User:SigmaJargon/math5110

Gregory Danner

MATH 5110

Prof. Borisyuk

Before anything else, let's go over a little about the eye, and how the pupil and iris work.

The pupil is an aperature in the middle of the iris that controls light entering the retina. In response to light (or the absense of light) on the pupil, the iris may expand or contract, allowing more more or less light into the pupil. This is very important, since if too much light enters the retina, it may be damaged, but if too little enters, it may not be able to form a good image.

The size of the human pupil ranges from a diameter of 3 to 4 millimeters when fully contracted, and widens to 5 to 9 millimeters when fully dilated (1). The standard unit of illuminance used for measuring light hitting the human eye is the lux, which is analogous to watts per square meter, but weighted by wavelength to fit a standardized model of brightness perception. About 50 lux is the brightness in a family living room, 320-500 lux is office lighting, 10,000-25,000 lux is ambiant daylight, and direct sunlight is at least 32,000 lux (2).

Let's start with light actually entering the pupil. The intensity of light entering the pupil

For a given illuminance I (in lux), the amount of light L (in lumens) actually entering a pupil of radius P (in meters) would be

$$L = I*\pi*P^2$$

Now, how does a pupil contract or expand when exposed to a certain amount of light? There is some threshold value for light entering the eye. Too far above this value and the amount of light is painful or damaging, but too far below it is insufficiently bright to properly see. The further from this threshold, the faster the eye will want to dilate or contract. Further, the eye will want to contract more slowly if it is already strongly contracted, and dilate more slowly if it is already strongly dilated. Let's say our maximum pupil size is $$P_{max}$$ and our minimum pupil size is $$P_{min}$$. Then, for a certain threshold t (in lumens), we can make a simple model:

$$dP/dt=k*(P-P_{min})*(P_{max}-P)*(t-L)=k*(P-P_{min})*(P_{max}-P)*(t-(I*\pi*P^2))$$

This is somewhat simpler than reality. For example, eyes actually contract about three times faster than they dilate (3), and I am presuming that change on rate from all of these factors is linear.

A quick few notes on this model. k is a constant, a scaling factor that can be adjusted to model pupils that dilate more or less speedily. $$(P-P_{min})$$ and $$(P_{max}-P)$$ are always positive, so the behavior (in terms of increasing or decreasing) is based solely upon the $$(t-L)$$ factor. If L is larger than t (that is to say, the expirienced illumination is greater than the threshold), then the eye will contract - if L is smaller than t, the eye will dilate. If the light is very dim (small I), then the $$(P_{max}-P)$$ term will diminish to zero as the eye dilates - if the light is dim enough, this term will go to zero before the $$(t-L)$$ term, indicating that the area is too dark to see in with the human eye. Likewise, with a sufficiently bright area (large I), the $$(P-P_{min})$$ might diminish to zero before the $$(t-L)$$ term, indicating that the area is too bright.

Now we come to the fun bit. If L is the equation listed above, it is easy to plot the above model on a time plane and find fixed points. However, in the problem we are given, L needs a little tweak. If P is large, it dilates so that it covers an additional light source, a small beam of light. For simplicity's sake, we'll treat this like a point, and have it communicate a certain number of lumens directly. Let us say this point is focused r (in meters) away from the center of the pupil, and it emits c lumens. Then:

$$L = \begin{cases} I*\pi*P^2, & \mbox{if }P<r \\ I*\pi*P^2+c, & \mbox{if }P\ge r \end{cases}$$

This breaks our normal consideration of single dimensional differential equations - that they be continuous. Without this, we cannot so easily say that the solution converges to a fixed point or diverges to infinity. Many of our normal analytical methods fail. However, numerical and graphical methods are still effective.

A lot of our constants fall within a fairly rigid fixed value range. Let's hash out the easy ones:

Minimum radius of pupil $$P_{min}=0.002$$

Maximum radius of pupil $$P_{max}=0.004$$

Scaling factor $$k=0.5$$. This one is fudged, but has no impact on the existance of oscillations

Threshold of the human eye $$t=0.75$$. For an average eye, this is brighter than ambiant daylight, but below direct sunlight.

And now for the ones that can reasonably change:

The ambiant lighting $$I=400$$. This can vary a lot - I picked a nice neutral classroom level of brightness

How far away from the center of the pupil to shine the light $$r=0.003$$. We are directed to aim a bit inside the edge, and we obey.

Brightness of the small light $$c=0.8$$. These are generally painfully bright, and thus above your threshold. It's possible to use a much less intense light, which will be discussed further below.

In this situation, the equations are

$$dP/dt = \begin{cases} ((7*p)/10 - 7/2500)*(p - 1/500)*(400*pi*p^2 - 26000), & \mbox{if }P<r \\ ((7*p)/10 - 7/2500)*(p - 1/500)*(400*pi*p^2 + 2000), & \mbox{if }P\ge r \end{cases}$$

See graph 1 for an illustration of the phase plane. Note that below r, the eye is dilating, and above r, the eye is contracting. Any starting position will move toward r, and then begin oscilating.

See graph 2 for an illustration of the phase plane for the same expiriement, except performed outside where I=12000. Note that the oscillations will be much more severe!

See graph 3 for an illustration of the phase plane for the same expiriement, except performed with a softer light of c=0.6. This does give us a significant change, since the light isn't strong enough to overcome the threshold, and no oscillation occur.

A sudden change in either I or c can throw the pupil into or out of oscillations. Oscillations will occur if I < t at r and I+c > t at r, and will not occur otherwise. Changing r does not affect this behavior as long as it is not moved beyond the edge of the pupil.

The other variables should not change suddenly. If they do, consult your optometrist immediately.

1. http://en.wikipedia.org/wiki/Pupil

2. http://en.wikipedia.org/wiki/Lux

3. Ellis, C. J. (1981). "The pupillary light reflex in normal subjects". British Journal of Ophthalmology 65 (11): pp. 754-759.