In B:CP the words “environmental function” and “environment function” occur four times, exclusively in the appendix.
k_e is not a variable, it is a constant. "The k-constants shown represent a linear approximation to the actual input-output relationships of the various functions involved. Thus for the input function, the constant ki relates the perceptual signal p to the input quantity q_i : P = k_i q_i " (p. 285; all page references here are to B:CP [2005]).
So k_e is a static quantity that “represents a linear approximation to the actual
input-output relationship” of the output quantity q_o to the input quantity q_i :
q_i = k_e q_o + k_d d
k_e is not something directly perceived in the environment, but k_e as a quantity controlled by the investigatory corresponds to some aspect of the environment. That aspect of the environment (whose effect is measured and quantified as k_e) is a one-segment environmental feedback path. It is the environmental feedback path through which the the effect which is measured and quantified as q_o traverses and becomes the effect upon the pertinent sensors, which are measured and quantified as q_i.
To answer again Warren’s prior question, the numerical value k_e is not present as such in the environment. It is a perception controlled by the person who measures it. The effect of q_o on q_i “is computable from knowledge of the environment function k_e which is observable” (p. 290). But k_e is in fact not directly observable. It can only be determined by measuring q_o and q_i and calculating the difference between them. Let’s look a bit more closely at how the formulae set that up.
“ki has units of signal units per physical unit, and ko has units of (other) physical units per signal unit. The environmental function ke has units of input per unit of output and the disturbance function kd has units of input per unit of disturbance” (p. 286).
So these constants have the property of transforming the ‘signal units’ of q_o into “physical units” and then transforming those “physical units” back into “signal units”, in addition to their quantitative function of stepping the large signal outputs to effectors into the signal inputs to receptors, which is typically a relatively quite small quantity unless push comes literally to shove.
But I believe that in practice—that is, in built and functioning models—a static value of k_e is found heuristically or perhaps given the value 1 (direct transfer of q_o to q_i). If anyone can show where models incorporate measurements in physical units, with k_o transforming signal units into those physical units and k_i transforming those physical units back inti signal units, then this challeng will have been refuted. Have at it.
Henry Yin and his students have measured signal units, but only at higher levels, not at the actual outputs directly at the individual motor fibers or the actual inputs directly from piezoelectric or photonic sensors. Arriving at a combination of e.g. all the motor nerve signals and ocular nerve signals involved in a tracking task surely has never been done so as to arrive at values of q_o and q_i in ‘signal units’.
As regards actual modeling practice these constants are idealizations which are important for any mapping from epistemic to ontological claims, but rather than actual measurements of properties of the environment they are inferred as a kind of mathematical fictions, imaginary perceptions controlled in the brains of modelers. Sometimes.
k_e is stipulated to be static in its influence upon the effect that q_o has upon q_i. So as regards the environment the formulae hold for laboratory conditions in which variability in that segment of the environment is controlled so as not to vary.
Isn’t it interesting that the disturbance d is not multiplied by k_e, even though the influence of the disturbance on q_i is also mediated through the environment. Instead, d is multiplied by a different constant, k_d. This k_d can therefore only be another environment function. So there are two functions representing properties of the environment mediating the influence of a variable upon q_i.
“The environmental function ke has units of input per unit of output and the disturbance function kd has units of input per unit of disturbance.” (p. 286).
Let’s make that more explicit, given the earlier account of what those units are.
“The environmental function ke has units of input [in signal units] per unit of output [in physical units] and the disturbance function kd has units of input [in physical units] per unit of disturbance [in physical units].” (p. 286).
The physical units for k_e and k_d must be the same, or else we could not "assume q_o and d to be linearly additive in their effects on the input quantity q_j so that
q_i = k_e q_o + k_d d "
(p. 286).
Somewhat different terms are used in Figure 5.2 on p.6:
In this representation, an important part is played by “physical laws”. Physical laws are quantitative relationships between measured aspects of physical phenomena which, at the specified scale of observation are currently understood by physicists (and others who understand them or merely believe them) to be inviolate. They are mathematical representations of rather abstract perceptions which can be placed in such relationship to present physical phenomena as to enable reliable predictions about them. They function as environmental feedback paths in the brain-internal environment of the higher-level control loops which employ them, but concepts and principles are not actually physically present in the environment among the “remote physical phenomena” depicted here.
Using the | character to represent the application of mathematical formulae to perceptions that the experimenter controls by measuring them in the environment (that’s how they become quantities):
k_o = “proximal results of muscle tension” | physical laws
k_i = k_e | “proximal physical stimuli”
k_e (here called “remote physical phenomena”) can only be more physical laws accounting for any difference between the output of (“proximal results of muscle tension” | physical laws) and the input of (k_e | “proximal physical stimuli”). True, it is also the locus at which the disturbance enters the loop, but it would be quite sensible, and simpler, to assign the physical laws entirely to the left and/or right of that entry point, set k_e=1, making it the only actual constant among the k family (except in strictly controlled laboratory conditions as noted).
Clearly, “physical laws” also intervene between “Cause of disturbance” and “remote physical phenomena”, though that is unstated.
Physical units persist through the physical laws and proximal physical stimuli on the left side of the diagram. The model term k_i is necessarily located in the input function. Recall the formulae for q_i and for p. This is entirely in physical units:
q_i = k_e q_o + k_d d
This transforms physical units to signal units:
p = k_i q_i
There is nothing in the environment that can do that; k_i is necessarily a property of the input function.
OK, so we have two segments of the environment which enable transmission of physical influences to receptors in the (peripheral) input function, where they are transformed from (k_e q_o + k_d d) physical units to (ki q_i) signal units. If the measurements that would yield values of k_e and k_d are never made in practice (my challenge above), then in practice the environment is ignored, and the effective formula is entirely in ‘signal units’
q_i = q_o + d
Wait a minute. You say d is in the environment? Then it must be in physical units. Well, yes, but d is only noticed as as an effect upon q_i. There is no measurement in physical terms and conversion from physical units to signal units. What? you say. But in a tracking model d is a variable quantity expressed as xy coordinates in the model. Yes indeed, in the same xy coordinate units as the perceptual signal p. I invite you to consider the relation of map to territory.
We should note that k_d is not a constant (except at the digital-computer fictional time t). It is a variable, and by obvious inference so is k_e. From p. 49:
The disturbance always calls for a response, even though as the organism moves about and as the environment changes, greatly different responses may occur as the effects of a given disturbance on the controlled quantity change.
Back to those two segments of the actual physical environment, one through which the control loop is closed from outputs measured in the aggregate as q_o to perceptual inputs to sensors measured in the aggregate as q_i, the other aggregating diverse paths through which unpredictable disturbances may affect the controlled input q_i. In experimental conditions which have been typical for building models, the investigator controls their perception of d to one path and one variable value. I will not belabor the problem that the aggregation of many into one is purely conceptual and not actual practice. Simplifying assumptions are necessary if one is to proceed at all. From p. 23:
The level of detail one accepts as basic must be consistent with the
level of detail in the phenomena to be described in these basic terms.
One can always, for other purposes, analyze further. If we wish to describe
the activity of the nervous system that correlates with the phenomena
of direct experience, and constitutes the inner component of
such behaviors as walking, talking, and execution of action patterns in
general, then it would be inappropriate to begin with an individual
neural impulse. No one neural impulse has any discernible relationship
to observations (objective or subjective) of behavior. Even if we
knew where all neural impulses were at any given instant, the listing of
their locations would convey only meaningless detail, like a halftone
photograph viewed under a microscope. If we want understanding of
relationships, we must keep the level of detail consistent and comprehensible,
inside and outside the organism.
The notion of p as a rate of firing is the same kind of necessary fiction. From p. 24:
As the basic measure of nervous-system activity, therefore, I choose to
use neural current, defined as the number of impulses passing through a
> cross section of all parallel redundant fibers in a given bundle per unit time.
The appropriateness of this measure depends on the maximum neural
current normally expected to occur in a given bundle of fibers. If the
maximum in a bundle of 50 fibers is 200 impulses per second in each
fiber, the maximum neural current will be 10,000 impulses per second,
and statistical variations will not be important at any level of neural
current in proportion to the whole normal range of operation (they
will be roughly 1 percent of the maximum, or less).
The use of neural current is appropriate, for example, in considering
the stimulation of a whole muscle, especially in terms of the forces
thereby developed on the tendons and thus on the bones. The impulses
going to the muscle arrive via hundreds of individual pathways,
each terminating on one tiny contractile fiber, but the net force developed
depends on all these parallel events, not on anyone of them. The
individual random twitches are averaged out.
The only quantity inside the nervous system that correlates with
the net force exerted by, say, the biceps muscle is the neural current
obtained by counting all the impulses reaching that muscle per unit
time. That is essentially the same as counting the impulses passing a
cross section of all the parallel motor-nerve fibers running from the
spinal cord to the biceps muscle. I am doing nothing more here than
formalizing a measure that is commonly used in neurology and physiology,
even if not instrumented with just this definition in mind.
Setting aside this necessary fogging of detail, let us return to the environmental referents of (q_o k_e) and d as the investigator’s controlled perceptions quantified aspects of the environment. They refer to realities in the environment. The subject control systems can also perceive these segments of the environment, and they can control their perceptions of them.
It is quite usual for a living control system to control their perception of that segment of the environment which transmits disturbances to their control, and they commonly follow that path of influence to its distal end to control a perception of the source of those disturbances.
It is also quite usual for a living control system to perceive a difference in the effectiveness of two environmental segments for reliably transmitting their q_o influences through the environment to their q_i controlled input, and to abandon one in favor of the other.
It is quite usual for a living control system to resist disturbances to the effectiveness or reliability of an environmental segment in transmitting their q_o influences through the environment to their q_i controlled input.
It is quite usual for control systems living in the same environment to use the same segments of the environment as means for transmitting their respective q_o influences through the environment to their respective q_i controlled inputs.
The aggregate effect of individuals’ resistance to disturbances to the effectiveness or reliability of an environmental segment in transmitting their respective q_o influences through the environment to their respective q_i controlled inputs is a very important aspect of collective control. On our mastery of this capability rests our survival as a species. Disclosing and enabling this mastery is in my opinion the most important task for PCT.
