Killeen's "electronics"

[From Bruce Abbott (960125.1035 EST)]

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

(1) B' = ------------,
            R + 1/a

which can be rewritten as

(2) B' = ------- * Bmax * zeta.
          R + 1/a

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

(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)

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:

(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.