Solving the scaling of the Motheral curves

[From Bill Powers (950714.2130 MDT)]

Bruce Abbott & other modelers of EAB phenomena

I've been concerned about how to get the model for the Motheral data to
scale up properly when the reference level is increased. I think we are
agreed that the two curves in question differ because the reference
level is different -- lower for the lower levels of deprivation. I've
been looking for some rational basis for the scaling, and I think I have
it.

When you plot the various fixed ratios as reinforcements per unit
behavior, they are straight lines drawn from the origin at various
angles. For each ratio, the actual behavior point must lie somewhere on
the corresponding line.

The line that passes through the peak for the lower curve also passes
through the peak for the upper curve. The scaling of the two curves is
radial around the origin. This says that the droop on the left begins at
the same ratio for both curves. Now all we need is some physical reason
for that to happen (other than sensing the setting of the apparatus!).

Let's define a "value" variable, V, as the rate of reinforcement R times
the size of each reinforcer, ks. Thus

V = ks*R

On a fixed ratio, R = B/m, where B is the behavior rate and m is the
ratio. So the value received per unit time is

V = ks*B/m

The cost C of behaving is the rate of behavior times a cost factor kc
per behavior:

C = kc*B

The net value per unit time is V - C:

V - C = B*(ks/m - kc)

This gives us a natural way to define the peak of the Motheral curve:it
is the point where V = C or V - C = 0:

0 = B*(ks/m - kc), or

ks/m = kc, or

m = ks/kc

So the peak will appear where cost just equals benefit, and that will
occur AT A SPECIFIC RATIO regardless of the reference level, the
reinforcement rate, and the behavior rate. This is exactly what is
needed to make the curves scale up correctly.

The parameter ks is the value of each reinforcement. The parameter kc is
the cost of each behavioral act in the same units (energy, for example).
If ks = 40 * kc -- that is, the value received from one reinforcement is
equal to the cost of 40 behavioral acts -- the droop will begin at m =
40.

All that remains is to find a rationale for an effect of V - C on the
output gain of the control system. This is almost self-evident, because
the motor output process draws on the resources that are increased by V
and decreased by C. There may be other kinds of "value" and "cost" -- we
don't really need to guess what they all are, because all we need is
some physical reason why the gain of the control system might begin to
drop when the cost of each action exceeds the value received from each
action.

Tomorrow I will modify the model to incorporate an effect of excess cost
on the output gain, computed as above, and see if we now will get a
curve that matches BOTH Motheral curves (for 85% and 98% normal body
weight) with only a change in the reference signal (after adjusting the
parameters to fit one of the curves). I'll bet that this one will work.

Best to all,

Bill P.