Fancy nonlinearities; gain control

[From Bill Powers (920722.0800)]

Martin Taylor (920720.1615) --

There is an ECS called Y whose output affects something in the outer
world called B that provides a sensory input b, which Y controls as the
percept y(b). There is another ECS called Z whose loop goes through B >and

separately through D. Z controls for the percept z(b+d). Take B >and D to
represent gain functions such that delta(b) = B delta (Oy + >Oz) where Oy
and Oz are the outputs of the two ECSs, and delta(d) = D >delta (Oz).

Can't picture it. Can you set up the equations for the two ECS in an
environment, or draw a diagram so I can do it? After my experience with
clever algebraic solutions I think I want to see a simulation of this.

My Degree of Freedom argument (920701 0240) suggests that gains must be
changeable, even if it is only in the binary sense that some ECS turns
off to allow another to take control

When you introduce gain changes, there are two ways to do it. One is to
have a control system specifically concerned with sensing and controlling
the gain of another system. The other is just to make the system nonlinear.
The first method is tricky, in that you have to decide what perception the
separate gain-controlling system is monitoring (how do you make gain a
controlled variable?). But Ve Haf Vays.

There is a way of turning control systems off in a nervous system that is
very simple. It starts with the realization that neural functions are
always one-way -- that is, neural signals can't go negative. In the
standard diagram, we have error = reference - perception. This means that
the perceptual signal is inhibitory, the reference signal excitatory. If
the reference signal is simply set to zero, there is no excitation of the
comparison neuron, and no amount of inhibitory perceptual signal will ever
make it fire. So this comparator will produce zero error signal if the
reference signal is zero, regardless of the amount of the perceptual
signal. The control system is turned off.

To get two-way control about zero, a pair of control systems is always
required in the nervous system. The pair of systems treats opposite
directions of change of the perception as positive, and the error signals
in the paired systems have opposite effects on the controlled variable. The
simplest example is a pair of opposed muscles and their associated spinal
control systems for controlling force. If the arm exerts a leftward force,
the left-controlling control systems sense and control a positive force to
the left. If the force is to the right, the right-controlling control
systems sense and control a positive force to the right. This much you'll
find in BCP.

To think of this pair of control systems as a single control system that
can exert a continuum of forces passing through zero, we must think of the
reference signals as a balanced pair. If the rightward reference signal is
nonzero, there is a force to the right. As this reference signal declines
toward zero, the rightward force declines toward zero. Then, just as the
rightward reference signal reaches zero, the leftward one begins to rise,
and the force begins to rise toward the left.

If both the rightward and leftward reference signals are zero, this pair of
systems is turned off. A disturbance may cause an inhibitory feedback
signal to arise, but because there is no excitatory reference signal
reaching either the leftward or rightward comparators, there will be no
error signal to drive either of the pair of outputs. The system will not
resist disturbances in either direction.

In order to achieve control of an arm in the state of zero net force, it's
necessary to add a common-mode signal to the pair of reference signals. Now
the "resting" state is that in which both control systems contain error
signals, causing the muscles to pull against each other. The net left-right
force is zero, but any force disturbance will cause one error signal to
decrease and the other to increase, so there is control in the vicinity of
zero NET force. Both control systems in the pair are receiving nonzero
reference signals now, with only the difference in magnitudes showing up as
a net left or right force.

The common-mode force is, of course, called muscle tone. A control system
that controls for muscle tone controls to keep the SUM of the two positive
force signals at a constant level. A second control system can then control
to keep the DIFFERENCE between the same two force-signals at another
reference level, which sets the net sideward force to left or right. The
difference-controlling system has to emit a pair of output signals that
vary in a complementary way; in fact it must also have a balanced pair of
comparators in order to handle positive and negative errors. The higher-
level muscle-tone control system can be a one-way system, because the sum
of the muscle tensions can never be less than zero.

In our work on the arm model, Greg Williams found a reference that provided
force-length curves for various muscles. These curves can be fitted quite
closely with a second-power function over most of the force range (below
the saturation level of tension). Muscle tension is produced by stretching
the passive component of the spring, so muscle tension goes very nearly as
the square of the driving signal and the amount of contraction.

When you oppose two such muscles, the net force as a function of length is
represented as the difference between two offset square functions.
Let c = common-mode contraction, and d = differential contraction. Then

F1 = (c + d/2)^2 and

F2 = (c - d/2)^2

As a result, we have

F1 - F2 = (c^2 + 2cd/2 + d^2/4) - (c^2 - 2cd/2 + d^2/4) or

F1 - F2 = 2cd.

This says that the differential force produced by a differential
contraction is proportional to the common-mode contraction: that is, the
output sensitivity of this force-generator is so determined. If the rest of
the system is linear, the loop gain of this force-control system is
linearly proportional to muscle tone, and the differential force at
constant muscle tone is a LINEAR function of the differential contraction
in the two muscles (until one muscle or the other totally relaxes).

This is why there is no control when you totally relax all your arm
muscles, which means setting muscle tone to zero. An external disturbance
will not produce any reaction; the arm will just give way and swing like a
dead fish.

In order to get control, you must raise the muscle tone from zero, so there
is some mutually-opposing force. Up to a point, the greater the muscle
tone, the higher the loop gain gets. That's why you tense your muscles when
you have to do something delicate. But if you tense them too much, the loop
gain will get too high, and you'll lose fine control again as system noise
gets amplified and also as dynamic instability approaches.

I think this principle of gain control may apply generally in neural
control systems. In one-way control systems, gain is zero when reference
signals are near zero, becoming high only when signals are increased into a
more linear part of the input-output functions. To get tight control of
neural signals near zero, the reference signals and control systems must be
present as balanced pairs, with a common-mode signal determining gain and a
difference-signal doing the controlling. Whether single-ended or double-
ended, a system is turned off when ALL reference signals associated with it
are set to zero.

In a balanced system, "zero reference signal" really means EQUAL reference
signals in a balanced pair of systems. In an unbalanced (one-way) system,
zero reference signal is zero reference signal.


David Goldstein (920721) --

I was surprised that Bill chose time to be the main measurement.

I didn't; Pat did.

Best to all,

Bill P.