JonEq6.txt

Jon Neumann
Physicist

Dear Jon,


"Part of the art and skill of the engineer and of the
experimental physicist is to create conditions in which certain
events are sure to occur."

Eugene Wigner

Subject: Four questions to clarify the behavior of the natural
eye under explicit scientific testing conditions.


You asked me to define our audience. I would suggest the
following.

I will talk to an individual pilot personally, and expect him
to do a considerable degree of analysis and before any actions is
taken by him. Only later will he act in concert with a group of
pilots who have gone through this analysis and have the same goal
in their life.

This must be the man who is going to gain the necessary
analytic skills during four years in an engineering college. By
having him answer the questions below, he will be able to start
the "thought process" that might encourage him to join our
preventive effort.

The pilot, after four years of college will be able to
correctly answer these engineering questions. Issues such a e =
2.718, as the "base of natural logs", will become easy for him.
Drawing correct conclusions about the behavior of the eye is
necessarily part of his analysis.



THE ENGINEERING EXPONETIONAL REPRESENTATION AND EQUATION

There are two forms of the ( e ^ -t/Tau ) function:

In Problem One, the minus lens is applied (as a step-input to
the system) but the lens is removed for each measurement. It is
necessary that we account for this fact in this analysis.

In Problem Two, I apply the "perturbation" to the system.
(For conceptual purposes this is a contact-lens can not be
detected.) Since you are measuring the eye with the "perturbation"
always in it, the equation takes the form as described in the
problem.


SOLVED PROBLEMS

I would expect that my book, "How to Avoid Nearsightedness",
will be read by the engineer-pilot before this test is taken. If
the man has difficulties solving these problems, then he should
refer back to this analytic reference book. I have not done
things perfectly, but this interactive process should help clear
the air.


CONSTANTS FOR THE NATRUAL OR NATIVE EYE

The eye must be considered to be a sophisticated auto-focused
camera -- for both the accommodation system and long-term
focal-status control.

The accommodation system is presented (simplified for
clarity) on my site as, "A Cybernetic Model of Accommodation."
This means that the accommodation reproduces the instantaneous
visual environment. The accommodation system must be understood
to be the "input", or average-value of accommodation in the
equations presented below.

The value of the two "constants", OFFSET = +1.5 diopters and
TIME-CONSTANT = 100 days, is taken from the fundamental primate
eye where BOTH the visual environment is controlled and the
refractive status of all eyes are measured.

All the problems below MUST use these two constants of the
eye's behavior.


Jon > You first have to get this analysis in a form that is
crystal clear. There are problems with many individuals
since they do not understand the concept of "e", the natural
base to the number system.


It is intended that these students will have a tutorial on
these types of problems. Crystal clarity should develop later.
We will do our best.


Jon > Where is an example problem by which they can follow the
analysis? You have to engage the audience. They do not
necessarily see where you are going with this. There has to
be a link and a concoction to your thesis, and their goal of
effective prevention.


Otis> That is the purpose of a tutorial. But I have not yet
provided that type of symposium. It might be true that the
pilot has no interest -- in which case we are wasting his
time. Below, I have included solved-problems as part of
this presentation.


THE AVERAGE VALUE OF ACCOMMODATION


Jon > Is the value for accommodation -1.0 D? Then state this.
You have introduced other information such as 16 hours a day
and 7 days a week which muddies the problem. The order in
which you state information is critical for a person's
understanding of the problem as they work to provide the
solution.


The INSTANTENOUS value of accommodation is determined by the
accommodation model presented in "A Cybernetic Model of
Accommodation". Initially, this value must be estimated from
observational measurements.

What can be done is to apply a -1.0 diopter lens. In which
case you will know the exact changed in the average value of
accommodation.


CLARIFICATION OF TIME-CONSTANT AND OFFSET

In Problem 1, the initial values are provided. The average
value of accommodation (-.8 diopters) was assumed for both groups.
The operative factor is that the average value of accommodation is
changed by -1.0 diopter. The average refractive status for the
entire groups was measured at +0.7 diopters.

After t-zero the control group has an average value of -0.8
diopters during this test, whereas the test group has a value of
-1.8 diopters -- after the application oof the -1.0 diopter lens.
This issue is to demonstrate that accommodation (as a signal)
controls its refractive status.

We can verify this by applying a "delta" in the average-value
of accommodation. The purpose is to establish a predictive model
for the behavior of the natural eye.


For day-zero the equation then looks as follows:

Long-
Term = Offset + Accommodation + Delta * [ 1 - Exp( -Time/Tau ) ]
Focus

Focus = +1.5 D + (- 0.8 D) + ( -1.0 D) * [ 1 - Exp ( - 0 / 100 ) ]

Focus = +0.7 diopters (before the test starts -- just
to verify the basic concept for the initial
conditions.)


After one day:

Focus = 1.5 + (-.8 ) + (-1.0) * [ 1 - 0.99]

Focus = +0.7 + (-1) * [ 0.01]

Focus = +0.7 - 0.01

Focus = +0.69 Diopters after one day of wearing a -1.0 diopter lens.


Otis> I hope the above clarifies your questions for a
solved-problem.

Jon > I have your book and I know where your to a certain extent,
but there are times that you are not clear.

Otis> I understand. That is why I suggest a tutorial on these
subjects. Many issues in fundamental physics are not clear
until after you have taken the course.

Jon > You have these problems so what is the instructional
objective to the student and what do you want them to know?

Otis> That by analysis they have a predictive mathematical model
for the behavior of all natural eyes.

Otis> Then can form a judgment of the effect the lens will have in
their goal to clear their distant vision from -1/2 diopter
(20/40) to 20/20.

Jon > Do you know their mathematical background?

Otis> On entry, I would expect good knowledge of basic math, and
the ability to solve an equation of this type. After four
years, I would expect the engineers to understand WHY this
equations represents the behavior of the natural eye as a
sophisticated system. This work does indeed does take time
which is the reason I judge that this study should be
restricted to only pilot-engineers who understand scientific
analysis of the eye's behavior.

Jon > If they do not have the background then you have to fill in
the blanks to give momentum to the concept.

Otis> My intention is to present the problems below as a method of
getting their minds focused on what they are going to be
doing one year after they evaluate themselves and their own
personal goals in life. Thinking is always easier that
acting. Success favors the prepared mind.

Jon > If the above are not clear then the your recommended
approach for effective prevention will not work either.

Otis> Correct! It is going to take a highly educated pilot or
engineer to even begin to understand the issues raised by
these questions.

Otis> I must rely on the intellectual ability of a capable
engineer. Without that skill, it would be virtually
impossible to conduct a preventive study of this nature.

Otis> A large number of optometrists believe that accommodation
has no effect on the eye's refractive status. However a
significant number believe that environment plays a profound
role in producing a change in the refractive status of the
nature eye. Here is the commentary by "Dr L". for your
thoughtful evaluation.

DrL > I do insist that a minus lens has no effect on the
refractive status of an individual when properly prescribed
and properly used. Scientists who run this type of
experiment would not disagree.

Otis> This is not true. There are many scientists, and a great
deal of experimental data that contradicts DrL's opinion.
The "second-opinion" by many scientists and optometrist is
that the refractive status of the eye will go "down" when
you place a strong minus lens on it.

Otis> Why don't you take the following test, and resolve this
issue by your own examination of scientific-objective facts,
themselves -- rather that relying on the opinion of others?

Initially we object to these questions. But the better idea
is to read them, put in the numbers, and crank out the correct
answers.

After that is done, we can back-track and discuss the reasons
for the questions, and the implications.

This is part of the fundamental approach used in the "hard"
sciences.


In all cases "focal status" means the following:

Using a trial-lens kit, read the Snellen eye chart at 6
meters.

If the person can read the character, then use plus in 1/8
diopter increments is added, until the .9 cm characters are "just
blurred", i.e., read only 3 out of 5 characters. This value is
recorded as the refractive status of the eye.

Similarly, if 1.8 characters are read, then increments of
-1/4 diopter will be used until the .9 ccm characters are read (4
out of 5 characters). The value of the negative lens is recorded
as the negative refractive status of the eye.

In all cases, the pilot shall operate the trial-lens case to
make these measurements. Monitoring of his work will be provided
but the accuracy will be certified by the pilots making these
measurements.

__________________________________________________


THE FOUR PROBLEMS

A test, or "questions" can often serve to clear-the-air, and
achieve a "clear mind" on a specific issue. The following
questions are designed to clarify the issues we have been
discussing.

I would expect that engineers and scientists will take the
following test and provide the correct questions.

Please answer the following multiple-choice questions.



PROBLEM 1

One hundred children ( 14 years of age ) have been maintained
in a distant visual environment of -0.8 diopters for one year.

We find their initial refractive status (for the entire
group) is +0.7 diopters

At this point, half the children begin wearing a -1.0 diopter
lens. The other half wear no lens. Both groups continue to live
in the same visual environment, but obviously "environment" is
-1.0 diopters "closer". Thus, the accommmodation system will be
"adjusted" for this change.

Use the following equation to answer the following questions.

Long-
Term = Offset + Accommodation + Delta * [ 1 - Exp( -Time/Tau ) ]
Focus


The "Delta" in this case equals the applied lens; which is
-1.0 diopters. The accommodation system will be -1 diopter
"closer" for 16 hours a day, 7 days a week.

1. What is the status of the test group after 1 day?

(a) .69 Diopters
(b) -1.50 Diopters
(c) .70 Diopters
(d) Since heredity controls the focal setting of the eye, both
groups will continue to have the same focal status. In all
cases, the refractive status of the control group will be
identical to the test group.

2. What is the focal status of the test group after 30 days?

(a) -0.781 Diopters
(b) 0.441 Diopters
(c) 1.700 Diopters
(d) 0.700 Diopters

3. What is the focal status of the test group after 200 days?

(a) -.200 Diopters
(b) .172 Diopters
(c) -.165 Diopters
(d) .700 Diopters

4. What is the focal status of the test group after 360 days?

(a) +3.250 Diopters
(b) -3.000 Diopters
(c) -0.273 Diopters
(d) 0.700 Diopters


PROBLEM 2

The human eye is in the process of growing. As it grows the
optical components of the eye continually change in value. Let us
assume that there is a sudden optical shift of +1.0 diopters.
(This would constitute noise in the system.) This change in total
power of the eye produces a new refractive status of =0.2
Diopters. For purposes of this these, the average value of
accommodation remains constant at -0.7 Diopters for both eyes.

The original focal status was +.8 diopters. Immediately
after the one-diopter focal perturbation the focal status is -.2
diopters.

This situation could be induced by the application of a +1.0
diopter contact lens. Please use the equation:

Focus = Offset + Accommodation - Perturbation * Exp(-t/Tau)

Focus = 1.5 D - 0.7 D - ( +1.0 D ) * Exp ( -Time / 100 )

1. What is the focal status of this eye after 1 day?

(a) -.190 Diopters
(b) 4.20 Diopters
(c) -.200 Diopters
(d) Since genetic information controls the optical components,
the eye will not recover from focal perturbations. The
focal status will remain at -.200 diopters. In all cases,
the refractive status of the control group will be
identical to the test group.

2. What is the focal status of this eye 30 days after the
optical change has occurred?

(a) .270 Diopters
(b) -.0592 Diopters
(c) 1.400 Diopters
(d) -.200 Diopters

3. What is the focal status after 100 days?

(a) +.200 Diopters
(b) +.397 Diopters
(c) +0.432 Diopters
(d) -0.200 Diopters

4. What is the focal status after 360 days?

(a) +.617 Diopters
(b) +.200 Diopters
(c) +0.772 Diopters
(d) -0.200 Diopters


PROBLEM 3

Eighteen monkeys are living in a caged environment. they
have an average visual environment of -1.8 diopters. at the start
of the test half of the monkeys are placed in a hooded (-2.6
diopter) visual environment. As a result, the environment "delta"
is -0.8 diopters. The refractive status of both groups (average)
at the start of the test is -0.300 diopters.

Using the following equation, calculate the refractive status
of the test group for the following days after the start of the
test.

Focus = Offset + Accommodation + Delta * [ 1 - Exp(-t/Tau) ]

1. What is the focal status of the test monkeys after 1 day?

(a) -.308 Diopters
(b) +1.300 Diopters
(c) -0.300 Diopters
(d) Since the eye's focal status is genetically determined,
the focal status of the test group will be identical to the
focal status of the control group.

2. What is the focal status of the test monkeys after 30 days?

(a) -.445 Diopters
(b) -.507 Diopters
(c) +.200 Diopters
(d) -.300 Diopters

3. What is their focal status after 60 days.

(a) -1.433 Diopters
(b) -6.661 Diopters
(c) -0.661 Diopters
(d) -0.300 Diopters

4. What is their focal status after 360 days.

(a) +1.078 Diopters
(b) -0.782 Diopters
(c) -1.080 Diopters
(d) -0.300 Diopters


PROBLEM 4

CHANGE IN REFRACTIVE STATE BETWEEN THE LEFT AND RIGHT EYES

The natural eye of primates have refractive states that are
very close to each other in terms of diopters. It is believed
that the eyes can maintain this accuracy because each eye controls
its refractive status to its accommodation system.

Since the environment of each eye is almost identical, it
should be possible to prove this thesis. The method is very
simple. Change the "environment" with an applied contact lens of
a reasonable negative value of say -2.0 diopters.

In this test the refractive state of both eyes is +0.8
diopters. The average visual environment has been maintained at
-0.7 diopters.

A contact lens of -2.0 diopters applied to the left eye will
change the refractive status of that eye to +2.8 diopters. The
"input" accommodation signal will change by -2.0 diopters.

Thus the "input" to the right eye is -0.7 diopters, and the
left eye is -2.8 diopters.

The right eye, with no "perturbation-lens" act as the
"control" eye.


The Challenge

Calculate the refractive status (with the contact lens in
place) for the following days after the "t = 0" perturbation is
applied.

The equation is:

Focus = Offset + Accommodation - Perturbation * Exp(-t/Tau)

Focus = +1.5 D - 0.7 D - ( -2.0 D ) * Exp ( -Time / 100 )

1. What is the focal status of the left eye after 1 day?

(a) +2.78 Diopters
(b) +2.80 Diopters
(c) +0.80 Diopters
(d) Since genetic information controls the optical
components of the eye -- the differential refractive
status of the eyes will remain at +2.0 diopters.

2. What is the focal status of the left eye 30 days
after the optical change has occurred?

(a) -3.50 Diopters
(b) +2.28 Diopters
(c) +1.40 Diopters
(d) +2.80 Diopters

3. What is the focal status of the left eye after 100 days?

(a) +0.80 Diopters
(b) -1.50 Diopters
(c) +1.53 Diopters
(d) +2.80 Diopters

4. What is the focal status of the left eye after 360 days?

(a) +0.80 Diopters
(b) -3.20 Diopters
(c) +0.85 Diopters
(d) +2.80 Diopters


__________________________________________


DETAILED TERMINOLOGY AND DEFINITIONS

The word "time-constant" refers to the dynamic response of a
control system. The time-constant of the primate eye (when
tested) is approximately 100 days.

The Heredity-offset, or "offset" is a design value. The
value, from the best experimental data is approximately 1.5
diopters. Better designed experiments could determine a more
precise value. I would agree that this value is a function of the
individual's heredity, and would explain why some individuals
develop a negative refractive state sooner than others.

For the problems below use 1.5 diopters for the offset, and
100 days for the time-constant, "Tau".

The concept of "perturbation", is that the natural eye must
have a self-correcting mechanism -- if is a sophisticated control
system. This "perturbation" could be a change in corneal
curvature, change in atmospheric pressure, and other random event.
In order to artificially simulate this perturbation, when can
place a +1.0 diopter contact lens on the natural eye.

For instance, if the eye has a measured refractive status of
+0.8 diopters, and we place a +1.0 diopter contact lens on the
eye, the measured refractive status will be -0.20 Diopters. If we
could not "see" this contact lens in the eye, we would measure the
refractive status to be -0.20 diopters. This is to "trick" the
eye into changing its refractive status -- as a control system.
It constitutes critical proof that this process must exist for all
natural eyes.


A CONTROL SYSTEM ANALYSIS

The short-term (accommodation) control of the eye is accurate
and effective. It is likely that this (averaged) signal is made
available to the long-term growth control of the eye for correct
positioning of the retina relative to the accommodation system.
This is the thesis of this presentation. A feedback control
circuit will insure that the retina is adjusted to the average
visual environment of the eye.

The Laplace transform of the eye's growth control system is:

1 / (TAU s + 1)

TAU = Eye's Time-Constant, Approximately 100 days

Applying a step input to this transfer function results in:


OUTPUT = INPUT * TRANSFER FUNCTION


V(s) = [ V(s) / s ] * [ 1 / (Tau s + 1) ]

Translating this function into the time domain gives:

V ( out ) = V ( in ) * [ 1 - EXP ( - t / Tau ) ]

Establishing initial conditions, we find that the equation
for the normal eye's behavior has a physiological offset of about
1.5 diopters.

Focus = Offset + Accommodation + Step Input * [ 1 - EXP ( - t /
Tau ) ]

Where:

Focus = The focal state of the normal eye.

Offset = The difference between the average value of accommodation
and the focal state of the normal eye -- considered over a
period of months. (For a population of normal eyes the
value
is +1.5 diopters.)

Accommo-
dation = Normal accommodation. By design, the accommodation
system's focal state is almost an exact replica of the visual
environment. The system is blur-driven and has a time-
constant of about 1/4 of a second.

Step-
Input = The step-input represents a sharp quantitative change in
the average value of accommodation.

Exp = Exponential function.

- t / Tau
e = Exp ( - t / Tau )

e = 2.718

t = Time, in days after the step change is induced in the
average visual environment.

Tau = The time-constant of a normal eye. All normal eyes have
a time-constant. (The typical value for the normal eye
is 100 days)