Collier data

[From Bill Powers (950811.1125 MDT)]

Bruce Abbott --

I've been going back over old posts trying to pull together the numbers
needed for the Test of the Collier et. al. data. I find that I have bits
and pieces of tables, but they don't seem to be from the same data sets.

What's needed is a table for each ratio of


The last column can be computed from the previous two.

This should be all we need to find an adequate control model to fit the
data. There's really no need for the formal Test; it's obvious that if
both meals/day and intake/meal remained constant as the ratio changed,
the intake per day would just be inversely proportional to the ratio: 1,
1/5, 1/10, and so on to 1/5120th of the FR-1 intake. No higher math
needed to show that the actual intake remains much higher than that.

Could you assemble these tables for our benefit? I'll give you three
M&Ms if you do.



Bill P.

[From Bill Powers (950812.0910 MDT)]--

Bruce Abbott (950810.1805 EST) --

Before you get too deeply into applying my diagram, don't bother. I have
become convinced that pressing rate is not the means of control in the
Collier et. al. task. After having spent a long evening and part of this
morning playing with numbers and plotting graphs, I have been unable to
see anything on which rate of pressing (i.e., meal frequency) depends
except the ratio or time. The primary -- it would seem the exclusive --
means of controlling intake is the amount eaten per meal. This is
essentially what you found when you decided just to let meal frequency
be an independent variable. The variable meal rates are nothing but a

What might be happening is this. The rat learns to press the bar rapidly
(but not at any particular rate) in order to make food accessible. The
controlled variable of that system is simply the accessibility of food,
a logical variable with only two states and a reference level of TRUE.
While that control system is operating, of course, the system for
controlling food intake has to have a reference level of zero, because
there's no way to get food and the rat is occupied elsewhere anyway.
When food become accessible, the output of the logical control system
switches the pressing-rate control system off and the eating control
system on -- with a reference level set by the level of deprivation,
presumably, although we haven't got to that yet.

I don't know why the meal frequency declines. Perhaps the logical
control system is simply insensitive to the disturbance caused by the
increasing ratio, since a logical condition doesn't depend on time.

The one relationship that does seem significant is between food intake
error and meal size.

Let ir = intake reference rate
    i = intake rate
    ms = meal size
    g = proportional gain

We can postulate a proportional control system of the form

   ms = g*(ir - i).

Since meal size ms and intake rate ir are known, we can fit the model to
the data by evaluating two constants, k and g. I did that roughly by
hand, and got what looks like a reasonable fit that should produce a
high correlation between error and action. The correlation between the
controlled variable (intake rate) and action (meal size) should,
properly, be far lower. I'll do it with numbers later today. You can
probably do this faster with your statistics program than I can.

I find that I've been very reluctant to accept the idea that pressing
rate is not varied as a means of control! It's simply turned on or
turned off, which is a means of control but not a continuous means. That
is, turning the repetitive pressing on does provide some control, in
that there would be no intake at all without some pressing rate, but
that is all the control there is. The output is on or off, and if
something else causes the actual output rate to vary, there is no
resistance to that change even though it may affect the state of a
controlled variable.

Once again I am impressed by your insight into the true explanation of
the curves in the Ettiger & Staddon paper and in the Motheral data. It
is so easy to accept that the "response rate" plotted on those curves is
actually what the rat is doing! Yet on reflection it is perfectly
obvious that in data presentations like these, the actual behavior of
the animal is not shown at all. What is called the response rate is
nothing more than the food or water delivery rate multiplied by the
ratio. The actual behavior consists of bursts of pressing at unknown
rates with scattered pauses of unknown duration -- and now we find,
varying rates of consumption of the "reinforcer." Just think what a plot
of actual _measured_ response rate versus reinforcement rate would look
like, on a point-by-point basis!

I wonder just how far the implications of your insight will spread. Is
it possible that rats actually have no ability to vary pressing rate as
a function of error? That ALL data relating so-called response rates to
so-called reinforcement rates are spurious? Even with my conviction that
this type of experiment is not a good way to study behavior, I find that
hard to believe. There must be situations in which that is the only
means of control -- for example, where fixed small bits of food are
obtained at a rate that depends strictly on the rate of acting, so the
wild card called "access" can't permit variations in amount consumed to
provide an alternate means of control. It may be that control exists in
this situation only for small ratios where loop gain is not too severely
degraded, and that for higher ratios the actual behavior rate simply
reaches a maximum while the ratio and pauses dictate the apparent
behavior rate. I can't yet give up the idea that there is _some_ range
of conditions within which an animal can control an input variable by
continuously varying its rate of behavior.

Considering the importance of your finding, I think that in our rat
experiments we should try hard to find evidence of real control via
variations in rate of pressing, and to discover the limits on the
conditions within which this kind of control can occur.


Final thought.

I can think of one way in which a pressing rate might be turned on and
off without any ability to vary the rate. Suppose we have a logical
comparator and output function that say "If no perception and no press
(is TRUE), emit one press." If there is no perception (of food) and no
press is under way, a reference signal to press down is set. As soon as
the pressing appears OR food appears, the logical condition for
producing a press disappears; the signal for pressing down is removed.
So we get one brief press.

On FR1, this will produce one press, which will produce a perception of
food and result in terminating the press until the perception of the
food disappears (which may take some time, since it can involve both the
sight and taste of the food). Then the logical error will reappear and
the next press will occur.

On FR2 or higher, the initial press will not produce a food perception,
but it will produce the perception of a press. That will turn off the
press-down reference signal, and the press will terminate. That will
result in an error again, and another press will occur. This cycle will
repeat until finally food is perceived, which will also terminate the
press, and create a pause until the sight and taste (etc) of the food
has dropped below a lower limit.

This model says that the running pressing rate is determined strictly by
how fast the logical comparisons and outputs, and their physical
effects, can change state. The rate itself is neither controlled nor
varied -- it simply reflects the maximum speed possible at a given time.
If the muscles tire, the rate will slow down, but not as part of any
control action. So this gives us logical control that entails the
appearance of a pressing rate, but no resistance to changes in
conditions that would affect the time it takes for the logical circuit
to switch from one state to another. In a computer simulation we would
have to insert finite transition times, or else the system would simply
operate as fast as the computer could compute.

Of course if we can find a situation in which rate is actually varied as
an explicit means of control, this model would not apply.

Bill P.