[From Bruce Abbott (950120.1630 EST)]
I've created a little simulation program called KILLEEN.PAS that implements
Killeen's equations for steady-state responding on fixed ratio schedules,
and it does reproduce the curves he fits to several example datasets when I
plug in the fitted parameters. The simulation produces discrete "responses"
at specific times and delivers discrete "incentives" at specific times,
whenever the ratio requirement is completed. This version is based on the
1994 BBS paper and does not include the food-level control system.
The program implements Killeen's equations by using interresponse and
interincentive times rather than the response and incentive rates as given
in Killeen's equations. As I noted in an earlier post, Killeen's equation
giving predicted response rate B as a function of incentive rate R, which is
a
(1) B = -----*R,
delta
can be reexpressed as
delta
(1a) IRT = -----*IIT,
a
where IRT = interresponse time and IIT = interincentive time. Delta is the
minimum duration of a response and thus the minimum IRT; "a" supposedly
represents the number of seconds of activity incited by each incentive delivery.
The predicted behavior rate must be adjusted to take into account the
"constraint" on response rate imposed by the minimum response duration,
which Killeen does as follows:
B
(2) B' = -----------, where B' is the adjusted response rate.
B + delta*B
I have shown that this is equivalent to
(2a) IRT' = IRT + delta, where IRT' = 1/B' and IRT = 1/B.
This implies that the minimum observed IRT' will be equal to delta, the
response duration (or minimum time between responses) when the IRT predicted
from equation 1a is zero. The observed response rate B' will then be
1/delta, which is Bmax, the maximum response rate.
The simulation begins by setting IRT' to some reasonable baseline level and
computing the IIT that would result from this given the schedule ratio (IIT
= IRT'*N). When each incentive is delivered, the time since last incentive
(IIT) is saved and used in equation 1a to compute a new IRT. This is then
plugged into equation 2a to give the new IRT'. A response is emitted as
each IRT' time-period elapses, and the response count is incremented. When
this count reaches N, the next incentive is delivered, the new IRT recorded,
and the new response rate computed. This cycle continues until the
simulated session ends.
Having seen how Killeen's S-R machine runs, I am puzzled by its logic and
have been trying to understand just what it actually represents. What is it
doing, _really_? Killeen interprets the parameter "a" as the number of
seconds of activity incited by each incentive delivery, yet after the
so-called adjustment is made to allow for the minimum response duration, I
wonder. It does seem somehow to reflect the "arousal" produced by each
incentive, but it is not at all clear whether, as implemented, this
parameter really makes physical sense. Perhaps there is another interpretation.
In describing equation 1, Killeen focuses on the ratio a/delta, which is the
number of seconds of arousal elicited by an incentive, divided by the
minumum interresponse time. This ratio would give the number of _responses_
that would occur in "a" seconds _if_ responses occurred at the maximum rate.
This ratio is multipied by R, the incentive rate, to give the predicted
response rate (before allowance for response duration). But what sense does
it make to convert a-seconds into an equivalent number of minimum-duration
responses?
Equation 1 can be "parsed" another way that may make more sense:
a 1
(1b) B = -----* R = ----- * a*R.
delta delta
Inverting both sides and reexpressing gives
(1c) IRT = delta * IIT/a.
The ratio IIT/a is a unitless quantity representing the interincentive
interval as a proportion of the "activation time" a. This representation
makes it clear that the predicted IRT is a proportion of delta, the minimum
interresponse time. The observed IRT is then
(2b) IRT' = delta + (IIT/a)*delta.
The observed interresponse time is equal to delta plus some proportion of
delta.
Alternatively,
(2c) IRT' = delta * (1 + IIT/a)
That is, the observed IRT will be some proportion of delta >= 1.0. Yet
another way to look at it is to invert equation 2c:
a
(2d) B' = Bmax * -------
a + IIT
When the interincentive time is zero, B' = Bmax. Equation 2d reveals that
observed response rate is a hyperbolic function of "a" with a limit of Bmax
as "a" goes to infinity and a limit of zero as "a" goes to zero.
The inverse of IIT/a is the seconds of activity incited by an incentive
expressed as a proportion of the interval between incentives. a/IIT can
also be expressed as a*R; both are unitless quantities. a*R is what Killeen
refers to as the the level of "arousal" produced by a given rate of
incentive delivery; the time from the end of one response to the beginning
of the next is an inverse function of the arousal level. That is, the lower
the arousal, the greater the time between responses.
After all this work I'm still not clear what "a" actually is, although I
know more about how B' varies with it and with IIT. If anyone has a
suggestion I'd like to hear it. It does not seem reasonable to me that the
arousal produced by an incentive delivery should produce a constant rate of
responding until the next incentive delivery, which is what is implied by
Killeen's equations. Nor does it seem reasonable that this activation
should exert its effect on the time _between_ responses (from end of one
response to the start of another) as a _proportion_ of response _duration_
or minimum interresponse interval (equation 1c).
At any rate, the constant response rate between incentives (even while the
system is still moving toward equilibrium) is not realistic. But perhaps
the equations Killeen gives for response rate are meant only to provide the
average rate between incentives. His earlier paper on arousal, showing a
rapid rise in activity following incentive delivery followed by slow
negative exponential decay until the next delivery, may provide some clue as
to how Killeen would handle the actual dynamics of incitement between
incentive deliveries.
By the way, Killeen's equations do handle the feedback of R on B' correctly,
as shown by my iterative simulation of the system, at least when response
rates stay constant between incentives as they do in this implementation.
It will be interesting to see what happens when variable rates between
incentive deliveries become possible via including dynamic "arousal" effects.
Oh, and one more thing. Killeen multiplies B' by zeta, the coupling
coefficient, to get the final predicted rate. Killeen gives the formula for
computing zeta for various Ns on fixed ratio schedules; KILLEEN.PAs does
include the effect of coupling to arrive at numbers that agree with
Killeen's fitted curves.
Regards,
Bruce