Â

Sixth, physically, the laws of motion by Newton
describe/prescribe the relation between force and acceleration but do not
describe/prescribe that for any given force, the acceleration shall be so that
speed is so that curvature will be so.

 Â

Seventh, biologically and empirically, the law is not “a must�
for each and every fly larva we studied. As we actually report in the paper: “A
few individuals did not comply with the law (…) corroborating that it is not
an obligatory outcome of our analyses� (L142).

Â

Eight, biologically and empirically, we have explored the
speed-curvature relation from data from other organisms (walking adult flies
and running mice, as well as swimming fish) and we have found huge deviations
from a power-law (data not shown in the paper).

Â

If none of the many arguments above is convincing due to its abstract
nature, the empirical demonstration that some flies do not show the power law
and all fish and mice test do not show the power law either should convince
anyone that: (i) the power law is not a trivial given result of the
mathematical relation between speed and curvature, and that therefore (ii) to
find it in human scribbling on paper and in the maggot, but not in other
organisms so far, is a relevant and interesting empirical result that opens new
perspectives to investigate the origin of such relation.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Wed, Aug 31, 2016 at 1:53 PM, rupert@perceptualrobots.com wrote:

So the velocity is that at which the organism is moving along the line? And it is found that organisms slow down on curves, is that right?

What is the angle alpha between?

On 26 August 2016 21:33:32 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alphaÂ

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.Â

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

[rupert]

Alex, you are ascribing statements to me that I have not made, and disagreeing with them, and making assumptions about what I do or do not believe. The statement you put in quotes is certainly not one of mine, where did it come from?

I was asking questions, not making statements, to see if we are understanding each other. Questions about a scenario which may, or may not, have bearing on the riddle you are exploring.

If you are able to answer them then we may be able to make further progress in the discussion.

Rupert

···

On Sat, Sep 3, 2016 at 7:42 PM, rupert@perceptualrobots.com wrote:

Alex, I’m afraid I didn’t understand your response, perhaps you could rephrase. Was the situation query I raised understandable? Rupert

On 3 September 2016 13:41:46 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

what if my way of keeping the pedal was shape beyond proximate circular causation so that the undulation of my eco-system had cristalized in evolutionary or developmental time into my way of controlling my perceptions? pct needs a wider Tinbergen-like perspective. the car has been driven for millions of years…

On Friday, 2 September 2016, rupert@perceptualrobots.com wrote:

Thanks.

A perhaps analogous situation comes to mind which might help me understand better where you’re coming from. Suppose you are driving a car and are doing so by controlling the angle of the accelerator pedal at a constant reference value; i. e. keeping it, and foot, in same position (let’s say mid-range). And you are driving over (straight) undulating terrain. What would an external person observe about the speed of the car? Would it be possible to determine a relationship between the changes in speed and the terrain? Would the changes in speed, or the relationship, be a property of the organism or the environment, or both?

On 2 September 2016 20:45:02 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Thanks for your questions Rupert, which allow to bring some clarity by answering to the point without “playing start on the PCT radio speech”:

 So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship

Yes, the EMPIRICAL finding is that flies slow down the sharper the turn in a way that there is a power law relation between these speed and curvature (which, I hope it is clear by now, is not a trivial mathematical nor physical constrain, but something observed in some living organisms).

 And you are saying that relationship is a property of the organism rather than the environment, is that right?

No, no, no. I am not saying anything of that (yet). Otherwise I would be immediately excomulgated from the purist PCTers. I am just saying (and that is what I have been saying since the very first email!) that I find this EMPIRICAL relation, which is not a trivial one, but something evolution may have pushed as a biological constraint, AND I wonder then what is the PCT interpretation of it, or better, what is the perceived quantity. For some reason curvature and speed are mutually constrained. I want to find out the origin of that.

Plus, the question does not need to be EITHER OR, namely, the power law relation (as a FACT) does not have to be either 100% illusion (what PCTers seem to call environment property) nor 100% true (what PCTers seem to call organism property). It can be, and most likely is, a MIXTURE of both! Why, because for sure the animal is trying to control for someting AND at the same time the properties of the substrate (friction, viscossity, etc) have an influence. So, again, not being a FUNDAMENTALIST or a CONTROL OF PERCEPTION CHAUVINIST, I am interested, as a biologist, to understand both the organism, the world in which it lives and the interaction between both.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Fri, Sep 2, 2016 at 8:49 PM, rupert@perceptualrobots.com wrote:

So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship between the slower speed and the speed along the straight path for the experimental occasion (given that the organism could began at a different speed).

Am I understanding correctly?

And you are saying that relationship is a property of the organism rather than the environment, is that right?

Rupert

On 31 August 2016 13:55:11 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Rupert, velocity is of course the velocity at which the organism moves (there is not a line “to move along to”; the organism simply locomotes and “traces” a line “behind” of a certain local curvature and at a certain instantaneous speed). Angle alpha is where the path is locally pointing to in space, so as alpha changes in time, it means that the animal is walking that path and changing direction.

Warren et al: Many arguments for it:

It is essential to realize that the constraint between speed and
curvature that we (and others) have empirically found complying with the formal
dependency of a power function (or power law) is actually not deducible from
mathematics or physics alone. Therefore, it must an emergent biological
phenomenon (whose origin still remains debated, and whose explanation might be
considered “trivial� but only after the fact, namely, once it has been
understood).

Â

First, intuitively, one can be aware that the shape of a path in
principle only restricts the direction of movement but not the speed at with
that path was traced. In other words, path curvature imposes “towards where� to
move (let’s say, a pen while scribbling or the path of a fly maggot) but is
agnostic so as to “how fast� to move.

Â

Second, mathematically, it is true that one can formally express
the curvature of a path as a function of velocity (or angular velocity) but such
formal expression precisely reflects how path curvature (whose natural argument
is arc length) can be parametrized as a function of time only when one adds a
specification of the speed profile. In other words, one could write the
expression of curvature in a coordinate system that only defines the shape of a
movement path and that is totally independent on the speed temporal profile
(see Huh and Sejnowski PNAS 2015 for an in-depth mathematical treatment).

Â

Third, mathematically, there is another way to see the
non-trivial relation between kinematics and geometry. In the expression of
curvature we see its explicit dependence on speed but also a term in the
denominator that contains a mixture of first and second derivatives, thus acceleration,
which cannot be specified by curvature, but are a property of the temporal
profile of speed, which can vary in different ways.

Â

Fourth, by simulations, one can actually synthetically create an
arbitrary path with a given local curvature and then change the time rate at
which that path is virtually drawn, and actually see how the power law does not
hold there.

Â

Fifth, more generally, curvature is by definition a geometric
quantity (only in space, that’s why we refer to it s “local� curvature) whereas
speed is obviously a kinematic quantity (in time, that is why we refer to it as
“instantaneous� speed) and so by basic notions of geometry and mechanics we can
be sure that geometry cannot dictate kinematics: purely spatial forms have no
way of acquiring temporal determination.

Â

As we briefly mention in the paper: L57: “The law is not
dictated by physical constraints: even when the path is imposed, as in
hand-drawing, movement speed could in principle vary in infinite ways.
Therefore, the law must result from physiological constraints�

Â

Sixth, physically, the laws of motion by Newton
describe/prescribe the relation between force and acceleration but do not
describe/prescribe that for any given force, the acceleration shall be so that
speed is so that curvature will be so.

 Â

Seventh, biologically and empirically, the law is not “a must�
for each and every fly larva we studied. As we actually report in the paper: “A
few individuals did not comply with the law (…) corroborating that it is not
an obligatory outcome of our analyses� (L142).

Â

Eight, biologically and empirically, we have explored the
speed-curvature relation from data from other organisms (walking adult flies
and running mice, as well as swimming fish) and we have found huge deviations
from a power-law (data not shown in the paper).

Â

If none of the many arguments above is convincing due to its abstract
nature, the empirical demonstration that some flies do not show the power law
and all fish and mice test do not show the power law either should convince
anyone that: (i) the power law is not a trivial given result of the
mathematical relation between speed and curvature, and that therefore (ii) to
find it in human scribbling on paper and in the maggot, but not in other
organisms so far, is a relevant and interesting empirical result that opens new
perspectives to investigate the origin of such relation.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Wed, Aug 31, 2016 at 1:53 PM, rupert@perceptualrobots.com wrote:

So the velocity is that at which the organism is moving along the line? And it is found that organisms slow down on curves, is that right?

What is the angle alpha between?

On 26 August 2016 21:33:32 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alphaÂ

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.Â

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

[rsmarken]

[From Rick Marken (2016.09.03.1230)]

···

On Sat, Sep 3, 2016 at 10:46 AM, Alex Gomez-Marin agomezmarin@gmail.com wrote:

AGM: All you say (and I mean, all of you) is understandable, but incomplete because it places a lot of emphasis on the organism the organism and the organism while ignoring that the organism is shaped by the world in many ways and at many timescales.

RM: PCT doesn’t ignore it; it shows that it’s an illusion. The environment does appear to shape (or select or constrain) behavior, but that’s not the way it works. There is a nice, short paper in LCS I why this is the case; why it’s the organism that shapes (controls) the environment and not the environment that shapes (controls) the organism. It’s called “The Asymmetry of Control” and it is very short – two pages – but very good.

Best

Rick

Â

So, evolution and development have enough time to make the brain-body-world confluence inseparable so that the animal integrates in its “behavior” aspects of the environment, and so it is not “only about discovering what is controlled and that’s it”.

–

On Sat, Sep 3, 2016 at 7:42 PM, rupert@perceptualrobots.com wrote:

Alex, I’m afraid I didn’t understand your response, perhaps you could rephrase. Was the situation query I raised understandable? Rupert

On 3 September 2016 13:41:46 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

what if my way of keeping the pedal was shape beyond proximate circular causation so that the undulation of my eco-system had cristalized in evolutionary or developmental time into my way of controlling my perceptions? pct needs a wider Tinbergen-like perspective. the car has been driven for millions of years…

On Friday, 2 September 2016, rupert@perceptualrobots.com wrote:

Thanks.

A perhaps analogous situation comes to mind which might help me understand better where you’re coming from. Suppose you are driving a car and are doing so by controlling the angle of the accelerator pedal at a constant reference value; i. e. keeping it, and foot, in same position (let’s say mid-range). And you are driving over (straight) undulating terrain. What would an external person observe about the speed of the car? Would it be possible to determine a relationship between the changes in speed and the terrain? Would the changes in speed, or the relationship, be a property of the organism or the environment, or both?

On 2 September 2016 20:45:02 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Thanks for your questions Rupert, which allow to bring some clarity by answering to the point without “playing start on the PCT radio speech”:

 So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship

Yes, the EMPIRICAL finding is that flies slow down the sharper the turn in a way that there is a power law relation between these speed and curvature (which, I hope it is clear by now, is not a trivial mathematical nor physical constrain, but something observed in some living organisms).

 And you are saying that relationship is a property of the organism rather than the environment, is that right?

No, no, no. I am not saying anything of that (yet). Otherwise I would be immediately excomulgated from the purist PCTers. I am just saying (and that is what I have been saying since the very first email!) that I find this EMPIRICAL relation, which is not a trivial one, but something evolution may have pushed as a biological constraint, AND I wonder then what is the PCT interpretation of it, or better, what is the perceived quantity. For some reason curvature and speed are mutually constrained. I want to find out the origin of that.

Plus, the question does not need to be EITHER OR, namely, the power law relation (as a FACT) does not have to be either 100% illusion (what PCTers seem to call environment property) nor 100% true (what PCTers seem to call organism property). It can be, and most likely is, a MIXTURE of both! Why, because for sure the animal is trying to control for someting AND at the same time the properties of the substrate (friction, viscossity, etc) have an influence. So, again, not being a FUNDAMENTALIST or a CONTROL OF PERCEPTION CHAUVINIST, I am interested, as a biologist, to understand both the organism, the world in which it lives and the interaction between both.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Fri, Sep 2, 2016 at 8:49 PM, rupert@perceptualrobots.com wrote:

So, you’re finding that the organism slows down on turns, and in such a way that there is a power law relationship between the slower speed and the speed along the straight path for the experimental occasion (given that the organism could began at a different speed).

Am I understanding correctly?

And you are saying that relationship is a property of the organism rather than the environment, is that right?

Rupert

On 31 August 2016 13:55:11 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

Rupert, velocity is of course the velocity at which the organism moves (there is not a line “to move along to”; the organism simply locomotes and “traces” a line “behind” of a certain local curvature and at a certain instantaneous speed). Angle alpha is where the path is locally pointing to in space, so as alpha changes in time, it means that the animal is walking that path and changing direction.

Warren et al: Many arguments for it:

It is essential to realize that the constraint between speed and
curvature that we (and others) have empirically found complying with the formal
dependency of a power function (or power law) is actually not deducible from
mathematics or physics alone. Therefore, it must an emergent biological
phenomenon (whose origin still remains debated, and whose explanation might be
considered “trivial� but only after the fact, namely, once it has been
understood).

Â

First, intuitively, one can be aware that the shape of a path in
principle only restricts the direction of movement but not the speed at with
that path was traced. In other words, path curvature imposes “towards where� to
move (let’s say, a pen while scribbling or the path of a fly maggot) but is
agnostic so as to “how fast� to move.

Â

Second, mathematically, it is true that one can formally express
the curvature of a path as a function of velocity (or angular velocity) but such
formal expression precisely reflects how path curvature (whose natural argument
is arc length) can be parametrized as a function of time only when one adds a
specification of the speed profile. In other words, one could write the
expression of curvature in a coordinate system that only defines the shape of a
movement path and that is totally independent on the speed temporal profile
(see Huh and Sejnowski PNAS 2015 for an in-depth mathematical treatment).

Â

Third, mathematically, there is another way to see the
non-trivial relation between kinematics and geometry. In the expression of
curvature we see its explicit dependence on speed but also a term in the
denominator that contains a mixture of first and second derivatives, thus acceleration,
which cannot be specified by curvature, but are a property of the temporal
profile of speed, which can vary in different ways.

Â

Fourth, by simulations, one can actually synthetically create an
arbitrary path with a given local curvature and then change the time rate at
which that path is virtually drawn, and actually see how the power law does not
hold there.

Â

Fifth, more generally, curvature is by definition a geometric
quantity (only in space, that’s why we refer to it s “local� curvature) whereas
speed is obviously a kinematic quantity (in time, that is why we refer to it as
“instantaneous� speed) and so by basic notions of geometry and mechanics we can
be sure that geometry cannot dictate kinematics: purely spatial forms have no
way of acquiring temporal determination.

Â

As we briefly mention in the paper: L57: “The law is not
dictated by physical constraints: even when the path is imposed, as in
hand-drawing, movement speed could in principle vary in infinite ways.
Therefore, the law must result from physiological constraints�

Â

Sixth, physically, the laws of motion by Newton
describe/prescribe the relation between force and acceleration but do not
describe/prescribe that for any given force, the acceleration shall be so that
speed is so that curvature will be so.

 Â

Seventh, biologically and empirically, the law is not “a must�
for each and every fly larva we studied. As we actually report in the paper: “A
few individuals did not comply with the law (…) corroborating that it is not
an obligatory outcome of our analyses� (L142).

Â

Eight, biologically and empirically, we have explored the
speed-curvature relation from data from other organisms (walking adult flies
and running mice, as well as swimming fish) and we have found huge deviations
from a power-law (data not shown in the paper).

Â

If none of the many arguments above is convincing due to its abstract
nature, the empirical demonstration that some flies do not show the power law
and all fish and mice test do not show the power law either should convince
anyone that: (i) the power law is not a trivial given result of the
mathematical relation between speed and curvature, and that therefore (ii) to
find it in human scribbling on paper and in the maggot, but not in other
organisms so far, is a relevant and interesting empirical result that opens new
perspectives to investigate the origin of such relation.

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

On Wed, Aug 31, 2016 at 1:53 PM, rupert@perceptualrobots.com wrote:

So the velocity is that at which the organism is moving along the line? And it is found that organisms slow down on curves, is that right?

What is the angle alpha between?

On 26 August 2016 21:33:32 BST, Alex Gomez-Marin agomezmarin@gmail.com wrote:

C is curvature indeed.
R = 1/C

A is rate of change of alphaÂ

A = d alpha / dt

also one could estimate A as V / R

a perfect cycle would have C constant all along the curve. but one could in principle trace it at any speed V or angular speed A.

again, important: curvature is a geometrical measure; it doesn’t know nor care about time. speed and angular speed are kinematic measures, so time is important and essential there.Â

the power law --be it an illusion, delusion or whatever pct judgement on its interpretation – is an a priori unexpected constraint between geometry and kinematics. one that physics alone does not impose and one that is found in biology.

thanks for your interest

On Friday, 26 August 2016, Rupert Young rupert@perceptualrobots.com wrote:

[From Rupert Young (2016.08.26 20.30)]

  I've no wish to stick a knife in open wounds, but I've eventually

found time to read this paper.

  Would you clarify the variables involved? C is a measure of the

curvature of a line at a particular point. So the tighter the
curve the higher the value of C. A radius R corresponds to the
value of C. Alpha is the angle between the tangent of a point on
the line and what? Would I be correct in saying that the angular
velocity A is nothing to do with the speed at which the organism
which produced the line was moving, but is a measure of how alpha
changes along the line?

  If so, and if we forget about organisms, what would be the values

of these variables for a perfect circle? I assume A would be zero?

  I may only be able to reply intermittently over the next couple

of weeks as I’m off on a trip,
https://www.justgiving.com/crowdfunding/cycle-for-humanity

Donations welcome!

Regards,

Rupert

On 06/07/2016 15:33, Alex Gomez-Marin wrote:

      Any

ideas why or how “the control of perception” may give rise to
this power law constraining geometry and kinematics in humans,
and now in fruit fly larvae?

http://biorxiv.org/content/early/2016/07/05/062166

Thanks,

Alex

Regards,

Dr Rupert Young

www.perceptualrobots.com

–

Sent from my Android device with K-9 Mail. Please excuse my brevity.

Richard S. MarkenÂ

“The childhood of the human race is far from over. We
have a long way to go before most people will understand that what they do for
others is just as important to their well-being as what they do for
themselves.” – William T. Powers