[Rick Marken 2019-06-08_11:26:21]

[Martin Taylor 2019.06.04.16.31]

`MT: I disagree with Rick's new understanding of what Bill was saying,`

though I stand to be corrected, because Rick could easily be right

about what Bill meant…Â

RM: I think what Bill meant can be gleaned from the context in which the letter was sent. Bill’s letter was in response to my first paper testing PCT. The paper described a further test of a fact described in Bill’s 1973Â *Science* and 1979Â *Byte* papers. Bill showed that, in a compensatory tracking task, the correlation between variations in cursor and handle position is near 0.0, while that between variations in the invisible disturbance and handle position is nearly perfect, on the order of -.99.Â Â

RM: Here’s Bill’s diagram of the situation in the compensatory tracking task. I’ll use the notation in this diagram to explain what Bill found and how it is explained by PCT.

RM: The basic finding is that the correlations between the variables in the causal path from d to h are always close to 0.0 while the correlation between d and h is close to 1.0. That is, the correlation between d and c and between c and h is ~0.0 while that between d and h is ~1.0.Â

RM: This is an astonishing finding and quite a challenge to conventional scientific psychology. My contribution to this “spadework at the foundations of scientific psychology” was a little experiment that showed that the correlation between variations in response outputs (h) on two trials with the same disturbance will be close to 1.0 while the correlation between corresponding variations in stimulus inputs © on those trials is close to 0.0. That is, the correlation between h1 and h2 (handle position variations on two different trials with the same disturbance on each) will be ~ 1.0 while the correlation between c1 and c2 (cursor position variations on the two trials, the only inputs that the subject can see) will be close to 0.0. This shows that there is no mathematical function of the cursor variations that will produce the observed variations in h on each trial.Â

RM: Of course, I wanted to show that PCT can account for this result. So I built a PCT model that performed the tracking task and, to my astonishment, the model didn’t behave like the subjects. Most surprisingly,Â the correlation between c and d for the model was 1.0 rather than the ~ 0.0 that was found for the subjects. And the correlation between c1 and c2 for the model was also 1.0, rather than ~ 0.0, as it was for the subjects.

RM:Â I thought adding noise would fix things up and it did, quite nicely. The models with noise behaved exactly like the subjects. I did feel a bit like I was cheating by doing this; but I figured that neural noise is a real phenomenon so it wasn’t really cheating. But in re-reading his old letter to me, I realized that Bill had shown how the fact that noise was needed to make the model behave like the subjects could be used to reveal the the existence of an internal reference signal in the subjects.Â The relevant part of Bill’s letter that led me to this realization is here:

RM: The important equation is delta h = - k delta c, which is true only when r is constant. So with r constant (as it presumably is in the compensatory tracking task), r falls out of the relationship between delta h and delta c. This suggested that there should be a correlation between delta c and delta h if variations in r were the source of the “noise” that caused the lack of correlation between c and h. So I looked at some tracking data that I had on hand and the results I got a shown in the first column of the figure below.Â

RM: The correlations between disturbance and cursor (d-c), cursor and handle (c-h) and disturbance and handle (d-h) are what Bill reported in the Science and Byte articles; correlations of ~0.0 for the causal path from d to h (d-c, c-h) and a correlation close to 1.0 between d and h. The most surprising one being the ~0.0 correlation between c and h. The next three are the correlations between the delta values of those correlations. Here, the most important one is the very high correlation between delta c and delta h, which is what was predicted by the formula in Bill’s letter: delta h = - k delta c.Â

RM: This result suggested that the “noise” that results in a lack of correlation between c and h comes only from the reference signal – the “other stimulus” for h. I tested this by adding noise (wide band) to the model at two different points in the causal loop; I added it to either r or p (which is equivalent to c since the input function is taken to be a multiplier of 1). The results are shown in the columns labeled “r noise” and “p noise”. These results are stunning. When noise is added to r, the model behaves just like the data; when noise is added to p the model behaves nothing like the data.Â

RM: I think this result shows rather clearly that the lack of correlation between c (“stimulus”) and h (“response”) that is observed in human tracking behavior (but not in a noiseless control model) is a result of “noisy” variations in r (the other “stimulus”). This shows that a reference signal variable, r, must be included in models of control behavior. The results of the modeling provides a kind of an existence proof for the existence of an autonomously set reference signal that specifies the desired state of the input, c. It also shows that nearly all of the variation in an input variable that is being kept in a constant reference state is due to random variations in the reference specification for and not in the perception of the state of that variable.Â

RM: I still have some more things to figure out about this but I think this could make a nice little paper. Comments andÂ suggestions are welcome.Â

Best

Rick

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–

Richard S. MarkenÂ

"Perfection is achieved not when you have nothing more to add, but when you

have nothing left to take away.â€?

Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â --Antoine de Saint-Exupery