The flash the eye counts

Light arrives in lumps made the case that light of a given intensity is a stream of photons, with brightness changing only how many arrive. The flicker that is half wave and half particle followed the consequence: a stream of independent lumps has a noise, and it is built into the light. Those essays treated the counting as a matter for photomultipliers and carefully prepared detectors. A remarkable measurement from 1942 used the human eye as the detector. Selig Hecht, Simon Shlaer and Maurice Pirenne sat three people in a dark room at Columbia University, adapted them to the dark for half an hour and flashed a weak green light onto a chosen spot on their retinas. Each time, the observer said whether a flash had been seen.

They varied the brightness of the flash. Their result is one of the most elegant in biophysics, and it is not simply that very little light is needed. Light at the threshold of seeing is so dim that its lumpiness becomes visible in the observer’s answers. A person looking at a succession of identical flashes sees some and misses others, and the pattern of those misses is not a defect in the person. To a very large extent it is the photon statistics of the light.

Why a threshold gives a curve #

Suppose first that a detector responds if and only if it absorbs at least $k$ photons. Each flash delivers a random number of photons. The number absorbed from a flash has a mean $m$, set by the flash’s brightness, and because each photon’s absorption is an independent event, the distribution around that mean is Poisson. So the chance of seeing a flash is

$$P(\text{seen}) = 1 - \sum_{j=0}^{k-1} e^{-m}\frac{m^{\,j}}{j!}.$$

Look at what is not needed in that formula. It has no noise in the observer. The curve comes from the source. A person with a perfectly sharp internal threshold, shown perfectly repeatable flashes, would produce this curve nevertheless. The classical wave account of light predicts nothing of the kind. If each flash delivered a definite amount of energy, a perfect threshold would give a step: never seen below some value, always seen above. Any softening of the step would have to be attributed to something in the observer. The physiologists of the time accepted as obvious that it was, and treated the sloping curves they recorded as a measure of a person’s biological variability.

The trick with a logarithmic axis #

The second step is the elegant one. The eye loses most of the light that reaches it. Some reflects off the cornea, some is absorbed in the lens and the fluid inside the eyeball and much of what reaches the retina passes between rods or through them without being absorbed. None of those could be measured directly in a living person in 1942. So how could one say how many photons a rod actually absorbs? It seems impossible to convert a measured flash energy at the cornea into anything inside the eye without knowing these losses.

But notice how $m$ enters the formula. If the loss multiplies the number at the cornea by a factor $a$, then $m = aI$, with $I$ the mean number arriving. On a logarithmic axis of $I$, a multiplication is a shift. Changing $a$ moves the curve bodily along the axis. It does not change its shape, because the shape depends only on $k$.

The observer’s data can then be plotted against the logarithm of the flash, and whichever $k$ gives the matching steepness is the threshold. No knowledge of $a$ is required. Hecht and his colleagues found that their three observers’ curves were best fitted by $k$ between 5 and 7 or so, and one with 8. That measurement of the eye’s absorbed photon count, in units of individual quanta, does not depend on anyone’s estimate of how much light survived the trip.

It is a beautiful example of what can be done with a known statistical shape. A loss factor appears in many measurements as an awkward unknown constant, and here the Poisson shape factors it out as a mere shift. The arrangement is reminiscent of the length nobody has to measure, in which a pendulum’s two periods give gravity without its length, by building the unknown out of the equations.

Fifty flashes do not give a number #

Of course, it is easy to read a curve off a figure when there is a smooth line through it, and much less so with real data. So it is worth simulating the experiment.

The fit is close, and it is not the truth. With three hundred and fifty trials from an observer with a threshold of six, the most likely threshold comes out as eight, with six through eleven all acceptable within a reasonable allowance on likelihood. That is honest and quite sobering: the method excludes a threshold of one or two very firmly, because those predict a much gentler curve, but it cannot tell six from eight. The original paper reported 5 to 8, and that is about what a data set of the kind they had can support. Real observers add noise of their own, which also softens the curve and pushes the estimate downward. So it is plausible that the true number is somewhat higher than the fit gives, by some amount that a fit of this kind cannot itself reveal.

The firm conclusion was and remains the qualitative one. The threshold is a small number, not a large one. The curve shape belongs to a count of few. Whatever the eye does with those few photons, the variation in what is seen from flash to flash is largely set by how many photons arrived.

A rod can do it alone #

Hecht’s group’s second conclusion came from combining that count with the geometry, and it was more startling.

Here they did estimate the losses, and the estimates agree well with the count. Now consider where those absorbed photons land. The flash was focused to a small spot of about ten minutes of arc, covering about 500 rods. Nine photons distributed at random among five hundred rods will almost never have two arriving at the same rod. So the rods must each be responding to single photons, with a signal big enough to be useful. Nothing else can account for the numbers. Hecht’s group wrote that in 1942, while no one could measure a single rod’s response. In other words, the physics of one absorbed quantum in one molecule of rhodopsin had to be visible, through a chain of amplification, in the output of a nerve.

It took until 1979 for Denis Baylor, Trevor Lamb and King-Wai Yau to confirm it. They sucked a single rod from a toad’s retina into a fine glass pipette and measured the current flowing in it. When they exposed it to dim light, the current made discrete steps of about a picoampere, each corresponding to a single photon. Each photon triggers one rhodopsin molecule to change shape; that activates a cascade of hundreds of enzyme molecules, which close ion channels in the rod’s outer membrane, and the single step of current follows. It may be the most sensitive detector of its size in nature. Its quantum efficiency, around 25 per cent for a rod, is competitive with that of the first photomultipliers. And since a photomultiplier’s output also comes in discrete pulses, the experiment a wave cannot pass and the rod’s records have something of a common look.

Why not one? #

The natural next question is why, if a rod can detect one photon, the brain’s threshold is at several. Why would evolution throw away the first few? The likely answer involves another of Baylor’s measurements, and it is a neat example of the noise every amplifier must add. Rhodopsin molecules occasionally change shape spontaneously from thermal energy, in a form indistinguishable from an absorbed photon. A rod contains a very large number of rhodopsin molecules, and in a toad at about 20 °C, it fires by itself about once every 40 seconds; monkey rods at body temperature, about once every 160 seconds.

So if the pooled output of those 500 rods fires on one count, the observer sees a flash about a quarter of the time, in pure darkness. Pooling is what the visual system does, as the retina’s neurons take in signals from many rods. So a threshold of a few counts removes most of the false alarms at modest cost. The observed threshold would be explained by a cost-benefit choice, a matter of keeping a false-alarm rate low given thermal noise in the pigment. Horace Barlow proposed the idea in 1956, before anyone had seen the single events, and he estimated the dark rate from psychophysics alone. Observers in the dark report a faint grey even with no stimulation, that “dark light” being treated as this noise. Barlow also noticed that animals with cold eyes might have a lower threshold and that this should show up, which turned out to be testable in cold-blooded species.

The point holds whether or not the explanation is the whole story. It shows that a threshold count of several does not imply an inability to detect one. The relevant question is how the decision is made. It also means that the threshold is partly a choice of the observer. Subjects told to report anything they might have seen report more flashes, with more false alarms, at a lower apparent threshold. Later signal-detection analyses, with confidence ratings, found that people do better than a simple threshold, at least partly distinguishing one or two absorbed photons from none.

Could a person see one photon? #

That question was tested directly by Alipasha Vaziri and colleagues in 2016, using a source of true single photons from a nonlinear crystal rather than a dimmed flash. Dimmed flashes have the Poisson spread that the 1942 experiment relied on. They have a real chance of containing two or three photons. A heralded source eliminates that, since each detected partner photon tells that exactly one has been sent. Three observers were asked in a forced choice whether a photon came in the first or second of two intervals, and over thousands of trials got it right 51.6 per cent of the time. That is marginally above chance, and the statistical significance was reported as moderately strong. The authors also found that the responses were better shortly after a previous photon, suggesting some cooperation over seconds.

The result has been received with caution. An effect of 1.6 per cent over a small sample, in a task with the possibility of unconscious biases, is not the kind of thing that is settled by one paper. Attempts to repeat it are under way and no consensus has formed. Taken at face value, however, it means that people have a very weak ability to register a single absorbed quantum, even if they cannot tell anyone they saw it.

What the same count does at ordinary brightness #

The photon count does not stop mattering when a scene is bright enough to see. Albert Rose argued in 1948 and Hessel de Vries in 1943 that the limit on seeing a faint feature in dim light should come from photon statistics. A patch that collects $N$ photons in the eye’s integration time has a random fluctuation of $\sqrt{N}$, by the shot noise already described, and a feature can only be seen if its contrast exceeds that fluctuation by some margin. So the smallest detectable contrast falls as the inverse square root of the brightness. That is the Rose–de Vries law, and it fits human contrast thresholds in dim conditions over several decades of brightness. Only in daylight does vision stop following it, and there the contrast threshold flattens out at about a per cent, a limit set in the brain and not in the light.

That line of reasoning was carried into engineering as the Rose criterion for imaging devices. A radiologist, for instance, wants the lowest possible X-ray dose, and it gives the number of photons needed per pixel in order to see a lesion of a given contrast, with a margin of about five. The grain in a dim photograph has the same origin, as does the speckle of a low-light image. These are unrelated to the grain that is in the light, where a coherent wave’s interference gives a graininess that exists at any intensity, though the two are often confused. An instrument can be told apart by testing how the graininess changes with brightness.

The reasoning extends down the size scale in biology. A person’s rods work at the limit, and so do the eyes of nocturnal insects: a nocturnal bee flying through a rainforest at night has an eye collecting only a few photons per facet per integration time, and it lengthens the time and pools from many facets to overcome the same arithmetic, trading speed and sharpness for count. Eric Warrant and his colleagues have studied these limits. They are also a good example of the comparison made in the eye a fly cannot afford, in which the size of a facet was a matter of diffraction. At night, photon count imposes a further limit that favours a larger aperture still. The two constraints are separate, and one has to choose between them.

There is a nice experimental footnote from 1942 concerning all of this. The apparatus required green light, a brief flash and a calibration of energy by a thermopile, and getting the absolute value of the energy at the cornea at those intensities was itself difficult. Hecht and his colleagues did it with a thermopile measurement of a much brighter beam, followed by attenuation with calibrated filters. A good share of the original paper is about that calibration. Modern workers would use photon counting detectors, and those did not exist then.

What the curve cannot see #

The Poisson form assumes independent photons. It would not hold for a light source of a different character, for example squeezed light or a single-photon source, as the noise pushed below the floor described. The same observer under those sources should give a different curve shape, and that has been proposed as a way of testing whether a person’s eye can distinguish quantum light, a test whose precision so far is modest.

The curves also attribute all variation to the photons. Any actual observer’s internal noise broadens the curve, and any variation in attention or a shift in criterion between sessions does too. That makes the fitted $k$ a lower bound if read naively, and an underlying threshold of some larger value is consistent with a real curve, if there is noise also in the nerves. Some later analyses found that for a fuller model of the same data, thresholds up to about ten or more could fit. The estimate from one curve is a figure for how much light-borne variation is needed to explain the data, and not a direct count of what the retina does.

Finally, the budget’s factors of 48 per cent and one-fifth came from dead eyes and estimates, and the numbers used vary from author to author, with roughly nine absorbed as a central figure and a true range of 5 to 14. The figure does not show the fact that the flash covers a small retinal area in a short time, chosen to match how rods integrate. Outside of this narrow window, at longer durations or larger areas, thresholds change in known ways, so these results do not apply to, say, seeing a dim star for many seconds. Comparing those conditions was how the experimenters found their best stimulus. The numbers belong to that stimulus: about a tenth of a second, a small spot 20 degrees off the eye’s centre, where rods are densest, and at 510 nm.

Still open: whether a brain can use one photon, and how #

The question left by the 1942 experiment has changed several times. It began with how few photons the eye can use, and it now asks how a brain decides. Recordings from retinal ganglion cells in mice and primates show that signals from a single absorbed photon do reach the optic nerve, though they are combined with a good deal of noise and a nonlinearity at the rod-to-bipolar-cell synapse that seems to throw some away. Experiments in which mice are trained to report dim flashes, while their retinal cells are being recorded at the same time, have shown that at some light levels the animal’s behaviour is limited by what the retina reports, and at others by something further on. Whether this nonlinear rejection in the retina is an optimal filter or a simple cost of having a neuron is a current question, as well as how much of it could be overcome by a person who has had years of practice in the dark.

A threshold of a few photons, with a Poisson source, gives a curve whose shape fixes the threshold and whose position alone carries the losses, so a flash at the cornea yields a count of about 5 to 8 absorbed photons without a measurement of what was lost; nine photons over 500 rods then show that a rod responds to one, and about 0.3 thermal false events per look make a threshold of one a poor choice. A person in a dark room was a quantum experiment for a decade before anyone built a rod recording. The habit worth keeping is to ask, of any uncontrollable constant in an experiment, whether a suitable axis can move it out of the way.