The ability of the cone visual system to regulate its sensitivity from twilight to bright sunlight is an extraordinary feat of biology. Here, we investigate the changes in visual processing that accompany cone light adaptation over a 5 log _{10} unit intensity range by combining measures of temporal sensitivity made in one eye with measures of the temporal delay *between* the two eyes in different states of adaptation. This combination of techniques, which provides more complete information than has been available before, leads to a simple model of steady-state light adaptation. At high light levels, visual sensitivity is maintained mainly by photopigment bleaching. At low-to-moderate light levels, it is maintained by trading unwanted sensitivity for speed and by an additional process that paradoxically *increases* the overall sensitivity as the light level rises. Each stage of the model can be linked to molecular mechanisms within the photoreceptor: The speeding up can be linked to faster rates of decay of activated molecules; the paradoxical sensitivity increases can be linked to faster rates of molecular resynthesis and to changes in channel sensitivity; and the sensitivity decreases can be linked to bleaching. Together, these mechanisms act to maintain the cone visual system in an optimal operating range and to protect it from overload.

^{2}from the level of noise to their response ceiling (e.g., Barlow & Levick, 1976; Shapley & Enroth-Cugell, 1984), the human visual system is able to operate over a >10

^{11}range of environmental light levels. It does this in part by depending on a more sensitive rod-driven

*scotopic*subsystem at low levels and on a less sensitive cone-driven

*photopic*subsystem at high levels (Parinaud, 1881; Schultze, 1866; von Kries, 1894, 1896). Within each subsystem, mechanisms of adaptation act to maintain the system within an optimal operating range as the illumination level is increased. In this article, we investigate the mechanisms of photopic light adaptation.

*cis*-retinal, to its all-

*trans*form, which rapidly causes a conformational change of the G-protein-coupled receptor-protein cone opsin (R) into the activated photoproduct R* (or metarhodopsin II). R* in turn activates the heterotrimeric G-protein transducin trimer (G

*α*-GDP-G

*βγ*) by catalyzing the exchange of GDP for GTP, which initiates the separation of the activated

*α*-transducin (G

*α** or G

*α*-GTP) from the trimer. G

*α** then activates the effector molecule, phosphodiesterase enzyme (PDE6*), which reduces the cytoplasmic concentration of cGMP by catalyzing its hydrolysis into GMP. The reduction in cGMP concentration results in the closure of the cyclic-nucleotide-gated (CNG) channels in the plasma membrane, which block the inward flow of Na

^{+}and Ca

^{2+}ions, precipitating membrane hyperpolarization and the initialization of the neural response.

*reduce*overall sensitivity independently of temporal frequency and are likely to have little effect on phase delay. Potential mechanisms are: (i) pigment bleaching (e.g., Boynton & Whitten, 1970; Burkhardt, 1994; Hecht, 1937) and (ii) response compression caused by the availability of fewer CNG channels as the light level increases (Baylor & Hodgkin, 1974; Dowling & Ripps, 1970; Matthews, Murphy, Fain, & Lamb, 1988). Lastly, a third category (Category C) contains mechanisms that are likely to

*increase*overall sensitivity in a way that does not depend on temporal frequency and probably have little effect on phase delay. Potential mechanisms are: (i) the increase in the rate of cGMP synthesis mediated by guanylyl cyclase (Hodgkin & Nunn, 1988; Koutalos, Nakatani, Tamura, & Yau, 1995; Koutalos, Nakatani, & Yau, 1995; Koutalos & Yau, 1996; Polans, Baehr, & Palczewski, 1996; Pugh, Duda, Sitaramayya, & Sharma, 1997; Tamura, Nakatani, & Yau, 1991) and (ii) the decrease in

*K*

_{1/2}(the half-activation concentration) for cGMP opening the CNG channels, which has the effect of making more channels available (Bauer, 1996; Chen et al., 1994; Grunwald, Yu, Yu, & Yau, 1998; Hsu & Molday, 1993, 1994; Rebrik & Korenbrot, 1998; Weitz et al., 1998).

_{10}units of intensity. On the basis of our new results, we propose a simple model of steady-state adaptation with two intensity-dependent parameters that can be plausibly linked to the underlying molecular mechanisms.

*μ*s and could produce sinusoidal modulations from 0% to 92%. Each was driven by computer-controlled programmable timers.

*combined*target and background.

*I*

_{max}−

*I*

_{min})/(

*I*

_{max}+

*I*

_{min}), is given in terms of M-cone excitation. Relative cone excitations were calculated using the Stockman and Sharpe (2000) cone spectral sensitivities. An alternative way of specifying threshold is in terms of the flicker amplitude, which is simply the difference between

*I*

_{max}and

*I*

_{min}. Amplitudes are given in units of log trolands.

*changes*in sensitivity between different levels of adaptation—which are what we actually model—are very similar (see Figure 6, upper panels). We speculate that the shape differences occur because M.M. has access to a residual low-frequency chromatic signal, arising from slight spectral sensitivity differences in his two expressed X-linked photopigments.

*I*/

*I*=

*k*) holds, and the proportional sensitivity to superimposed lights is maintained as the luminance level is changed. This occurs mainly at low frequencies and at high intensities.

*I*

_{max}−

*I*

_{min}) as a function of frequency, as shown in Figure 4 for M.L. (left panel) and M.M. (right panel). This way of plotting the data emphasizes those levels between which the change in background has no effect on the amplitude required for detection (i.e., when Δ

*I*=

*k*). Given some extrapolation to frequencies above the measurable range, such behavior, which is known as “high-frequency linearity,” is found to occur between 0.42 and 2.79 log td for M.L. and to occur between 1.05 and 3.39 log td for M.M. Comparable data have been reported before (e.g., De Lange, 1958; Kelly, 1972; Roufs, 1972a).

*appear*to be consistent with high-frequency linearity, the phase data suggest that linearity fails (see below).

*relative*phase delays in degrees between M-cone flicker presented to the left and right eyes of M.L. (upper panel) and M.M. (lower panel). For each subject, the luminance level in the right eye was fixed at 4.16 log td, whereas that in the left was varied (see key). A phase delay of 0° implies that the phase delays in the left and right eyes are the same (or a factor of 360° different). Overall, the phase delays for M.M. and M.L. are very similar. In general, increasing the adaptation level in the left eye advances the flicker signal, whereas decreasing the adaptation level delays it. For both subjects, the responses in the left eye are delayed at levels below 4.16 log td because the eye is

*less*light adapted than the right eye, whereas they are slightly advanced above 4.16 log td because the left eye is

*more*light adapted.

*independent*of frequency (so that curves analogous to those in our Figure 5 would all be parallel). We suspect that their results, particularly those obtained at low frequencies, reflect the distortion of suprathreshold sinusoidal flicker producing unwanted signal components at higher frequencies. Unfortunately, however, very few methodological details were provided in their article. Intriguingly, though, some years later, Cavonius, Estévez, and van der Tweel (1992) reported that their counterphase dichoptic flicker produced visible second harmonic flicker.

*qualitative*model of adaptation that requires one or two intensity-dependent parameters. In light of the clear differences between the data measured in the regions above and below 4.16 log td, we treated data from those two regions separately. Below circa 4.16 log photopic td, the amplitude and phase-delay data for both subjects show frequency-dependent changes, which are broadly consistent with a speeding up of the visual response and a shortening of the visual integration time. Above 4.16 log photopic td, adaptational changes are consistent with multiplicative scaling of the amplitude thresholds, as would be produced by bleaching. (In this region, the estimated time constants are in any case too short for adaptational changes to have a significant effect on the shapes of the amplitude and phase data in the visible range of frequencies.)

*τ*) of one or more (

*n*) cascaded leaky integrating stages (or buffered RC circuits), which we also refer to as filters (see Watson, 1986). This approach (see the Discussion section), which was first proposed long before the details of the phototransduction cascade were understood, is still relevant in the context of cascade processes because leaky integrators are comparable to first-order biochemical reactions. In the filter, the response to a pulse decays exponentially with time, whereas in the reaction, the concentration of the reactant decays exponentially with time. The formula for the amplitude response,

*A*(

*f*), of

*n*cascaded leaky integrators is

*P*(

*f*), is

*f*is the frequency in cycles per second (hertz) and

*τ*is the time constant in seconds. There are two important properties of the cascade, which can be inferred from consideration of Equations 1 and 2. First, when the frequency

*f*is high relative to 1/(2

*πτ*) (the so-called cutoff or corner frequency of a low-pass filter in hertz), the amplitude and phase are effectively

*independent*of changes in the time constant, because in the case of

*A*(

*f*),

*A*(

*f*) ≈ (2

*πf*)

^{−n}, and then in the case of

*P*(

*f*),

*P*(

*f*) =

*n*× 90°. Thus, a cascade can obey “high-frequency linearity.” Second, when the frequency is low, the loss of sensitivity is proportional to the shortening of the time constant raised to the power of the number of integrators. Thus, a cascade that obeys high-frequency linearity can also, in principle, obey Weber's law, given the appropriate intensity-dependent changes in

*τ*.

*absolute*phase delays of the visual system under these conditions. We therefore restrict our modeling to account for the

*changes*in phase delay and the

*changes*in log threshold amplitude between the six successive levels from 1.60 to 4.16 log td, which are shown in Figure 6 for M.L. (left panels) and M.M. (right panels).

*n*(which we assume is independent of intensity) and

*τ*(which is intensity dependent), the time constants of each of the

*n*filters were varied together, thus altering the threshold amplitudes according to Equation 1 and the phase delays according to Equation 2. Allowing the time constants of each filter to vary independently yielded extra parameters but did not significantly improve the predictions of the model. We therefore yoked the time constants together, but we recognize that, in reality, the time constants of different stages are unlikely to be identical. The phase and amplitude data were weighted so that their influence was approximately equal (otherwise, one or other sets of data would dominate). When

*n*was allowed to take on noninteger values, the best fitting models for both subjects lay between

*n*= 2 and

*n*= 3. We have chosen to show model predictions for

*n*= 3 because

*n*= 3 is also consistent with the subsequent model in which sensitivity scaling is also allowed. We emphasize, however, that

*n*is poorly constrained by the fit because increases in

*n*can be offset by decreases in

*τ*and vice versa. The data, however, require that

*n*≥ 2 because the phase changes with adaptation are greater than 90°, the maximum change for one filter. The model fits for

*n*= 3 are shown by the dotted-dashed lines, which are color coded in the same way as the symbols. We refer to this as “Model 1: time constant only” because the only intensity-dependent parameter is

*τ*.

*single*intensity-dependent parameter,

*τ,*the predictions are, we believe, impressive. Although this single intensity-dependent parameter model predicts the data moderately well, there are some discrepancies. We next tried to reduce them by extending the model.

*cumulative*sensitivity scaling assumed in Model 2 is shown in the middle panel of Figure 7 by the open circles (for details above 4.16 log td, see the next section). Importantly, the scaling

*improves*sensitivity between 1.05 and 2.79 log td for M.L. or between 1.05 and 3.39 log td for M.M. (see Figure 7). The sensitivity improvements are by factors of more than 8 and 4 for M.L. and M.M., respectively. Again, we emphasize that

*n*is poorly constrained. The predicted time constants for versions of Model 2 with

*n*= 2, 3, and 4 are compared in Figure 9 (see the Discussion section).

*I*

_{0}) of 4.30 log td and the fraction of unbleached pigment,

*p*=

*I*/(

*I*+

*I*

_{0}) (Rushton, 1963, 1965). Thus, the effect of bleaching on sensitivity starts to become significant above circa 4.16 log photopic td. Accordingly, we modeled the sensitivity losses between those levels as a multiplicative scaling of the amplitude thresholds (i.e., as vertical shifts of the logarithmic threshold amplitude functions without change of shape), which is consistent with the effects of photopigment depletion. To estimate the logarithmic shifts, we used two related methods, which gave results within 0.01 log

_{10}unit of each other. Using a least squares fitting criterion, the data from 4.75 to 5.69 log td were shifted to vertically align with the data for 4.16 log td. The aligned data were then either (i) averaged to produce a mean function or (ii) entered into a curve generation program to produce an arbitrary mean template shape. Thereafter, the individual data were vertically shifted to fit either the mean function or the mean template. Carrying out these procedures iteratively did not significantly change the fits. The calculated shifts, plotted cumulatively, are shown as the open circles in the middle panels of Figure 7 (above 4.16 log td).

*n*leaky integrators (A), which,

*for simplicity,*are assumed to shorten together with adaptation (gray diamonds, Figure 7) but, in reality, are unlikely to be yoked together. The second intensity-dependent parameter is multiplicative sensitivity scaling (open circles, Figure 7), which we have subdivided into sensitivity scaling that reduces sensitivity (B) and sensitivity scaling that increases sensitivity (C). Sensitivity scaling that reduces sensitivity has been further subdivided into photopigment depletion or bleaching (B

_{1}), response compression (B

_{2}), and other neural factors (B

_{3}). We show sensitivity scaling due to response compression as a separate element in the model (B

_{2}) for completeness. However, its effects, if any, cannot be distinguished from those of other factors (B

_{3}).

*n*) is poorly constrained by the model fits. Although we chose to use three filters in our model, we could have chosen between two and four filters with proportionally shorter time constants for fewer filters. The trade-off between filters and time constants is discussed in the next section. The remaining components of the model can be linked to (i) pigment bleaching and (ii) response compression (from Category B), which decrease sensitivity, and to (i) the increase in the rate of cGMP synthesis and (ii) the decrease in sensitivity of the CNG channels (from Category C), which increase sensitivity. Given, however, that the model just provides an estimate of the

*overall*scaling, we can only estimate the effects of each of these different underlying mechanisms.

*in addition to*photopigment depletion could be due to bleaching desensitization, the additional loss of sensitivity caused by bleaching photoproducts (see Fain, Matthews, & Cornwall, 1996; Fain et al., 2001; Lamb & Pugh, 2004; Leibrock, Reuter, & Lamb, 1998; Pepperberg, 2003), to pigment depletion being higher than predicted by standard equations (e.g., Burns & Elsner, 1985, 1989; Mahroo & Lamb, 2004; Reeves, Wu, & Schirillo, 1998; Smith, Pokorny, & van Norren, 1983), or to other factors.

_{2}). We link this to response compression from Category B caused by a reduction in open CNG channels. We speculate that this mechanism may play only a small role in our experiments because they were carried out under steady-state conditions using near-threshold flickering targets. Response compression would presumably play a much greater role if the illumination changes were large and rapid.

*n*) and the time constant of each filter (

*τ*) because the two parameters interact in the model fits: An increase in one parameter can partially offset a decrease in the other. We can explore the relationship between the two by fitting versions of Model 2, the time-constant-and-scaling model, with different values of

*n*to the amplitude and phase data. Figure 9 shows the best fitting values of

*τ*for

*n*= 2 (open inverted triangles),

*n*= 3 (filled diamonds), and

*n*= 4 (open triangles) filters for M.L. (upper panel) and M.M. (lower panel) plotted in double-logarithmic coordinates. It is immediately apparent that, for each set of predictions, the relationship between log(

*τ*) and log luminance, which we refer to as log(

*I*), is approximately linear. Least squares linear regression provides the following estimates of the slopes: −0.68, −0.46, and −0.34 for

*n*= 2, 3, and 4, respectively, for M.L. and −0.60, −0.41, and −0.30 for M.M. Thus, the mean slopes across

*n*are −1.37/

*n*for M.L. and −1.21/

*n*for M.M. These slopes provide an approximate

*general*solution for how yoked time constants shorten with light adaptation for different

*n*values.

*n*= 2, 3, and 4, respectively, and

*A*(0) ∝

*I*

^{−1}(i.e., the amplitude of the steady component at 0 Hz would be inversely proportional to the mean luminance and, therefore, constant). Because Weber's law does hold (see Figure 7), the

*increases*in sensitivity caused by scaling must therefore approximate to

*A*(0) ∝

*I*

^{0.36}for M.L. and

*A*(0) ∝

*I*

^{0.20}for M.M. to compensate for the losses in excess of Weber's law caused by the shortening time constants. Given that Weber's law holds so precisely, despite differences in time constants between observers, there must presumably be some control system that compensates for the losses in excess of Weber's law and restores it.

*same*level of 2.79 log td (e.g., the 1.60 log td data were adjusted to 2.79 log td using the predicted differences between 1.60 and 2.20 log td

*and*between 2.20 and 2.79 log td; the 2.20 log td data were adjusted to 2.79 log td using the predicted differences between 2.20 and 2.79 log td, etc.). Second, a mean smoothed template was derived for each subject to describe all his amplitude data adjusted to 2.79 log td and then another template was derived to describe all his phase data adjusted to 2.79 log td. The templates were derived using a curve discovery program (TableCurve 2D, Jandel Scientific). They are the smooth functions fitted to the 2.79 log td data in Figures 10 and 11, the formulae for which are provided in the 1. Finally, the model predictions were used to adjust the smoothed templates for 2.79 log td back to each of the intensity levels (e.g., the smoothed 2.79 log td template was adjusted to 1.60 log td using the predicted differences between 1.60 and 2.20 log td

*and*between 2.20 and 2.79 log td, etc.). Above 4.16 log td, only scaling is assumed in the model; thus, in this range, the amplitude template is fixed in shape and vertically shifted. The templates adjusted for each level are shown in Figures 10 and 11 as the continuous lines (we attach no special significance to formulae for the template functions).

*gains*. These failures are much less conspicuous in the amplitude data because the sensitivity gains shift the amplitude threshold curves

*together*so that they still seem to converge in the way required of high-frequency linearity. However, the convergence occurs at temporal frequencies too low to be consistent with the phase-delay data, which should themselves converge to zero (such a convergence is predicted by the model but at higher frequencies than we can measure).

*changes*in high-pass filtering. Such a feature is found in the model of Kelly (1961b). Third, our data do not require a change in the number of integrators with adaptation.

*does not change*with adaptation (lowest curves, left panels of Figures 10 and 11), which selectively attenuates steady signals generated by backgrounds.

*a*is the amplitude threshold in log td,

*p*is the phase delay in degrees, and

*f*is the frequency in hertz, are as follows: