[From Bruce Abbott (2014.02.13.1450 EST]
Rick Marken (2014.02.09.1600) –
RM: Note that for the EP model limb angle moves continuously toward the new equilibrium point after disturbance; there is no evidence of “push back” against the disturbance. (This is the behavior that is seen in the continuous graph of limb angle in Bruce’s EP model each time the disturbance is increased by 1 kg). The response of the P control model is quite different; limb angle is initially increased by the disturbance but the model “pushes back” to get the limb back as far as it can toward the reference angle (of zero). The gain of the P control system is so low that it doesn’t get the limb angle back very close to the reference but it is clearly working toward that end; there is push back.
RM: I’ve set the simulation up so that the P control system brings the cv (limb angle) back to a limb angle that is equivalent to the equilibrium point limb angle achieved by the EP model (ep). This makes it easier to see that the behavior of a P control model is quite different than that of an EP model, which suggests strongly that the EP model is not a control model at all.
RM: The fact that the EP model is not a control model is even more evident when one compares the behavior of the EP model to that of a control model that better represents what actually happens when increasing step disturbances of weight are applied to a limb. This is shown by the yellow line (labelled icv to indicate that these are the variations in limb angle that result when limb angle is controlled by an integral control system). Except for the brief “jerks” that occur at the points where the step disturbance increases, the control system keeps the limb angle right at the reference angle (0 in this case) protected from the increasing step disturbances. This corresponds to the behavior you would actually observe in a human. You could see this by by having someone hold a bag in their hand at a fixed angle from their body and then drop one pound weights one at a time into the bag. I think you will find that the behavior of the person’s arm angle over time will looks a lot more like the yellow plot (icv) than the green one (ep).
RM: This, by the way, is the correct way to test the models to see if they match real behavior.What you want to do is see if the model (EP or control) controls like the human does. So far the EP model has not been tested to see if it controls like humans. Indeed, the tests that have been conducted on the EP model have asked the participants to try not to control. As Bruce Abbot said in a recent post regarding the tests of the EP model in the Lan/Zhu paper:
BA: The participant has been instructed not to attempt to hold the initial joint
angle, but to allow the joint angle to change as it will. The angle at which
the forearm stops is a measure of the equilibrium point.
RM: I read this in the Lan/Zhu paper myself and I was gobsmacked; I couldn’t believe what I was reading. They are not testing to see if the model controls; they are testing to see if the model acts like a mass on springs whose spring constants are set by a machine at a control panel – a control panel that is presumably inside the participants who would otherwise be inclined to control the position of their limb. So the methodology used to test the EP model confirms (without my having had to have done all this testing) that it is not a model of control (of how people “hold the initial joint angle”, for example). The EP model is a model of the behavior of purposeless (non-control) systems. So the only mistake the EP theorists are making is in applying their model to the behavior of living systems. An easy mistake to make,apparently;-)
When I suggested that the EP model may be behaving like a low-gain proportional controller, the proportional controller I had in mind was one that acts through the same environment that the EP model does. In the EP model, a change in R produces a torque on the joint. This torque, divided by the moment of inertia, produces an angular acceleration, which is integrated with each time-step to get the velocity, which is integrated to get the position. In your spreadsheet simulation, you could have simulated the effect of these integrations on the stability of the system by employing our usual leaky integrator output. This would permit the controller to have a gain higher than 1.0.
Be that as it may, your simulation does show that the proportional controller begins to resist the initial effect of the disturbance in the next time-step and does “push back,” as you say, although not by much due to the low gain. The muscles of the EP model just stretch under the increased load – but not entirely passively (see below).
The EP model does have a feedback loop; in essence it determines something equivalent to the spring constant of the muscle. Without feedback the muscle would simply stretch passively when an external torque was applied to the limb. Feedback from the muscle spindles increases the activity of the alpha motor neurons, increasing muscle tension. The muscle will stretch under the external torque until the increasing counterforce developed by the muscle is enough to prevent further stretching.
In the EP model, changing R alters the lambda value of each muscle, the threshold length at which the muscle will begin to resist further stretching. When no external torques are present, the torques developed by the muscles will move the limb to the specified angular position if there is no co-contraction. Changing lambda effectively changes the zero-point of the spring, the point at which no counterforce is generated by the muscle, equivalent to the resting position of a spring that not being stretched.
I haven’t gotten far enough along in my research to know anything about proposals for the systems that set R and C. What if the physiologists have it right and the muscles are in fact contracted by changing alpha motor neuron thresholds as the EP model proposes? If that were the case, then a PCT model would have to act through the same mechanism. I can imagine a level 2 system that would set the R and C values while receiving its own proprioceptive input from the muscles, tendons, and joints. Such a system would automatically compensate for the springiness of the muscles by employing R and C values that brought the joint to the specified angle against external loads, so long as the loads were not overpowering.
With respect to the methods used to determine equilibrium points, those were specifically designed to determine whether the changes in joint angle following various degrees of unloading could be understood in terms of simple changes in alpha motor neuron thresholds, changes necessitated by the requirement to maintain position against the external torque prior to partial unloading. The change in joint angle after partial unloading is supposed to reflect only the involuntary reflex adjustments, which is why participants were told not to attempt to voluntarily resist the joint movement. Voluntary movement presumably involves participants changing R and C values via higher levels in the nervous system.
Until I learn more about how the EP model has been applied (e.g., to deal with voluntary control of joint position), I can’t say whether it is a mistaken model or a reasonable portrayal of how things work at the physiological actuator level. If the latter, then any model – command and compute, PCT, or something else, will have to deal with exerting control over joint angle via this EP mechanism.*
*That said, there are other models out there that propose somewhat different mechanisms at this level. The EP model may not be the right one to capture the actual physiology.
Bruce