[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

Q.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.

Regards,

Bruce