[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