[From Bill Powers (2000.05.07.1147 MDT)]
First, an aside:
Here's one for Gary Cziko. The seed pods of the mountain mahogany bush
(growing in our back yard) are fuzzy spiral helixes (large diameter at the
top, small at the bottom, a little over 1 turn) with a sharp pointed seed
at the bottom. They are blown by the wind and descend like spinning
helicopters, somewhat like maple seeds. When they hit the ground the sharp
seed end hits first, resting on the ground while the pod tilts so the
spiral part also rests on the ground. Then humidity changes make the spiral
part expand and contract, as a result of which the seed pod SCREWS ITSELF
INTO THE GROUND.
Here's another remark for Gary. Gary Cziko's latest book, _The things we
do_, turns out not only to be about his "universal selection theory," but
about all the advances in PCT that have taken place since the first book
was published, including all the most important developments in the 10
years of existence of the internet discussion group (CSG-L at first, then
CSGnet). I know of no other place where so much of current thinking about
PCT is preserved in a mainstream book. And Cziko has put PCT into a far
larger historical context with its roots in the thinking of Bernard and
Darwin. For those who grasp the message, this book ought to provide an
inspiration (inspiration: a gasp).
···
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Muscle model:
In the Shadmehr and Arbib (1992) paper, a model of the areflexive muscle is
constructed, starting on p. 470. This development ends with a _linear_
muscle model: a force generator in parallel with a spring and a dashpot,
all in series with another spring. Without the force generator, the model
consists of two springs in series, the spring constants being about equal
and one spring having a dashpot in parallel with it (which resists motion
in proportion to velocity). This arrangement seems to have been used
strictly to account for the "quick-release" experiment, in which a
stretched isolated muscle is suddenly released from a fixed catch and
subject to a small tension after that. The muscle typically shortens very
rapidly after the release (that is the series spring shortening), after
which the shortening approaches a somewhat shorter limit via a negative
exponential curve (the dashpot slowing the shortening of the parallel spring).
The parallel spring, however, is active only when stretched beyond its
resting length. The authors cite McMahon as saying that the resting length
(of the two springs in series) corresponds to "the length that a muscle
occupies in the body," but they don't say whether this is the maximum ,
average, or minimum length (for example, the length of the biceps with the
elbow extended or fully flexed). I assume it is the maximum length. This is
important because if the resting length corresponds to the muscle length
when the joint is fully open, then no external process can possibly stretch
the muscle further _in vivo_. For the series elastic element this makes no
difference, but for the parallel element, consider this: "The forces which
develop at lengths smaller than [the resting length] are solely due to the
action of the active component." If no external force can lengthen the
muscle further, then the muscle can only contract during normal actions,
and the parallel spring element will always be shorter than its resting
length. It will never come into play at all. So the _only_ reason for
including the parallel spring element in the model is to take care of the
case in which a muscle is removed from the body and artificially stretched
beyond the length it would ever have in the body. We can therefore ignore
it in our models.
This being so, we need only a model of the contractile element in series
with an elastic element which stretches in proportion to the log of the
force applied to it.
An odd thing happens in the S&A paper. On page 469, the discussion is about
the stretching of a non-reflexive muscle, with reference to data showing
that the force is an exponential function of the change in length of the
series elastic element (this arises from the fact that the stiffness,
delta-force over delta-length, rises linearly with force). That is
equivalent to observing that the change in length of the series elastic
element under isometric conditions varies as the log of the muscle-applied
force. One would expect that the next thing would be to find the function
that turns neural signals into muscle tension, so that the tension can be
transformed by the nonlinear series spring into a displacement proportional
to the log of the tension.
Instead, the authors turn to a _linear_ muscle model (their fig. 8a), the
same one McMahon and others cite, in which both the series and the parallel
spring elements are linear rather than exponential. They then proceed to
derive a model for conversion of a neural signal of a given frequency into
a force to be applied in parallel with the parallel spring element (which
has, remember, no effect in the body). The force-generation model is taken
from data by Inbar and Adam (1976). After a number of manipulations, this
nonlinear model of force generation proves to create an exponential
relationship between stretch of the whole muscle and the force generated by
the whole muscle. So even though the nonlinear series elastic element is
made linear, the mathematical analysis manages to come up with a linear
increase of stiffness with force, and thus an exponential dependency of
force on stretch (or a log dependency of stretch on force).
This nonlinear model of force generation also makes the dashpot
superfluous, since the model integrates a series of impulses which build up
a force through time along a decelerating exponential curve -- the same
effect that the dashpot is supposed to create (Fig. 9). The dashpot simply
adds its own damping to that which is already present in the
force-generation model. While it can be left in the equations, as the
authors do, it is redundant.
Actually, the nonlinearity of this force-generation model comes from the
fact that overlapping impulses do not continue to add linearly as the rate
of twitching increases, but approach a limiting amount of effect called, if
I interpret it right, "tetanus." It is this falloff from linearity that
makes the driving signals less and less effective in producing more output
force as the input frequency rises, and that in turn is what creates the
appearance of a nonlinear force-displacement function via the _assumed
linear_ series elastic element. Only now, of course, there would be only a
_linear_ stretch as a result of applying an external force to the end of
the muscle. So the model fails to account for the data cited at the start
of its development but abandoned without any particular notice being given.
I will not swear that I understand the authors' mathematical development
correctly. However, it seems to me that they have really merged two models,
one that begins with a nonlinear series spring element, and then a second
one in which the nonlinear element is made linear and the required
nonlinearity is introduced -- again -- via the function that converts
neural signal frequency into muscle force. Also, there is redundancy in the
dashpot introduced to create the correct time constant for growth and decay
of force, because the neural model of force generation also creates a time
constant of the same sort.
I hope that someone else will check out this article and the math in it,
preferably someone who is more adept than I, to see if I have misconstrued
what is going on. If I am right, then I think we can come up with a much
simpler model starting with the Inbar and Adam data that covers all the
effects we expect to see _in vivo_. If anyone wants to put the parallel
spring back in, to account for the stretch-release data _ex corporum_, feel
free to do so, but it will have no effect on behavior of the model with the
muscle in its normal position (and resting length) in the body.
Isaac, if you want to ask the opinion of your professors or friends about
all of this, feel free to do so. I'd like to know what they say.
Best,
Bill P.