**Thirty years ago, Mollon, Stockman, & Polden (1987) reported that after the onset of intense yellow 581-nm backgrounds, S-cone threshold rose unexpectedly for several seconds before recovering to the light-adapted steady-state value—an effect they called: “transient-tritanopia of the second kind” (TT2). Given that 581-nm lights have little direct effect on S-cones, TT2 must arise indirectly from the backgrounds' effects on the L- and M-cones. We attribute the phenomenon to the action of an unknown L- and M-cone photobleaching product, X, which acts at their outputs like an “equivalent” background light that then inhibits S-cones at a cone-opponent, second-site. The time-course of TT2 is similar in form to the lifetime of X in a two-stage, first-order biochemical reaction A→X→C with successive best-fitting time-constants of 3.09 ± 0.35 and 7.73 ± 0.70 s. Alternatively, with an additional slowly recovering exponential “restoring-force” with a best-fitting time-constant 23.94 ± 1.42 s, the two-stage best-fitting time-constants become 4.15 ± 0.62 and 6.79 ± 1.00 s. Because the time-constants are roughly independent of the background illumination, and thus the rate of photoisomerization, A→X is likely to be a reaction subsidiary to the retinoid cycle, perhaps acting as a buffer when the bleaching rate is too high. X seems to be logarithmically related to S-cone threshold, which may result from the logarithmic cone-opponent, second-site response compression after multiplicative first-site adaptation. The restoring-force may be the same cone-opponent force that sets the rate of S-cone recovery following the unusual threshold increase following the offset of dimmer yellow backgrounds, an effect known as “transient-tritanopia” (TT1).**

*direct*S-cone input to the luminance pathway (e.g., Schrödinger, 1925; Luther, 1927; Walls, 1955; de Lange, 1958b; Guth, Alexander, Chumbly, Gillman, & Patterson, 1968; Smith & Pokorny, 1975; Boynton, 1979; Eisner & MacLeod, 1980; Lee & Stromeyer, 1989; Stockman, MacLeod, & DePriest, 1991; Ripamonti, Woo, Crowther, & Stockman, 2009).

_{1}) when the input shifted towards either extreme than when L+M and S inputs are more balanced (ΔS

_{2}). It is the effect on increment detection when the L+M signal predominates that concerns us in these experiments. Note that there is also a “restoring force” shown by the red arrow that acts to reduce any persistent polarization towards either L+M or S, which we will discuss next. (The particular way in which the second-site adaption is implemented is not critical in this context.)

*offset*of long-wavelength adapting backgrounds of less than about 5 log

_{10}Td (Stiles, 1949; Mollon & Polden, 1975; Mollon & Polden, 1976; Augenstein & Pugh, 1977; Mollon & Polden, 1977). The restoring force shown in Figure 1 with a time constant of

*τ*was a force postulated by Pugh and Mollon (1979) to account for the roughly exponential recovery of log

_{S}_{10}S-cone sensitivity in TT1 following the offset of the long-wavelength background. Probably the best psychophysical measurements of the time course of recovery of TT1 are those by Augenstein and Pugh (1977), who in a series of measurements on two observers found mean time constants ±1

*SE*of 15.45 ± 1.05 and 27.57 ± 3.28 s (see their table 1). These time constants may be relevant to our second model, below. The lower illustration in Figure 1 will be described later.

_{10}quanta s

^{−1}deg

^{−2}) as a function of the time after the onset of various 6.5° diameter, 581-nm backgrounds. The background retinal illuminances, which are indicated by the different symbols shown in the key, ranged from 3.45 to 5.58 log

_{10}photopic trolands (Td). For clarity, the data in both figures, which have been reproduced to facilitate comparisons between the different models, have been vertically shifted in successive steps of 0.5 log

_{10}unit relative to the data shown by the green diamonds, which are plotted correctly with respect to the ordinate. The unshifted data can be seen in the original publication.

_{10}Td and brighter, after which S-cone threshold rises to reach a peak after about 5 s, and then falls monotonically towards a steady-state value. By contrast, after the onset of backgrounds of lower illuminances, the peak sensitivity loss, which is typically less than about 0.5 log

_{10}unit, coincides with the onset of the adapting field and recovers monotonically over time. This latter pattern is more similar to classical data obtained with achromatic flashes and backgrounds of comparably low illuminances of 3.70 log

_{10}Td or less (Baker, 1949).

_{10}units less sensitive to the background light of 581 nm than to the 436-nm targets to be detected (Stockman, Sharpe, & Fach, 1999). Consequently, TT2 must depend primarily upon on an interaction between the L- and M-cones and the S-cones (Mollon et al., 1987). Additionally, given the cone spectral sensitivity differences between the 436-nm target and 581-nm background wavelengths, the detection of the 436-nm targets in these experiments should be

*mediated*predominantly by S-cones, although S-cone sensitivity will be

*modulated*by the L- and M-cones by way of second-site interactions. We can estimate the relative cone sensitivities to these lights from standard cone spectral sensitivity functions (Stockman & Sharpe, 2000). At 581 nm the S-cone quantal spectral sensitivity has fallen by 4.12 log

_{10}units below its peak sensitivity at 440 nm. By contrast, the L- and M-cone quantal sensitivities have fallen by only 0.02 and 0.23 log

_{10}unit, respectively, below their peaks; whereas at 436-nm, the L-, M-, and S-cone quantal spectral sensitivities have fallen 1.34, 1.17, and 0.03 log

_{10}units, respectively, below their peaks. Thus, the brightest 581-nm background used in these experiments of 5.58 log

_{10}trolands, which is 11.76 log

_{10}quanta s

^{−1}deg

^{−2}is (after factoring in the relative insensitivity of the S cones to this wavelength) equivalent to a background of 7.64 log

_{10}quanta s

^{−1}deg

^{−2}for the S-cones at 436 nm. This is only slightly above the dimmest detectible 436 nm target flash in this condition and cannot possibly account for the large initial rise and fall in threshold after background onset.

_{10}Td (Hayhoe, Benimoff, & Hood, 1987; Hayhoe, Levin, & Koshel, 1992), and subtractive adaptation may follow a comparably rapid time course (Geisler, 1978; Hayhoe et al., 1987), although some evidence suggests that it might be delayed by 200 ms and take as long as 10–15 s to complete (Hayhoe et al., 1992). Neither of these processes, however, is likely to be the direct cause of the delayed S-cone sensitivity loss found in TT2 either because they are too fast or because their recovery is monotonic. It is perhaps worth noting that while multiplicative and subtractive adaptation are convenient descriptors, it is not clear how they are implemented neurally. It seems likely that multiplicative adaptation is predominantly caused by shortening time-constants in the visual pathway. Shortening time constants reduces temporal integration and increases the relative sensitivity to higher temporal frequencies, producing briefer, smaller flash responses (de Lange, 1958a; Kelly, 1961; Roufs, 1972; Stockman, Langendörfer, Smithson, & Sharpe, 2006). Subtractive adaptation, on the other hand, associates light adaptation with high-pass filtering; i.e., relative increases in low-frequency attenuation (e.g., Rider, Henning, & Stockman, 2016). Low-frequency attenuation implies a biphasic flash response (Watson & Nachmias, 1977), but also causes an earlier, lower peak flash response which, in terms of flash detection, would be indistinguishable from a change in multiplicative gain.

_{10}Td or higher. These backgrounds bleach a significant amount of photopigment (more than 74% of the L- and M-cone photopigments in the steady state; see Tables A1 and A2). A reasonable supposition, therefore, is that the suppression is related to the effects of a bleaching photoproduct or intermediary, such as, inactivated forms of metarhodopsin II, metarhodopsin III, or opsin (e.g., Lamb, 1981; Okada, Nakai, & Ikai, 1989; Cornwall & Fain, 1994; Matthews, Cornwall, & Fain, 1996; Leibrock, Reuter, & Lamb, 1998; Zimmermann, Ritter, Bartl, Hofmann, & Heck, 2004). For reviews, see Lamb (2004) or Reuter (2011). Indeed, the rise and fall in S-cone threshold is reminiscent of the production of metarhodopsin III in human rods (see figure 10 of Alpern, 1971)—albeit about 15 times faster.

*in vivo*human cone photopigment generation in retinoid cycles that involve either the retinal pigment epithelium or Müller cells (Reuter, 2011; Sato & Kefalov, 2016), we will concentrate in this section on defining the dynamics of the lifetime of this hypothetical, but mystery photoproduct (which we call “X”).

*τ*; and X grows according to the depletion of A and decomposes to the final product, C, with a time constant,

_{A}*τ*. (The time-constant,

_{X}*τ*, is the time taken for the decay to reach 1/

*e*or 36.8% of its original value or, in the case of growth, to 1 − 1/

*e*or 63.2% of its maximum value.)

*t,*s) according to

_{0}is the initial concentration of A (at

*t*= 0), and

*τ*is the time-constant, in seconds, of the first reaction. [X], the concentration of X, changes according to

_{A}*τ*is the time-constant of the second reaction.

_{X}*I*, required to detect the 436-nm flash against the background and in particular whether the relation between [X] and Δ

*I*is closer to being linear or logarithmic. We discuss the relation between [X] and Δ

*I*further below, but initially we take an empirical approach, based on the form of the data and using Equation (3) for [X], and treat the relation as approximately logarithmic:

^{−1}deg

^{−2},

*c*is the unknown factor that includes both [A]

_{X}_{0}from Equation (3) and some gain factor, and

*v*is a vertical logarithmic shift of the function (which corresponds roughly to the ultimate baseline steady-state sensitivity of the S-cone system). We discuss this logarithmic relation below—it has been assumed before in describing dark adaptation data (Dowling, 1960; Rushton, 1961).

*τ*, from the fit, thus at the lowest levels:

_{A}*R*as a measure of goodness-of-fit is problematic—e.g., Kvalseth, 1985; Spiess & Neumeyer, 2010—so that as well as giving

^{2}*R*values in Tables A1 and A2, we also give the standard error of the residuals, which can be used to assess the accuracy of the predictions. The residuals are plotted in the lower panels of Figure 2.) The fitted parameters are given below as the parameter ±1

^{2}*SE*of the fitted parameter.

*τ*

_{A}and

*τ*

_{X}_{,}were applied across all background illuminances and observers. The best-fitting parameters and their standard errors are given in Table A1 in the Appendix. Given the simplicity of this descriptive model, the fits are remarkably good with an

*R*goodness-of-fit of 0.982, an adjusted

^{2}*R*of 0.980, and a

^{2}*SE*of the residuals of 0.010 log

_{10}threshold units.

_{10}Td level, between 7 and 15 s at the lower background levels, and between 40 and 60 s at several levels. For AS, the predictions are worse: Discrepancies similar to those for JDM occur between 10 and 20 s in the 5.58 log

_{10}Td data set, but there are other discrepancies between 1 and 20 s in the 4.75 log

_{10}Td level. The model consistently underestimates the data at 100 ms for both observers and overestimates the data after 50 s. Nonetheless, this simple model provides a not unreasonable description of the time course of TT2.

*τ*

_{A}= 3.09 ± 0.35 s and

*τ*= 7.72 ± 0.70 s. It should be noted that the order of time constants and thus the order of the reactions is constrained by our assumption that

_{X}*τ*alone determines threshold after background onsets of lower illuminances. Without this assumption, the order of

_{X}*τ*

_{A}and

*τ*is reversible (i.e., we do not know which of A→X or X→C has the shorter time constant).

_{X}_{10}Δ

*I*on 1 −

*p*, the proportion of bleached pigment, where

*p*is the proportion of unbleached pigment. The arguments are relevant here because our models depend on assumptions about the link between 1 −

*p*, the resulting photoproducts, and Δ

*I*. According to Dowling (1960) and Rushton (1961), during

*dark*adaptation the logarithm of threshold, log

_{10}Δ

*I*, is proportional to 1 −

*p*. More recently, however, Lamb (1981) has suggested that instead the linear threshold, Δ

*I*, is proportional to 1 −

*p*, except for nearly complete rod bleaches. Similarly, for cones, Pianta and Kalloniatis (2000) have also suggested that in dark adaptation the relation is linear rather than logarithmic. In general, if 1 −

*p*∝ log

_{10}Δ

*I*and the unbleached pigment recovers exponentially (i.e., according to first order kinetics), then a plot of log

_{10}Δ

*I*against time (i.e., the same log-linear axes as in Figures 2 and 3), should follow an exponential decay, whereas if 1 −

*p*∝ Δ

*I*the decay of log

_{10}Δ

*I*should follow a straight line. Over their

*whole*range, dark adaptation curves plotted on log-linear plots are clearly better described by an exponential decay rather than by a straight line (e.g., Pugh, 1976). Yet, by assuming that dark adaptation measurements reflect successive exponential recoveries of several photoproducts, each of which elevates threshold

*linearly*, the recovery curves can be fitted by overlapping line segments (e.g., Lamb, 1981; Pianta & Kalloniatis, 2000). In practice, however, there is often little difference between fits of exponentials and fits of overlapping line segments to recovery data either for rod (see figures in Lamb, 1981) or for cone (see figures in Pianta & Kalloniatis, 2000) vision. We could similarly fit monotonically falling pieces of the data in Figures 2 and 3 with overlapping line segments, but without some theoretical underpinnings such fits would be largely meaningless.

*light*adaptation rather than

*dark*adaptation measurements, and some are made at levels that cause significant photopigment bleaching (see Tables A1 and A2). Thus, the question of whether Δ

*I*is proportional to [X] or not may be moot, since nonlinearities introduced by the processes of light adaptation (e.g., Stockman et al., 2006) will affect thresholds in addition to any putative effects of [X]. As introduced above, such processes could include multiplicative adaptation, subtractive adaptation, and response compression. Multiplicative adaptation, for example, is complete within 50–100 ms (Hayhoe et al., 1987; Hayhoe et al., 1992), and thus is much faster than the time constants associated with X. Consequently, if X alters sensitivity

*prior*to the multiplicative stage, its effects will be partially compensated for by multiplicative adaptation. Our model suggests that

*if*[X] is indeed proportional to Δ

*I*, then these nonlinear adaptational processes can be approximated by a logarithmic transform. Thus, plotted on log-linear coordinates as in Figures 2 and 3, light adaptation follows the negative-exponential rise and exponential fall shown by the fitted curves in Figure 2. (It should be noted that the data in Figures 2 and 3 are inconsistent with linear versions of the same model in which thresholds rise rapidly but with a decelerating slope and then fall linearly.)

*after*multiplicative and perhaps subtractive processes of light adaptation. The logarithmic relation between [X] and Δ

*I*might then reflect a subsequent logarithmic compression. In terms of the sequential light adaptation model of Hayhoe, Benimoff, and Hood (1987) in which the order of adaptation processes is multiplicative, subtractive, and compressive, the effect of [X] would apply after the multiplicative and subtractive stages but before the compressive (saturating) nonlinearity. We favor the proposal that [X] acts after the multiplicative process (and perhaps after the subtractive process), since that order explains why the effect of [X] is not attenuated by multiplicative and subtractive stages.

*τ*

_{A}and

*τ*to vary across background illuminances and observers. This difference led to relatively unstable solutions with large standard errors, yet neither

_{X}*τ*

_{A}nor

*τ*varied systematically with background illuminance at the higher levels that produce the TT2 phenomenon. Indeed, we found that we could simplify the model by fixing

_{X}*τ*

_{A}and

*τ*across background illuminances and observers, and still obtain a good fit that only reduced the

_{X}*R*goodness-of-fit from 0.996 to 0.982 and doubled the

^{2}*SE*of the residuals from 0.005 to 0.010 log

_{10}threshold units. We infer therefore that the time constants

*τ*

_{A}and

*τ*are approximately independent of background illuminance. But, given that

_{X}*τ*

_{A}and

*τ*are independent of background illuminance, how might [X] relate to the rate of photopigment bleaching and the initial production of the active metarhodopsin II that initiates the transduction cascade?

_{X}*p*is the proportion of unbleached pigment,

*I*is the background illuminance, and

*I*is a constant that is equal to the background illuminance at which 50% of the pigment is bleached in the final steady state. Given that

_{0}*p*≈ 1 at background onset, the rate of change of

*p*at early times is approximately

*I*, for small

*t*. The solution of Equation 6 for

*p*is:

*I*of 4.30 log

_{0}_{10}Td, the time constants for JDM for background illuminances between 4.77 and 5.54 log

_{10}Td that show a clear TT2 effect should change from 30.4 to 6.53 s, and the time constants for AS for background illuminances between 4.75 and 5.58 log

_{10}Td that similarly show a clear TT2 effect they should change from 31.4s to 5.98 s. By contrast, the time constants for the lifetime of X are roughly constant across these illuminances at 3.09 and 7.73 s, which correspond to the time constants at the highest illuminances.

_{10}Δ

*I*. We additionally assume that it only reduces sensitivity if it exceeds the effect of other processes of light adaptation, which could be first-site or second-site processes. We find that these other unknown, slow processes can be summarized by another exponential decay with a fixed time constant, which we refer to as

*τ*

_{S}. The model is described by:

*c*

_{S}scales the slow exponential decay and the “max” function returns the greater of

_{10}photopic Td for JDM and at and above 4.75 log

_{10}photopic Td for AS.

*τ*

*,*

_{A}*τ*

_{X,}and

*τ*

_{S}across background illuminances and observers, and allowed only

*c*

_{X},

*c*

_{S,}and

*v*to vary. As for the first model, this model was simplified to a single exponential recovery at lower background levels to avoid instability in the fits. This was achieved by setting

*c*

_{X}in Equation (8) to 0, and was necessary only at the two lowest levels for JDM and at the three lowest for AS. The best-fitting parameters and their standard errors are given in Table A2 in the Appendix. The fit of the model suggests that

*τ*

*= 4.15 ± 0.62 s and*

_{A}*τ*= 6.79 ± 1.00 s and that

_{X}*τ*= 23.94 ± 1.42 s to account for slow overall recovery. Note that the order of

_{S}*τ*

*and*

_{A}*τ*is reversible (i.e., we do not know which of A→X or X→C is faster).

_{X}*R*goodness-of-fit of 0.995, an adjusted

^{2}*R*of 0.995, and a standard error of the estimate of the residuals of 0.005 log

^{2}_{10}threshold units. The lower panels of Figure 3 show the residuals. The model predictions for JDM are particularly good and characterize the delayed sensitivity loss for backgrounds from 4.27 to 5.54 log

_{10}photopic Td level remarkably well. The predictions for AS are also good, except for the 4.75 log

_{10}photopic Td level (green diamonds), where there are substantial deviations at shorter times. According to the fits, the peak sensitivity loss occurs at 5.2 s for both observers. (Note that we excluded the 100-ms points from the modified fit at the four lowest luminances for JDM and the three lowest for AS. These points distorted the fits, and we suspect that at low illuminances the immediate loss of sensitivity is due mainly to effects of masking operating over a window of approximately 100 ms rather than to the effects of bleaching or adaptation; see, for example, Bachmann & Francis, 2014.)

*τ*= 23.94 ± 1.42 s could reflect subtractive adaptation, which Hayhoe, Levin, and Koshel (1992) have reported to take up to 10–15 s to compete, but is more likely to reflect the hypothetical restoring force of the cone-opponent site postulated by Pugh and Mollon (1979) to explain the exponential recovery of S-cone sensitivity found in TT1 after the offset of yellow backgrounds of less than about 5 log

_{S}_{10}Td (see above). In fact, our estimate of

*τ*of 23.94 s falls nicely between the estimates of the time constant for two observers by Augenstein and Pugh (1977) of 15.45 and 27.57 s.

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