# Stability Analysis of an Equilibrium Point Model (Lan & Zhu, 2007)

[From Bruce Abbott (2014.01.02.1515 EST)]

I’ve obtained a copy of an article that performs a stability analysis of one version of the λ version of the equilibrium point model for a single joint. (There are other EP models out there.)

The article reference is Lan, L., & Zhu, K. Y. (2007). Biomechanical stability analysis of the lambda-model controlling one joint. International Journal of Neural Systems, 17(3), 193-206. These authors conduct a mathematical analysis of this system and then present the results of simulations based on that analysis. Below is a block diagram of that version of the EP model:

In this version of the EP model, the “reciprocal commands” represent changes in the opposite direction of λ for the opposed flexor and extensor muscles, whereas “coactivation commands” change λ in the same direction. Changes in R change the equilibrium point function, yielding different final joint angles. As C increases, the opposing muscles pull more strongly against each other, increasing muscle stiffness.

The next figure shows the dynamic responses of the system:

This is a stable system in that all starting points converge to a single point. This behavior can be seen in a graph of joint-angle change as a function of time:

The system does a bit of “ringing” or “hunting” around the final joint angle for some combinations of R and C. Note that these oscillations disappear with higher values of C. Different values of R result in different final joint angles.

The authors did not evaluate the model against empirical data as that was not the purpose of their analysis. They simply wished to conduct of rigorous evaluation of the stability of the target EP model.

Bruce

[From Rick Marken (2014.01.03.0820)

···

Bruce Abbott (2014.01.02.1515 EST)–

BA: I’ve obtained a copy of an article that performs a stability analysis of one version of the ë version of the equilibrium point model for a single joint. (There are other EP models out there.)

The article reference is Lan, L., & Zhu, K. Y. (2007). Biomechanical stability analysis of the lambda-model controlling one joint. International Journal of Neural Systems, 17(3), 193-206. These authors conduct a mathematical analysis of this system and then present the results of simulations based on that analysis. Below is a block diagram of that version of the EP model:

The system does a bit of “ringing” or “hunting” around the final joint angle for some combinations of R and C. Note that these oscillations disappear with higher values of C. Different values of R result in different final joint angles.

The authors did not evaluate the model against empirical data as that was not the purpose of their analysis. They simply wished to conduct of rigorous evaluation of the stability of the target EP model.

RM: The EP model described by Lan and Zhu here is a control model. The controlled variables are not clearly specified although they are some combination of joint angle and joint angle velocity. They also don’t show external disturbances, which would be torques entering the joint dynamics box from outside the system. I have done my own little experiments, lowering my forearm slowly from 0 to 60 degrees and the model behavior that fits my performance best is the curve in Figure 6 labeled R=60, C = 80; my forearm moves directly from 0 to 60 with no overshoot or ringing; I have good control of my limb movement. I also performed the same behavior while my wife applied varying downward pressure (torque) as I moved my arm down. I’m sure their model would perform the same (using the R=60, C = 80 settings for the opponent control system references) as I did if they added torque disturbances.

So if Lan and Zhu want to call their model an EP rather than a control model that’s the way it goes. But I think we could pretty easily implement this model as a computer program and analyze it from a PCT perspective. If this is what EP theory is about then it is, indeed, the same as PCT, just not as clear (doesn’t include the perceptual functions) or complete (no disturbances).

Best

Rick

Richard S. Marken PhD
The only thing that will redeem mankind is cooperation.

``````                                               -- Bertrand Russell
``````

[From Bruce Abbott (2014.01.03.1425 EST)]

Rick Marken (2014.01.03.0820)

Bruce Abbott (2014.01.02.1515 EST)–

BA: I’ve obtained a copy of an article that performs a stability analysis of one version of the λ version of the equilibrium point model for a single joint. (There are other EP models out there.)

BA: The article reference is Lan, L., & Zhu, K. Y. (2007). Biomechanical stability analysis of the lambda-model controlling one joint. International Journal of Neural Systems, 17(3), 193-206. These authors conduct a mathematical analysis of this system and then present the results of simulations based on that analysis. Below is a block diagram of that version of the EP model:

BA: The system does a bit of “ringing” or “hunting” around the final joint angle for some combinations of R and C. Note that these oscillations disappear with higher values of C. Different values of R result in different final joint angles.

BA: The authors did not evaluate the model against empirical data as that was not the purpose of their analysis. They simply wished to conduct of rigorous evaluation of the stability of the target EP model.

RM: The EP model described by Lan and Zhu here is a control model. The controlled variables are not clearly specified although they are some combination of joint angle and joint angle velocity. They also don’t show external disturbances, which would be torques entering the joint dynamics box from outside the system. I have done my own little experiments, lowering my forearm slowly from 0 to 60 degrees and the model behavior that fits my performance best is the curve in Figure 6 labeled R=60, C = 80; my forearm moves directly from 0 to 60 with no overshoot or ringing; I have good control of my limb movement. I also performed the same behavior while my wife applied varying downward pressure (torque) as I moved my arm down. I’m sure their model would perform the same (using the R=60, C = 80 settings for the opponent control system references) as I did if they added torque disturbances.

I did a little of that sort of testing myself!

RM: So if Lan and Zhu want to call their model an EP rather than a control model that’s the way it goes. But I think we could pretty easily implement this model as a computer program and analyze it from a PCT perspective. If this is what EP theory is about then it is, indeed, the same as PCT, just not as clear (doesn’t include the perceptual functions) or complete (no disturbances).

Absolutely.

It bears mentioning that there are several versions of EP model out there, some of which antedate the one analyzed by Lan and Zhu, and that each version (as well as the whole EP approach) has had its critics on one ground or another. Furthermore, we don’t see any specification here as to how the R and C “commands” are generated. Lan and Zhu’s Figure 1 (which I didn’t reproduce) shows a hierarchy of three levels, with feedback between the levels. (However, they don’t concern themselves with the upper levels in their paper.) Some theorists have proposed that these “commands” are formulated based on high-level “programs” or “planning” that involves the use inverse kinematics to determine how the joints should move to accomplish some task such as reaching. In this approach, fast movements to target are executed open-loop, with feedback coming into the picture as the target is reached, to reduce any error produced during the fast execution phase. I can’t help but think that hierarchical control similar to that implemented in the inverted pendulum would provide a simpler and more adequate model that does not require all that planning and switching between open-loop and closed loop modes.

If any of our younger CSGnetters would like to take a stab at developing a PCT model of this, I encourage you to give it a try. This is still an area crying out for a better model!

Bruce

[From Bruce Abbott (2014.01.02.1515 EST)]

Iâ€™ve obtained a copy of an article that performs a stability analysis of one version of the Î» version of the equilibrium point model for a single joint. (There are other EP models out there.)

The article reference is Lan, L., & Zhu, K. Y. (2007). Biomechanical stability analysis of the lambda-model controlling one joint. International Journal of Neural Systems, 17(3), 193-206. These authors conduct a mathematical analysis of this system and then present the results of simulations based on that analysis. Below is a block diagram of that version of the EP model:

<image001.jpg>

In this version of the EP model, the â€œreciprocal commandsâ€? represent changes in the opposite direction of Î» for the opposed flexor and extensor muscles, whereas â€œcoactivation commandsâ€? change Î» in the same direction. Changes in R change the equilibrium point function, yielding different final joint angles. As C increases, the opposing muscles pull more strongly against each other, increasing muscle stiffness.

The next figure shows the dynamic responses of the system:

<image002.jpg>

This is a stable system in that all starting points converge to a single point. This behavior can be seen in a graph of joint-angle change as a function of time:

<image003.jpg>

The system does a bit of â€œringingâ€? or â€œhuntingâ€? around the final joint angle for some combinations of R and C. Note that these oscillations disappear with higher values of C. Different values of R result in different final joint angles.

The authors did not evaluate the model against empirical data as that was not the purpose of their analysis. They simply wished to conduct of rigorous evaluation of the stability of the target EP model.

Bruce

[From Rick Marken (2014.01.04.1300)]

···

On Sat, Jan 4, 2014 at 1:15 AM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Bruce and Rick, this is great to see. I have some queries.

Bruce, with some editing from Rick, would you be interested in writing about these models and their relationship with PCT, as well as how PCT would advance them in terms of parsimony, disturbances, etc, for the new edited book? I think it would make an extremely relevant chapter that would really show how PCT can advance contemporary theories in the behavioural sciences.

RM: I think this is a great idea. But to do it right it might take a while. For example, I think we should have a working version of the EP model (as described in the Lan/ZHu paper) so that we can see for ourselves what variables matter in terms of replicating their results and seeing what variables matter to performance (Lan/Zhu don’t report some of the variables that must be involved, like the gain, slowing and transport lag). I also think we have to see if Feldman (who I think is the developer of the EP model) endorses this version of EP; and whether he has his own simulations of an EP model of forearm angle control.

But I would love to work with Bruce on this. I’ll start cobbling together a spreadsheet model asap.

Â

RM: Could I check too… How does threshold reference get translated into dynamic threshold. Is this what we would call the reference values for the rate and direction of change of threshold?

RM: it looks to me like their “dynamic threshold” is just the dynamically changing error signal that results from comparing their threshold reference to the perceived muscle length/rate of change in length perception. This “dynamic threshold” becomes the reference for the lower level length rate of change control system.

Â

WM: If so, yes I see it is approximating to the inverted pendulum model. But the inverted pendulum model is somewhat at odds with the B:CP model in its specifics within PCT, so see below…

RM: You mean it’s at odds with the specific proposed levels of types of perceptions controlled? I think we have to stop worrying about whether the hierarchical models we build to account for specific behaviors control perceptions that correspond to the types in the hierarchy. I see the hierarchy Bill proposed as kind the kind of map we might get once we have done a lot of research on perceptual control. It would be kind of like the PCT version of the periodic table; and you couldn’t build the periodic table until a lot of chemistry had been done based on an atomic model of matter.

WM: I see both the omissions and complications that Rick sees which ultimately mean it misses the point of perceptual control despite implementing it, but I also see some ‘domain-specific specifications’ that are helping PCT with its selection of physical constraints and perceptual functions for the limb motor system in particular.

RM: That may be true; certainly to threshold (one way) control and opponent process aspect of the model corresponds better to the way things actually work in the NS. But we’ll know more once we build the model. Indeed, we should compare this EP model of limb movement to the PCT model as implemented in the Little Man. I’m sure that would be a way to say some nice things about this version of the EP model; it’s more true to the physiology.

WM: PCT was not developed to explain limb motor control specifically but perceptual control generally in any behavioural system across species, robots and even aliens! So I don’t see a problem here in magpie-ing some domain-specific clues.

What do people think?

RM: Again, super idea!! I’m for carrying on our discussion of work on the model on CSGNet so people can see what we’re up to. But doing it via personal communication is fine with me to.

Best

Rick

Â

Warren

Warren

Sent from my iPhone

On 2 Jan 2014, at 20:15, Bruce Abbott bbabbott@FRONTIER.COM wrote:

[From Bruce Abbott (2014.01.02.1515 EST)]

Â

Iâ€™ve obtained a copy of an article that performs a stability analysis of one version of the Î» version of the equilibrium point model for a single joint. (There are other EP models out there.)

Â

The article reference is Lan, L., & Zhu, K. Y. (2007). Biomechanical stability analysis of the lambda-model controlling one joint.Â International Journal of Neural Systems, 17(3), 193-206.Â These authors conduct a mathematical analysis of this system and then present the results of simulations based on that analysis.Â Below is a block diagram of that Â version of the EP model:

Â

<image001.jpg>

In this version of the EP model, the â€œreciprocal commandsâ€? represent changes in the opposite direction of Î» for the opposed flexor and extensor muscles, whereas â€œcoactivation commandsâ€? change Î» in the same direction. Changes in R change the equilibrium point function, yielding different final joint angles. As C increases, the opposing muscles pull more strongly against each other, increasing muscle stiffness.

Â

The next figure shows the dynamic responses of the system:

Â

<image002.jpg>

This is a stable system in that all starting points converge to a single point.Â This behavior can be seen in a graph of joint-angle change as a function of time:

Â

<image003.jpg>

Â

The system does a bit of â€œringingâ€? or â€œhuntingâ€? around the final joint angle for some combinations of R and C.Â Note that these oscillations disappear with higher values of C.Â Different values of R result in different final joint angles.

Â

The authors did not evaluate the model against empirical data as that was not the purpose of their analysis.Â They simply wished to conduct of rigorous evaluation of the stability of the target EP model.

Â

Bruce

Richard S. Marken PhD
Â Â Â Â Â Â Â Â Â Â Â Â
The only thing that will redeem mankind is cooperation.

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â – Bertrand Russell

Ok great. I still think that for a book chapter a conceptual review is sufficient. This would set the stage for the empirical paper. Bruce? But the comparative modelling would make a great journal article that we could even collaborate with the competing authors on. It’s great Rick that you see it worthy to do some actually modelling!

I will contact the other authors and see what they say!

Cheers,

Warren

···

On Sat, Jan 4, 2014 at 1:15 AM, Warren Mansell wmansell@gmail.com wrote:

WM: Hi Bruce and Rick, this is great to see. I have some queries.

Bruce, with some editing from Rick, would you be interested in writing about these models and their relationship with PCT, as well as how PCT would advance them in terms of parsimony, disturbances, etc, for the new edited book? I think it would make an extremely relevant chapter that would really show how PCT can advance contemporary theories in the behavioural sciences.

RM: I think this is a great idea. But to do it right it might take a while. For example, I think we should have a working version of the EP model (as described in the Lan/ZHu paper) so that we can see for ourselves what variables matter in terms of replicating their results and seeing what variables matter to performance (Lan/Zhu don’t report some of the variables that must be involved, like the gain, slowing and transport lag). I also think we have to see if Feldman (who I think is the developer of the EP model) endorses this version of EP; and whether he has his own simulations of an EP model of forearm angle control.

But I would love to work with Bruce on this. I’ll start cobbling together a spreadsheet model asap.

RM: Could I check too… How does threshold reference get translated into dynamic threshold. Is this what we would call the reference values for the rate and direction of change of threshold?

RM: it looks to me like their “dynamic threshold” is just the dynamically changing error signal that results from comparing their threshold reference to the perceived muscle length/rate of change in length perception. This “dynamic threshold” becomes the reference for the lower level length rate of change control system.

WM: If so, yes I see it is approximating to the inverted pendulum model. But the inverted pendulum model is somewhat at odds with the B:CP model in its specifics within PCT, so see below…

RM: You mean it’s at odds with the specific proposed levels of types of perceptions controlled? I think we have to stop worrying about whether the hierarchical models we build to account for specific behaviors control perceptions that correspond to the types in the hierarchy. I see the hierarchy Bill proposed as kind the kind of map we might get once we have done a lot of research on perceptual control. It would be kind of like the PCT version of the periodic table; and you couldn’t build the periodic table until a lot of chemistry had been done based on an atomic model of matter.

WM: I see both the omissions and complications that Rick sees which ultimately mean it misses the point of perceptual control despite implementing it, but I also see some ‘domain-specific specifications’ that are helping PCT with its selection of physical constraints and perceptual functions for the limb motor system in particular.

RM: That may be true; certainly to threshold (one way) control and opponent process aspect of the model corresponds better to the way things actually work in the NS. But we’ll know more once we build the model. Indeed, we should compare this EP model of limb movement to the PCT model as implemented in the Little Man. I’m sure that would be a way to say some nice things about this version of the EP model; it’s more true to the physiology.

WM: PCT was not developed to explain limb motor control specifically but perceptual control generally in any behavioural system across species, robots and even aliens! So I don’t see a problem here in magpie-ing some domain-specific clues.

What do people think?

RM: Again, super idea!! I’m for carrying on our discussion of work on the model on CSGNet so people can see what we’re up to. But doing it via personal communication is fine with me to.

Best

Rick

Warren

Warren

Sent from my iPhone

On 2 Jan 2014, at 20:15, Bruce Abbott bbabbott@FRONTIER.COM wrote:

[From Bruce Abbott (2014.01.02.1515 EST)]

Iâ€™ve obtained a copy of an article that performs a stability analysis of one version of the Î» version of the equilibrium point model for a single joint. (There are other EP models out there.)

The article reference is Lan, L., & Zhu, K. Y. (2007). Biomechanical stability analysis of the lambda-model controlling one joint. International Journal of Neural Systems, 17(3), 193-206. These authors conduct a mathematical analysis of this system and then present the results of simulations based on that analysis. Below is a block diagram of that version of the EP model:

<image001.jpg>

In this version of the EP model, the â€œreciprocal commandsâ€? represent changes in the opposite direction of Î» for the opposed flexor and extensor muscles, whereas â€œcoactivation commandsâ€? change Î» in the same direction. Changes in R change the equilibrium point function, yielding different final joint angles. As C increases, the opposing muscles pull more strongly against each other, increasing muscle stiffness.

The next figure shows the dynamic responses of the system:

<image002.jpg>

This is a stable system in that all starting points converge to a single point. This behavior can be seen in a graph of joint-angle change as a function of time:

<image003.jpg>

The system does a bit of â€œringingâ€? or â€œhuntingâ€? around the final joint angle for some combinations of R and C. Note that these oscillations disappear with higher values of C. Different values of R result in different final joint angles.

The authors did not evaluate the model against empirical data as that was not the purpose of their analysis. They simply wished to conduct of rigorous evaluation of the stability of the target EP model.

Bruce

Richard S. Marken PhD
``````                                               -- Bertrand Russell