This web page is inspired by the work of Hung and Ciuffreda, in particular "A Unifying Theory of Refractive Error Development" (Bulletin of Mathematical Biology (2000) 62, 1087-1108). That paper describes a model that accounts for many observations of growth in the normal and myopic eye. It has the interesting feature that the growth of the eye is regulated only by the magnitude of the refractive error, and not its sign.
Under that model, growth to normal sight is possible, but undercorrecting myopia does not work. This seems counter-intuitive to many people, including me. This web page provides an example of a very simple model in which undercorrection may not work either, showing that this effect can indeed occur, even in very simple systems. It then discusses some more general models. showing that, if the eye did indeed sense only the magnitude of its refractive error, some part of the susceptibility to myopia would be an almost unavoidable consequence of a single underlying problem; that the mechanisms controlling the growth of the eye can tell when the eye is out of focus, but not whether it is long-sighted or short-sighted.
Unfortunately, the real-world relevance of this article is doubtful. In "Homeostasis of Eye Growth and the Question of Myopia", (Neuron, Vol. 43, 447-468, August 19 2004), Wallmann and Winawer provide a large amount of information about research into the controlling mechanisms of eye growth. They describe a study by Schaeffel and Diether on chicks, in which chicks were made to wear either positive or negative lenses in a restricted environment, with the accommodation mechanism paralysed, in such a manner that the magnitude of the refractive error was the same in both cases; only the direction differed. The eyes of both groups of chicks compensated in the correct direction, suggesting that the mechanism controlling this must have been sensitive to the sign of the refractive error, as well as its magnitude. If this information is available to the human eye, as it is to the chick eye, the main work in this writeup is of no practical importance whatsoever.
It is reasonable to assume that the growth of the eye is not completely controlled by genetic "dead reckoning", but tracks some signal that tells it how to grow to produce and maintain good vision. Variations in size much smaller than those seen in the growth of almost any other body part would completely destroy vision. Furthermore, experiments on chicks show that the back of the eye can be made to grow to produce myopia by various manipulations, including attaching diffusers in front of the eye to blur the vision. This response to the visual environment continues when many of the nerves in the eye are severed; it appears to be produced very close to the area of growth, at the back of the eye, presumably by the retina.
It is reasonably easy to believe that such a mechanism can tell whether the eye is in focus or not, but without communication with the focussing mechanism of the eye, it is difficult to see how such a mechanism could work out whether the eye is long-sighted or short-sighted. (Wallman and Winawer describe mechanisms that require assumptions about the environment, or knowledge of aberations in the eye that give rise to different blur patterns depending on the direction of the error). Hung and Ciuffreda discuss a mechanism that does not need this information; it is sensitive to changes in the amount of blurring, and in their computer simulation it is stimulated by a square wave of 0.1Hz (that is, with a period of 10 seconds). They use two forms of growth. A genetically programmed growth, perhaps affecting the lens of the eye and its general growth, tends to make it more long-sighted. Growth in response to changes in the amount of blurring tend to make it more short-sighted. Almost all eyes are long-sighted at birth, and grow to become less long-sighted (so more short-sighted). Under Hung and Ciuffreda's model, a very long-sighted eye experiences small changes in blurring, and tends to grow to become less long-sighted. As it grows into better, and more consistent, vision, changes in the amount of blur seen over periods of seconds are much rarer, and the drive to become less long-sighted and more short-sighted decreases. Hung and Ciuffreda show that their model fits the observed data by computer simulation. This web page shows a drastically simplified variant, that can be understood with a single picture, backed up by mathematics taught to me in grammar school. It then goes on to discuss more complex models.
A very simple model of eye growth in response to the magnitude of the refractive error is as follows:
dy/dt = ab - a|y + z|Here y is the refractive error, with positive error long-sightedness and negative error short-sightedness. || of course means absolute value, so if z is zero and y is either -10 or +10, |y| is 10. dy/dt is the rate of change of the refractive error, in response to the difference between ab, which represents a genetically programmed drive to long-sightedness, and a|y + z|, which represents a drive to short-sightedness in response to some absolute measure of refractive error. z can be the effects of introducing a corrective lens. I will ignore it a lot of the time. The constant term ab is written slightly oddly as a product of two factors, to make things neater later on.
Here a is not zero (or else no change would ever occur). If a is less than zero, then a sufficiently large long-sighted error is never corrected, but grows more and more extreme. Since we want to model an eye that is long-sighted at birth and becomes more short-sighted in an attempt to correct itself, we will have a > 0.
If b is less than zero, then dy/dt is always less than zero, and the eye becomes steadily more and more short-sighted. Since we want a model that includes normal eyes that attain a reasonably stable point of reasonably good vision, we will have b ≥ 0.
Because this is only a first order differential equation, the future values at any time (ignoring z) are entirely determined by the current value of y. We don't need to know how y got there to predict future growth. We can draw the entire behaviour of this equation with a few lines on a graph; to see what will happen given any particular value of y, move to a point on the graph where the curve takes up that value of y, and follow it on from there. To see what happens when some error input disturbs this path, simply take the value of y you have reached, add on the effect of the error, find the line on the graph that reaches the new value of y, and follow it on from that point. Here is such a graph (produced using Minitab), for a = b = 1. The lines parallel to the axis mark special values of y discussed later.
This graph is only intended to show the general idea. Even with this simple model, it can be distorted in various ways by changing the parameters a and b. The three curved lines on the graph are three different paths that the growth of the eye might take, without an error term or corrective lenses. We can move between graphs only by invoking the error term to change the growth (for example, as the result of sustained exposure to near work), or by using corrective lenses.
The top curved line is when everything goes right. The eye starts off long sighted and gradually grows to a stable state that is still slightly long sighted, but good enough, especially when the eye is young and can focus itself. The middle curved line shows that this system is reasonably robust; if a transient error changes it to be mildly short-sighted, it can recover and grow towords the original stable state.
The lowest curved line shows myopia that will progress indefinitely unless something is done. The only way to move off this line would be to remove whatever error got us on there, and use sufficient corrective lenses to move up to one of the other two curved lines, preferrably the middle one, so that the eye becomes slightly less myopic.
Suppose we write that s = at. Then we have dy/ds = dy/dt . dt/ds = dy/dt / a. Then dy/ds = b - |y + z| and we have made parameter a go away by measuring time in a different way. So our parameter a affects the speed at which the eye changes, but not the characteristics of that change, and if that change stabilizes in the long term, the position of that stable solution is not affected by a at all.
Ignoring z we have dy/dt = ab - a|y|, so if y = +/-b, then dy/dt = 0 and we have no change from then on (ignoring z). If b > 0, then y = b is stable; values of y just above b cause it to decrease towards b, while values of y just below b cause it to increase towards b. This is the long-term state of the normal eye. It does not attain an ideal value of y = 0, but is slightly long-sighted. The ability of the eye to focus itself, and the photographer's "depth of field", provide good, if not perfect, vision. As the eye ages, its ability to focus itself decreases and the person requires reading glasses.
If b > 0, then y = -b is unstable; values of y just above this cause y to increase, while values of y just below this cause it to decrease. In this theoretical model, negative values of y less extreme than -b will correct themselves (that is, move towards y = b); this allows the eye to recover from even quite large departures from its theoretical line to stability. Values of y below -b cause a myopia that continually increases, and is not cured by under-correction, unless that correction, z (≥ 0), is enough to make y + z > -b. In this case, the eye would grow to a new stable value of b - z, in which the vision when wearing corrective lenses is the same long-sighted b value that would have been attained, without lenses, had the excursion below y = -b not occurred.
If b = 0, then dy/dt = -a|y|, so that values of y either just above 0 or just below zero cause it to decrease; this point is only stable to deviations in one direction, and so in practice is not stable at all.
If we were to pick a value of b for ourselves, we would be faced with a trade-off; large values of b produce a growth process that is very unlikely to fall into y < -b and hit permanent myopia, but they also produce a long-sighted stable state. Small values of b produce much better corrected eyes if they reach a stable state, but increase the risk of permanent (and steadily increasing) myopia, when the error term becomes large enough to cause y < -b. One option would be to decrease b over time; this might fall out naturally from the decrease of the different rates of growth over time, or from the decrease in the ability to accommodate. The existence of a trade-off between high and low values of b might explain why myopia has not been evolved away, especially if environmental causes in modern life, such as reading and other near work, expose the eye to a much larger range of accommodation than that regularly enountered during evolutionary history.
To solve this equation, we need to get rid of the absolute value term by handling the cases of y ≥ 0 and y < 0 separately. We start from y = y0 at t = 0, for various values of y0.
If we have dy/dt = ab - ay, then y = (y0 - b)e-at + b gives us dy/dt = -(y0 - b)ae-at = ab - ay, and y = y0 at t = 0.
If we have dy/dt = ab + ay, then y = (y0 + b)eat - b gives us dy/dt = (y0 + b)aeat = ab + ay.
This confirms the long-term behaviour deduced from the states with dy/dt = 0. If y ≥ 0, we have y = (y0 - b)e-at + b giving us y -> b as t increases, and y converges to b from either direction, always remaining ≥ 0.
If y0 < -b, then y = (y0 + b)eat - b with both terms negative and y decreases further and further. In this model the decrease is explosive, but in practice the decrease in y would be limited by the maximum possible growth rate of the back of the eye.
If -b < y0 < 0, then y = (y0 + b)eat - b where the first term is positive so y actually increases towards 0 with t until it reaches y = 0. At this point it flips over into the solution for positive y, y = (y0 - b)e-at + b, and y continues to increase, continuing stably towards b.
In dy/dt = ab - a|y + z|, z accounts for some factor that causes a bias in the error fed back into the control loop. A positive value of z causes the eye to receive information that the eye is more long-sighted than it actually is. This would be the expected result of wearing spectacles that correct for short sight. Let w = y + z, so y = w - z. Then dy/dt = dw/dt so we have dw/dt = ab - a|w|, which is the same equation as before. Introducing positive z has two main effects:
I can think of two ways that near work could cause myopia under this very simple model, which I will call the medium-term theory and the short-term theory. Both of these effects are due to the large range of accommodation required to handle both near work and more distant vision, rather than near work itself.
The short term theory is again inspired by Hung and Ciuffreda, and their idea that the trigger for myopic growth is not so much blur on its own, as changes in the amount of blur. Increasing the range of distances that the eye has to deal with is likely to amplify this signal. Consider again dy/dt = ab - a|y|. How might that |y| term be derived? Suppose that for an object at distance n away from the average distance, the blurring effect is (y - n)2. This blur need persist for only a short time while the eye accommodates to the object after resting on it. As we look between objects at distance +n and -n from the average distance, the difference in blurring effects is |(y - n)2 - (y + n)2| = |-4ny|. So we have a more detailed model for the term -a|y| in dy/dt = ab - a|y|, with the prediction that a increases as the range over which the eye has to adapt increases. What about the ab term? This models the genetically programmed growth of the eye towards short-sightedness, so the term ab as a whole will not change with near work. Since a increases, this means that our b term decreases. Referring back to the graph, reducing b increases the chance of drifting below -b and falling into runaway myopia.
The medium term theory follows from a look at the graph, and the fact that at different times we focus our eyes at different distances. Growth towards myopia is triggered by refractive error outside the range of [b, -b]. If the range of accommodation required is sufficiently extreme, then at least one of the distances required, if not both, will force the eye into a state that corresponds to growth towards myopia. If the myopic growth in this state is sufficiently rapid, it will overwhelm growth towards long-sightedness at other times, even assuming such other times exist. This appears to require quite small values of b, relative to the ability of the eye to accommodate, but b is not fixed by this model to any pre-specified value determined by the shape and size of the eye; it may derive from a tradeoff established in evolutionary time between the benefits of a more finely tuned eye and the risks of runaway myopia. If so, it is not unreasonable that a very recent increase in the range of accommodation normally performed has increased the amount of short-sightedness.
If you try the model dy/dt = ab2 - ay2, you don't have to worry about the absolute value function ||, so at grammar school level you can just turn the handle and churn out a solution, although the algebra is slighly more annoying. The broad features of the solution are very similar; for sensible parameter choices you get a stable equilibrium at y = b, an unstable equilibrium at y = -b, and runaway myopia below it. Are all the equations like this?
Let us move off the syllabus offered by Down High School c. 1979 to "Dynamic Models in Biology", by Ellner and Guckenheimer (Princeton University Press, 2006), section 5.1 (Geometry of a Single Differential Equation). If we have a differential equation of the form dy/dt = f(y), where y is a simple scalar variable, then we can illustrate this by drawing a line for y and drawing arrows on that line, like this:
--------<<--------<-----------X-----<-------------<<-----------
The points on the line correspond to the possible values of y. The arrows show the sign of dy/dt = f(y), and so the direction that y will move with increasing t. Here X is an attempt at a stable point representing perfect sight at y = 0. We approach this from the far right, which I take to be initial long sight. I have put double arrows in on the far right, then a single arrow, to show dy/dt = f(y) decreasing as we approach perfect sight. We know from the case dy/dt = - |y| that we can make this work, at least as we approach from the right, but we also know that there is a catch, at least in that case, and this catch does indeed appear in the general case.
We have a biological constraint; the function f() does not know the sign of the error, so we must have f(y) = f(-y). This accounts for the arrows to the left of our X point y = 0, which is not stable in general; even minute deviations to negative values of y result in more and more movement to the left, and we fall into runaway myopia. Under the constraint f(y) = f(-y) we cannot have a single stable equilibrium point at 0. If we have a stable equilibrium at b != 0, then dy/dt = f(b) = 0 = f(-b), so we have at least two points with f(y) = 0. Here is such a diagram.
----<<-----<----(-b)--->-----0--->----(b)----<--------<<--------
We start off long-sighted, at the far right. I have put an equilibrium at y = b, so let's have dy/dt = f(y) decrease towards b. Since it is an equilibrium, we have f(b) = 0. For this to be stable, we must have increasing long-sightedness for points just below b, to bring us back up to b from just below it. To keep things simple, I have made this continue right the way down to 0. This fixes f(y) for y ≥ 0. Our biological constraint is that f(y) = f(-y), so we have now fixed f(y) for y < 0, as implied by the arrows in the diagram; for -b < y < 0 we have f(y) > 0, to match 0 < y < b, and for y < -b we have f(y) < 0 to match y > b.
This is the mixture as before, in a more general setting. If we have a stable point at b, then f(b) = 0. f(y) = f(-y) forces -b to be an equilibrium, since f(-b) = f(b) = 0, but an unstable one, because the arrows above and below it are pointing in the wrong direction. We have long-sightedness moving into a stable point at which we are only slightly long-sighted. We can tolerate errors that take us off the beaten path. Even errors that make us slightly short sighted don't divert us from the path to stable (but slightly long-sighted) vision. But sufficiently large errors in the direction of short-sightedness lead to runaway myopia, for which the natural mechanism cannot compensate, because it is working in the wrong direction. Furthermore, there is a trade-off between large values of b (the stable destination is quite long-sighted) and small values of b (the chances of falling into runaway myopia are high). Changing b over time does not completely remove this trade-off, as it is still possible to get runaway myopia at the last minute, although the effects of this would be much less, because there is less time left for the eye to become more short-sighted before growth stops with age.
If we add more points to the diagram we can at least stop runaway myopia heading off to infinity, but there is a constraint on this, too. We need the region between our initial long-sightedness at birth and our stable destination b to be clear of anything that would stop us arriving at b. We don't gain anything by putting points between b and zero, because the arrows there are already in the right direction. We could improve on nature by putting a equilibrium at a value +/-c, stable for -c, where c is much larger than the initial long-sightedness, to catch eyes that would otherwise fall into runaway myopia and progress indefinitely, but there seems little point to this; in a state of nature, anybody that short sighted is probably already dead.
Since this more general model predicts runaway myopia, it also predicts that sufficiently large values of under-correction can increase the progression of myopia, compared with the correct prescription. Suppose that we are at the myopic point M.
----<<--M---<----(-b)--->-----0--->----(b)----<--------<<--------
Myopia (movement to the left) will continue to progress unless our corrective lens takes us at least to the right of (-b). In fact, in this model, the target for correction should be point 0. We know that dy/dt = f(-b) = f(b) = 0, so there is a pretty good chance that movement to the right (back towards long-sightedness) is greatest at y = 0. Extreme over-correction (back to points to the right of b) can also allow myopia to progress. Note that the 'medium term theory' predicts progression of myopia, as long as near work persists, regardless of correction, for a very large class of functions f() in dy/dt = f(y).
It is important, under this model, to have a clear picture of what near work is, and the accommodation required of the eye to handle it. Under the 'medium term theory', we need to have a feel for the increase in the daily range of accommodation imposed by the introduction of periods of near work. Under the 'short term theory', we need to have a feel for the size of the range of accommodation required when moving our gaze between a book and the back of the desk on which it rests, relative to the range of accommodation required to shift our view from the skyline to a fellow hunter, or from a cricket ball in flight to the scoreboard. When considering reducing near work, it would be useful to know the difference, in terms of accommodation, between a book at 15cm (the distance from my eye if I read it in bed), and a computer screen at 60 cm. We might suspect that near work introduces large amounts of accommodation if we have ever manually focused a lens on a camera, and remember the difference between the amount of travel required to focus at short distances and that required to focus at larger distances.
Simple geometrical optics tells us that 1/f = 1/u + 1/v, where f is the focal length, and u and v are differences to the object and to its image, respectively. This can be justified by considering rays passing through the centre of the lens without change, and rays parallel to the axis of the lens, which are diverted to pass through the focal point of the lens. Both these types of ray must originate from a point on the object and travel to form the corresponding point on the image.
In the eye, most of the focussing effort works by changing f, but it is still useful to consider the situation when f is fixed and differences in u lead to differences in v. This is because the amount of blur introduced when the u provided does not match the f and v chosen depends on the distance between the u we actually have and the correct value of u; it is introduced by cones of light that have their base at the pupil of the eye, and would terminate in points on the image if the correct value of u was used, but in fact intersect with the retina to form circles. We have 1/f = 1/u + 1/v. If we differentiate this with respect to u, while holding f constant, we have 0 = -1/u2 - 1/v2 dv/du. Rearranging this, we have dv/du = -v2/u2, so the amount of blur produced, for instance, by objects at small distances from the plane that the eye is focussing on, is proportional to 1/u2. Note that, in the eye, v is almost unchanged at all distances, as focussing is performed by changing f, so we can ignore v from now on as long as we are only interested in the relative amount of blur introduced.
One measure of the amount of accommodation required when an object changes its distance can be obtained by integrating 1/u2 over that distance. This charges each small portion of distance with a measure of the blur that it introduces for objects just in front and just behind that portion, when the eye focuses on the distance. Alternatively, suppose that you decide on an acceptable amount of blur, and divide the distance out from a near point to infinity into a number of steps, where each step is just long enough that focussing on its near boundary takes its far boundary to the limit of the acceptable blur. If your acceptable blur is very small, then the measure obtained by integrating 1/u2 amounts to counting the number of such steps. Integrating 1/u2 takes us back to 1/u, so the charge incurred as we travel from distance a to a larger distance b is 1/a - 1/b. To show the accommodation required for near work, and how this changes with distance, I have taken the accommodation required to go from my near point of 15cm to infinity, and divided it into 8 parts. This is equivalent to taking the line between 1/15 and 0 with the usual metric, and dividing that into 8 parts, so what we are seeing is the reciprocal of 8/120, 7/120, 6/120, and so on.
| Distances separated by equal amounts of accommodation cost | ||||||||
| 15cm | 17.14cm | 20cm | 24cm | 30cm | 40cm | 60cm | 120cm | infinity |
This table confirms that the increase in accommodative range involved in (for example) working with points as close as 15cm, as opposed to points no closer than 60cm away, might plausibly drive effects in which increased accommodative range can produce growth towards myopia in susceptible individuals (in our terms, those with small b parameter). Check: in a 50mm Pentax lens, the focussing scale is marked with distances infinity, 15m, 5, 3, 2, 1.6, 1.2, 1, 0.85, 0.7, 0.6, 0.55, 0.5, and 0.45, with most of the travel at the near end. The distance between the markings for 0.45m and those for 0.5m is one of the largest. On an Olympus Macro lens there is a similar effect, but the situation becomes more confused; for closer distances the travel becomes so extreme that the distance between film and object changes very little, with the lens moving between the two.
Our mathematical model examined the simple equation dx/dt = f(x), subject to f(x) = f(-x). The eye may in fact be controlled by a more complex equation, so it is worth looking to see if these effects persist there too. The state space of dx/dt = f(x) is 1-dimensional; the state of the system is entirely contained in the single variable x. Our original control law was linear in |x|. Anybody who has successfully focused binoculars has in fact been able to bring a similar system into a stable state, by using a more complex control law in a larger state space; typically I move the focus wheel in one direction and look to see if the picture gets sharper or less sharp. Then I move the focus wheel in the sharpening direction until the picture starts to get less sharp again. My memory of the direction to move, after my initial experiment, would be represented by an extra dimension. I am also usually in a simpler situation than the eye; the only thing changing is my manipulation of the focus wheel, and there is very little 'noise'. Even then, I do sometimes get confused and end up moving the focus wheel in the wrong direction. I think this leaves the question of what might happen in N dimensions open, at least until we start doing some mathematics.
Let us write the state of the eye, as it grows, as a vector split into two components, x and w, where x is the refractive error and w is a sub-vector with a fixed, possibly large, number of dimensions. In the binocular-focussing analogy, my memory of the early, experimental, stage, is stored as one or more components of w. We have the following:
d(x, w) / dt = f(x, w)f() is now vector-valued, and is subject to the constraint f(x, w) = f(-x, w). We will assume that f is smooth enough that we can approximate it by a linear function for small values of x and w, as long as the area of approximation does not cross the boundary at x = 0. Can we have a stable state at x = 0? Without loss of generality, we can assume that w = 0, because if there is a stable state at f(0, w) for some other w we can always redefine f() to move it to f(0, 0).
If we have a stable state at (0, 0), then we must have f(0, 0) = 0. As we did with f(x) = ab - a|x|, we consider the two cases of x < 0 and x > 0 separately. Because of our assumption about f(), by considering points sufficiently close to the supposed stable point at (0, 0), we can approximate f() by a linear function. In other words, we can linearise the equation to d(x, w) / dt = A (x, w)', where A is a matrix which differs on either side of the x = 0 boundary. Because of f(x, w) = f(-x, w), the two A matrices are the same, except that its first column changes sign.
The solutions of dx/dt = Ax, for multi-dimensional x, are made up of linear combinations of primitive solutions, of the form x = vp(t) exp(at), where v is a vector, p(t) is a polynomial, and exp() is the usual exponential function. a is an eigenvalue of the matrix A. If any a is > 0 then there is a straight line path from the origin to infinity, so the solution is not stable. Since we can change v to -v we can always find such a path without crossing the boundary x = 0; if the path lies along the boundary we can just continue to follow it there. If we have a zero eigenvalue, then the situation is only neutrally stable, which is of little use, because it is vulnerable to drift from any source of noise or error. For stability, we need a < 0 for all eigenvalues a, of both the original matrix A and that same matrix with the sign of its first column changed. Where a is complex, we need its real part to be negative.
The eigenvalues of A are the roots of the equation |A - xI| = 0. This expands out as follows:
| a00 - x, a01, a02, ... | | a10, a11 - x, a12, ... | | a20, a21, a22 - x, ... | |...
And we can write this out as the sum of two determinants
| a00, a01, a02, ... | | -x, a01, a02, ... | | a10, a11 - x, a12, ... | + | 0, a11 - x, a12, ,,, | | a20, a21, a22 - x, ... | | 0, a21, a22 - x, ... | |... |....
If we expand out the determinants we get +/- f(x) + xg(x). The term +/- f(x) represents the first determinant, and its sign changes when we change the sign of the first column of the A matrix. The term xg(x) represents the second determinant, which has no constant term but has a higher power of x than f(x). What happens when x = 0?
When x = 0 the determinant reduces to +/- f(0), so for at least one of the two versions of A we have |A| > 0. If f(0) = 0, then 0 is an eigenvalue and the system is at best neutrally stable, which is no good to us.
If the polynomials +/-f(x) + xg(x) are of odd degree we know that they each have at least one root, because as x varies between -infinity and +infinity they decrease from +infinity towards -infinity (if you look at the determinant you can see that the highest term in xg(x) is -xn). If f(0) > 0 then there is a root of f(x) + xg(x) between x = 0 and x = +infinity. If f(0) < 0 then there is a root of -f(x) + x(g)x between x = 0 and x =+infinity. Therefore for odd n, at least one of the two A matrices has an eigenvalue greater than 0 and our system is unstable.
If the polynomials +/-f(x) + xg(x) are of even degree then we know that they go towards +infinity as x goes to +/-infinity. If f(0) < 0 then there is a root of f(x) + xg(x) between 0 and +infinity again. If f(x) > 0 then there is a root of -f(x) + xg(x) between 0 and +infinity. So again at least one of the two A matrices has an eigenvalue greater than 0 and our system is unstable.
Therefore, even in the n-dimensional case, there is no way we can have an unconditionally stable state with 0 error, using a system d (x, w) /dt = f(x, w) subject to the conditions that x is smooth enough to be linearised away from the boundary x = 0 and f(x, w) = f(-x, w). Our ability to focus binoculars may be due to using an f() which breaches these conditions, or to a state space in which the initial experimental phase in practice provides us with a w large enough that we do not encounter unstable solutions that might be associated with very small values of w (perhaps the occasions when we become confused about which way to turn the focus wheel).
With n dimensions and f(x, w) arbitrary (except for f(x, w) = f(-x, w)), there isn't much we can say about the situation; there are a huge number of possibilities, from something like d(x, w)/dt = ab - a|x| - w to a hugely complicated f(x, w) corresponding to a human focusing a pair of binoculars.
If we make the strong assumption that the system is linear, so that f(x, w) = A (|x|, w) + b, then the situation depends on the solution of A (x, w) + b = 0. If this solution has x < 0, then there is no equilibrium at all. If this has x = 0, then there is an equilibrium at x = 0, which we know cannot be stable. If the solution has x > 0, then there are two equilibriums, at (x, w) and (-x, w). The stability of these solutions depends on A, and on A with its first column sign-inverted; we know from the previous discussion that at most one of them is stable.
I don't think the following is biologically at all plausible, but it shows, by its very existence, that simply setting f(x, w) = f(-x, w) doesn't rule out the existence of large domains of attraction to a stable solution for both x < 0 and x > 0.
Our building block is the linear system x' = -s(x - b), where x is a two-dimensional vector and s is a scalar; b is therefore a two-dimensional vector. The solution to this is x(t) = exp(-st)(x0 -b) + b, which means that x converges directly to b from everywhere, moving in a straight line. in particular, we can lay out a polygon and a point, so that if we start off inside the polygon, we end up converging towards the point, in a straight line.
Here is some more Ascii Art.
-----------------------------------
+ |\ /|
| \ / |
| ----------------------------- |
| | | |
| | | |
| | | |
| | | |
| | X | |
| | | |
| | | |
| | | |
| ----------------------------\ |
| + \ |
-----------------------------------
+
This is a two-dimensional diagram for a two-dimensional system. Within each region in this diagram, the state of the system (x, y) evolves according to a system linear differential equation of the form
dx/dt = -s(x - a)but the system as a whole is non-linear, because the different regions obey different linear equations. Each '+' on the diagram represents an (a, b) to which the states within some region is drawn. The two '+'s outside the main box are the targets for the long thin regions pointing towards them. The '+' in the L-shaped region is the target for the L-shaped region. The 'X' marks a saddle point for the inner box, and I don't care what happens outside the outer box. Any point in the two long thin regions will move towards its '+', but will never get there, because it crosses a boundary into the L-shaped region in its journey, and then converges to the '+' in the L. The points in the central box will (except for the unstable equilibrium in the 'X' itself) be drawn out of the box into one of the border regions and end up at the L '+'. So almost all points in the inner or outer box end up at the '+' in the L,
Now imagine this system reflected in the y axis, to see what happens when the refractive error is negative. The main region contained a saddle point, and we only really cared that points in main region ended up crossing its boundaries. It turns out that in two dimensions, exactly one of each pair of reflected equilibrium points is a saddle point. Therefore we can arrage for this region, in the reflected version, to be stable; we have produced a system with large stable regions capable of correcting both negative and positive refractive error, subject to the constraint that f(x, w) = f(-x, w), by accepting that we cannot produce a perfect correction (and, in this case, by using a rather implausible function f().