[From Bruce Abbott (960221.0855 EST)]

Bill Powers (960220.2300 MST)

When you measure behavior in VI schedules with a closed loop system, the

closed loop will modify the actual distribution induced by the irregular

schedule to produce a different apparent distribution. If you're just

trying to characterize the feedback function (which I think is the main

point), all you need to do is supply various steady rates of behavior

(variance of zero) and measure the mean rate and distribution of

reinforcement, for each VI schedule. That will give you the true

character of the feedback function itself.

Using a steady response rate would be rather uninformative, I believe. For

example, imagine using a steady rate of 1 resp/s, with dt = 0.1 s. A

response will be generated every tenth iteration. Meanwhile the VI interval

timer is decrementing toward zero; when it reaches zero a reinforcer is set

up. It may be that this coincides with one of those MOD 10 responses, in

which case the reinforcer gets delivered immediately. Or it could happen on

the next iteration, or the next, etc, out to nine iterations. Since the VI

setup is random with respect to the response generator, the probability that

setup will occur on any given iteration is constant, yielding a rectangular

distribution of delays between setup and collection with a mean of 10/2

iterations or 0.5 s. This, I believe, is the limit under a zero-variance

IRT: VId = VIp + 0.5 IRT.

If you set up the rat's behavior open-loop as an independent random

variable with a particular mean and distribution, and run that through

the schedule, you'll get the product of two stochastic variables. This

may be a good way to measure the effective feedback function, assuming

that the rat's variability in rate of pressing is due entirely to

irreducible noise in the rat's output. However, this is a shaky

hypothesis when you are using a variable schedule, because even with a

perfectly regular rat function (with no internal noise), the statistical

variations in the feedback function will lead to random-appearing

variations in the system output. Unless you can show that these same

variations would be present with a perfectly regular rate of

reinforcement, you can't assign them to behavior as if they were an

inherent part of the rat's actions. Nor is the resulting product

representative of the feedback function.

I agree with your main point, which is that variations in the rat's output

are likely to be as much a product of variations in the schedule feedback

function as intrinsic to the rat. My interest in assessing how the feedback

function changes as a function of response distribution came about initially

because a plot of IRT versus VId for McSweeney's data revealed a rather

"scattered" set of points. When I superimposed Killeen's function and the

linear function on these points, all but one point fell below Killeen's

line; the general trend followed the linear function, but with quite a bit

of variability. That got me to wondering how the function-lines actually

vary with response distribution. Normal variation around the mean IRT

(Killeen) and an exponential distribution of IRTs around the mean IRT

(linear) cover a good range of different possibilities. With less irregular

data it might be possible to work backward and infer something about the

general distribution of IRTs from the obtained schedule function.

I kept meaning to bring this up way back in the battle over the E. coli

model, and kept forgetting. By generating behavior using a probabilistic

output function (constant probability of a press per unit time, etc.),

you are making a claim not only about observed mean rates, but about

observed variances in the behavior rate. If the actual behavior is not

as variable as the model's behavior, a different means of generating the

behavior has to be found. Your model of E. coli showed much too much

random variation in the measured dependence of tumbling delay on rate of

change of concentration, as measured by Koshland using perfusion of

tethered bacteria. The actual response was quite regular.

Good point, and an excellent test of model fit; one has to be as concerned

with matching variation of model and data as with matching the changing

values of the cv. But of course, my e. coli model was not intended as a

serious model of real e. coli.

What news on prospects for getting the rat experiments started?

I will be ordering the rats this week.

Regards,

Bruce