[From Bruce Abbott (960125.1035 EST)]

Killeen's (1994) "behavior" equation was given as

zeta*R/delta

(1) B' = ------------,

R + 1/a

which can be rewritten as

R

(2) B' = ------- * Bmax * zeta.

R + 1/a

This equation bears a formal similarity to the following equation from basic

electronics:

R(l)

(3) V(l) = ----------- * V(s),

R(l) + R(s)

where V(l) is the voltage across the load, V(s) is the source voltage, R(l)

is the load resistance, and R(s) is the source resistance.

An ideal voltage source would have no resistance, i.e., R(s) = 0.0, in which

case V(l) = V(s). However, any real voltage source has an internal

resistance R(s) that limits V(l) to something below V(s). For a given R(s),

V(l) approaches V(s) from below as R(l) increases. When R(l) equals R(s),

V(1) will be exactly half the value of V(s).

A voltage source is said to be "stiff" if the load voltage remains nearly

constant as the load resistance increases. The "stiff" region is defined as

beginning at that point where the load voltage reaches 99% of source

voltage. This will occur when R(l) is approximately 100 times greater than

R(s). When R(l) equals infinity, the load voltage equals the ideal (source)

voltage.

In Killeen's equation we can let R play the role of load resistance, 1/a

play the role of source resistance, Bmax play the role of V(s), and set zeta

= 1.0. Applying the same analysis, we can say that B' (the observed

behavior rate) will be "stiff" to the effect of reinforcement rate R when R

is approximately 100 times 1/a. When R = infinity, B' will equal Bmax.

As 1/a becomes smaller, the region of stiffness begins at progressively

smaller values of R. In addition, variation in 1/a is expected to have a

progressively larger impact on variation in B'. In Killeen's system, a is

the "specific activation" provided by an incentive under a given level of

drive. The higher the product of drive (e.g., hunger) and the intrinsic

incentive value to the organism of the incentive, the higher a becomes, the

smaller 1/a becomes, and the sooner B' enters the region of stiffness (99%

of Bmax). If we have Killeen's fitted value for a for a given dataset, we

can easily calculate the region of stiffness for reinforcement frequency.

For example, a = 123 sec/inc for the Powell (1968) data; the region of

stiffness begins at about 0.81 inc/sec or about 49 incentives per minute.

For Mazur's (1982) data, a = 44 sec/inc or about 136 incentives per minute.

I would assume that these are unadjusted values that do not take collection

time into account.

When 1/a = R, responding reaches half Bmax. For the Powell (1968) data,

responding reaches half Bmax when R = 0.49 incentives/minute; for Mazur's

data the value of R is 1.36 incentives/minute.

There is another form of the electronic equation for a voltage source that

also is illuminating with respect to Killeen's formulation:

V(s)

(4) I(l) = -----------

R(s) + R(l)

Here, I(l) is the load current. Equation 4 tells us that in an ideal

voltage source (i.e., R(s) = 0.0), as the load resistance R(l) approaches

zero, the load current I(l) approaches infinity. Killeen's "uncorrected"

formula for B has this same problem, with B approaching infinity as delta

approaches zero. Imagine that delta is R(s) and 1/B is R(l). If the source

resistance takes on some finite value, I(l) no longer approaches infinity as

R(l) goes to zero; rather, it approaches V(s)/R(s). Similarly, in Killeen's

system, if delta takes on some positive value, B no longer approaches

infinity as 1/B goes to zero, but approaches 1/delta instead. So delta can

be viewed as playing the role of the fixed resistance and 1/B (the inverse

of the predicted behavior rate) the role of the load resistance, which

becomes progressively smaller as a and R become larger.

Regards,

Bruce