[From Bill Powers (950914.2025 MDT)]

Bruce Abbott (950914.1840 EST) --

I don't wish to quote Bolles more extensively than I already have,
but the thrust of the position he outlines is that body weight, fat
storage, etc. may not be regulated around some set point at all,
but may result instead from a kind of dynamic equilibrium at which
the animal settles under given conditions. Thus, reduced body
weight follows when food is adulterated with quinine, because
palatibility is one of the factors that determines the equilibrium
value. In such a system, the search for a comparator and a
physical reference signal would be futile.

Don't let's get hung up on words. "Settling point," "dynamic
equilibrium," and other such terms are just ways of talking about
reference levels. Basically you have two variables and two functions
relating them through different paths. If you look at the equations for
a pendulum, you find that an applied force causes the pendulum to move
to one side (one relationship) and at the same time the restoring force
depends on the amount of deviation (the second relationship). Put these
two relationships together and you get a system of equations (in this
case differential equations) from which you can deduce the state of the
system, either stationary or oscillating.

A control system is simply one of these systems of equations in which
one of the functions is particularly steep at one point within its

···

* /
* /
* /
* /
/ *
/ *
/ *
/ *
/ *

---------------------------------------------

... you have this:

************* /
/
/
/ *
/
/
/ *
/
/

---------------******************************-

If there is a small change along the x axis in the first case (the
asterisk line), there is a small drop along the y axis. In the second
case, the same small change along the x axis causes a very large change
along the y axis. The first case represents a physical equilibrium; the
second case requires some kind of amplification and a power source to
drive it. That's the main difference. The second case is how a control
system behaves.

On the net, we have discussed several times the fact that most
physiological control systems have to be one-way systems: in the nervous
system, neural signals can't go negative in frequency, and in
biochemical systems, concentrations can't go negative. To get two-way
control, therefore, two opposing control systems are required, one
handling negative errors and the other positive errors. As Avery Andrews
(950915) points out, even the muscles are one-way devices; they can pull
but not push. (Of course if the reference signal is always considerably
positive, two-way control may not be required).

The muscle response to driving signals closely follows a square law. As
a result, if the reference signals for two control systems using opposed
muscles are set slightly above zero, the curves of the opposing output
tensions will have an overlap region. Within this region, the _combined_
output functions are linear, with a slope that depends on the degree of
overlap. So the gain of the composite system can be varied by varying
the mean reference signal, while the position of a limb can be varied
linearly by increasing one reference signal and decreasing the other
around this mean value (which has been called muscle tone). This kind of
model requires treating all reference signals and perceptual signals as
balanced pairs, one increasing while the other decreases. In my
(unpublished, naturally) model of oculomotor control systems, I used
this balanced-pair concept throughout, with all control systems acting
in one direction only.

Combining two opposed control systems can result in a smooth transition
from one direction of control to the another, as in the case of muscle
systems, or it can leave a dead zone between the active regions. The
dead zone limits the accuracy of control near the zero value of the
controlled variable, but it can also prevent a waste of resources when
there are small perturbations always present. Dead zones are used in
spacecraft stabilization systems to keep from wasting propellant in
correcting small attitude variations due to crew movements and
mechanical operations, most of which will eventually average out to
zero. We would probably find dead zones in some human control systems,
too, having much the same function.

When the dead zone is wide enough, the combined system operations look
like a limit-avoidance system, with no action being taken until the
controlled variable has reached some fairly large deviation from the
center. In living control systems, it often happens that very different
output mechanisms are involved in the two directions, as for the body
temperature control systems which use different mechanisms for
increasing and decreasing body temperature -- even though the same
variable is being controlled in either direction (temperature).
-----------------------------
It would indeed be futile to look for a box labeled "comparator" and a
signal labeled "reference signal" in the real organism. The basic
control-system block diagram is meant to represent functional
relationships among the physical variables. There are endless specific
circuits that could create exactly the same relationships. For example,
suppose that the perceptual signal, instead of entering a separate
comparator, entered an output function directly. Then the output (in a
linear system) would be

o = k*p.

We could now add a reference signal that also directly enters the output
function, its effects subtracting from those of the perceptual signal:

o = k1*p - k2*r

This is in fact how the reference signals and perceptual signals meet in
the motor nuclei of the brain stem; the reference signals are inhibitory
and they enter the motor output function directly.

This is the same as

o = -k2*(r - p*k1/k2),

which is our standard comparator and output model if we just redefine
the perceptual signal into slightly different units such that k1/k2 = 1,
and measure o on a scale that is pointed in the right direction to get a
negative sign.

Similarly, we could say that the reference signal adds a negative
(inhibitory) effect into the input function, with the perceptual signal
directly driving the output function. It's just a matter of a few
algebraic manipulations and rescalings to show that the resulting system
is exactly equivalent to our standard diagram. So the standard diagram
actually covers a multitude of physical arrangements which all have the
same net effect.

In some cases, for example the CO2 detectors in the carotid artery, the
reference signal may be a built-in physical property of the sensory
neurons. Remember that the operational definition of a reference level
is that level of the input variable at which the output just becomes
zero (approached from the appropriate side). The CO2 detectors have a
certain sensitivity to CO2, but also a threshold. There is no signal
until the CO2 partial pressure rises above some threshold amount, and
that threshold is the reference level, exactly as in the operational
definition. Reactions against further rises in CO2 do not begin until
the CO2 level has risen above that reference level. In this case,
different properties of the sensory neuron are acting as the input
function, the comparator, and the reference signal. By representing
these properties appropriately, we could use the same standard control-
system diagram to represent this system. The reference "signal" would
simply be a physical property of the sensory neuron -- but even so, it
might turn out that the threshold level is affected by chemical inputs,
temperature, or pressure, so there is a real reference signal after all.

It is possible to measure the characteristics of apparently two-way
control systems and find that there are different parameters that apply
to the different directions. I have done a little in that direction for
the tracking experiments, and have found some small differences in my
own systems. I don't know if they're meaningful. In general, we would
expect to find different parameters, because physically different
systems have to be involved. The basically one-way nature of neural and
biochemical systems, and muscles, requires this.

I mention all this just to reassure anyone who wonders about it that we
have considered far more elaborations than are contained in the simple
models we generally use. But it is unrealistic to think that we are in a
position to propose and test models with that degree of detail. Before
that can happen, PCT has to acquire serious support and be in a position
to attract and pay full-time researchers from many disciplines. We can
see what it might be like to fly, but at present we had better
concentrate on learning to walk.
------------------------------------
Bolles, apparently, shares a prejudice with most biologists who don't
like the idea of set-points or reference signals and are trying to avoid
control theory. Somehow if they can reduce the phenomena to a mere
equilibrium between opposing forces, they feel more comfortable with an
explanation. But they don't ask the right questions. How come one of
those forces changes so rapidly right at the "sharp-cornered ceiling?"
Why is that ceiling where it is? It is always at the same level? Is this
merely a meeting of blind forces, or is one of the forces part of an
active, amplified, control process? I don't know why this idea gives
biologists and psychologists the heebie-jeebies, but it clearly does.

Such limits could take simple forms, as when (to use one of
Bolles's examples) a cat grows heavier until its excessive weight
begins to limit the number of mice it can catch.

By not learning about the properties of control systems, Bolles has
deluded himself. As the cat's weight increases, its mouse-catching
abilities do not gradually decline at the same time (as he evidently
imagines). They stay essentially the same until finally the cat can't
increase its efforts enough to counteract the effects of its increasing
weight, and then its mouse-catching goes into a steep decline. The
thirty-five-year-old wide receiver may have lost a step, but when he
turns thirty-six he loses it all and has to retire. You can hold your
arm outstretched for x minutes, keeping it in exactly the same position.
In the middle of the x+1th minute, the arm starts to sag and you have to
give up. The onlooker doesn't know, but you can feel the effort
continually increasing all during those long minutes. When the ability
to drive the muscles still harder finally reaches its limit, the result
is collapse. Bolles is telling us what he imagines will happen according
to his model of how behavior _should_ work. But he's not reporting what
actually does happen. That's the trouble with making up examples to
support a belief.
-----------------------------
Control theory is based on a straightforward analysis of real systems.
It's just systems analysis applied to systems that happen to be control
systems. We never need to force any issues here, or engage in rhetoric
designed to persuade ourselves that we're looking at a control
phenomenon. Either we are or we aren't, and a proper analysis of the
system will show us which. The outcome doesn't matter; it would be just
as bad to conclude incorrectly that we're seeing control as to conclude
incorrectly that we're not. By actually doing a systems analysis of the
experiments you've brought before us, we have managed to show that there
is control of intake in those experiments, but it is accomplished by
varying meal size and not, as I originally thought, by varying rate of
pressing. There may be situations where rate of pressing is varied as a
means of control, but this was not one of them.

When we can analyze behavior in this way, what does it matter what we
think or what Bolles thinks is "really" going on? Forming any idea in
advance is unnecessary; the analysis will show us what is going on. Of
course we can get a head start if we suspect correctly that control is
going on, but if we do the analysis right, we can find out when our head
start was in the wrong direction. The main thing to keep in mind is that
the outcome DOESN'T MATTER, as long as we do the analysis in such a way
that we can be confident of the results.
-----------------------------------------------------------------------
Best,

Bill P.