Killeen's model: arousal

[From Bruce Abbott (960109.1250 EST)]

Bill Powers (950609.1545 MST) --

Perhaps what you want to say that there is a quantity that builds up
with every incentive, and that it can produce responses for some time
after the incentive (God, these words!). We can say that the amount of
the quantity is Q, and that it increases by m with each incentive. Q
also decays at some rate, which we can call c. So with R and B in terms
of rates,

dQ = (m*R - c)*dt, and

B = Kb*Q
where Kb is an arbitrary constant expressing responses/sec per unit of

Now if m is large enough, a single jolt of reinforcer will produce
enough Q so that the behavior rate does not fall to zero for some time
afterward, and multiple responses occur.

This may not be what you want to propose, but at least it expresses a
similar idea. Note that in my interpretation of your idea, the duration
of behaving after a single incentive depends is m*dt/c, and the number
of responses depends on Kb*Q.

I'm still waiting for Killeen's 1994 BBS paper to arrive, but meanwhile I've
obtained a copy of an earlier paper by Killeen, Hanson, and Osborne (1978)
in which the concept of arousal receives mathematical treatment. It may
provide a hint of how Killeen handles arousal in the 1994 description of his
"behavioral mechanics." The Killeen et al. (1978) paper deals with changes
in pigeons' activity-level following the delivery of a few seconds of access
to grain, as recorded by switch-closures of six hinged floor-panels which
formed the floor of the chamber. Activity rises quickly following the end
of grain presentation and then falls off gradually to produce a
right-skewed, hill-shaped curve. Killeen shows that the declining limb of
this curve is well-described by an exponential decay function of the form

  R = A1*exp(-t/alpha),

where A1 (A-one) is the intercept of the function at time t = 0, t is the
time since the feeding, and alpha is the time constant. So, Killeen assumes
that the ability of the feeding to instigate behavior (arousal) falls off as
an exponential function of time since incentive delivery.

Page 575 presents a section entitled "cumulation of arousal," in which
Killeen says the following:

    If feedings are delivered more frequently than every 30 min and if
    arousal cumulates, we would expect to see a higher level of activity
    with each successive feeding. The quantity, An, denoting the arousal
    on the nth feeding, will have decayed by the factor exp(-T/alpha) (where
    T is the interval between successive feedings) when the next impulse of
    arousal, A1, is added on trial n + 1:

        An+1 = An*exp(-T/alpha) + A1.

I believe this is how your model proposed above for "Q" behaves. Killeen
also includes a second exponential function representing inhibition of
activity which is supposed to occur briefly following each feeding; this
subtracts from the initial instigation produced by the feeding; since this
effect decreases rapidly, the result is a fast-rising level of arousal
immediately after a feeding, followed by the slower exponential decay, thus
"accounting for" the right-skewed activity curve actually observed.
However, the model would not behave much differently in terms of average
rates of behavior if the inhibitory effect were to be ignored for the
simulation: the arousal function just rises a bit more slowly to its maximum
after each feeding rather than increasing instantaneously. By the way, in
the 1995 paper Killeen notes that v, the "value" of the incentive, will be
some function of m, which brings his formulation of arousal even closer to
your suggestion.

Thus my guess that Killeen probably uses a decay mechanism such as you
propose to prevent unlimited cumulation of arousal gains support from his
treatment of the issue in the 1978 article. Our librarian tells me that the
1994 paper is in the mail and that I should receive it in a day or two; thus
I should be able to provide a definitive answer to the question soon.