[From Bill Powers (990928.1140 MDT)]

[For Isaac Kurtzer but copied for general interest]

Hi, Isaac --

Me:

Let perception of any objective (physical) stimulus measure S be

proportional to the logarithm of S.

P = s*log(S)

where s represents the coefficient appropriate to the stimulus S.

You:

i don't understand the first statement. It it worded the way you

intended.."stimulus measure S proportional to the logrithm of S"?

should'nt it be P is proportional to the log of S?

The sentence is "Let perception .... be proportional to the logarithm of

S." The prepositional phrase "of any objective physical stimulus measure"

refers to the subject, perception. You missed the "be" in the sentence, or

connected it to the wrong grammatical element.

after having read the following, which i am happy so say i followed

mathematically, i am not sure if i get the concepetual point. Is it if

two

signals are made to match that have similar input functions, logrithm vs

exponential, but different mulitpliers, then any common

relation between the two drops out and there are no terms that correspond

actual value of the signal P in expressing N.

That is it. Stevens thought he was measuring the true relationship between

the objective stimulus and the subjective perception, which is what

psychophysics is about. But he was only getting people to compare one

perception with another. This, of course, has also been a problem in all

other psychophysics experiments.

So if you are to lift one load 1 and then we'll call that one "UGA" heavy

then at first you can give very reliable assessments of other weights

according to a power law.

Let me reprise a bit and then carry the analysis a little further.

In the following, I'm going to assume that the magnitude of a quantitative

perception of any kind is equal to some constant times the log of the

actual magnitude of the stimulus. I will show that from this basic law,

which is like the Weber-Fechner law, we can show that the reported

magnitude of a stimulus, in numbers, will be proportional to a power of the

actual stimulus magnitude, the power being the constant just mentioned.

Let's say that perception of weight W is Pw, following the law

Pw = w*log(W).

Estimates of weight are given in numbers, N; the subjective magnitude of a

number N is, according to the same assumed law,

Pn = n*log(N)

Following the derivation in my previous post, when the subject vocalizes a

number N whose magnitude is perceived as the same as the magnitude of the

weight perception, we have Pw = Pn and therefore

w*log(W) = n*log(N), or

N = W^(w/n) (using ^ to indicate that the following expression is an

exponent).

This is the observed power law relating the reported numbers and the

perceived magnitude of the numbers Pn. The power in the exponent is w/n,

although we can't measure n and w separately.

At the same time, let Pe be the perceived effort that is related to the

signals entering the muscles (whether derived from reafferent signals or

from other sensory data). The new equation is

Pe = e*log(E),

where E is the actual effort.

Note that if (for example) effort were sensed from Golgi tendon organs and

weight were sensed from skin-pressure sensors, the physical stimuli for Pu

would be different from those for Pe, and we would expect different

calibration factors to apply. The measurement of E in particular entails an

unknown calibration factor.

To continue, we now have

Pe = e*log(E) and

Pn = n*log(N).

When the subject vocalizes a number N whose magnitude is perceived as the

same as the magnitude of the effort perception, we have Pe = Pn and therefore

n*log(N) = e*log(E), or

N = E^(e/n).

Note that log(E) is a constant times log(N): therefore a log-log plot of E

versus N will be a straight line of slope e/n. Likewise, a log-log plot of

W versus N will be a straight line with a slope of w/n. You mention that

these plots, under certain conditions, will be parallel to each other,

implying that the two straight lines are separated from each other.

If they are separated but parallel, we can say two things: first, the

parallelism implies that (e/n) = (w/n) and hence that e = w, and second,

that one of the plots must have a constant added or subtracted to make it

coincide with the other plot. Let's see how this would be done.

First, start with

n*log(N) = e*log(E)

Construct a constant C such that e times the log of the constant is equal

to the separation of the two parallel lines, and add this quantity to the

right side:

n*log(N) = e*log(E) + e*log(C), or

n*log(N) = e*[(log(E) + log(C)], or

n*log(N) = e*log(C*E)], or (to skip a couple of steps)

N = (C*E)^(e/n).

Let's call the two reported numbers Ne and Nw, for effort and weight. They are

Nw = W^(w/n), and

Ne = (C*E)^(e/n).

The constant C, if calculated as above, makes Ne equal to Nw, so

W^(w/n) = (C*E)^(e/n)

The exponent 1/n drops out, so we have

W^w = (C*E)^e, or

W = (C*E)^(w/e)

Note that perceived weight and perceived effort are also related by a power

law. However, if E is calibrated as suggested, w = e, and we get

W = C*E

The weight is some constant times the effort.

This would make sense if the perceived effort were calibrated so it is

equivalent to the upward force being applied at the load. Being sensed at

some place other than at the load, it would not be objectively of the same

magnitude, however. If the mechanical disadvantage from tendon to load were

8:1, an actual effort sensed at the tendon would be 8 times the force

applied at the load, but of course the magnitude of the aggregate sensory

signal would depend on the number of sensors and their individual

sensitivities. There is no direct way to determine C.

Now to the final problem: what happens after exercise fatigues the system.

What you report as being observed is that the ratio of perceived Effort to

perceived Weight increases, while both Effort and Weight perceptions increase.

Fatigue, it is known, reduces the response of muscles to driving signals.

If actual tension in a tendon were being sensed, there would be no

difference in sensed effort, because actual muscle forces are simply

increased as required to hold the load in position. The driving signals,

however, would be larger because of the reduced muscle response. Thus this

increase in the ratio of perceived effort to perceived weight might be

explained if effort is sensed in the imagination mode -- that is, if the

driving signals are sensed via what have been called re-afferents,

perceptual signals derived directly from motor output signals rather than

from their effects. I believe you said that this is the currently accepted

explanation.

The other possibility is that sensed Weight might be decreased due to

habituation of skin-pressure sensors (or whatever sensors are involved).

This would not, however, affect muscle tension because the load is still

maintained in the required position. Thus it appears that the present

analysis supports the explanation that effort is perceived via

re-afferents, or some equivalent path.

How does this square with the concurrent increase in sensed weight? The

implication is that sensors directly affected by the weight of the load

become _more_ sensitive after repeated lifts. But this is not the usual

effect of habituation; the usual effect is in the opposite direction.

One answer might come from the process you call "hefting" a load:

accelerating it up and down as a way of measuring its mass (as opposed to

its weight). If the muscles fatigue, they will be less effective in

accelerating the load, a phenomenon which would also result if the muscles

remained as sensitive as before but the load's mass increased. If the brain

assumes constant muscle strength, it will perceive an increase in mass and

hence infer an increase in weight.

if the procedure is not telling you the absolute relation than what is it

conveying?

It's telling you about a relationship between two perceptions, rather than

a relationship between an objective stimulus and a perception. You start by

estimating the number of EFFs per UGA. Then you fatigue the system, and

estimate the number again. The ratio has changed. But you still don't know

how big an EFF is in subjective units. You're saying that one perception

has changed relative to another one, or perhaps relative to its previous

value. That's all you can say without physically measuring impulses per

second over all pathways affected by a given stimulus. Even that would

leave open the question of just how big X impulses per second appears to

the rest of the brain.

Best,

Bill P.