The slowest clock is chemical

The afterimage and the clock on chromatic adaptation are one mechanism described twice: a gain with a time constant, in two pools of different spatial extent. Treating them as one removes a free parameter, and it leaves something named and absent — a third gain, with a third time constant, sitting upstream of both.

It is the pigment. A cone’s sensitivity depends on how much photopigment is present; light consumes pigment; the pigment is remade chemically at a rate that has nothing to do with the nervous system. That is a state variable, it is slower than anything else in the eye, and no appearance model on this site or anywhere else has it.

The claim #

The eye’s slowest state variable is not in the nervous system, and it is invisible to every steady measurement.

• The clocks are one second, sixty seconds and a hundred and twenty, with the rods at four hundred. The two fast ones are chromatic adaptation, are in the adaptation-clock machinery, and are the whole of what this site’s clock essays are about. The slowest is chemical and was not modelled at all.
• A steady bleach cancels exactly. A bleach is a diagonal gain in the cone basis; chromatic adaptation is a diagonal gain in the same basis; so an eye that has settled at the light which bleached it is — to 6 × 10⁻¹⁴ ΔE00 — an eye that was never bleached. The effect is real, it is large, and no steady measurement can see it.
• And a white bleach is not white. The three cone classes catch different numbers of photons from the same light, bleach by different fractions, and leave an unbalanced gain: ΔE00 10.4 on a neutral patch immediately after a minute on snow, still above half a unit nearly two minutes later.

The model, which is one equation #

Bleaching has a standard photochemical form and the matching machinery already has its steady state:

$$\frac{dp}{dt} = \frac{I(1-p)}{I_{\frac12}\,\tau} - \frac{p}{\tau}$$

with p the fraction of pigment bleached, I½ the half-bleaching retinal illuminance — 10^4.3 trolands, quoted, and imported rather than restated — and τ the regeneration time constant.

Its steady state is p∞ = I/(I + I½), which is exactly the form bleachedDensity already used, so the two files agree at rest by construction and assertItAgreesWithTheSteadyState requires them to at five light levels.

What the time axis adds is the asymmetry. At constant light the equation is linear in p, so the solution is an exponential approach to the steady state at a rate of (1 + I/I½)/τ. That rate rises with the light: a bright field bleaches fast. The recovery rate does not — with the light off the second term is all that remains and the rate is 1/τ, the same τ it always was. Bleaching is fast and unbleaching is slow, and it is one equation rather than two mechanisms.

Sensitivity is taken as proportional to the pigment remaining, which is the simplest of the several relations in use, is stated rather than fitted, and scales every result here without reordering any of them.

Why nobody has needed it #

This is the part worth the essay, because it explains a two-century absence rather than complaining about one.

A bleach leaves a diagonal gain in the cone basis — three numbers, one per class, multiplying the responses. Chromatic adaptation is also a diagonal gain in the cone basis; that is the whole content of von Kries, and it is what every white-point transform on this site does.

Two diagonal matrices in the same basis multiply, and normalising to the white removes the product. So an observer who has settled at one light level and adapted to its white is, exactly, an observer who was never bleached — 6 × 10⁻¹⁴ ΔE00, which is floating point.

The effect is large and no steady experiment can detect it. That is the same structure as the two eyes, where one person’s eyes differ by a whole ΔE00 from macular pigment alone and adaptation cancels it exactly, which is why nobody notices and why nothing in colorimetry has a term for it.

There is a wrinkle worth having, because it decides which arithmetic is right. The cancellation is exact in CAT16 and is not exact in CIELAB’s own white-point argument — dividing XYZ by the white is a diagonal in XYZ, which is a different basis, and it leaves ΔE00 0.66 behind. That is a wrong-von-Kries residue of exactly the kind the four-transforms essay is about, arriving in a place nobody would look for it.

What survives: the transient #

If steady states cancel, what is left is the interval during which they have not settled — and the intervals do not match, because the clocks do not.

Walk in from bright snow. The pigment is 94 per cent bleached and returns over minutes, which is far slower than the gain a viewing condition settles at. The neural pools relax over seconds and a minute. For the first two or three minutes indoors, the neural gains have finished adapting to the room and the pigment has not, so the cancellation is incomplete and what is left is visible.

And what is left has a colour. Under D65 the three cone classes take 46.8, 41.0 and 12.2 per cent of the total photon catch respectively, which is not the balance an equal-energy light would give them — so the effective bleaching illuminance differs per class, the three bleach by different fractions, and the gain left behind is unbalanced. On a neutral test patch, immediately after, that is ΔE00 10.4. At sixty seconds it is 1.2. At two minutes it is 0.48, and still above half a unit at a hundred and ten seconds — against a slow neural constant of sixty.

What it does not change #

The essay would be dishonest if it stopped at “large and invisible” without saying where the boundary is, because a mechanism this size, if it were active indoors, would move numbers all over this site.

It is not. At a display’s few hundred trolands the pigment loss is 1.5 per cent and the tint left on a neutral patch is a fifth of a ΔE00 — smaller than the residual between this site’s median observer and the CIE’s standard one, and a fifth of the tightest tolerance any specification is written to. In a well-lit room the loss is 4.8 per cent and the tint is under seven tenths of a unit, and both figures are for a reader who has been staring at a uniform field of that brightness indefinitely, which nobody does.

assertItChangesNothingIndoors pins both, and it is written the way this site’s absences are written: not as a disclaimer but as a check that would fail if the model started predicting an indoor bleaching effect. Every essay on this site computed without this file is therefore still correct, and the reason is a measurement rather than an assumption.

The prediction that could be wrong #

The model makes one claim that nothing else on this site makes and that is not obviously true: the strongest after-tint is at neither end of the light range.

At low light nothing bleaches and there is nothing left behind. At very high light everything bleaches — all three classes head for the same place, and the imbalance between them collapses again. The maximum is in between, at about 1.3 × 10⁵ trolands, which is roughly an overcast sky, at ΔE00 11.9; by 10⁷ it has fallen back to 10.6.

That is a testable statement of the kind this site likes: it says the worst afterimage comes from a bright day rather than from staring at the sun, and it follows from saturation rather than from anything fitted.

assertTheStrongestTintIsNotAtTheBrightestLight requires the maximum to be interior. If a future revision moved it to the end of the range, the site would stop building — which is the only way to keep a prediction from quietly becoming a description.

The four steady states come out of one line #

Before anything else, the steady-state figure is worth checking, because everything downstream rests on it and it is the one part of the model that can be recomputed from the essay’s own text.

With the half-bleaching illuminance at 10^4.3 = 19,953 trolands, the steady state p∞ = I/(I + I½) gives 1.5 per cent at 300 trolands, 4.8 per cent at 1,000, 60.1 at 30,000 and 93.8 at 300,000. Those are the four figures quoted for a display, a lit room, an overcast sky and snow, to the digits they are quoted at — and the second of them independently fixes what this collection means by a lit room, at a thousand trolands, which is the same figure its room-settling essay uses.

The asymmetry can be given a number too. The approach rate is (1 + I/I½)/τ, so on snow the bleaching time constant is 7.5 seconds against 120 for recovery — a factor of sixteen from one equation, and under an overcast sky a factor of two and a half. Bleaching is fast and unbleaching is slow is a ratio that depends on the light, and on the brightest case in the essay it is larger than the word fast suggests.

The tint decays faster than the pigment does #

Three values are published for the after-tint: 10.4 units immediately, 1.2 at sixty seconds, 0.48 at a hundred and twenty. Read as a decay, they do not run on the pigment’s clock.

A pure exponential at the pigment’s 120-second constant would take 10.4 to 6.31 at sixty seconds and 3.83 at a hundred and twenty. The published figures are a fifth and an eighth of those. Solving for the implied constant on each interval gives 27.8 seconds over the first minute and 65.5 over the second — both faster than 120, and the second close to the neural pool’s sixty.

That is not an error and it does need an explanation the essay does not give. A quantity driven by one exponential state variable should approach that variable’s rate once the departure is small, and this one is still running at half the pigment’s constant at two minutes. Either the tint is a strongly nonlinear function of the pigment imbalance well below the levels where saturation is obvious, or something else in the figure is decaying alongside it.

It also softens a comparison the essay draws. Still above half a unit at a hundred and ten seconds — against a slow neural constant of sixty invites the reading that the chemical clock is what keeps the tint alive. The tint’s own measured constant over that interval is 66 seconds, which is the neural number rather than the chemical one, so the sentence is true and the arithmetic behind it does not obviously belong to the pigment.

The interior maximum is real and its stated reason is not #

The one prediction the essay offers as falsifiable is that the strongest after-tint is at neither end of the light range, peaking near 1.3 × 10⁵ trolands at 11.9 units and falling back to 10.6 by 10⁷. The reason given is that at very high light all three classes head for the same place, and the imbalance between them collapses again.

Under the model as stated they do not. The surviving gain in each class is g = I½/(I·s + I½) with s the class’s own share factor, so the ratio between two classes is (I·s₂ + I½)/(I·s₁ + I½) — which starts at 1 in the dark and is monotone, approaching s₂/s₁ rather than returning to unity. Taking the essay’s own photon-catch shares of 46.8, 41.0 and 12.2 per cent, the long-to-short gain ratio runs 0.95 at a thousand trolands, 0.33 at the reported peak, and settles at 0.261 — and stays there however bright the light gets.

So the imbalance does not collapse. It saturates, at its largest value, and it is still at its largest value at 10⁷ trolands.

The falling tail is very likely real and needs a different cause. The most plausible one is in the reporting rather than the mechanism: at 10⁷ trolands almost all the pigment is gone, so the patch a bleached eye is matched to is very dark, and ΔE00’s chroma and hue terms both shrink towards zero as lightness does. The tint would then be turning over because the stimulus is disappearing, not because the classes are coming back into balance.

That matters because the essay’s assertion checks the maximum is interior and the closing section notes it is the prediction most sensitive to the response relation. If the turnover is a lightness artefact of the difference formula, it would survive a change of response relation and would not be a statement about the pigment at all — and the check would keep passing for a reason the essay does not intend. Separating the two is one line: report the gain imbalance alongside the ΔE00, and see which of them turns over.

A much smaller bleach is the ordinary case — a bright room rather than snow — and it says whether the time constant depends on how much pigment went.

Where the model stops #

One time constant for three classes. Regeneration is quoted near two minutes for cones as a whole; the three pigments regenerate through the same retinal cycle and there is no strong reason to give them different constants, but there is also no measurement here that says they share one.

The gain is linear in the pigment remaining. Several relations are in use — some make sensitivity proportional to the pigment, some to the log of it, some fold in a response compression — and the choice scales every number here. What it does not do is change any ordering, because every result is a ratio of two states of the same model.

The per-class split is a modelling decision and it is the only one in the file. The half-bleaching constant is quoted for the luminance path; splitting it three ways needs an assumption about what each class is catching, and the assumption made is that each class’s effective illuminance scales with its share of the photon catch relative to an equal-energy light. Another assumption would move the imbalance and would not remove it.

And there is no rod pigment in the tint. Rhodopsin regenerates in about four hundred seconds — three times slower again — and rods are absent from the fovea, which is why a bright bleach’s peripheral afterimage outlasts its central one. Nothing here computes that.

Staring for three minutes rather than one is past where the bleach saturates, which is the check that the tint is set by the bleach and not by the dwell.

The generalisation #

The sentence worth carrying is: a mechanism that cancels at steady state is invisible to steady measurements and shows up only in transients, and this site keeps finding them.

The two eyes cancel. The bleach cancels. The pigment’s density change with light level is a different observer and cancels for the same reason. In every case the effect is real and large, the apparatus that would detect it takes a steady reading, and the effect is arranged — by adaptation, which is doing its job — to leave no steady trace.

The methodological consequence is uncomfortable. A model that only ever predicts steady states will fit every steady measurement perfectly while omitting arbitrarily large mechanisms, and there is no way to find out from within the steady data. The only lever is a transient, which is why this collection keeps arriving at another clock.

The surprising connection is with the lamp. A luminaire takes about four hundred seconds to reach its working temperature, which is longer than any of the eye’s three clocks. So in the first minutes after a switch is thrown, the light is moving, the neural gains are moving and the pigment is moving, on three different time constants, and every appearance judgement made on this site assumes all three have finished.

Who found it, and when #

Rushton measured pigment bleaching in living human eyes in the 1950s and 1960s, by retinal densitometry: shine light in, measure what comes back out, and infer how much pigment is present. The half-bleaching constant and the regeneration time constants this file quotes descend from that work.

The relation between bleaching and sensitivity is older and more contested. The idea that sensitivity should follow the pigment present is Hecht’s; the measured relation is not proportional over the whole range and the departures are known as the Rushton–Dowling law, which is logarithmic in the bleached fraction over a wide middle range. This file uses the proportional form and says so.

The afterimage’s chromatic character has been observed for as long as anybody has looked at a bright light, and it is usually explained by neural adaptation alone — which is right for the ordinary case, since an afterimage from a display bleaches almost nothing. The pigment contribution needs sunlight, and sunlight is the case nobody runs an experiment in.

And the cancellation argument is the reason the pigment stage stayed out of the appearance models. CIECAM and its predecessors are built from matching and appearance data taken under steady conditions, where the stage contributes nothing measurable. Leaving it out was the correct decision for the data available, and it becomes a limitation only when somebody asks a question with a time in it.

What the pictures cannot show #

They cannot bleach the reader. Every patch on this page is at a few hundred trolands, which takes about one and a half per cent of the pigment — an amount the essay’s own assertion requires to be negligible. The phenomenon under discussion needs a light four orders of magnitude brighter than this page can produce.

And the tint patches are the usual afterimage compromise. Each is what an unbleached eye would have to be shown to match what a bleached one makes of a neutral. It is not what a bleached reader would see, because a bleached reader looking at the figure would apply their own gains to it on top of the ones already drawn in.

Where the ladder goes next #

The nearest unfinished piece is the join proper. the adaptation-clock machinery has two pools with a stated local share; the pigment is local by construction, since one piece of retina’s pigment is not pooled with another’s. A three-stage model would therefore predict that a bleached afterimage has a sharper edge than an adapted one, which is a difference somebody could look for.

The second is the rods. Four hundred seconds, absent from the fovea, and responsible for the long peripheral tail of any bright afterimage. Everything needed is in the site’s own retinal machinery already and the constant is quoted above.

And the third is the response relation. Proportional sensitivity is the simplest choice and the literature’s is logarithmic over a wide range. Substituting it would scale the tint magnitudes and — this is the interesting part — would move the interior maximum, so the one prediction this file makes that could be wrong is also the one most sensitive to the modelling decision it is least sure of.