[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.