Page 8
PUBLISHED IN:
The 5th Annual Engineering in Medicine & Biology Symposium, Columbus, Ohio
IEEE-EMBS, 1983
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A Heuristic Model for Tonic Accommodation
Otis S. Brown, General Electric
Karel Montor, Leadership and Law
( Paper 8 )
ABSTRACT
The normal eye has a competently designed lens control system. The existence of optical deadband, or depth-of-field, causes a fundamental non-linearity in this operation of an automatically focused
camera. The lens support muscles generate a noise signal which produces a back-and-forth motion in
the lens across the optical dead band. To better understand the functioning of this device, we have
designed an analog computer which predicts the eye’s focal status as a function of depth-of-focus, noise
in the muscles, and the diopter value of the eye’s environment.
The focal status of the lens is controlled by blur, as long as the retina can sense blur. When the
eye is in darkness, the lens is driven to a “standby”, or tonic accommodation position. Experimental
evidence has been presented which demonstrates that the eye’s focal status will show a servo response
to a long-term change in the visual environment. It is probable that tonic accommodation will exhibit a
similar response. This heuristic model yields focal status predictions of tonic accommodation as a
function of time, and the individual’s long-term environment.
INTRODUCTION
This model represents the servo control activity of the lens, as the eye views distant objects in
sunlight. The predictions of this model do not apply to eyes that are assumed to be defective.
To keep the model relatively simple, slight transient and focal bias effects from the convergence
system are not modeled. Computer simulation models for these effects have previously been discussed.
(1) (2)
This paper presents and analog computer developed from currently accepted block diagrams of
the normal accommodation system. In addition, the analog computer simulates the focal activity of two
long-term systems; the eye’s focal status relative to the accommodation system, and the eye’s longterm control of tonic accommodation.
EYE RESOLUTION
A camera or eye can be limited in resolution by diffraction effects caused by a small aperture, or
by granularity effects due to the response characteristics of the film or retina.
The normal eye has a resolution of 6 minutes of arc. The eye has an approximate length of 25
mm. By trigonometry, the separation of two points of light must be at least 0.044 mm on the retina to
be perceived as two distinct points of light.
The human eye would have a diffraction limited resolution of 6’ of arc if the aperture were 0.35
mm. Since the smallest aperture for the un-drugged human eye is approximately 2 mm, we can
conclude that the resolution of the human eye is limited by the “granularity” effects of the retina, rather
than by diffraction effects caused by a small aperture of the iris.
DEPTH-OF-FOCUS
Because of granularity effects of the retina, we can change the focal power of the eye's lens
without changing the sharpness of the image formed on the retina. This range of focal status values is
called the depth-of-focus. Specific values are calculated by the following equation: (Encyclopedia
Britannica, Optics, 1970)
FG = d * I / a
Where:
FG = Depth-of-focus
I = focal length
d = Blur circle
a = aperture diameter
The depth of focus is inversely proportional to the diameter of the aperture. With an aperture
of 8 mm, the human eye has a depth-of-focus of +/- .15 diopters. In sunlight the iris will have an
aperture of approximately 2 mm, which gives us a calculated value for depth-of-focus of +/- .6 diopters.
FIGURE 1. A BLOCK DIAGRAM OF THE ACCOMMODATION AND LONG-TERM FOCAL CONTROL SYSTEMS.
The accommodation system is shown within the dotted lines in Figures 1 and 2. Figure 2 shows
the analog computer for accommodation.
The light rays that pass through the lens ultimately produce blur on the retina. The blur is
sensed; this then generates a neurological signal which changes the focal status of the lens. The lens is
the summing point for the input signal versus the feedback signal. This is the essential definition of a
servo-control system.
ANALOG COMPUTER IMPLEMENTATION
The visual environment is simulated by making one volt equal to one diopter. The
accommodation system is implemented by using four operational amplifiers
1.
2.
The error detector amplifier compares the feedback signal (from the retina) with the input
visual (environment). An output signal is produced which is proportional to the image blur.
The dead-band operational amplifier duplicates the retina's inability to sense blur while the
image is within the dead-band.
3.
4.
Motor Simulation. This amplifier changes the neurological signal into movement by
controlling muscles and ligaments attached to the eye's lens.
Optical stop simulation. This is implemented by use of a buffer amplifier whose output state
(voltage) is proportional to the focal state of the eye. This limit simulates the measurement
of the optical state of the eye.
FIGURE 2. AN ANALOG COMPUTER SIMULATING THE FOCAL STATUS OF THE LENS OF THE EYE.
DEAD-BAND SIMULATION
A silicon diode has a knee of 0.6 volts. By using two back-to-back diodes in conjunction with an
operational amplifier, we can simulate the effect of dead-band in a servo system. With this simulation
there will be no output from the amplifier until the input exceeds +/- .6 volts. The alternative approach
is to use a comparator, in which the width of the dead-band can be better adjusted and controlled.
THE LENS-PLANT TRANSEFER FUNCTION
Previous studies have shown that the plant (lens support muscles) has a ( 1/ TAU s + 1 ) transfer
function. This type of plant transfer function causes a “lag of accommodation” in the operation of the
lens control servo. We are using an operational amplifier with a gain of ten and an time constant of 1/3
second to simulate the plant dynamics.
NOISE IN ELECTRONIC SERVO SYSTEMS
There are numerous sources of noise in a mechanical servo system. The development of these
random perturbations are due to thermal effects, noise in the amplifier, and back-lash in the gears. This
mechanical system produces a signal which is continually perturbated about the desired, or command
signal. This is called control system jitter.
The output position of a noisy servo system can be described by a statistical distribution in
terms of frequency spectrum, mean, and root-mean-square values. Systems are normally designed to
minimize the effect of noise. Paradoxically, a designer can intentionally use the existing noise in a
system to produce a scanning motion in the lens and thereby minimize the effect of dead-band on the
output of a control system.
Noise will always exist in a physiological control system, and the output position of these
systems can be measured as random perturbations around the mean of the command signal. If the
input visual environment is constant, the output signal will be exclusively due to the effects of
measurement errors and internal noise in the system.
NOISE SIMULATION
Dr. A. Suzumura has determined that the noise in the muscles has a characteristic frequency
spectrum of from 1/4 to 4 hertz. (3) The output of the analog computer shows a noise signature which
is close to the dynamic noise characteristics of the human eye.
THE SYSTEM'S DYNAMIC RESPONSE
Blur control can only occur as noise drives the lens beyond the threshold of the optical deadband. A recording of the eye's focal status is shown in Figure 3. The eye's focal status never rests and
must be described by a statistical-distribution, with the skirts extending beyond the edge of the optical
dead-band, as show on Figure 4.
The focal status of the lens is equal to the diopter value of the environment. The standard
deviation of this distribution will be perhaps +/- 0.6 diopters. Recent measurements, made by Dr. R.
Yamaji using an infrared optometer, have shown servo noise response characteristics of the human eye
which are similar to the voltage output of the analog computer. (2)
A BLOCK DIAGRAM FOR DYNAMIC LONG-TERM CONTROL
Based on Dr. Francis Young's experimental data, we show a block diagram that obtains its
control from the accommodation signal. (3) Since the stop-to-stop lens travel is ultimately limited by
the optical state of the eye, the block diagram shows the long-term system controlling the physical stop
position of the lens of the eye, as shown in Figure 1.
MEASURING THE EYE'S FOCAL STATUS
The limiting effect caused by the eye's optical state (or length) can be simulated in the analog
computer by using an operational amplifier with a diode in the feedback path, as shown in Figure 2. As
long as the lens voltage is above the reference value (linear region), the output will be a replica of the
input. When the input exceeds the reference value, the output will remain clamped at the reference
voltage, thus simulating the measurement of the optical length of the eye.
THE TONIC ACCOMMODATION SYSTEM
When the eye is in darkness, the retina is unable to sense blur. The eye (accommodation
system) will then select an alternate source for the reference signal. Under tonic accommodation
control the lens is not part of the feedback loop. The status of the tonic control can be determined by
measuring the focal status of the eye with an infrared optometer. A block diagram of the tonic
accommodation control system is shown in Figure 5.
CONCLUSIONS
This model for the normal eye's focal status was constructed on the principle of using the
smallest number of physiologically justifiable elements that still give a reasonably precise fit to the
experimental data. The model presented is probably the simplest model that can be built that can
accurately describe the behavior of the tonic and long-term focal control characteristics of the eye.
In a number of cases the experimental record is fragmentary or nonexistent. Where no
experimental data exists, we have had to infer control activity based on our judgment of prior
experimental data. In any event, the predictions of this model may be confirmed or denied by
comparing the predictive accuracy of this model against other theories that yield explicit numerical
predictions for the long-term focal status of the normal eye.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
Hung, G., and Semmlow, J., “STATIC BEHAVIOR OF ACCOMMODATION AND VERGENCE:
COMPUTER SIMULATION OF AN INTERACTIVE DUAL-FEEDBACK SYSTEM”, IEEE Trans. Biomed.
Eng., Vol. BME-27, pp. 439 – 447, Aug. 1980
Hung, G., and Semmlow, J., “A QUANTITITAVE THEOR OF CONTROL SHARING BETWEEN
ACCOMMODATIVE AND VERGENCE CONTROLLERS”, IEEE Trans. Biomed. Eng., Vol. BME-29, pp
364 – 370, May 1982
Suzumura, A., "ACCOMMODATION IN MYOPIA", Proceedings of the Second International
Conference on Myopia, Prefecture Hall, Yokohama, Japan, pp. 55-67, May 1980
Yamaji, R., "TREATMENT OF ACQUIRED MYOPIA (PSEUDOMYOPIA)", ibid., pp. 91-100
Brown, O., Young, F., “THE RESPONSE OF A SERVO CONTROLLED EYE TO A CONFINED
ENVIRONMENT”, The 18th Annual Rocky Mountain Bioengineering Symposium, pp. 41 – 44, April
1981
Young, F., “THE EFFECT OF RESTRICTED VISUAL SPACE ON THE PRIMATE EYE”, American Journal
of Ophthalmology, Vol. 52, Prt II, Nov. 1961
Brown, O., Young, F., Berger, R., “MEASURING THE EYE’S FOCAL ACCURACY: A HEURISTIC
APPROACH”, The 3rd Annual Conference of the IEEE/EMBS, pp. 64 – 68, Sept. 1981
ACKNOWLEDGEMENT
I would like to acknowledge the assistance and review by Dr. Peter Greene.
Special thanks also for Dr. Francis Young for the many years of effort he has spent conducting
carefully controlled experiments. This meticulous work allows a clearer appraisal of the dynamic servo
control characteristic of the normal human eye.