To:  Dr. Scott Steadman,

From: Otis Brown

Subject:  Some questions for you concerning mathematical
     modeling of the eye's long-term behavior
     as a physiological control system.

Scott> Your challenges to this community to answer questions
      regarding a simple exponential decay suggests that you do
      not quite understand the difference between data and a
      model, as does your offering of the Flitcroft modeling
      effort when challenged for data.

Otis > A resistor-capacitor will have a "decay" as you state.  However
      an active control system will also display a "time-constant",
      depending on "damping".  An over-damped control system
      exhibits the type of time-constant we are talking about.
      Your statement suggests you do not understand this distinction.

Otis > Most of the group does not have the mathematical modeling
      background to answer the questions I posed.  Since you
      state that you are an expert on the subject matter, I
      challenge YOU to answer the questions posted below.

Scott> I hope that 100% of my students walk away from my
      physiology class understanding what a model is, but I know
      they don't, as the concept is not the easiest to grasp in
      one semester.

Otis > I hope you grasp the concept of answering direct an honest
      question concerning the behavior of the eye as a control
      system.    I look forward to reviewing your answers so we can
      build a better mathematical model -- to more accurately
      represent the dynamic behavior of the natural or native
      eye.

Scott> Of all the criticism I'm offering here, this is one I can
      let you slide on -- many people have the same problem.

Otis > I am not certain what "problem" you are talking about here.
      I never asked you to let me "slide" on mathematical
      modeling.  What I am asking you to do is to stop making
      criticisms, if you have nothing better to offer.  But
      equally, am asking you to help me improve the mathematical
      model representing the behavior of the eye as a control
      system.    I hope you do not "slide" by this issue, and avoid
      answering the questions posted to you.

__________________________

Dear Scott,

Re:    I've made no statement.  I just chuckle at the way you
      compare yourself to a Kuhnian revolutionary.  I chuckle for
      a number of reasons.  Scott

    I appreciate the fact that you like to "chuckle" about
various issues.

    But the issue I asked you concerned physiological modeling of
a control system, and specifically the fact that the natural eye
changes its refractive state as the visual environment is changed.
(Incidentally, this is not just with neo-natal primate eyes -- you
seem to believe that statement.  Unless you clarify pregnant
primates as "neo-natal".)

    You proceeded to tell me about your extensive academic
background, and did not answer the questions concerning the proven
behavior of the natural eye -- as a control-system.

    I always believe in looking a person in the eye and asking
him questions which he should be able to answer if he has the
professional background to do so.

    Rather than "attacking me", could you please answer the
following questions concerning the behavior of that natural eye as
a control system.

    After you provide the correct answers we can then discuss
your belief (it seems) that the natural eye does not behave as a
control system.

    I have provided some "text" to explain the nature of the
questions, and a mathematical model of the system.

    I look forward to reading your honest effort, and look for
your support to further develop a mathematical model to represent
the behavior of the eye.

    Four representative "case studies" with their associated
questions are posted below.

Best,

Otis

______________________________

Dear Scott,

    "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.

    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.

Comment> 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 of 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.

Comment > 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-engineer 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

Comment > 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 "test" -1.0 diopter lens.  In which
case you will know the exact changed in the average value of
accommodation.    You can them measure the consequence change in the
eye's refractive status.  If this does not happen than that would
prove that the natural eye is an "open-loop" system.

          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 of 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.

Comment > 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.

Comment > 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.

Comment > 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.

Comment > 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.

Comment > 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 natural 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.

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 cm 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 PROBLEM GROUPS

    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 accommodation 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 adolescent 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 measured refractive status of the 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 average 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)