This discussion paper seeks to reshape the contemporary understanding of visual adaptation. Received wisdom says that input luminance is scaled down in the retina. There is, first, a near-logarithmic compression described by the Naka–Rushton equation and, second, a control of gain (better attenuation) by feedback from the output of each ganglion cell that is equivalent to modifying the half-saturation constant in the Naka–Rushton equation. The reinterpretation proposed here asserts the following instead: (a) the scaling down in the retina is accomplished by receptive fields of different areas, which function over different ranges of luminance, ranges inversely proportional to the area of the receptive field. (b) The visual pathway is differentially coupled to the physical stimulus, so that the maintained discharge increases only as the square root of the luminance. (c) The Naka–Rushton equation describes merely the saturation of neural response as input increases; when a neuron is overloaded, output tends to regularity and onward transmission is blocked by a subsequent stage of differential coupling. Three existing studies of the relation between input to and output from retinal ganglion cells are reinterpreted in the light of this alternative view of visual adaptation.
© 2013 Optical Society of America
Human vision functions over a luminance range of . In part this is accomplished by two sets of receptors, rods and cones. The rods handle the lowest four decades, while the cones function over the highest seven, the two systems overlapping by one decade . But neurons have a much smaller response range, no greater than [2, p. 273]. So, how does cone vision function over seven decades?
Received wisdom [2–7] says that the output of each retinal unit is scaled down in inverse proportion to the ambient luminance, so that differences in output match contrast in the physical world. Recent studies include [8–11]. The scaling down is accomplished by a near-logarithmic compression described by the Naka–Rushton equation, with the half-saturation constant in that equation modified by nonlinear negative feedback from the output. Since the reinterpretation suggested here is radically different, I begin with a short sketch to emphasize that difference. The sections following set out the experimental evidence that shows how the cone system can accommodate seven log units of luminance. Then I examine the existing idea of retinal gain control; it conflicts with a number of well-established experimental findings. Finally, I reinterpret three electrophysiological studies that have looked at the relation between the input and output of individual retinal ganglion cells.
A. A Reinterpretation of Visual Adaptation
To preempt any confusion, I am concerned here with adaptation to the ambient illuminance. Different retinal ganglion cells have receptive fields of widely differing areas. Under a common illumination, the input to each cell is proportional to the area of its receptive field. Putting this the other way round, each cell operates within a functional range of illuminances inversely proportional to its area; that is, cells with the smallest receptive fields function at the highest illuminances and support the finest visual acuity. The range of receptive field areas spans four log units.
The interface between the visual stimulus and retinal ganglion cells is differential, as also is the coupling between successive stages of visual analysis. This impacts on visual adaptation in two ways. First, the maintained discharge (half-wave rectified quantal noise) increases only as the square root of the illuminance. Perception depends on small perturbations about that maintained discharge, so an internal response range of would accommodate four log units of illuminance. Second, at sufficiently high illuminances, the output of a neuron saturates. Differential coupling between subsequent stages of visual analysis suppresses saturated responses, freeing perception from masking by the output from saturated neurons. In short, differential coupling [assertion (b) in Abstract] is the essential insight on which everything else [assertions (a) and (c)] depends.
Both explanations of visual adaptation, received wisdom and the reinterpretation offered here, are functional in the sense that adaptation is expressed as an operation performed on a model of the physical stimulus. Of course, that operation needs to be realized in neural processing, and the traditional view has prompted a large number of electrophysiological investigations, three of which are reexamined here. But those studies do not amount to a physiological explanation. Present understanding of the physiology permits only statements such as “this feature of retinal ganglion cells is deemed salient and relates to visual adaptation by realizing a near-logarithmic compression (or, as the case may be, differentiation) of the stimulus.” The principal argument has to be functional. Other aspects of the physiology are ignored because their contribution is implicitly deemed to be small.
B. Differential Coupling of Visual Input
The arguments that follow depend, repeatedly, on differential coupling between the stimulus and visual analysis. There are a wide variety of phenomena that point to a differential interface [12, Chap. 5; 13]. Two must suffice here; thereafter differentiation will be taken for granted.
It is widely accepted that the visual system contains parallel channels optimally sensitive to different wavenumbers (or spatial frequencies) of gratings. These channels appear to have a true band-pass characteristic with a bandwidth at half-peak transmission variously estimated at 0.5 to 1.5 octaves, depending on the model used . Blakemore and Campbell  obtained a value of one octave from the threshold elevation for gratings of different wavenumbers following adaptation to a high-contrast grating. If these channels are indeed band-pass, they have zero admittance at zero wavenumber; that is, visual stimuli are differentiated on input.
Again, since observers are able to detect very small (0.0025) modulations of luminance and discriminate even finer differences in contrast (0.001; ), the visual analysis of a grating must depend on an accelerated nonlinear transform of the contrast—not of the local stimulus luminance, but of the contrast abstracted from that luminance . That is to say, the contrast is stripped off from the mean luminance and, mathematically speaking, that is differentiation.
Figure 1 illustrates a model of differential coupling (with specific application to the detection of increments, below). It represents the input to a retinal ganglion cell as a function of time. The stimulus, including an increment of duration [Fig. 1(a)], enters half into the positive (excitatory) and half into the negative (inhibitory) input [Fig. 1(b)]. The negative input is delayed by a constant interval (here ). Hughes and Maffei  examined the retinal response to flicker and reported phase shifts at different frequencies corresponding to a delay of approximately 65 ms. In the interest of simplicity, that delay is here taken to be fixed, but, of course, it is not so in nature. The negative input is simply a Poisson process of the same density as the positive, but taken negatively, and on combination the mean inputs () cancel [Fig. 1(c)]; the half increments also cancel where they overlap, but the quantal noise combines in square measure because the two components of the input are statistically independent. Retinal ganglion cells (and subsequent sensory neurons) receive both positive and negative inputs, but transmit action potentials of one polarity only. This is modeled as half-wave rectification; it generates a maintained discharge from the positive excursions of the quantal noise [Fig. 1(d)], with the high-frequency components of the Poisson noise attenuated in recognition of the low-pass character of receptive field units.
The noise [Fig. 1(c)] is approximately Gaussian, and provides an immediate explanation of Weber’s law . A Gaussian noise process can be specified by just two entities, its power and its autocorrelation function . A Poisson process has zero autocorrelation, and the autocorrelation of the sensory process derives from the inertia and parallel capacitance of the tissues through which it propagates (it is the autocorrelation of the impulse response function; see ). So noise processes of different power, deriving from stimuli of different luminance, have a common autocorrelation function and differ solely in power; that is, they differ solely by a linear scale factor (the square root of power) and are geometrically similar. A discrimination between two samples of Gaussian noise of powers and is likewise similar to a discrimination between noises of powers and , for any . Weber’s law follows from this principle of geometric similarity.
The noise in the positive and negative inputs in Fig. 1 has been taken to be statistically independent on the assumption that it derives from independent quantal absorptions. This may not be so. Recent research  suggests that some quantal absorptions may contribute to both inputs, generating a positive cross correlation between them. This does not invalidate the argument because correlated noise is, functionally, a part of the stimulus. Moreover, Weber’s law follows from the assumed invariance of the autocorrelation function, cross correlation included.
The model in Fig. 1 is oriented towards the psychophysical detection of increments. As a model of a retinal ganglion cell it has two defects: first, it is out of the question that each and every receptive field has an equal number of positive and negative inputs. Second, a static stimulus (e.g., a grating) does not produce a permanent increase in output, but a fading response [22,23], so that electrophysiologists have to drift a grating across the receptive field . Both of these defects could be addressed by making the differential coupling consequent on some equilibration process within the cell.
2. VISUAL ADAPTATION
Studies of retinal ganglion cells have shown their receptive field sizes to vary. For example,  reported on-center retinal ganglion cells in the cat (from many different animals) with estimated areas of the center component ranging from 0.25 to (), and  reported central radii ranging from 0.15 to 3 deg ( in area). More recently  has measured the dendritic field areas of retinal ganglion cells in the owl monkey and reports standard deviations of c 0.5 log unit for cells near the fovea (eccentricity ). Finally,  reported “large local variation in dendritic field size” of midget cells in human retina. This means that the idea of visual adaptation by selective use of parallel pathways with different sizes of the receptive field is more than just plausible. The question is whether the variation of receptive field size in man is great enough to support cone vision over seven log units of luminance.
A. Parallel Channels in Vision
It is widely accepted that the visual system contains parallel channels optimally sensitive to different wavenumbers. I assume, as a matter of convenience, that retinal receptive fields of different areas are morphologically similar, differing only with respect to a linear scale factor. This enables the wavenumber sensitivity of a channel to be transposed into a measure of the size of its associated receptive field; specifically, a channel tuned to wavenumber will have scale factor proportional to and input proportional to .
An extensive set of measurements by  provides the raw data for the following analysis. Figure 2 reproduces part of van Nes’ data for the detection of sinusoidal modulations of green light (). The stimulus was produced by transilluminating a photographic grating, and the depth of modulation was diminished (from a maximum of 73%) by mixing with a veiling illuminance from a second monochromator. The grating measured 4.5 deg horizontally and 8.25 deg vertically against a dark surround, and luminance was varied horizontally according to the equation2 are the geometric means of van Nes’ “just noticeable” and “just not noticeable” depths of modulation.
Contrast threshold decreases approximately as the square root of illuminance (gradient with respect to the double logarithmic coordinates) up to a transition illuminance, , and thereafter remains approximately constant. To emphasize this relation, the thresholds for each wavenumber have been fit with a rectilinear characteristic. The estimates of , both for the data in Fig. 2 and for two other sets of threshold measurements with red () and blue () light, are shown in Fig. 3, where they are each compared with the best-fitting straight line of gradient 2, corresponding to the relation2 are limiting values for the proportionate variation in luminance that can be detected, a constant value corresponds to Weber’s law (with respect to luminance, not contrast) and gradient corresponds to a threshold increasing as the square root of illuminance. So the same stimulus, subject only to a difference in luminance, issues in one distribution of output below and a different distribution above. The different values of betoken different wavenumber-sensitive channels and the range revealed by van Nes’ data (0.5 to ) is sufficient to span four log units of luminance.
The luminance profile of a contrast stimulus [Eq. (1)] translates proportionally into the density of a Poisson process of quanta absorbed in the retina. The stimulus mean cancels on differentiation, but some part of the contrast passes through, because the positive and negative inputs are differently disposed in space and impose different attenuations on the sinusoidal modulation . That transmitted contrast (amplitude ) needs to be picked out from background noise of power (rms scale factor ). If now the luminance is increased to some arbitrary value and the contrast is reduced to , the signal-to-noise ratio is preserved . This can be seen in Fig. 1(c). The half increments are of size , giving a signal-to-noise ratio of . If the luminance is increased to (rms noise to ), it is sufficient to increase the increments to ; thereafter the two increments are equally detectable relative to their respective backgrounds. No subsequent processing [e.g., the half-wave rectification in Fig. 1(d)] can modify the predicted square-root law (except possibly some level-dependent constraint such as saturation of neural transmission). As it stands, the model generates a square-root law for detection with respect to mean luminance.
The Weber’s law characteristics in Fig. 2 must therefore implicate some process prior to the stage of analysis represented in Fig. 1. That process must replace the Poisson input with a noise process of power proportional to the luminance, so that thereafter all thresholds conform to Weber’s law. This process could be a differentiation at the level of horizontal or bipolar cells, which communicate by slow potentials, feeding a noise process only to the retinal ganglion cells, or, alternatively, it could be realized in the retinal ganglion cells themselves. If so, the center/surround structure of retinal ganglion cell receptive fields does not contribute to the wavenumber sensitivity of the parallel channels, which is determined at a cortical level. Wavenumber sensitivity of cat retinal ganglion cells terminates at about , which does not match the range exhibited in Fig. 2 (though neither does the species).
The model presented in  provides a good account of the phenomena associated with the detection of gratings within a matched surround of equal mean luminance. It invokes two successive stages of analysis structured as Fig. 1, with the inputs from the two eyes coming together between the two. This arrangement is dictated by experiments presenting different gratings to each eye [31–33]. The second stage, however, is binocular. This suggests that the first stage corresponds to that part of the visual pathway up to and including the simple cells located in layer IVb of the striate cortex, since these cells are predominantly monocular , but the complex cells are not. Although the lateral geniculate nucleus receives inputs from both eyes, those inputs appear to be transmitted independently through both layers IVc (circularly symmetric units) and IVb, before any significant interaction takes place. It also suggests that the wavenumber sensitivity of different channels is determined by the receptive fields of simple cells, in which case the retinal ganglion cells may do no more, functionally speaking, than replace an input Poisson process by differentially coupled Gaussian noise. That replacement leaves only the quantal noise as a cue to discrimination. Weber’s law follows immediately from the principle of geometric similarity, and no subsequent processing can reinstate a square-root law. So, how does the square-root relation at low illuminances in Fig. 2 arise?
B. De Vries–Rose Law at Low Luminances
Differential coupling of the physical stimulus requires quantal absorptions in the positive input [Fig. 1(c)] to be opposed by absorptions in the negative. If the Poisson density is sufficient, so that perturbations resulting from successive absorptions overlap, the mean inputs cancel. But at a sufficiently low luminance the quantal absorptions occur one at a time, one in the positive input, now one in the negative (Fig. 4). In this case the positive inputs are not canceled by any negative input and are transmitted uncanceled. Half-wave rectification deletes the negative input from the model in Fig. 1 and reinstates a square-root law threshold relation based directly on Poisson statistics.
The failure of cancellation illustrated in Fig. 4 is apparent in the results of [37, Fig. 4]. Figure 5 displays “thresholds” for test spots of varying diameters centered on the receptive field of an on-center ganglion cell in the retina of a cat, superimposed on steady backgrounds of (green symbols) and (red symbols). “Threshold” here means adjustment of the stimulus luminance “so as to produce the weakest reliably detectable perturbation of the maintained discharge as played over a loudspeaker” [37, p. 739].
As a matter of convenience in calculation, I assume the shape of the receptive field to be circularly symmetric bivariate Gaussian [after 22], with standard deviations and in the center and surround components, respectively. This has the convenient consequence that the quantum catch from a test spot of radius and luminance , centered on the field, increases as .
At the lower background luminance the surround component seems to have no effect. Assuming that is so low that the surround fails to cancel the center (Fig. 4), “threshold” reflects the central input only and, to be detected, the stimulus of whatever size (and shape) simply has to deliver a minimum additional flux to the center. That additional flux is ), and the green line in Fig. 5 has equation4) reflects the idea that passage through the tissue of the retina attenuates the noise in relation to the mean. The parameter estimates and the MS error are set out later in Table 1.
Barlow et al. [38, Figs. 1 and 4] report two similar experiments with off-center ganglion cells, comparing “thresholds” for flashes of light of different diameters centered on the receptive field against a lighted background and in darkness. As the stimulus covers more of the center, the “threshold” falls, as in Fig. 5. Against a lighted background, “threshold” then rises again as further increases in the size of the stimulus engage the surround, but in darkness that rise is not seen:
“A spectacular demonstration…using a stimulus light which falls near the edge of the dark-adapted receptive field. In the dark-adapted state this stimulus elicits a discharge at the same phase of illumination as one falling on the centre—at “on” in an on-centre unit and at “off” in an off-centre unit. Upon light adaptation a similarly placed stimulus may produce the opposite effects, giving “off” discharges and inhibition at “on” in an on-centre unit, and “on” discharges with inhibition at “off” in an off-centre unit; furthermore, these responses may occur when the stimulus intensity, though stronger than that required in the dark, is not much higher than that required to excite the centre of the light-adapted receptive field” [38, p. 342].
This difference cannot be ascribed to the Purkinje shift. At an intermediate background level, red and blue–green flashes have approximately the same threshold for all sizes of the stimulus, even though the red flashes excite the ganglion cell though cones and the blue–green chiefly through the rods. In darkness, quantal absorptions in the center are no longer canceled by inputs through the surround. Instead, the unit behaves as though the surround were no longer there.
Barlow and Levick  summarize their results as follows: “from a state of complete dark adaptation up to a luminance of about (viewed through a pupil) the mean rate of on-centre units increases.… Above …. On units often decrease and then increase again at a higher level.… Over the range where mean rate increases monotonically with adaptation level, it is shown that the surround of on-centre units does not inhibit and off responses cannot be elicited. When the monotonic increase is slowed or reversed it becomes possible to elicit responses from the surround” [39, p. 699; see also their Fig. 4].
This is precisely what one should expect on the basis of Fig. 4. The initial increase with luminance corresponds to a failure of cancellation between center and surround. When (at a sufficient luminance) cancellation kicks in, a surround response can be observed, and the mean rate is reduced, because the maintained discharge now reflects the rms noise value instead of, as previously, the mean. “It is as if the ganglion cells at adapting levels above about … acquired a differential input” [39, p. 716]. With further increase in luminance, the rms noise increases further, overtaking the previous peak discharge rate from the center alone. The surround becomes effective when the total quantum catch of the receptive field exceeds some critical value () and the corresponding luminance, of course, depends on the area of the receptive field.
The difference between threshold behavior above and below is acutely apparent in . Thresholds were measured for brief white flashes (20 and 200 ms duration; 1.7′ to 1.3 deg diameter) presented on a 10.4 deg field to which the observer was carefully adapted, and compared with the same measurements 0.2 s after the adapting field had been switched off. Except for the lowest field intensities, flash thresholds increased as Weber’s law, indicating an aggregate input in excess of (cf. Fig. 2), but thresholds measured 0.2 s after extinction of the field increased approximately as the square root of adapting illuminance. After extinction, the bleached rhodopsin (density proportional to adapting illuminance) continues to regenerate, producing an equivalent background proportional to the adapting illuminance , but at a level now less than . At that level cancellation (Fig. 4) fails and thresholds obey the De Vries–Rose law (cf. Fig. 2 again).
C. Transition Illuminance
As luminance increases, the change from square root to Weber’s law is not immediate, but gradual. For quantum catches clearly less than the critical value there is no cancellation (Fig. 4), while for quantum catches clearly in excess differentiation is complete. This leaves an intermediate region where some of the positive input is canceled and some passes through unopposed. A transition illuminance is estimated by extrapolating the square-root and Weber’s law characteristics in Fig. 2. The estimates are set out in Fig. 3; they increase with respect to wavenumber approximately as Eq. (2).
If the receptive fields of different channels are morphologically similar, differing only with respect to a linear scale factor, a grating of wavenumber will be most sensitively detected by a channel of linear dimension , where5), so, substituting from that equation and subsuming the constant within the definition of , 3 presents estimates of transition illuminances for wavenumbers from 0.5 to 48, a range of in linear dimension and nearly in area. Under a uniform luminance the smallest, highest-wavenumber, channels will absorb only as much light as the largest. That is, different parallel channels with different areas of projection into visual space can accommodate at least four decades of the range (seven decades) over which cones function.
Differential coupling [Fig. 1(c)] cancels mean input, leaving only quantal noise of power proportional to . Half-wave rectification [Fig. 1(d)] passes only the positive excursions of that noise as a maintained discharge. That discharge has the scale of the standard deviation of the noise, that is, . Perception depends on small perturbations [e.g., Fig. 1(d)] to that maintained discharge. If the range of responses from a single neuron is as great as [2, p. 273], then each neuron can, by itself, accommodate a luminance range of . Combining this with the scaling of luminance input consequent on the different sizes of receptive fields of different channels gives a total range of possibly . This is sufficient to accommodate the seven decades of cone vision.
D. Suppression of Saturated Neural Responses
Differential coupling makes a second contribution to visual adaptation through the suppression of saturated neural responses. Transmissions through different wavenumber-sensitive channels are reassembled in perceptual space, “out there.” If saturated responses were included in that reassembly, perception would be masked, as is sometimes experienced, with the glare from oncoming automobile headlights in the dark.
To demonstrate the reassembly of transmissions through different wavenumber-sensitive channels, Fig. 6 reproduces part of the data for one observer in a study by Campbell and Robson [42, Fig. 4]. The open circles are modulation thresholds for sinusoidal gratings of various wavenumbers from 0.2 to . The open squares show the corresponding thresholds for square-wave gratings of the same wavenumbers. Thresholds were determined with the grating switched on and off at 0.5 Hz. The Fourier decomposition of a square-wave grating of contrast is6 are the square-wave thresholds depressed by . These depressed values agree with the sine-wave threshold down to , showing that over this range detection of the square-wave grating is contingent simply on detection of its fundamental. In addition, a square-wave grating can be distinguished from a sine-wave of the same wavenumber and times the contrast (to equate the amplitudes of the respective fundamentals) when the third harmonic of the square-wave grating is itself detectable . But below this equivalence breaks down, and below there is no further decrease in the square-wave threshold.
At low wavenumbers, contrast sensitivity increases at most in proportion to the wavenumber (continuous straight line of gradient 1 in Fig. 6), while the amplitudes of the harmonic components in the Fourier decomposition [ in Eq. (7)] decrease in exact inverse proportion. This means that as wavenumber decreases, additional harmonics become detectable in their own right, but each individual harmonic never becomes easier to detect than the fundamental, however low the wavenumber of that fundamental may be. Compound gratings are more detectable than simple ones, roughly in proportion to the fourth root of the number of components . So, as wavenumber decreases, the square-wave grating (viewed as a compound) acquires an advantage from its harmonics, relative to the sine wave, in inverse proportion to the fourth root of the wavenumber. This calculation is shown as the straight dotted line in Fig. 6. That dotted line manifestly underpredicts the threshold for square-wave gratings below .
To this point the calculation has taken account of all the information implicit in the harmonics of the square-wave grating except relative phase. So phase information is also relevant; that is, the throughputs of different channels must ultimately be reassembled to recreate (a weighted transform of) the square-wave grating. To put this another way, there is an upper limit to the threshold of a square-wave grating at low wavenumbers; that limit is provided by detection as the eye moves across an edge. But, for that threshold to be independent of wavenumber, the contributions of the different channels must be brought together in a correct phase relationship.
Now, the superposition of outputs from different channels, each with input in proportion to the squares of their most sensitive wavenumbers, will ordinarily compound inputs at different scales, and some of those inputs must surely overload the capacity of neural transmission. Yet vision is not impeded at high luminance. If saturated responses were not somehow blocked, they would mask perceptual information. I suggest, first, that overloading leads to regular spike trains and, second, that regular spike trains are blocked by differential coupling at some subsequent level.
Figure 7 shows the peak response to a 0.5 deg test spot flashed for 500 ms once every 5 s in the center of an on-center receptive field of a ganglion cell in the cat retina in a study in . The ordinate shows the mean number of discharges during the first 50 ms (to avoid the complications of the different time courses of response at different adaptation levels), less the number that would have been emitted in the maintained discharge. This corresponds to the initial increment from the center in Fig. 1, unopposed by the surround. The experiment was repeated at six different background levels, with up to seven different test intensities at each level. The curves fitted (by eye) to the data points represent the Naka–Rushton equation ,7. Incremental output is linear for small test intensities (), then evolves into an approximately logarithmic relation (), and finally saturates (). Undistorted transmission of information requires inputs in the linear region of this characteristic.
Rodieck  has reported a wide variety of patterns of maintained discharge from ganglion cells in the cat retina, many of them showing a quasi-periodicity. But the incremental discharge rates in Fig. 7 rise to 10 times the background rates (c ) reported by Rodieck, and I conjecture that the saturation evident in Fig. 7 results from a refractory phase within the cell. A refractory phase would also tend to generate regular trains of discharges. The quantal noise in Fig. 1 is transmitted precisely because it is statistically independent in the positive and negative inputs. Take that independence away, and the two inputs cancel. Regular spike trains would pass poorly, if at all, through a subsequent differential interface. So differentiation plays a second important role in visual adaptation, preventing perception at high luminances being masked by input through channels with large quantum catches (sensitive to low wavenumbers).
3. RETINAL GAIN CONTROL
The account of visual adaptation set out above must now be compared with existing ideas. Received wisdom says that the output of each retinal unit is scaled down in inverse proportion to the ambient luminance, so that it matches contrast in the physical world. This suggestion originated with Rose . It was developed by Rushton with respect to dark adaptation in the rods, and his theory is most succinctly set out in .
After a full bleach, rhodopsin recovers exponentially with respect to time (but see , below) and the log threshold for a flash reduces with the same exponential time course ; that is, log threshold in darkness is a linear function of the percentage of rhodopsin still unregenerate. Absolute threshold depends, of course, on the diameter of the test flash, and  showed that the relation between threshold and flash size with a fixed proportion of rhodopsin bleached could be mimicked by the same set of measurements against a suitable background illuminance, the eye being otherwise fully dark adapted. This means that the increase in threshold consequent on bleaching some proportion of the rhodopsin was equivalent to superposition on a suitable illuminance (the equivalent background).
In light of these findings, Ruston proposed that visual input was scaled down, not in the rods themselves, but at the level of an adaptation pool (which might be a retinal ganglion cell). The effect of bleaching might be mediated in either of two ways: (a) by a signal from the bleached rods indistinguishable from the absorption of light quanta (the “equivalent background,” which would have to increase as the exponential of the proportion of rhodopsin bleached) or (b) by a separate bleaching signal directly into the adaptation pool (a signal that could be strictly proportional to the bleach). Rushton [46, Fig. 8] favored (b). However, this poses an immediate problem. An afterimage in darkness is bright. But, if Crawford’s equivalent background is consequent on a bleaching signal from the rods scaling visual input down, there is no input to generate the bright afterimage and, even if there were, it would be scaled down by the bleaching signal. The equivalent background has to result from spontaneous photoisomerisations (“dark light”) in the rods. Lamb  has since pointed out that the regeneration of rhodopsin after bleaching is more complicated than  envisaged. It appears to involve three intermediate products, and one of the transitions is rate-limited. The equivalent background does result from photon-like events occurring in the course of regeneration; it is linearly related to the process of recovery, and is indistinguishable from incident light.
Rushton’s  theory has since been adapted to daylight vision (see  for a review) with this change in emphasis. “We believe…that adaptation keeps the retinal response to contrast invariant with changes of illumination, and thereby achieves…constancy of the visual perception of reflecting objects” [2, p. 267]. This is achieved by adjusting the gain of each retinal unit to the reciprocal of the incident illumination, a scaling that gives Weber’s law. As a consequence the eye is able to handle a very wide range of illuminances. Shapley and Enroth-Cugell  used the Naka–Rushton equation [Eq. (8) above] modified by a gain factor , which, in turn, depends on the present and recent past values of visual input. Equation (8) is modified to give10) is already near-logarithmic (cf. Fig. 7), so that Weber’s law readily follows from an appropriate choice of as a function of preceding illumination [49,50].
But there are problems with this view of visual adaptation:
- (i) Equation (10) describes a monotone increasing transform from illuminance to internal response, when the interface is demonstrably differential.
- (ii) Revisiting one argument (above), observers are not only able to detect very small (0.0025) contrasts, but can also discriminate even finer differences (0.001; ). The visual analysis of near-threshold grating stimuli depends on an accelerated nonlinear transform of contrast , a scaling that would not give Weber’s law.
- (iii) The scaling implied by Eq. (10) gives Weber’s law for all configurations of the stimulus levels. But over the range 0.1–1000 td, the data from  in Fig. 1 reveal either a square-root relation or Weber’s law depending on the wavenumber. Moreover, differential coupling already generates a natural explanation of Weber’s law .
- (iv) The scaling implied by Eq. (10) creates an internal representation that mimics physical contrast in the outside world. This might appear to present a simple explanation of contrast constancy, but who perceives that internal representation of contrast? It does but invoke Ryle’s  “ghost in the machine.”
- (v) The data in Fig. 6 show that if the channel tuned to wavenumber delivers sufficient information, a periodic pattern is perceived. If the channel tuned to wavenumber 3g likewise delivers sufficient information, it can be seen that the pattern is square-wave rather than sinusoidal. Perception is the interpretation of sensory information, projected “out there.” It is not otherwise necessary for the world outside to be modeled in the brain, and it is not clear how an internal response proportional to contrast would be interpreted.
- (vi) Figure 8 is the well-known simultaneous contrast figure; the four small squares are all 50% black, but look different. Retinal gain control has a difficult, and quite unnecessary, task of explaining why contrast gain control fails in this instance. Much simpler is (a) the interface between the stimulus and visual analysis is differential, so that a comparison between the four small squares has to be constructed from the changes in luminance at the boundaries; (b) assessment of those changes is no better than ordinal; and (c) the assembly of those ordinal changes is systematically biased. In short, people cannot do arithmetic with contrasts .
- (vii) Even more problematic are the brightness matches reported by Heinemann [54,55]. In substance, the observer was asked to adjust the luminance of the right-hand small square in Fig. 8 (taking the black to signify no surrounding luminance), seen in one eye, to match the perceived brightness of the left-hand small square, judged against the white surround, seen in the other. The matching relation is not monotonic. A (white) surround luminance less than that of the (gray) square slightly enhances its perceived brightness, while if the luminance exceeds that of the enclosed square by more than 0.1 log unit (26%), no match is possible. The (gray) square then appears darker than complete darkness in the other eye, notwithstanding that the enclosed square can be made to appear even darker by further increase in the (white) surround luminance . The idea that brightness equates to some level of activity in the visual system is not tenable.
4. STUDIES OF THE RELATION BETWEEN INPUT TO AND OUTPUT FROM RETINAL GANGLION CELLS
Rushton’s  theory has spawned studies of the change in retinal ganglion cell response as a function of stimulus illumination. I examine three:
A. Sakmann and Creutzfeldt 
Figure 7 shows that as background illumination is increased, so also is the response to the onset of a flash in the center of the receptive field. Shapley and Enroth-Cugell [2, p. 275] cite this as an example of gain control (“curveshifting”) in cat retinal ganglion cells. These data pose two problems. First (for retinal gain control), why is the lateral displacement not in proportion to the background luminance? The difference in location between the curves fitted (by eye) to adapting luminances of and is only 2.3 log units, and [23, p. 176] reports generally that threshold increases only as (adapting luminance) , where “n varies between 0.45 and 0.75 in most units.” Second (for the present argument), why is there any “curve shift” at all?
The discharge rate in Fig. 7 is calculated over the first 50 ms of the response only, which means that it corresponds to the initial increment in Fig. 1, with the center unopposed by the surround. The stimulus is substantially smaller than the receptive field and engages only a proportion, , of the center component. The quantal input given stimulus luminance is therefore only (relative to the adapting luminance), which explains why the curves in Fig. 7 lie so much to the right of the adapting luminances (indicated by the arrows at the top of the diagram). The rate actually plotted is the difference between the rate recorded (during the first 50 ms) and the mean maintained discharge. Finally, differential coupling tells us that the mean maintained discharge equates to , where is the adapting luminance. Inserting these details into the Naka–Rushton equation [Eq. (8)],11) shows, first, that the maximum is reduced by the existing maintained discharge and, second, that, functionally speaking, the half-saturation constant is replaced by consequent on subtracting the maintained discharge. The parameter transposes into the location of the curves in Fig. 7 relative to the abscissa (2.3 log unit shift). However, the question of whether Eq. (11) gives an adequate account of the increases in discharge rate cannot be resolved without further data from the experiment.
B. Barlow et al. 
Barlow and Levick  endeavored to study retinal scaling via a quantum/spike ratio, that is, the number of additional quanta that needed to be absorbed in the receptive field to produce one additional discharge. The number of quanta absorbed could not be known with sufficient precision, so Barlow and Levick argued as follows: the absorption of quanta is a Poisson process. If some number, , of quanta were needed to produce an additional discharge, the interval between successive discharges would be a gamma variable of parameter , a value that can be estimated from the ratio of the interval distribution . They found that increased with adaptation level, as one might expect, but, on the other hand, could assume values less than 1 at low luminances. Figure 9 from  is a case in point.
Figure 9 presents a histogram of the intervals between successive discharges from a ganglion cell in the retina of a cat in darkness. These discharges result from photon-like events arising as the rhodopsin regenerates  and, except for a gradual decline in density as regeneration proceeds, those events should constitute a Poisson process. The output (see Fig 9) does not. However, selecting exactly every third discharge from the sequence represented in Fig. 9 gave a close conformity to an exponential distribution of intervals (the empirical manifestation of a Poisson process). Accordingly, Barlow et al. suggested that in this case each photon-like event in the regeneration of rhodopsin gave rise to three discharges.
The curve in Fig. 9 reflects a different reading of the data. There is, first, a Poisson process of photon-like events from the regeneration of the rhodopsin. Let that process have density . Each such event creates a transitory perturbation of cellular potential resulting from passage thorough neural tissue. Model that perturbation with a simple R-C low-pass filter, with impulse response function . The negative exponential has the property that following a discharge, which may itself follow a photon-like event after some small interval, the residual tail of the perturbation is always . So the distribution of interdischarge intervals can be obtained in this manner:
Suppose the interval (0, ) since the last output to be split into subintervals of length . The probability of an output in the interval is , and the probability of no output at all in the complete interval (0, ) is
The density function in Fig. 9 follows upon differentiating with respect to :9 is plotted against a logarithmic ordinate, with respect to which the equation becomes 9, and the equation actually shown (an approximation, fitted by eye) is 14) has been ignored.
The expression 1.3–2.7t (with respect to logarithmic ordinate) is the long-term asymptote of Eq. (15). It represents the Poisson process of photon-like events from the regeneration of rhodopsin. Not every such event needs result in a discharge; represents the proportion that do. The expression is the impulse response function of an R-C filter 3 dB down at 3.34 Hz. It represents the proportion of photon-like events that generate more than one output. The expression matches the gradient of Eq. (15) at small (but not the smallest) intervals and represents the combined effects of the R-C filter and the Poisson input. The deficit of very short intervals () is attributed to refractoriness and is ignored.
The important conclusions here are, first, that the idea that each quantum absorbed gives rise to three discharges is by no means necessary—the shape of the interval histogram can be modeled by a simple RC low-pass filter—and, second, that noise energy is lost in such filtering, so that, as assumed in the analysis of the data in Fig. 5, the effective noise level is less than the luminance L. There is a further implication that ganglion cell discharges are not the simple transmission of quantal impulses through the cell, so that the quantum/spike ratios calculated by Barlow and Levick do not stand up. On the other hand, RC low-pass filtering of a Poisson input necessarily gives an interval histogram of the shape shown in Fig. 9, while [39,59] have reported (approximate) gamma distributions of interspike intervals. It is not known, however, whether the input levels in these latter cases were within the linear range supporting perception or reflected the onset of saturation of neural response (cf. Fig. 7).
C. Barlow and Levick 
Figure 10 shows further “thresholds” from the same experiment as Fig. 5. The stimuli in Fig. 10 are a 0.4 deg test spot and an annulus with internal diameter 2.1 deg, centered on the receptive field. The adapting field was with, in addition, an adapting disc at (the higher background luminance in Fig. 5), again centered on the receptive field. The point of interest in Fig. 10 is the variation of “threshold” with the diameter of the adapting disc. The stimuli were presented for about 1 s (manually controlled), repeated about every 3 s.
Barlow and Levick’s  Fig. 4 includes peri-stimulus-time (PST) histograms for the data points surrounded by haloes in Fig. 10. All these histograms show the maintained discharge adjusting rapidly to a stable level following the presentation of a stimulus. Since the stimuli were presented at intervals of about 3 s, I shall assume that in the interim the maintained discharge adjusted to an equilibrium value determined by the quantal noise only.
The data in Fig. 5 suggest that there should be cancellation of the center by the surround for large background spots (at ), but not for small ones (uniform background at ). Happily, this makes little numerical difference to the calculations for the 0.4 deg test spot, which reflect a signal/noise discrimination at all diameters of the adapting disc. Background noise comes from two sources: first, the uniform background at and, second, the adapting disc at contributing for all radii of the adapting disc. Putting these two contributions together, “threshold” should be proportional to the rms value of the combined noise, and the equation fitted to these data in Fig. 10 is3), (4), and (16) were estimated by minimizing the sum of squared deviations for all three sets of data, analyzed to this point, simultaneously. The constants are reinterpreted in Table 1 as more familiar parameters of the receptive field. All other parameters are scaling constants calculated to give the best fit to the data.
The “thresholds” for the annulus present some profound problems of interpretation—at first sight, they appear to defy the constraints of signal-to-noise ratio—and what follows here is conjecture going way beyond the data. The calculations take the parameters set out in Table 1 as given.
In the first place, the annulus thresholds relate specifically to a suppression of output when the annulus is switched off. Barlow and Levick’s Fig. 4 is labeled “Spot on,” but “Annulus off.” But the annulus also has an on response, which, at zero adapting-spot diameter, could have been detected 1 log unit below the published threshold.
Second, the processes represented in my Figs. 1 and 4 are realized at preganglion cell levels; that is, cancellation between center and surround inputs and the attenuation of noise occur independently in different parts of the receptive field. So, when the adapting spot overlaps part of the annulus, it is necessary to distinguish that part of the annulus that is overlapped from that part that sees only a uniform adapting luminance of . These two components of the annulus input are scaled differently. (Note that this issue does not arise with the 0.4 deg test spot, which is always centered on the adapting disc).
The 0.4 deg test spot presents a net central input equal to 0.035 of the weight of the center, while the annulus presents a surround input equal to 0.696 of the surround. The log difference is 1.30. So, for the largest diameter of adapting disc (covering the whole field), the “threshold” for the annulus should be 1.3 log units below that of the test spot. (The reported difference is 0.95 log units.) At lesser diameters of the adapting disc, that part of the annulus input outside the adapting disc sees only the adapting luminance and is transmitted without cancellation. To fit the data I have assumed that absence of cancellation increases sensitivity to the annulus by 1 log unit. The curve in Fig. 10 for adapting disc diameters greater than 0.5 deg is a weighted average, weighted in proportion to the overlap of the annulus by the adapting disc, of two values; one is 1.3 log units below the threshold for the test spot, and the second is reduced by one further log unit.
For the smallest diameters of adapting disc the off-response from the annulus would have produced a complete suppression of ganglion cell output. For that suppression to be detected, there would need to be a certain minimum input to be suppressed. Figure 5 implies that at these smallest diameters there is no cancellation between surround and center. For the off-response to be detectable, the annulus needs to provide a contribution to the center component of the receptive field to increase the aggregate input up to that minimum level. The dotted purple curve in Fig. 10 links the annulus luminances needed to increase the existing center input up to a fixed value (equivalent to an adapting input of log units). It provides a possible account of thresholds for the six smallest adapting spots (using two free parameters, one to match the overall level vis-à-vis log luminance and one to match the range).
The conventional view of visual adaptation by retinal gain control is in conflict with several well-established experimental findings. The interface between luminance and visual analysis is differential, not logarithmic. It purports to explain Weber’s law in terms of scaling by the Naka–Rushton equation, when there is a natural explanation of Weber’s law consequent on the differential interface; moreover, the explanation provided by the Naka–Rushton equation applies everywhere, even where a square-root law is observed. It is confounded by the phenomenon of simultaneous contrast (Fig. 8) and especially by visual stimuli that appear darker than complete darkness. In short, the idea of retinal gain control must now be abandoned.
There is, of course, a scaling down of luminance input in the retina; it is accomplished by receptive fields of different sizes. The smallest fields function at the highest luminances and provide the finest visual acuity. The relationship of visual acuity to illuminance has not otherwise been mentioned, but is implicit in Figs. 2 and 3. Adaptation to luminance and the enhancement of visual acuity are products of the same mechanism. The problem of matching luminance input to the response range of neural units is eased by differential coupling, consequent on which the maintained discharge increases only as the square root of input. A neuron with a response range of could handle four log units of luminance. Finally, the Naka–Rushton equation describes, not the rescaling of input, but merely the saturation of neural response as input increases. When the input is overloaded, the output tends to regularity, whereafter further transmission is blocked by a subsequent stage of differential coupling.
The argument in this paper has concerned cone vision only, because that is the span of illuminances covered by van Nes’ thresholds. But retinal ganglion cells take input from rods as well as cones, and the properties of grating detection in general are found in purely rod vision in achromats [60,61]. It is conceivable that further measurements of grating thresholds in rod vision, similar to those in Figs. 2 and 3, would support the extension of this argument to the whole range of visual function.
If electrophysiological observations are to be correctly interpreted, there must first be an overarching functional map of the visual pathway. It needs to be said that retinal gain control was conceived with that purpose in mind. If we except passing suggestions in earlier papers , the idea of retinal gain control predated the systematic study of visual gratings  and of differential coupling . It was proposed at a time when, under the impact of signal detection theory, Weber’s law was center stage and a wide variety of models were being proposed. But subsequent discoveries have exposed the limitations of retinal gain control as an explanatory concept, and it is time to let it go.
The idea of retinal gain control, quite naturally, prompted studies of the relation between luminance input and retinal output. The fact that there is such a relation does not of itself validate retinal gain control. Three such studies have been reinterpreted here in terms of the model illustrated in Fig. 1. That model was conceived as a model of sensory discrimination at the psychophysical level; moreover, it was devised to admit reasonably simple calculations. As a model of cellular function, applied to retinal ganglion cells, it has two serious defects. Nevertheless, it is capable of development by the incorporation of some process of equilibration, to provide, first, differential coupling as a general principle and, second, adaptation to static inputs. Testing and validation of such developments will require electrophysiological experiments specifically for that purpose. Existing studies are suggestive, but not sufficient.
The conceptual change underlying all of this concerns the notion of “inhibition” as the “arrest of the functions of a structure or organ, by the action upon it of another, while its power to execute those functions is still retained” . While that concept of inhibition might be applicable in some parts of the nervous system, the observations summarized by [38, p. 342] and [39, p. 699] compel the view that in retinal ganglion cells “inhibition” is simply negative excitation [63,64], and is effective only when there is sufficient positive input to be canceled. In spite of its present defects, the model of Fig. 1 provides the framework within which to begin again.
I thank William Levick for the raw data replotted in Figs. 5 and 10 and Floris van Nes for the data in Figs. 2 and 3. I also thank Bruce Henning and Adam Reeves for their comments on an earlier version of this paper.
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