[From Bruce Abbott (950501.1335 EST)]
Bill Powers (950501.0715 MDT) --
Bruce Abbott (950430.1710 EST)
I don't suppose it occurred to Pavlov that if he was collecting the
saliva to measure it, that same saliva couldn't have been performing its
normal function in the mouth, so he wasn't seeing the system in its
normal state.
There are several salivary ducts inside the dog's mouth (five or six, I
believe); Pavlov's operation redirected the output of only one of these,
leaving the dog with plenty of saliva with which to lubricate the food. At
worst, the operation only _reduced_ the amount of saliva involved in the
feedback loop and thus decreased the system gain a bit.
With more experience the UR tends to occur only near the end of the
CS-US interval.
Whoa! Now we have to put a timer into the model. This probably means we
have to consider at least a two-level model.
Time clearly is an important variable in classical conditioning; as Pavlov
observed, dogs will learn to salivate just prior to food delivery in the
absence of an explicit CS if the food is delivered at regular intervals
(e.g., every minute on the minute). However, such timing may not be
required for short CS-US intervals. The important consideration is which
perception (CS or time) provides the most reliable indicator of the time of
US onset. For short intervals, transport lags and other delays may obviate
any need for timing--by the time the output changes, the US is already arriving.
One more comment: the "strength" of response to the CS is always
some fraction of the "strength" of response to the US (at least for
salivary conditioning) Thus, if food is capable of eliciting, say,
30 drops of saliva over the course of 15 seconds, then a fully
conditioned CS would always produce less than 30 drops in the same
time period.
If this rule holds true, it tells us that the US is being treated by the
organism as if it is the CS.
You mean the other way around, don't you?
But doesn't this leave some quantitative
questions unanswered? What happens if you ring the bell twice as loudly
as usual? What if you reduce the amount of US? Could it be that the size
of the response is dictated by how much action is needed to counteract
the effect of the US?
The effect of CS intensity has been investigated. The current view, I
believe, is that for the most part, any effect of CS intensity is due to the
larger difference between the CS and other "contextual" sensory inputs. A
larger difference facilitates detection, or the capturing of attention
("salience"). Reducing the "intensity" of the US once a stable CR has
developed will be followed by a reduction in UR. And as for the last
question, yes.
A big problem is that the response to the US makes sense for the
organism only if the CS occurs. When you terminate presentations of the
CS, pretty soon the US becomes superfluous because there's never any
disturbance of "liquidity". Of course during the "extinction" phase, the
CS produces a response that is considerably larger than what would be
appropriate for the (nonexistent) US. So this raises the question, what
would happen if you halved the size of the US -- would the response to
the CS initially be too large, and slowly decline?
Yes. By the way, this effect is predicted by the Rescorla-Wagner model I
described several moons ago. Rescorla-Wagner, as you may recall, is just an
integrator model which uses trials rather than time. The strength of
conditioning to the CS is assumed to change, following each CS-US pairing,
in proportion to the difference between the maximum level of conditioning
supportable by the US (which depends in part on US intensity) and the
current level of conditioning. After the CS has approached its asymptote,
cutting the US intensity would be expected to produce a reduction in the CR
across trials (decreasing negative exponential curve), approaching the new
maximum from above. If you change the US intensity to zero you have no US,
and you get another decreasing negative exponential curve with a zero
asymptote: extinction.
These questions don't
arise if you think of CS, US, and UR as "events" that just happen or
don't happen. When you look on them as quantitative processes, a whole
new batch of questions arises, and the only way to find the answers is
to do the experiments and record the right data.
Actually, the parameters of classical conditioning have been examined
quantitatively. What has been missing is not so much the quantitative
manipulations as the proper way to model the system. I agree, however, that
the control systems model will probably suggest recording different
variables, or at least reporting them differently (as time functions in the
single participant). I've started reviewing the salivary conditioning
literature to see what's there.
In a system with finite gain, finite lag, and having variables
subject to finite rates of change, the disturbance changes
propagate around the loop at finite speed. The transformations get
complex as the uncancelled remnants of the disturbance signal
recirculate like reflected and re-reflected waves, producing a
complex pattern of interference with their own "reflections" and
the current disturbance signal.
This is close to the right picture. But remember that there are also
integral lags, which mean that the output action of the system may
change by only a small amount in the time it takes for a perturbation to
make its way completely around the loop. A "response" may just be
getting started after the first "wave" is complete, and become a little
larger on the next "wave", and so forth. Likewise, a disturbance may
just have appeared by the time the response starts to oppose it, and may
get only a little larger during the next cycle, and so on, so that both
the disturbance and the output action change on a time-scale measured in
many "loop trip times."
The integral lags are part of what I had in mind when I wrote "having
variables subject to finite rates of change."
The method of Laplace transforms, used to solve the differential
equations of control systems, handles time lags in a very simple way. If
the Laplace variable is s, a pure time lag of length t appears in the
equations as a multipler exp(-ts). This automatically takes care of any
number of "reflected and re-reflected waves", correctly giving their
effect summed over all of past time. The remainder of the terms in the
equations remain exactly as they would be if the lag were zero. The lag
does have an effect on the solutions of the equations, but the amount of
this effect declines gracefully to zero as the amount of lag declines to
zero (and exp (-ts) tends toward 1).
Thanks for the tutorial on Laplace transforms. I've been encountering them
in some of my recent reading, but they're pretty tough going for a guy with
the limited math background I have. How do you get from the Laplace
transform equations for a system to a set of equations you can put into our
iterative, discrete-time computer models? (Or is this a nonsense question?)
I see you've gotten caught up in your mail, so you must have seen my
eyeblink simulation by now. Comments?
Regards,
Bruce