# Trilling, escape

[From Bill Powers (930118.1000)]

Rick Marken, Gary Cziko, Avery Andrews (930117 ...) --

This is getting entirely too complicated. What we're talking
about here is frequency response. If you vary the reference
signal for a finger-position control system up and down in a sine
wave, the finger will move up and down in a sine wave. With a
constant-amplitude reference signal sine wave of very low
frequency (one cycle per 10 seconds), the finger will move up and
down so the perceived position matches the reference signal at
every point on the sine wave. If you now start raising the
frequency of the reference sine wave while keeping its amplitude
constant, the perceived finger position will still track the
sine-wave reference signal faithfully up to some frequency.

Above that frequency, the finger sine wave will begin to get a
little smaller in amplitude, and lag a little in phase behind the
reference sine wave. The higher the frequency, the smaller the
finger oscillations, the more the phase lag, and the greater the
error signal in the position control system. The reason is that
the loop gain is falling off because one or more components in
the loop are not responding proportionally, but in a way that
becomes less and less at higher frequencies.

If you stop raising the frequency when the amplitude of the
perceived finger position has fallen to 0.707 times the amplitude
of the reference signal, the phase of the finger position will be
lagging the reference signal by about 45 degrees. You can now
bring the finger sine wave back to the original (low-frequency)
amplitude by increasing the amplitude of the reference sine wave
by a factor of 1.414. There will still be a largish error in the
position control system, and a phase lag of 45 degrees, but
you're compensating for the drop in amplitude by increasing the
amplitude of the reference signal sine wave.

Clearly, you can continue raising the frequency of the reference
signal changes, maintaining the amplitude of the finger movements
the same as it was at the lowest frequencies by means of
increasing the amplitude of the reference signal sine wave. You
will eventually run into a limit when, with increasing frequency
of the reference signal, the error signal becomes as large as it
can physically get or the finger-moving muscles are producing the
maximum possible peak forces. That is the maximum frequency of
finger movement.

With practice, the muscles become stronger and capable of
accelerating the finger faster; it may be that more parallel
neural pathways develop so larger error signals can be supported.
But whatever the degree of practice, the response of the position
control systems will always fall off as frequency increases, and
there will always be some limit beyond which increasing the
amplitude of the reference signal variations can no longer make
up for the loss of loop gain in the position control systems.

This phenomenon can, therefore, be adequately modeled without
bringing in any qualitative changes, as from open-loop to closed-
loop. There is a continuum of changes as frequency is increased,
with tight control for low frequencies and progressively looser
control as frequency increases. A higher system can compensate
for the loosening of control, to some extent, by increasing the
excursions of the reference signals.

ยทยทยท

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Avery Andrews (930118.1839) --

On a completely different topic, I stumbled across a Camhi
article from 1985 producing some evidence for closed-loop
control of cockroach esacpe direction; yet when the exchange
with Beer was on a year or so ago, he said that the escape was
known to be open-loop.

Beer was wrong. He was talking, I think, about the speed of
escape, not the direction. I saw the Camhi thing, too, I think.
It was clear that both the distance moved and the direction of
movement were related to the nearness and direction of the scarey
stimulus. It's been my experience that people who insist on open-
loop explanations often distort the facts or speak in qualitative
terms where there are actually quantitative relationships. Note
that the term "escape" has a built-in directionality that's taken
for granted.
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Best to all,\

Bill P.