Comments on control theory/differential equation problem with speculative application to Myopia?

I wonder if anybody can correct, extend, or comment on the following. In
particular, does the math here in fact follow from the biological
assumptions made?

The eye is long sighted at birth and normally grows to become correctly
sighted, or possibly slightly long sighted, as is revealed in old age
when it loses much of its ability to change focus. Experiments (e.g. on
chicks) suggest that this growth is under feedback control, at least
some of which is quite local, occurring in the back of the eye, and
persisting when neural connections from elsewhere are severed. A few
biological mechanisms have been suggested (involving local sensing of
contrast, or change of contrast) that could provide a means of sensing
refractive error to drive the control loop. Many of them suggest that
the mechanism can sense the magnitude of the refractive error, but not
its sign.

What does this constraint alone tell us?

I have only considered the situation for first order differential
equations, with a 1-dimensional state space. Consider choosing f(y) in
dy/dt = f(y), subject to the constraint that f(y) = f(-y). We wish to
bring the eye from an initial large positive error of y_0 to a stable
equilibrium, at which f(b) = 0. For this to be stable we need f(b) to be
negative just above b and positive just below b. f(y) = f(-y) rules out
b = 0, so our destination is likely to be slightly long-sighted,
unless we can somehow cheat and have b < 0. If we do that, we need f(y)
0 for y < b, which means that f(y) > 0 for y > -b, which means we
never get below -b. So we have b > 0, f(y) < 0 for y in (b, y_0], f(y) >
0 for y just below b, and we might as well set f(y) > 0 for y in [0, b).

Here is some ascii art, with arrows pointing along the state space to
show the direction of f(y) = dy/dt

----<<---<----- -b --->--- 0 --->--- b ---<--------<<----- y_0 -----

This system is reasonably robust; it can recover even from short
excursions into short-sightedness (y < 0) as long as y > -b. When y does
drop below -b, myopia increases continually. There is a trade-off; large
values of b are very robust but produce a rather long-sighted
equilibrium. Small values of b produce better near vision in
equilibrium, but are vulnerable to errors which cause excursions below -
b, leading to steadily increasing short sight. The best (or least bad)
trade-off might depend on the environment to be selected for; making b
decrease with time might help a bit, too.

A simple example of this is dy/dt = 1 - |y|. We can solve this from an
initial y_0 to get three different curves:

For y_0 >= 0 we have y = 1 + (y_0 - 1) exp(-t) which converges to y = 1
(from either direction).

For y_0 < -1 we have y = (y_0 + 1) exp(t) - 1 which heads towards -
infinity and never stops.

For -1 < y_0 < 0 we have y = (y_0 + 1) exp(t) - 1, but since y_0 + 1 >
0, this heads upwards until it reaches y = 0, at which point it changes
into the first curve and continues stably on to approach y = 1.

How might this go wrong? Well, we don't spend all our time looking at
the same distance. Suppose that (even after allowing for the effects of
our eye's ability to focus itself) the error signal seen from moment to
moment varies by more than 2b. Values of y outside the range [-b, b]
cause the eye to become more short-signed. If the range over which the
eye is asked to focus is so large as to cause error signals outside [b,
-b] much of the time, then the eye will grow steadily more short-
sighted, regardless of the fitting of corrective lenses (if they are
single vision lenses worn all the time). So if we take an animal evolved
for a situation that requires to it look mostly into the distance
(perhaps a hunter-gatherer lifestyle on the african plains) and put it
in a situation where it is also required to do a great deal of near work
(for example a school) we should expect myopia, at least in those
specimens where b is small.

What is near work?

Consider the equation of geometrical optics: 1/f = 1/u + 1/v, where f is
the focal length of the lens, and u and v are the distances from the
lens to the object and the image respectively. The eye focuses mostly by
changing f, unlike a camera, which focuses mostly by changing v. I would
like to find a measure of nearness that is likely to be simply related
to our y. For that, I need to worry about the depth of field, because
the error seen by the control loop is likely to have some relation to
blurring.

Suppose that the eye has focused on an object, thus fixing f. Consider
other objects at distances close to, but not the same as, the target
object. The images of these objects will be out of focus; the light
reaching the retina from them will take the form of cones with base at
the pupil and point just in front of, or just behind, the retina. So a
measure of how blurred those objects are is the change in v for small
changes in u with fixed f. From
1/f = 1/u + 1/v
we get
0 = -1/u^2 - 1/v^2 dv/du
or
dv/du = -v^2/u^2
To make a distance scale out of this I can integrate it, placing the
point at infinity at 0, so that equal distances on this scale correspond
to equal exposure to blur. I treat v as fixed and ignore it, because it
is set by the dimensions of the eye. This of course integrates back to
the reciprocal, so a measure of the distance of an object, under a
metric that reflects its likely impact on our control loop, is the
reciprocal of its tape-measure distance.

As an example, let me fix a near point of 15cm, which is the distance my
eyes might be from an object if I was reading in bed. I will divide the
line out to infinity into eight equal sections, that is, equal under
this reciprocal scale. I get the following simply by taking the
reciprocal of n/120 for n = 0, 1, 2.. 8:

15cm 17.14cm 20cm 24cm 30cm 40cm 60cm 120cm infinity

The prediction from this is that relatively minor changes to near work
might have large effects; an animal evolved to work at distances of 60cm
or larger (but with a capacity to accommodate larger than this,
especially when young) would be in foreign territory if it read at 15cm.
This would hit it two ways. The accommodation required, over the medium
term, to cover both reading at 15cm and far sight at infinity is about
four times as much as before. The accommodation required in the very
short term, or resultant blurring, to work at distances of 15cm when not
everything is at exactly 15cm from the viewer, is likely to be larger
than that required to work at longer distances.

The suggested strategy is to perform near work at the longest practical
range; quite large changes in distance can be made when switching e.g.
from a paperback book read in bed to a soft-copy version read on a
computer screen, but the metric appears to show that even apparently
minor changes, e.g. from 15cm to 20cm, might be worthwhile. Arranging
for as bright a light as possible is probably synergistic with this; it
should increase contrast and may make longer distances practical.

(This has been posted under a valid email address, which I will retain
and monitor until it becomes choked by spam).
--
A.G.McDowell
.

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