VI schedule function

[From Bruce Abbott (960221.1710 EST)]

Bill Powers (960217.0100 MST) --

    Bruce Abbott (960216.1625 EST)

    The reason that the error between programmed and obtained
    reinforcement rate is rather large at VI 15-s and decreases
    progressively as the schedule interval size increases is that this
    is the result of the limits imposed by the feedback function at the
    rates McSweeney's rats tended to respond on these schedules.

Beautiful, Bruce! Can you work this in reverse? That is, for each rat,
find the steady rate of responding that gives the best fit to the data
by using your formula? This should make the model fit the data even more

Below are the response rates (resp/min) implied by the observed
reinforcement rates using the linear function (VId = VIp + IRT)

    Rat 15 30 60 120 240
    161 12.0 8.5 9.0 2.0 0.6
    162 10.0 8.0 9.1 2.0 0.2
    163 10.3 12.2 15.8 3.6 0.4
    164 6.4 7.1 6.1 5.7 0.4


    MEAN 9.7 9.0 10.0 3.3 0.4

These response rates give the EFFECTIVE rate of responding -- how fast the
rat would have to respond on average IF it responded with constant
probability in each successive time increment. Actual rates usually
differed from these values, which implies that the IRT distributions may
have varied within schedules (across components), across schedules, and from
rat to rat. However, as viewed from the perspective of the effective rate
of responding, the data do reveal some consistencies. First, none of the
rats responded at a high effective rate on the 240-s VI, and three out of
four showed a large decline in effective response rate on the 120-s VI as
compared to the rates sustained by the shorter intervals. There is no clear
pattern of rate-change from VI 15 to VI 60. For two rats the VI 15-s
schedule produced the highese effective rates, for one the VI 30-s schedule
did and for the remaining rat the VI 60-schedule did. From VI 15-s to VI
30-s increasing, level, and decreasing trends can be found; on average the
rates across these schedules was nearly constant at 9-10 resp/min. Rat 164,
which had the lowest effective response rates on these schedules, was the
only rat to maintain nearly the same effective rate on the VI 120-s
schedule; the other rats decreased their response rates considerably.

Killeen's formula, which assumes normally distributed IRTs, produces lower
effective response rates, but the general patterns are the same. For
example, Rat 162's effective response rates calculated on both the linear
and Killeen formulas were as follows:

              15 30 60 120 240
   linear 10.0 8.0 9.1 2.0 0.2
   Killeen 5.6 4.3 4.6 0.9 0.1

If you can do this fitting, you can then do it for each 5-min segment of
the experiment, including the "satiation" regions, to get a picture of
how the actual rate of responding varies with time, for each rat. This
in turn might allow us to extrapolate to the total amount of obtained
food that would result in zero response rate -- the reference level for
each rat.

I think this can be done, but I would not be as simple as it should be owing
to the variations in _scheduled_ reinforcement rate across 5-min segments.
This becomes more of a problem for larger VIs. McSweeney's schedules
comprised 25 intervals each. On VI 15-s it required 25*15 = 375 s to
complete one cycle through the schedule if each reinforcer were collected as
soon as available; this is greater than the 300 s component length, so a
cycle cannot be completed in one component and the intervals programmed over
that time may not average to 15-s, depending on the sequencing of the
intervals. For VI 240-s it would require 6000 s of session time to complete
one cycle through the schedule, which permits much more variability in
programmed reinforcement rate across components. Still, here is an example
assuming VI 15-s actual average per component (for Rat 164):

Rft/5 min 7.0 14.4 14.6 13.6 11.8 10.2 6.8 6.4 4.8 2.2 2.8 2.6
Ex Rsp/ min 2.2 10.3 10.8 8.5 5.8 4.2 2.1 1.9 1.3 0.5 0.7 0.6
Ob Rsp/ min 17.1 17.9 12.6 8.6 7.9 4.2 3.1 1.5 1.3 1.4 1.5 2.0

The expected responses/min were computed using the linear formula. The
large initial difference between expected and observed rate probably
reflects a "warm-up" effect: the rat probably did not respond at all during
the first half of the component and then at a high rate thereafter. If a
long interval were being programmed then, the excess rate would not
translate into reinforcer deliveries. After the initial components, most of
the observed rates are not far from those prediced from the linear function,
even though the actual scheduled rates in each component were not taken into
account in the computations.

Re: Lab

I have obtained a video camera with which to photograph the activities
during each session and will have a few leftover rats from a colleague's
study available for my use in about two weeks. Between now and then I hope
to have the programs developed and tested that will schedule reinforcers and
record all responses and reinforcer-deliveries, along with the times at
which they occur.