For Jan -- The definition of a "step-function" for a control system

Discussion in 'Optometry Archives' started by Otis Brown, Jul 26, 2004.

  1. Otis Brown

    Otis Brown Guest

    Dear Jan,

    Re: Your question, " ... what is a step-input", or "step-function".

    This defines the meaning of a "step-input" for a control system.

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

    For instant, applying a -3.0 diopters to population of natural
    primate eyes constitutes an input "step-input" for the system.
    Their measured refractive status would be the "output" for the system.

    Also supplied, the definition of a "thought experiment"
    defining the behavior of the natural (not defective eye).







    By Otis Brown

    Science is the attempt to make the chaotic diversity of our
    sense-experience correspond to a logically uniform system of

    Albert Einstein


    It is important that we conduct a thought experiment before
    we commit ourselves to doing a full scale measurement study of the
    natural eye's dynamic behavior. If the eye is identified as
    natural, we can then presume that it is continuously adjusting its
    long-term focus by a dynamic control process, with respect to the
    accommodation signal.

    We will exclude the medically-defective eye from the present
    analysis. We will concentrate only on natural eyes that change their
    refractive status as the visual environment is changed.

    In this chapter I will present a conceptual scheme for the
    natural eye's behavior and in the next chapter the actual factual
    data that confirms this specific behavior of the natural eye.

    The equation developed from this analysis is an order of
    magnitude more accurate than the Helmholtz-heredity theory in its
    power to predict experimental results and future refractive states
    of the natural eye.


    The short-term (accommodation) control of the eye is accurate
    and effective. It is likely that this (averaged) signal is made
    available to control the control of the eye for correct
    positioning of the retina relative to the accommodation system.
    We will report only direct measurements (i.e., refractive status).
    No attempt will be made to extrapolate refractive status into
    either "defect" or length. Such extrapolations would require that
    we assume that the eye is proven to be a rigid system. It is
    manifestly clear that the this is not the case.

    This is the thesis of this presentation. (1) A feedback
    control circuit will insure that the retina is adjusted to the
    average visual environment of the eye. (Figure 1)

    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:


    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 ) ]


    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

    Accommodation = 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.

    e ^ ( - t / TAU )

    e = 2.718

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

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

    Does this equation represent the fundamental long-term focal
    behavior of the natural eye? Only a direct factual test will
    confirm the accuracy of this equation. To verify this, we will
    subject the human eye to a fundamental experiment.

    The primary technique for the evaluation of two theories in
    science is to compare their predictive capabilities. The theories
    must be stated in mathematical terms, and must yield quantitative
    predictions. Both the Helmholtz-passive and Helmholtz-dynamic
    theory of the natural eye's behavior meet this requirement.

    Both theories yield explicit numerical predictions that can
    be checked against reality by actual test. The theory that yields
    the most accurate prediction becomes the controlling theory, and
    will be relied on for further analysis.

    The first question is this: Does the equation predict the
    refractive control characteristic of the natural human eye; day
    after day, year after year?

    Using 231 days as the value for time, and - 0.9 diopters
    (estimated) as the average value of accommodation, we find that
    the normal eye sets its long-term focus to + 0.6 diopters. Actual
    measurement of monkeys shows that their eyes have a focal status
    of + 0.57 diopters when their eyes are maintained in an "open"
    visual environment. (2)

    The transfer function correctly describes and predicts this
    dynamic behavior characteristic of the natural eye in an open
    environment. This is the first test of any theory that
    characterizes the natural eye's long-term behavior, in a steady,
    or constant visual environment. This refractive stability is a
    result of continuous small refractive "corrections" relative to
    the (averaged) accommodation signal.


    If we can make a sudden shift in the eye's visual input, we
    can check the difference between these two theories of the eye's
    long-term behavior. A negative lens may be used to achieve this
    goal. This question tests for an "open-loop" system, (i.e., no
    change in refractive state will develop), versus a "closed-loop"
    system, where a change in refractive status of the test group
    MUST DEVELOP relative to the control group.

    The negative lens causes parallel rays of light (from
    infinity) to be diverged. This causes the accommodation mechanism
    to change as though the object is much closer. All objects closer
    than infinity will be moved correspondingly closer to the eye.
    (Figure 2) (i.e, for a -1.2 diopter lens, objects at infinity will
    be moved in to approximately 33 inches, and objects previously
    "read" at 20 inches, (-2.0 diopters) will not be optically at -3.2
    diopters, and the accommodation system will change by this
    corresponding amount. Thus the accommodation AVERAGE signal is changed by
    -1.2 diopter on a 16 hour a day, 7 days a week.


    In order to test these two Helmholtz theories of vision we
    will subject them to a hypothetical test. We will select 100
    children (eight years of age) and verify that they have normal
    vision. (A positive focal state of + 0.6 diopters -- average.
    The spread of refractive states will be from 0.0 diopters
    to +2.0 diopters.)

    Fifty of these children will begin wearing a - 1.2 diopter lens (33
    inch focal length). The other fifty children will be the control
    group. The focal state of their natural eyes will be measured at
    seven day intervals using the Snellen eye chart and
    a trial-lens kit. A FORTRAN generated graph, using the
    numerical predictions of the Helmholtz-passive (open-loop) and
    Helmholtz-dynamic (closed-loop) theories are shown in figure 3.


    The "delta" accommodation is zero diopters for the control group.

    Focus = Offset + Accommodation + Delta * [ 1 - EXP ( -t / TAU ) ]

    Focus = ( 1.5 D ) + ( - 0.9 ) + ( 0 ) * [ 1 - EXP ( - 231 / 100 ) ]

    Focus = + 0.60 Diopters, after 231 Days.


    The "delta" accommodation is - 1.2 diopters for the test group.

    Focus = ( 1.5 ) + ( - 0.9 ) + ( -1.2 ) * [ 1 - EXP ( - 231 / 100 ) ]

    Focus = - 0.48 Diopters, after 231 Days.

    Actual testing will help us choose between these two concepts
    of the natural eye's behavior. If the test group does not show a
    change in its focal state, the passive theory of the normal eye's
    behavior will be confirmed. If the test group demonstrates a
    change, the dynamic concept for the normal eye's behavior will be


    After 231 days the conceptual test of the normal eye's
    behavior was terminated. The test need not be carried to this
    extent to demonstrate the greater predictive accuracy of the
    dynamic theory. Since the heredity of the control group is
    identical to the heredity of the test group, both groups should
    continue to maintain their positive focal state, if the
    Helmholtz-passive theory is correct.



    -1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6
    ---- ---- ---- ..........................................
    0 .60 .60 L <-----------------------<< F
    7 .52 .60 L Step Change 1.2 D F H
    14 .44 .60 L (Negative Lens) F H
    21 .37 .60 L . F H
    28 .31 .60 L . F H
    35 .25 .60 L . F H
    42 .19 .60 L . F H
    49 .14 .60 L . F H
    56 .09 .60 L . F H
    63 .04 .60 L .F H
    70 .00 .60 L F H
    77 -.04 .60 L F. H
    84 -.08 .60 L F . H
    91 -.12 .60 L F . H
    98 -.15 .60 L F . H
    105 -.18 .60 L F . H
    112 -.21 .60 L F . H
    119 -.23 .60 L F . H
    126 -.26 .60 L F <-----<< Test H
    133 -.28 .60 L F . Group H
    140 -.30 .60 L F . H
    147 -.32 .60 L F . H
    154 -.34 .60 L F . H
    161 -.36 .60 L F . H
    168 -.38 .60 L F . H
    175 -.39 .60 L F . H
    182 -.41 .60 L F . H
    189 -.42 .60 L F Control >>--------> H
    196 -.43 .60 L F Group . H
    203 -.44 .60 L F . H
    210 -.45 .60 L F . H
    217 -.46 .60 L F . H
    224 -.47 .60 L F . H
    231 -.48 .60 L F . H
    -1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6

    In this thought experiment, the fifty children who wore a
    -1.2 diopter lens have undergone a -1.14 diopter net change in
    their normal eye's focal state, while the control group continues
    to maintain their + 0.6 diopter focal state. The conclusion is
    that the Helmholtz-dynamic theory is an order of magnitude more
    accurate than the heredity theory in representing the fundamental
    behavior characteristic of the normal eye.

    The Helmholtz-dynamic theory leads to an exact mathematical
    analysis and representation of the performance of the eye. (3)
    The Helmholtz-heredity (open-loop) theory produces inaccurate
    predictions about the long-term behavior of the natural eye.

    There is a logical continuum between an accommodation
    (feedback) control system and a long-term focusing (feedback)
    system. There is a sharp logical break between a precisely
    accurate accommodation system and a Helmholtz-heredity theory of
    the natural eye's behavior.


    1. Brown, O., Berger, R. A NEARSIGHTEDNESS COMPUTER, (343-346)
    Proceedings of the 7th New England Bioengineering Conference
    (1979) (Additional references can be found in this paper.)

    2. Young, F. A. Unpublished study of 375 monkeys who were
    maintained in an open visual environment. Their mean focal
    status was +0.577 diopters.

    Proceedings of the IEEE, Vol. 54 #6, (858-872) June, 1966
    Otis Brown, Jul 26, 2004
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  2. Otis Brown

    Guest Guest

    Major snip in a lot of blablablabla

    Your feedback mechanism needs a repair Otis.
    Roland asked, not Jan

    Otis usual manouvre, stepping out of a thread

    This is my question to Otis as formulated in another thread which he refuses
    to answer besides his incapability to proof he is right in advising people
    to use a pluslens to prevent myopia.

    Speaking of control systems, is it possible to switch on your memory?
    As you stated before you are more than willing to give direct answers on
    direct questions.
    Are you familiar with the following control systems
    The negative feedback mechanism.
    The positive feedback mechanism

    Jan (normally Dutch spoken)
    Guest, Jul 26, 2004
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  3. Otis Brown

    Otis Brown Guest

    Dear Friends,

    Subject: Correction to "graph" posted for Paper 25

    The posted graph is a smooth "curve" of the
    function e ^ (-t / Tau) function.

    This curves moves from +0.6 diopters to
    -0.48 diopters (nearsighted) in about
    231 days.

    The applied step-input is -1.2 diopters,
    and the resultant change is eyes that
    are natural-and-normal is -1.14 diopters.

    If you have a "graphing" calculator
    you can re-create this graph.

    The software on "google" obviously
    compresses "spaces" to reduce the
    size of the posted "bytes".


    Otis Brown, Jul 26, 2004
  4. Otis Brown

    Roland Izaac Guest

    On 25 Jul 2004 22:01:41 -0700, (Otis Brown) wrote:

    Otis, what are we talking about here-------Are we talking about a
    correctly prescribed lens for the myopic patient or are we talking
    about over prescribing. No one delibrity over-prescribes a myope
    unless there is some binocular vision problems. Therefore if you are
    talking about over-prescribing (ie a stronger minus lens that is
    necessary) then you are wasting your effort. I repeat, this is not
    what we do----we do not overprescribe. Why worry about the effect of
    something when that something does not take place. Besides, I doubt
    you will ever get an aproval to conduct human trials if your aim is to
    cause harm by inducing myopia.

    If however we are talking about correcting the myopia with an
    apropriate lens, then you have not been listening.
    I agree.
    I don't agree here.

    The lens places the virtual image of an object at infinity, at the far
    point of the myope (when correctly prescribed) When the myope looks at
    an object at his far point, no accomodation is necessary.
    When this myope (wearing his correctly prescribed spectacles) veiws an
    object at near, say 33cm., he will need to accomodate less as compared
    to an emetrope, not more. Interestingly, a hyperope will have to
    accomodate more at near with his correctly prescribed lenses when
    compared with an emetrope (no specs) or a myope (with specs). Sorry
    to burst your bubble.

    Your step-imput is therefore opposite to what you thought.

    Roland J. Izaac
    Roland Izaac, Jul 26, 2004
  5. (Otis Brown) wrote in
    That's not a thought experiment. A thought experiment is when you put out
    a protocol, and think about what your possible outcomes are, and how a
    possible outcome would effect your hypothesis.

    Scott Seidman, Jul 26, 2004
  6. Coupla thoughts. First, your tacit point is well taken--its tough to
    come up with a set of hypotheses that spans the whole solution set, and
    obviously its possible that more than one hypothesis is true.

    Note the "myopiagenesis follows the equation he is using" sentence, and
    compare that to all your other hypotheses. This one is phenomelogical,
    and tells you little or nothing about mechanism. The others are
    mechanistic, and are asking sound questions.

    I have alot of respect for Dr. Ciuffreda-- he's actually used the
    alternative medicine resources at NIH, I believe, and backs his opinions
    with studies and publications. Look at his CV, and talk about his
    published papers---those are the ones that count. He talks about
    vision therapy and abnormal nearwork-induced transient myopia, and
    convergence insufficiency. He doesn't talk about vision therapy and
    general all around myopia.

    My understanding is that AAPOS says that VT has no proven efficacy for
    dyslexia or other learning disabilities, and have published that opinion in
    Pediatrics. That's why I asked my original
    question-- I understand VT has been recently found to work for some
    conditions, and not work for others, in a published, peer reviewed work.
    Now, we can start narrowing down what conditions fall into which list.
    Silly me, I deleted that message that came in on the listserve I monitor,
    and now can't recover it. I was hoping that as a VT advocate and list
    organizer, you'd be able to point me to that paper.

    I agree, meaningless debate is to be avoided. That's why scientists
    generally limit debate to published, peer-reviewed studies. It cuts out
    alot of the noise. We pay attention to posters at conferences and
    abstracts to stay cutting edge, but we become quite wary if that work
    doesn't find publication within a few years.

    Web-sites are generally best avoided as primary source references.
    There is nothing controlling their accessibility in the future, and
    nothing preventing them from changing the sites willy nilly. Published,
    peer reviewed stuff. That's where its at. If an academic could take
    your n=1 case, demonstrate that you started with run of the mill myopia,
    sans accomodation problems, and that this myopia went away in your case,
    and then demonstrate the same in 10 other patients, get it published in
    a peer reviewed journal, and list methods sufficient for others to
    duplicate the work, I'd have no trouble believing that VT is useful for

    Scott Seidman, Jul 26, 2004
  7. Otis Brown

    Otis Brown Guest

    Dear Francine and Scott,

    Subject: The function of a "thought-experiment" in pure science.

    With all due respect ...

    The work that I have done follows Dr. Jacob Raphaelson,
    and the "problems" he had with the general public that
    walked into his office. He reported a "success" with
    the plus -- at the threshold (say -1/2 diopter and 20/40).

    Since Jacob was an honest man, and used the plus on
    his own children, I had to respect that. Even so,
    that result was HARD TO BELIEVE.

    I wondered if the nature of the behavior of strictly
    the natural eye, using basic INPUT / OUTPUT analysis,
    and studying STRICTLY only eyes that were natural.

    Thus the work is to prove or dis-prove the concept
    that the natural eye CHANGES ITS REFRACTIVE STATUS,
    if the accommodation SIGNAL, is changed.

    In the above sense, then the concept only deals
    with the behavior of the natural eye, where
    the refractive states of the natural eye
    have gaussian distribution, and the mean
    is established by natural eye's kept in a
    "open pen" enviroment.

    Thus, the order of study is to first establish
    whether this fundamental eye is controling its
    refractive stats (direct measurement), to
    the accommodation-signal, which is directly
    related to the "environment". In this
    sense NO DEFECT is caused. What you measure
    is REFRACTIVE STATUS, and you determine whether
    the refractive-status changes as the average-value
    of accommodation is changed in a negative direction.

    It is expected that the natural eye will exhibit
    this change (or not).

    A closed-loop system will (auto-focused camera, where
    accommodation time-constant = 1/5 second, and long-term
    control has a time-constant = 100 days.) exhibit this
    type of natural behavior.

    This type of testing only verifies that the natural
    eye is "dynamic" in the above sense. It says nothing
    about "defect" in any sense of the word.

    People who append "defect" or "emmetropia" (refractive
    status exactly 0.0 diopters) "ametropia" (all other refractive
    states) are simply jumping to a conclusion.

    Attempting to use these outdated words and antique understandings
    simply produces a profound mis-understanding of this



    Otis Brown, Jul 27, 2004
  8. It wasn't on that listserv. It was on a professional listserv.

    Scott Seidman, Jul 27, 2004
  9. I thought I knew what a step function was before reading this. Now, I must
    conclude that I never knew or that Otis does not know. Which path do you
    think my mind will take?

    Repeating Rifle, Jul 27, 2004
  10. Otis Brown

    andrew Judd Guest

    This is a very reasonable statement. Its so reasonable that it would
    seem that anybody involved in improving myopia could easily meet the
    requirements suggested.

    I would dearly love to find an academic who would be prepared to
    monitor what i am doing so that something could be published.

    The kind of work i am doing is by its nature long term work, but all
    that I require from the academic is the before and after refractions
    and his authority that this was done correctly. I can do the rest.
    I can do all the work via the internet, or by phone if they will pay
    and keep that documented/recorded. All the clients can be from
    within his own practice or home location. The academic need do
    nothing once the study has begun.

    By its very nature proven reductions in myopia would be impressive.

    This is such a simple experiment but so far i have no academic. I
    have a few optometrists who are interested but none want to put their
    name to what i am doing for fear of a professional backlash.

    However i am inclined to believe that once I had done the 10 I would
    then be expected to do a more thorough study involving 1000

    andrew Judd, Jul 27, 2004
  11. Otis Brown

    Otis Brown Guest

    Dear Bill,

    One of the purposes of my discussion about the
    behavior of the natural eye as a control system,
    is to get our minds off the stero-type of the
    eye as a box-camera, where only a refractive
    status of 0.0 diopters is normal. (Yes,
    that is the OD theory.)

    In electrical engineering you use a "step input"
    to "shock" the system.

    If the system is "open-looped" then the output
    (refractive status) simply does not change.

    If the system is "closed-looped" then you can
    get several responses.

    1. The output is a "damped sine wave", typical
    of a two-pole control system.

    2. If the sytem is unstable, you will get
    growing oscillations.

    3. If the system is over damped, you get
    a "exponetial time-constant". NO this is
    not a resistor-capacitor response. It is
    an active control-system response.

    But how does the natural eye (not defective) eye

    Only DIRECT TESTING will reveal the correct response.

    Againk this is absolutly engineering and has
    nothing to do with medicine.

    It is abstract intellectual testing and verification.



    Otis Brown, Jul 28, 2004
  12. Otis Brown

    Dr. Leukoma Guest

    (Otis Brown) wrote in

    ....and it has nothing to do with eyes.

    Dr. Leukoma, Jul 28, 2004
  13. In (to my own horror) my defence of Otis, doctoring IS engineering. It is
    the application of science to the solution of mans' problems albeit in a
    biological field. But it does include much art. Nevertheless, there is good
    engineering practice as well as not so good or poor practice.

    In most engineering, concepts are often represented by simplified models of
    the process. The art is to know how to approximate so as to keep the essence
    of the problem while throwing out confusing factors. That is done all the
    time in medicine. Remember the four humor model that is now discarded.

    What simplifies much of engineering and physics is that parts of the world
    behave linearly. That makes the concept of step functions useful. Whatever
    else Otis may say, eyes do not behave linearly in a mathematical sense. Some
    behavior is approximately linearly. For example, eyes can see multiple
    objects so that in a sense, one can say that the visual response to two
    visual objects is the sum of the responses to the individual objects, and
    thus linear.

    It also turns out that the most interesting effects are due to
    nonlinearities. For example, listening to two pure tones can produce a
    response that mimics the presence of a tone that is not acutally sounded.

    Repeating Rifle, Jul 28, 2004
  14. Not quite. This is a well known test for efference function in the inner
    ear. The hair cells attached to the tectorial membrane are actually
    active, and can shorten (really quickly, almost w/ piezoelectric timing).
    When you play two perfect tones in the ear canal under the right
    conditions, the shortening of the hair cells makes the tectorial membrane
    vibrate at a third frequency, which is some function of the two stimuli
    frequencies. When you put a small microphone in the ear, you can actually
    record that third frequency (because the membrane is playing it, much like
    a guitar string). So the two tones don't -mimic- a third tone; rather the
    two tones stimulate the active properties of the hair cells to actually
    -produce- a third tone. It's really there. It called the DPOE, short for
    Distortion Product Oto-acoustic Emission.

    Scott Seidman, Jul 28, 2004
  15. Otis Brown

    Dr. Leukoma Guest

    I have been an engineer, and I have been a doctor. I disagree with Otis on
    a number of things. Otis has been an engineer, but he has never been a

    Dr. Leukoma, Jul 28, 2004
  16. My use of the term *mimic* is not technially accurate. The nonlinearity
    produces a tone that is not present in a linear system.

    Repeating Rifle, Jul 29, 2004
  17. Otis Brown

    Otis Brown Guest

    Dear DrL,

    I do not consider a detailed analysis of the behavior
    of the natural eye to be "doctoring" in any sense of
    the word -- and I would object in strongest terms
    if anyone said that -- or jumped to that conclusion.

    In fact -- in the discussions I do have with pilots -- I make
    that situation absolutly clear. I consider a refractive state
    of -1/4 diopter to be no more "medical" than a refractive
    status of +1/4 diopters.

    Clearing vision -- by moving all objects to infinity
    is an engineering issues -- not a medical issue.

    A pilot who does this -- and verifies that his he can
    clear his distant vision from 20/30 to 20/20 is
    doing SCIENTIFIC work -- and Dr. Stirling Colgate
    did it.

    As far as objective facts are concerned -- you need
    to think for yourself. No one can do that for you.

    We disagree about the effect the minus lens has
    on the refractive status of the natural or
    fundamental eye -- not on any "defects" at

    Sorry you do not understand that specific issue.

    The "graph" in paper #26 shows how close the behavior
    of the natural eye is correlated to shifts (or step-function)
    "inputs" to the natural eye. The "output" or refractive
    status of the population of natural eyes shows
    the characteristic e ^ (t/Tau) of a control-system.

    The are direct, objective, scientific facts.

    The correlation coefficient is 0.97.

    I would suggest that others confim this
    calculation if you doubt this result -- or
    stop arguing about it.



    cc: Engineers who do understtand the nature
    of mathematical analysis of the natrual eye's behavior.

    Otis Brown, Jul 29, 2004
  18. Otis Brown

    Dr. Leukoma Guest

    (Otis Brown) wrote in

    It is an optical issue, a physiological optical issue to be precise, and I
    doubt that you have any training in that field.

    Dr. Leukoma, Jul 29, 2004
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