[From Bill Powers (2000.05.06.1619 MDT)]
Isaac thought our discussions of muscle properties were on CSGnet (it was
actually a limited-circulation post), and after thinking it over, I decided
what the heck, we modelers have just as much right to CSGnet as people who
want to talk about cooperation, and nobody is required to comment on things
they aren't interested in, so why not?
The problem to which I refer below is an apparent exponential dependence of
muscle force on stretch. When I tried to model it, I got an awful result,
with lots of lag and hardly any loop gain before oscillations set in. I've
been beating my brains out over an article by Shadmere, R. and Arbib, M. A.
(1992) in Biological Cybernetics,( a mathematical analysis of
force-stiffness characteristics of muscles, Biol Cybern _66_, 463-477, if
anyone wants to look it up). Very complicated and detailed, and in places
beyond my bandwidth. But further study has shown me that I used the
equations the wrong way. I haven't got the final answer yet, but I can see
the way now...
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Hi all. I think the solution is at hand. The problem is that the way
stiffness is tested is not the appropriate way. At least it completely
misled me. As stiffness increases, the force generated by a standard amount
of stretch increases. But this assumes that the external agent doing the
stretching simply alters the length of the muscle by a preset amount no
matter how much (or how little) resistance the muscle offers. So of course
as the muscle gets stiffer, the external agent simply increases the applied
force as much as necessary to get the same amount of stretch. That gives
the exponentially increasing curve of force as a function of stretch.
However, stretch does not cause force; instead, force causes stretch.
When a muscle contracts or when loads are applied to a limb, what is
applied is a force, not a fixed increment of muscle length. As the average
force increases, the muscle gets stiffer, and the result is to _decrease_
the amount of stretch for a given change in force. A given increment in
force is transmitted to the attachments through the nonlinear spring
through smaller and smaller increments in the length of the muscle as the
force increases. The coupling of force to the attachments gets tighter.
External loads on a limb are generally applied as forces, not as fixed
displacements. The directly appropriate way to measure muscle stiffness
would be to vary the stretching _force_ and measure the amount of
displacement that results. Then we would see how much displacement of a
limb is caused by an applied force, rather than how much force must be
applied to produce a fixed displacement. Of course the appropriate term
would then be "compliance," the reciprocal of stiffness. Compliance
decreases as the level of force increases. The result is simply the same as
the authors' plot of fig. 7, but turned on its side and then flipped left
for right.
The Golgi tendon receptors measure the mechanical stretching of the tendon,
or the capsule in the tendon. If the tendon itself is a linear spring at
the place where the Golgi receptors are, then it measures directly the
force due to muscle contraction (muscle length is irrelevant). If the
tendon shares the same nonlinear character as the muscle's overall series
elastic component, then the Golgi signal would indicate the log of applied
force. Either way, this is a heck of a lot better than what I had assumed
before, that the Golgi signal was an exponentially increasing function of
the muscle tension.
I think I can come up with an appropriate model now. A few more days...
Best,
Bill P.
