A band in the lens reads as age or as macula

The matches carry their own lens found that one session of asymmetric colour matches — surfaces under daylight matched against the same surfaces under a phosphor LED — fits an observer’s rod signal weight, lens age and macular pigment together. A lens of the wrong shape lands in the macula then asked what the fit does with a lens of the right density at 400 nanometres and a different slope. The fit read the difference mostly as macula, moved the rod weight by six tenths of its standard error, and could not tell from its own residuals. Fitting the edge’s position as a fourth parameter repaired it: the bias vanished, the precision barely changed, and the edge came back.

That essay closed on the other thing an older lens does. The model’s lens is one exponential, scaled by age; a real lens yellows by accumulating pigments that absorb in a band just past 400 nanometres, which puts a shoulder on the curve rather than making it steeper. The prediction was that the new edge parameter would take up a shoulder near 420 nanometres, since both change the lens’s density most just past 400 and the lamps sample that region coarsely, and that a shoulder near 450 would be fitted partly as macula whatever the lens model, because there the lens band overlaps the macula’s.

The edge takes none of it #

A tenth of an optical density of extra lens absorption, fifteen nanometres wide, centred at 410, 420 or 430 nanometres, is read by one session as 3.7 to 6.4 years of extra lens age — three to five of that parameter’s standard errors. At 420 the edge parameter takes a fiftieth of one of its standard errors. Centred at 440, 450 or 460, the same band is read as 0.034 to 0.097 of an optical density of macular pigment, two to six standard errors, and survives fitting the edge. Either way the rod weight moves: by 0.4 to 1.4 of its standard errors in one session with the edge fitted, up to 1.8 in two. The misfit stays under 1.1 standard deviations of the noise in one session and 1.4 in two.

• The first half of the prediction fails. A shoulder near 420 is not a change of slope, and the edge parameter does not see it.
• The second half holds, and more strongly than predicted: from 440 nanometres the band is read as macula, not partly but almost wholly.
• The weight moves at every wavelength the band can sit at, in opposite directions either side of 440.
• No session can tell, and the lamp pairs that are safe from a band are the ones that measure the weight worst.

What a band in the lens looks like #

The model’s lens at sixty has an optical density of 0.97 at 400 nanometres, falling by a factor of e every 68. The bands put on top of it are Gaussian, a tenth of a density high and fifteen nanometres from centre to the point where they fall to 0.6 of their height — the kind of hump the yellow chromophores of an older lens add to its absorption, stated here rather than measured. At 420 nanometres the band sits where the lens is already steep and adds about a seventh to its density there; at 450 it sits where the lens has fallen to under half a density and adds more than a fifth, and where the macula, a wider band centred at 460, is near its peak.

The macula matters because the fit has a parameter for it. The macular is a band, not a filter is the collection’s account of why macular pigment is modelled as a band rather than a filter: it absorbs within about forty nanometres of 460 and nowhere else. A band of lens absorption near 450 is, to a colour match, almost the same thing — a dip in the light reaching the cones around 450 — and the only difference between the two is that the lens band also sits in front of the rods, while the macula does not.

Where the band goes, wavelength by wavelength #

The fit sees the band only through the matches, and it explains the matches with whatever combination of its four parameters best reproduces them. Up to 430 nanometres it chooses the lens age: a band at 410 is read as 5.0 of the lens’s standard errors, a band at 420 as 4.7, a band at 430 as 2.9. An older lens absorbs more at every wavelength, most at the shortest — the absorption that the eye stops at the lens found setting the short-wave limit of vision itself; a band just past 400 adds absorption at the shortest wavelengths the lamps carry, and the difference between a band and a denser lens is out where the lamps have little light to show it.

From 440 nanometres it chooses the macula: 1.9 of its standard errors at 440, 4.0 at 450, 4.8 at 460, with the edge fitted. The lens age goes the other way, a year or two younger, because the fit needs less absorption at the short end to go with the macula’s extra absorption at 460.

The edge parameter takes little anywhere: under half of its standard error from 410 to 430, and up to 1.2 at 450. It was built to express a lens that is steeper or shallower throughout, and a band is neither. The earlier essay’s repair was specific to the defect it was built for.

What the band is taken for, in years and densities #

A band of a tenth of a density at 410 nanometres is read as 6.4 years of lens age; at 420, 5.9; at 430, 3.7. For comparison, the observer has no age found the lens the one part of the eye that changes steadily with it, at about a fiftieth of a density a year at 380 nanometres. A pigment band that some sixty-year-olds carry and others do not would spread a group’s fitted lens ages by several years, all of it attributed to the lens’s overall density.

At 450 nanometres the same band is 0.075 of a density of macula, at 460 0.097 — comparable to the spread of macular density between people, which the collection’s population model puts at a standard deviation of 0.13 about a median of 0.35 — and a field size is two changes is where the same pigment’s fall from the fovea to ten degrees does half that again. The crossover between the two readings is near 440 nanometres, where the band is read as a little of each: 0.55 years and 0.034 of macula.

This is the prediction’s second half, and it is not “partly”. The fit has two parameters that each describe absorption in the blue, one broad and one a band, and a new absorption is given to whichever it resembles more. Near 450 that is the macula almost entirely.

Either way the weight moves #

The rod weight is the parameter that pays for whatever the others cannot absorb. A band from 410 to 430 pulls it down, by 0.40 to 0.84 of its standard error from one session; a band at 450 or 460 pushes it up, by 1.5 to 1.7. The reason for the sign change is where the band sits against the rods. The rod signal enters the S pathway with a peak near 500 nanometres as seen through the lens, so a lens band at 450 takes more from the rod signal’s short-wave flank than the macula — which is not in front of the rods — can mimic, and the fit makes up the difference with more rod weight. A band at 420 takes more from the S cones than from the rods, and the fit makes up that difference the other way.

In the weight’s own units the shifts are not small. The observer’s true weight is one half. With the edge fitted, one session reads it as 0.46 for a band at 410 nanometres and 0.45 at 420, and as 0.58 at 450 and 0.58 at 460 — a band a tenth of a density high, which a lens can carry without anyone noticing, moves the measured rod contribution to the blue–yellow pathway by up to a sixth of itself. That is two thirds of the difference between the two readings of the rod signal’s size that age can size the rod signal set out to distinguish at forty-five, where one reading predicts a weight of 0.5 and the other 0.62.

Fitting the edge moves the curves and brings none of them to nought. From one session the weight’s shift with the edge fitted runs from 0.37 to 1.39 standard errors across the six centres. Two sessions, which fit the weight twice as precisely, shift it by up to 1.8 of their smaller standard errors with the edge fitted and up to 3.0 without. The pattern the earlier essay found for the lens’s slope — the sharper design is the more exposed — holds for bands too.

Which lamps a band misleads #

Tungsten against the LED and tungsten against the three-emitter lamp are the most exposed to a band at 450: they move the weight by 4.4 and 3.9 of its standard errors. The three-emitter lamp’s pairs with daylight and with the tube move it by 2.7 and 2.8; daylight against the LED, the design the earlier matching sessions used, by 1.4.

Two pairs are indifferent to it. Tungsten against the tube moves the weight by four hundredths of its standard error and daylight against the tube by two tenths. They are indifferent for the same reason they are poor measurements of the weight in the first place. Between a tungsten lamp and a tube, the light at 450 nanometres against the light near 500 barely differs, so a change in what the eye absorbs at 450 changes both lamps’ share of the blue alike and the match between them does not move — and the rod signal, which is read through the same short-wave flank, moves the match little for the same reason. Tungsten against the tube fits the weight to ±0.345, more than five times daylight against the LED’s ±0.061.

So the safety is not a property to be designed for. A pair that sees the rods sees the blue, and a pair that sees the blue sees whatever is in the lens there.

Whether the session warns #

It does not warn loudly enough to act on. After fitting weight, lens age, macula and edge, the residual one session is left with stands at 0.3 to 1.1 standard deviations of the chi-square its own noise produces; two sessions reach 1.4. The largest misfits are for bands near 430 to 450, where neither the lens nor the macula fits the band well; they are still well short of the two standard deviations at which an experimenter would stop.

This is the second time in two essays that a lens defect of a plausible size has moved the rod weight by around a standard error while leaving the fit’s residuals looking healthy. The matches carry their own lens reported standard errors that are correct for its model; what these two essays add is an error budget for the model itself, which the standard errors do not include.

What an experiment on older observers needs #

A lens model with a band in it, or a stated allowance for one. The first is available: the yellow chromophores of the ageing lens have known absorption spectra, and a lens model that carries one of them with a free amplitude would do for a band what the edge parameter does for a slope — if the band’s position is right. The fit cannot find the position itself, because a band’s position near 440 trades against the macula, and the separation between them rests entirely on the rods seeing through the lens and not the macula.

Failing that, an allowance. If sixty-year-olds’ lens bands vary by a tenth of a density, the rod weight fitted from one daylight-against-LED session carries an extra error of about one of its own standard errors, and a group study of the weight against age has to count it as between-person spread rather than measurement error. It does not average away within a person across sessions, because it is a property of the person’s lens.

How the bands were fitted #

The observer is the model’s sixty-year-old — pigment templates at their median peaks and density, a lens of density 0.5 + 0.02 × (age − 20) at 380 nanometres falling as exp(−(λ − 380)/68), the standard macula, the rod signal added to each cone at a tenth of its peak with the S cone’s at weight one half — with a Gaussian of height 0.1 and standard deviation 15 nanometres added to the lens density, in front of cones and rods alike. Each match is the CIELAB difference of one of forty-two surfaces under the two lamps, adapted through CAT16. The fit’s sensitivities are finite differences at the model observer, including the edge parameter of the earlier essay; each setting carries 0.5 ΔE split across three axes; the shift in each parameter is the least-squares projection of the band’s effect on the matches, and the standard errors and misfit are from the Fisher information and the residual. Bands are centred every ten nanometres from 410 to 460.

What this leaves out #

The band’s shape and height are stated. Real lens chromophores have their own spectra, broader on the long-wave side than a Gaussian, and their density varies between people; a tenth of a density is a size to reason with, and the shifts scale in proportion to it.

The rods see through the lens and not the macula in this model, which is the anatomy: rods are absent from the fovea, where macular pigment is. That asymmetry is the only thing that lets a fit tell a lens band from macula at all, and it is modelled rather than measured for the field the matches would be made in.

One observer, one age. A band’s effect at seventy, with a denser lens underneath, would be a little smaller in the blue where less light reaches it, and its trade with the macula the same.

Still open: whether the rods can tell a lens band from the macula #

The rod signal is the one thing that sees the lens and not the macula, so it is the one thing that could separate them. The matches here use it only as a small addition to the cones. A match made at a lower light level, where the rods carry more of the signal, would weight the lens more heavily against the macula.

The calculation is this fit with the rod signal’s fraction raised from a tenth of the cones’ peak to a third — the mesopic end of the range — and the band at 450 nanometres fitted with and without a free band amplitude. The prediction is that at a third the macula and a lens band at 450 separate: the band moves the rod signal and the macula does not, so the fit can tell them apart, and the rod weight’s shift falls below half a standard error. If that holds, the experiment on older observers should include a dimmer session, not to measure the rods better but to measure the lens through them.

Two parameters for one absorption #

The habit is about what a fit does when the world contains a thing the model has no parameter for but two parameters that each resemble it.

The fit had a broad absorption in the blue, the lens, and a band at 460, the macula. A band at 420 or 450 is neither, and the fit gave it to whichever it resembled more — the lens below 440, the macula above — and made up the rest with the rod weight, the parameter nearest to hand. Nothing in the fitted values said a band was there; each came out as a plausible observer.

The failure mode is to trust a fit’s decomposition of an effect into its named parameters when the effect has a form none of them has. The names are the model’s, and a real eye assigns absorption to whatever pigments it has.