Apparent Stiffness: EP versus Powers (1999)

[From Bruce Abbott (2014.02.19.1155 EST)]

Powers (1999) extended the “Little Man” demo to include kinesthetic control of joint angle via the muscles. The bottom-level system was modeled as itself having two levels arranged in a hierarchy, the inner loop controlling the perception of force (sensed via the Golgi tendon organs) and the outer loop controlling the perception of muscle length (sensed via the muscle spindles), the latter and acting by setting the reference for the force system. The model of opposing muscles acting on a single joint was simplified by representing the actuator as if it were a single muscle that can act both directions rather than two opposing muscles each acting only in the direction of contraction. In the full model, higher-level systems set the references for muscle length to accomplish movement of the shoulder around two rotational axes and movement of the elbow around a single axis.

In his analysis of this system, Bill examined the behavior of the muscle system across a single joint (e.g., the elbow joint). On page 471 he states “The control model as developed so far, with stretch and tendon reflexes alone, behaves like a person with an injury that cuts off brainstem control. If reference signals Rg and Ra are set to hold the arm out strait and level and gravity is then switched on, the arm will slowly sag as much as 45 degrees, a condition known as “waxy flexibility.” Then on page 472 he continues as follows: “The control system so far consists entirely of spinal reflexes, with position being controlled only through sensing muscle length. . . .the parameters which optimize the reflex loops for fast control leave position control against sustained disturbances rather weak. The effective return spring constant is only Kg/Kt + 1/(KoKt). A reasonable number of KoKy is 200, and Kg/Ko is approximately 3, leaving an effective spring constant about 3 nM per radian of deflection. This is why the spinal arm sags under gravity.

The effective spring constant, sometimes also referred to as the apparent stiffness of the muscle, is measured in torque per unit of angular deflection. So, how does the EP model compare in terms of apparent stiffness? I modified my EP model demo to produce a constant “gravitational” force regardless of joint angle (that is, a constant torque on the joint) and recorded the steady joint angle for each of the weights (1 to 5 kg) and for different settings of the C-command (0, 5, 10, 20, 40, & 80). R was set to 0 so that the recorded joint angles can be taken as the amount of deviation of joint angle from zero. I then computed the stiffness at each of these values in Newton-meters per degree. When C = 0 (no co-contraction), stiffness varied with the applied torque from 0.17 for the 1 kg weight (torque = 5.9 Nm) to 0.46 for the 5 kg weight (torque = 29.4 Nm). Bill’s figure for his model of 3 nM per radian converts to 0.052 Nm per degree. So the EP model apparent stiffness with no co-contraction is higher than the apparent stiffness of Bill’s model by over a factor of three at the lowest torque and by a factor of almost 9 at the highest. Stiffness rises to about 3.3 Nm per degree when co-contraction = 80 and becomes nearly constant over the range of torques tested.

I conducted this comparison with Rick Marken’s assessment of the EP model in mind, in which he pointed to the amount by which the simulated forearm sags under added weight as a reason to reject the EP model. It would appear that the Powers model is even more compliant (before a higher level of control is added).

Both models treat the muscles as analogous to damped springs. Both models make the muscles behave as springs by including a feedback path that senses muscle length via the spindles and apply feedback, via the alpha motor neuron, that opposes stretching. Neither model by itself (that is, without higher levels of control to set the references) does a very good job of opposing externally applied torques.

The EP model includes an input for setting co-contraction of the opposing muscles thus allowing stiffness to be adjusted as needed, but this ability is absent in Bill’s model due to the simplification of representing the actuator as a single bi-directional mechanism. Because co-contraction can substantially raise the stiffness of the system (as needed, for example, when rapidly moving the arm from one position to another over a relatively large distance), and is known to be employed in the real systems being modeled, Bill’s model will need to be revised to represent the opposing muscles and their unidirectional action so as to permit co-contraction as called for by the higher-level systems during movements such as reaching and pointing.

Bruce

[From Rick Marken (2014.02.19.1600)]

···

Bruce Abbott (2014.02.19.1155 EST)

BA: In his analysis of this system, Bill examined the behavior of the muscle system across a single joint (e.g., the elbow joint). On page 471 he states “… If reference signals Rg and Ra are set to hold the arm out strait and level and gravity is then switched on, the arm will slowly sag as much as 45 degrees, a condition known as “waxy flexibility.”

RM: And shortly thereafter, on p. 471, Bill says: "This [waxy flexibility] occurs when the parameters are adjusted so that repetitive square-wave changes in the reference signal

are followed accurately by the arm with a time constant of about 0.1 s." [Emphasis mine]. In other words this waxy flexibility only occurs for parameter settings of the first level kinesthetic control systems that let these systems respond quickly to reference signal changes from the second level (pitch, roll, yaw) control systems.

You also left out this from p. 477. " Figure 7 shows the behavior of the elevation angle with both levels of kinesthetic control active and a square-wave reference signal being sent to the second level reference input. The response [to the change in reference signal] is a little slower. Now, however, switching gravity on and off makes no perceptible difference in the plots."

So with a slight change in the parameters of the kinesthetic systems – the one’s now responding a little slower – we get a slightly slower response to a change in reference signal from the higher level systems but no detectable effect of gravity.


BA: I conducted this comparison with Rick Marken’s assessment of the EP model in mind, in which he pointed to the amount by which the simulated forearm sags under added weight as a reason to reject the EP model. It would appear that the Powers model is even more compliant (before a higher level of control is added).

RM: It’s not clear to me that it is the addition of the higher level control systems that takes away the “waxy flexibility”. But I admit that it is somewhat ambiguous. But based on what Bill said above when he was just talking about the first level systems it sounds to me like the “waxy flexibility” was eliminated from these kinesthetic (muscle tension, length) control systems.

RM: By the way, in that paper there is a whole section comparing the control model to the mass-spring model (which is basically what the EP model is). Bill suggests ways to compare these models. But I think we can compare them nicely by comparing the EP Model to Bill’s first level kinesthetic control model in that 1999 paper.

Best

Rick


Richard S. Marken PhD
www.mindreadings.com
The only thing that will redeem mankind is cooperation.
– Bertrand Russell

[From Bruce Abbott (2014.02.20.0820 EST)]

Rick Marken (2014.02.19.1600) –

Bruce Abbott (2014.02.19.1155 EST)

BA: In his analysis of this system, Bill examined the behavior of the muscle system across a single joint (e.g., the elbow joint). On page 471 he states “… If reference signals Rg and Ra are set to hold the arm out strait and level and gravity is then switched on, the arm will slowly sag as much as 45 degrees, a condition known as “waxy flexibility.”

RM: And shortly thereafter, on p. 471, Bill says: "This [waxy flexibility] occurs when the parameters are adjusted so that repetitive square-wave changes in the reference signal

are followed accurately by the arm with a time constant of about 0.1 s." [Emphasis mine]. In other words this waxy flexibility only occurs for parameter settings of the first level kinesthetic control systems that let these systems respond quickly to reference signal changes from the second level (pitch, roll, yaw) control systems.

BA: The reference signals being sent a repetitive square wave are those of the first-level system. This system has six adjustable parameters, which according to the text were adjusted for optimal performance (section 3.1 page 476).

You also left out this from p. 477. " Figure 7 shows the behavior of the elevation angle with both levels of kinesthetic control active and a square-wave reference signal being sent to the second level reference input. The response [to the change in reference signal] is a little slower. Now, however, switching gravity on and off makes no perceptible difference in the plots."

So with a slight change in the parameters of the kinesthetic systems – the one’s now responding a little slower – we get a slightly slower response to a change in reference signal from the higher level systems but no detectable effect of gravity.

BA: The sag was eliminated by adding the second level control systems and applying the square-wave manipulation to the references at that level rather. Bill didn’t say anything about changing the parameters of the first-level system to get this result. The only change he mentions is adding in the second level control system – which controls joint angle. This makes sense, because the relatively slow sag permitted by the first-level system would be counteracted by the second-level system through an adjustment of the references of the first-level system. To the second-level (joint-angle control) system, the sag is just another disturbance, which it deals with as it does any disturbance to joint angle.

For the same reason, I expect that adding a second-level joint-angle control system to the EP first-level model will produce a similar result.

BA: I conducted this comparison with Rick Marken’s assessment of the EP model in mind, in which he pointed to the amount by which the simulated forearm sags under added weight as a reason to reject the EP model. It would appear that the Powers model is even more compliant (before a higher level of control is added).

RM: It’s not clear to me that it is the addition of the higher level control systems that takes away the “waxy flexibility”. But I admit that it is somewhat ambiguous. But based on what Bill said above when he was just talking about the first level systems it sounds to me like the “waxy flexibility” was eliminated from these kinesthetic (muscle tension, length) control systems.

BA: I rather doubt that this was the case; no mention is made of readjusting the first-level system parameters. The fact that “the response is a little slower” (as Bill says) is what you expect of a second level control system, relative to the first whose references it manipulates.

RM: By the way, in that paper there is a whole section comparing the control model to the mass-spring model (which is basically what the EP model is). Bill suggests ways to compare these models. But I think we can compare them nicely by comparing the EP Model to Bill’s first level kinesthetic control model in that 1999 paper.

BA: Yes, I read that. Bill’s first-order system is also basically a mass-spring model. As Bill points out in the paper, the “apparent” spring constant is almost entirely a due to the actions of the two control systems he places in the first-order system, overwhelming any effect of the spring shown in the muscle diagram (unless the gain is too low). What Bill seems to have missed is that the “mass-spring model” he appears to be talking about (I assume it’s the EP model) acts as a mass on a spring for the same reason – the springiness in the EP model is not the passive springiness of the muscle but rather the result of feedback from muscle spindles to the alpha motor neuron.

BA: There are of course important differences between the two models, but this isn’t one of them.

Bruce

[From Rick Marken (2014.02.20.0940)]

···

Bruce Abbott (2014.02.20.0820 EST)

BA: The sag was eliminated by adding the second level control systems and applying the square-wave manipulation to the references at that level rather. Bill didn’t say anything about changing the parameters of the first-level system to get this result. The only change he mentions is adding in the second level control system – which controls joint angle. This makes sense, because the relatively slow sag permitted by the first-level system would be counteracted by the second-level system through an adjustment of the references of the first-level system. To the second-level (joint-angle control) system, the sag is just another disturbance, which it deals with as it does any disturbance to joint angle.

RM: Yes, on re-reading I see that you are correct. The sag would be corrected by the angle control system.

BA: For the same reason, I expect that adding a second-level joint-angle control system to the EP first-level model will produce a similar result.

RM: Yes, it I think it surely would.

RM: By the way, in that paper there is a whole section comparing the control model to the mass-spring model (which is basically what the EP model is). Bill suggests ways to compare these models. But I think we can compare them nicely by comparing the EP Model to Bill’s first level kinesthetic control model in that 1999 paper.

BA: Yes, I read that. Bill’s first-order system is also basically a mass-spring model. As Bill points out in the paper, the “apparent” spring constant is almost entirely a due to the actions of the two control systems he places in the first-order system, overwhelming any effect of the spring shown in the muscle diagram (unless the gain is too low). What Bill seems to have missed is that the “mass-spring model” he appears to be talking about (I assume it’s the EP model) acts as a mass on a spring for the same reason – the springiness in the EP model is not the passive springiness of the muscle but rather the result of feedback from muscle spindles to the alpha motor neuron.

RM: So we’re back to the EP model being a control model? Controlling muscle length?

BA: There are of course important differences between the two models, but this isn’t one of them.

RM: Well, we shall see. Again, I think the crucial test of the two models will be in terms of how they deal with varying torque disturbances. Bill’s model apparently handles such disturbances (resulting from the limb movements themselves) quite nicely. But, again, this may be the result of the operation of the level 2 systems. So I look forward to seeing what you come up with.

Best

Rick


Richard S. Marken PhD
www.mindreadings.com
The only thing that will redeem mankind is cooperation.
– Bertrand Russel