Recently, we reported measurements of heterochromatic flicker photometry (HFP) in 22 young observers, with stimuli that (nominally) modulated only L- and M-cones and were kept at (approximately) a constant multiple of detection threshold. These equiluminance settings were represented as the angle in the (L, M) cone contrast plane, with the *greenish* peak of the flicker in quadrant II and the *reddish* peak in quadrant IV; equiluminance settings were reported as the greenish angle. The mean equiluminance angle was 116.3° (an M:L cone contrast ratio of −2 at equiluminance), but individual differences in the settings were substantial, with the variation across individuals almost five times larger than the within-subject precision in the settings. In the present study we sought to determine the degree to which we could account for our observers’ HFP settings by plausible variations in the macular pigment optical density (MPOD), the lens pigment optical density (LPOD), the cone photopigment optical densities (PPOD), and serine/alanine polymorphism in L-cone opsin (λ_{max} shift). Most of the range of our measured equiluminance angles could be accounted for by these factors, although the largest two angles (smallest |ΔM/M: ΔL/L| ratio at equiluminance) could not. Individual differences in HFP have sometimes been taken to indicate variations in the ratio of L:M cone number; our results suggest that most of the individual differences in HFP might be equally well ascribed to physiological factors other than cone number. Simple linear models allow predictions of equiluminance angle, cone adapting level, and artifactual S-cone contrast from the values of the four factors considered here.

*greenish*peak of the flicker in quadrant II and the

*reddish*peak in quadrant IV; equiluminance settings were reported as the greenish angle (Figure 1a). In the present study we focus on the method that we found to be most precise and reliable, heterochromatic flicker photometry (HFP). In the HFP experiment, the grey background was set to 75 cd/m

^{2}at chromaticity coordinates (0.289, 0.315), and produced an M

_{adapt}/L

_{adapt}ratio of 0.86. The test field was a 2° × 2° square patch, sinusoidally flickering around gray (75 cd/m

^{2}) at 10.63 Hz. Observers set the equiluminance angle at two different contrast levels, a total of 20 HFP settings per observer divided over two sessions (four observers only completed one of the sessions).

_{HFP}= 116.3°

^{1}(Figure 1a). This equiluminance angle can be transformed to a luminosity function as the black curve in Figure 1b (see detailed discussion in He et al., 2020), which was obtained by summing the weighted L- and M-cone fundamentals. Consistent with prior results (e.g., Gibson & Tyndall, 1923; Sharpe, Stockman, Jagla, & Jägle, 2005; Sharpe et al., 2011), our observers showed substantial individual differences in their HFP settings, with observers having α

_{HFP}ranging between 99.3° and 146.2°, corresponding to M:L cone contrast ratios of −6.1 and −0.7. The variation across individuals was almost five times larger than the average within-subject precision in the settings.

_{max}shift)—are the focus of the present work. We aim to understand how these factors may have contributed to the individual differences observed in our study, specifically, and for other observers and stimuli more generally. To do this, we begin by making a provisional assumption—that all observers are identical postreceptorally (with regard to their HFP settings) and differ only in their cone fundamentals (due to plausible variations in MPOD, LPOD, PPOD, and L-λ

_{max}shift). We name this the “neural equiluminance constancy” assumption. This assumption means that if we model the response of the luminance mechanism for

*observer i*as a weighted sum of cone contrasts

_{1}/k

_{2}is the same for every observer, such that at equiluminance (zero luminance response)

*i*is used to indicate that the cone contrasts are based upon the observer's own cone fundamentals. The equiluminance angle in the second quadrant of observer's own (∆L/L, ∆M/M) plane is

_{1}/k

_{2}ratio) might reflect neural factors in the retina, lateral geniculate nucleus, or cortex, but a common interpretation of variations in HFP settings is an anatomical one: that individual differences in HFP are due to differences in the numbers of L- and M-cones (e.g., Brainard et al., 2000; Gunther & Dobkins, 2002; Kremers et al., 2000; Rushton & Baker, 1964). Our primary goal is to study how much of the variation across individuals can be accounted for by the neural equiluminance constancy assumption alone. This assumption may be interpreted as a purely computational convenience: it allows us to calculate the changes in equiluminance that can be achieved

*without*changes to k

_{1}/k

_{2}(

*any*equiluminance angle can be achieved by changing the k

_{1}/k

_{2}ratio); compare Bieber, Kraft, and Werner (1998), who fixed the L/M ratio for similar reasons. We are agnostic as to whether there might be changes in k

_{1}/k

_{2}in addition to changes in cone fundamentals, and also as to whether, if there are changes to k

_{1}/k

_{2}, they would reflect variation in cone number or in other neural factors. The implications of our modeling for understanding effects of relative cone number, and of possible interpretations of neural equiluminance constancy, are briefly discussed in the Conclusions section.

_{HFP}for our 22 observers is 116.3° (standard deviation = 12.4°). Using that mean angle in Equation 3 implies a relative cone contrast weighting of k

_{1}/k

_{2}= 2.03 ≈ 2, such that L-cones contribute twice as much as M-cones to the response underlying HFP. Specifying a particular k

_{1}/k

_{2}, along with the neural equiluminance constancy assumption, permits us to quantitatively estimate the contributions of the individual difference factors that determine the shapes of the cone fundamentals.

_{A}and observer O

_{Z}represent the maximum and minimum values of MPOD, LPOD, PPOD, and L-λ

_{max}shift, producing extreme equiluminance angles in different directions. The values of their

*prereceptoral factors*are represented as

*PRF*and

_{A}*PRF*in the first column, respectively. Similarly, the prereceptoral factors for the standard observer O

_{Z}_{SS}are represented by

*PRF*.

_{SS}*individual cone contrasts*(S-cone contrast is zero for Observer SS). The cone contrasts produced by the flicker at the HFP settings are indicated by the double-headed colored arrows; these all have −2:1 M:L in their individual cone contrasts (an application of the neural equiluminance constancy assumption). The RGB settings corresponding to that 2:1 ratio are shown as three dots in RGB space, in the yellow plane representing all the stimuli used in the experiment. The green arrow lies in the plane of the page because experimental stimuli produce zero S-cone contrast for the standard observer. This is not necessarily the case for nonstandard observers. Red and blue arrows are tilted out of the plane of the page to suggest the production of S-cone contrast by the RGB settings of the nonstandard observers. The double-headed arrows and dots are drawn to attempt to clearly illustrate the ideas and are not intended to accurately represent the cone contrasts or RGB settings (a more accurate rendition of the RGB plane is given in Figure A2).

_{i}are transformed to the standard LM cone contrast plane using the Stockman-Sharpe fundamentals, as done by He et al. (2020) in analyzing experimental results. The three example equiluminance angles are indicated by the red, green, and blue lines in quadrant II of the standard cone contrast LM plane at the far right. These

*LM*cone contrast angles are what we reported as equiluminance settings (He et al., 2020); in the current study we examine the influence of the preretinal factors on these settings. In addition, we analyze the unwanted S-cone contrasts produced for these observers by stimuli that were designed to have zero S-cone contrast for the

_{SS,i}*standard*observer.

*M*/

*M*)

_{i}/(Δ

*L*/

*L*)

_{i}= −2 for a given observer. The other plane in RGB is defined by (Δ

*S*/

*S*)

_{SS}= 0 for the standard observer, illustrated in yellow in Figure 2. The intersection of those two planes is a line; the RGBs along that line satisfy the neural equiluminance assumption for that observer as tested with the experimental stimuli.

*mean*or adapting level of cone responses, and these effects are not captured by contrasts. High values of both MPOD and LPOD will lower the average light level reaching the cones and thus reduce their adaptation level. On the other hand, increases in PPOD will, by increasing quantal catch, tend to have the opposite effect. We calculate the adaptation state for each cone type for all the model observers. To the degree that cone-independent (von Kries) adaptation applies to our conditions (likely a high degree, given the constant gray background of approximately 2.6 log Td), these changes in adaptation are not very important, but we report them for completeness in the Results section (see

*Adaptation state*section and Figure 11, and Equation A3).

_{max}) may be produced by cone opsin gene polymorphisms. The most common polymorphism is the substitution of alanine for serine at position 180 in the L-cone photopigment gene, which leads to a shift of approximately 3 nm for normal subjects (Neitz & Jacobs, 1990; Sharpe et al., 1998). The Stockman-Sharpe fundamentals, used as our standard, are based upon pooling observers with both the ala180 and ser180 polymorphisms. Therefore, the range we adopted for λ

_{max}shift of the L-cone fundamental was minus 1.5 nm (left-shifted) to plus 1.5 nm (right-shifted), such that the midpoint (no shift) corresponds to the Stockman-Sharpe L-cone average (545 nm). The differences between the extreme cases and the standard curve are shown in Figure 6. Less common polymorphisms are not considered here.

_{HFP}range accounted for by each factor

_{HFP}range that each factor alone can account for. With MPOD, LPOD, PPOD, and L-λ

_{max}shift in turn being the only source of variability, the ranges they can explain are 14.75°, 7.25°, 16.00°, and 7.25°, respectively. The top row of Figure 7 shows nine example α

_{HFP}produced by values of each factor within the selected range. The green cross represents the α

_{HFP}calculated from the standard value of the factor, and the blue and red dots represent the upper and lower bounds of the α

_{HFP}range. The other colored dots represent other intermediate α

_{HFP}. As the value chosen for the factor (density or λ

_{max}) increases, α

_{HFP}decreases for all factors. The change is nearly linear, with PPOD and MPOD having the steepest decreasing slopes. The bottom row of Figure 7 represents these ranges in the LM plane of the standard Stockman-Sharpe cone contrast space. In each panel, the standard observer is represented by the green central line. The red line represents the lower bound of the range the factor can explain, which has a smaller angle compared to the standard; whereas the blue line represents the upper bound, which corresponds to a larger angle. The angles falling within the range covered by the red and the blue lines represent the HFP settings this factor can account for. Figure 8 plots these ranges as bars, with the standard angle (α

_{HFP}= 116.3°) being represented as green dots. The two vertical axes indicate the angle in QII and the M:L cone contrast ratio at each tick mark. Changes in MPOD and PPOD account for larger changes in α

_{HFP}while LPOD and L-λ

_{max}shift produce relatively small changes. Note that only the two extreme values of L-λ

_{max}shift refer directly to the serine/alanine polymorphism; the intermediate values are calculated for completeness.

_{HFP}.

_{max}mimic the effects of increases in relative L-cone input (the M:L cone contrast ratio) into a luminance mechanism. The density increases reduce the observer's sensitivity to the blue and green primary lights relative to the red light; the increases (right shift) in L-cone λ

_{max}promotes the sensitivity to the red light. Both increases therefore cause the model observer to move in a deuteranopic direction (a deuteranope would set α

_{HFP}at 90°). Decreases in the values of the four factors from the standard have the opposite effects, moving the α

_{HFP}in a protanopic direction.

_{HFP}range accounted for by all factors

_{1}/k

_{2}ratio has to be altered (Equation 3). Of course, variations in the k

_{1}/k

_{2}ratio can produce any equiluminance angle, including the two most extreme that we observed. To produce the largest angle we observed (146.2°), while keeping the four factors at the minimum values of the ranges specified in Table 1, requires changing the k

_{1}/k

_{2}ratio from 2.0 to 1.185.

_{max}shift in our model is restricted to the serine/alanine L-cone polymorphism at position 180, so a left shift of 1.5 nm produces the largest possible equiluminance angle. However, if an observer had zero lens density (an extreme and hypothetical situation), the equiluminance angle would be 144.75° with other factors fixed at the lowest of the selected ranges. An implausibly low PPOD can produce angles as low as 151.25° in combination with the extreme values for the other factors shown in Figure 7. The lowest LPOD and PPOD, with the other two factors fixed at the lowest of the selected ranges, result in an equiluminance angle of 159.25° (Table 2, bottom row). This angle is the largest equiluminance angle (lowest absolute M:L cone contrast ratio) that can be produced after assuming a fixed k

_{1}/k

_{2}of 2. Thus, our model can produce the entire range of angles observed in He et al. (2020) by use of extreme, and perhaps implausible, values of LPOD and PPOD.

_{HFP}and the values of MPOD, PPOD, LPOD, and L-λ

_{max}shift (−1.5 to +1.5 nm) as

^{2}= 0.99). The linear model can account for angles from 87.95° to 132.01° using the chosen ranges of the variables, without computing cone fundamentals at all. The values of the coefficients depend on our assumed k

_{1}/k

_{2}= 2 ratio and our monitor primaries, but the general form is likely to be generalizable to other k

_{1}/k

_{2}ratios and primary stimuli. The degree of linearity between the combination of these four factors and the equiluminance angle is remarkable and surprising.

^{2}= 0.98). Note that L-λ

_{max}shift makes no contribution.

*Chromatic transforms and the neural equiluminance constancy assumption*section, variations in MPOD, LPOD, and PPOD can alter the mean quantal catch levels of the cones. These mean changes affect both the numerators and denominators of the cone contrasts, and so cancel out, mimicking the effects of cone-specific adaptation. However, substantial changes in cone adaptation do alter visual processing and, using the same 6561 model observers, we have calculated these changes for our primaries for all three cone types (see the Appendix for details). Example cone fundamentals are shown in the top row of Figure 11. The fundamentals are not normalized in this figure to illustrate the vertical shifts that produce the major changes in cone adaptation level. The calculated adapting levels themselves, produced by multiplying the primary lights with the cone fundamental and integrating, then adding the results for the three primaries (Equations. A3 and A5), are shown in the bottom row of Figure 11. The adapting levels are in arbitrary units; here they are scaled relative to the standard observer, which is shown as a black dot in each panel. For MPOD and LPOD, increasing the density moves the cone fundamentals down, especially for the S-cone fundamental, thus reducing total quantal catch, whereas higher PPOD shifts the cone fundamentals up and thus increases the quantal catch for all three cone types. These changes in cone adaptation state are generally small, with only the effect of macular pigment on S-cones being appreciable (approximately one log unit) over our chosen ranges.

_{max}, while being agnostic about whether individuals

*also*differ in L/M cone weights (k

_{1}/k

_{2}, Equation 1). Individual differences in equiluminance based upon HFP have often been interpreted as being due to differences in the L:M cone number in the retina (Brainard et al., 2000; Gunther & Dobkins, 2002; Kremers et al., 2000; Rushton & Baker, 1964), which in principle could alter k

_{1}/k

_{2}(Equation 1), although those cone weights might also reflect postretinal factors. Clearly, extreme variation in the relative numbers of L- and M-cones must affect the equiluminance setting; in the limit imposed by dichromacy, α

_{HFP}= 90° for a deuteranope and 180° for a protanope, and less extreme changes could also be reflected in k

_{1}/k

_{2}. In color normals, L:M cone number ratios measured by adaptive optics imaging or retinal densitometry agree well with electroretinogram (ERG) measurements (Brainard et al., 2000), and ERG and HFP estimates also provide similar ratios of L:M response (Gunther, Neitz, & Neitz, 2006; Kremers et al., 2000). Rushton and Baker (1964) reported a high correlation between the L/M ratios estimated by HFP and retinal densitometry (in observers who have extreme L- or M-cone sensitivities). They also found a more modest, but statistically significant, correlation between HFP and Rayleigh matches. This latter correlation is consistent with the neural equiluminance constancy assumption, since changes in cone fundamentals produce changes in Rayleigh matches. Rushton and Baker (1964) discounted the HFP/Rayleigh match correlation, focusing instead on the larger one between HFP and densitometry, but both effects were found.

_{HFP}from 91.5° to 134°, corresponding to measured cone contrast ratios of −38.19 to −1.04, without varying the relative amount of L- and M-cone input into the luminance mechanism (k

_{1}/k

_{2}, Equation 1). One implication of our results is that without independently estimating the levels of MPOD, PPOD, LPOD, and cone polymorphisms for individual subjects, it will be difficult to determine the relationship between relative cone numerosity and HFP settings. Similar but more limited conclusions were made by Bieber et al. (1998) based upon modeling and by Carroll, McMahon, Neitz, and Neitz (2000) based upon ERG evidence. In contrast, very recently, Lee et al. (2020) argued that L/M ratio could be estimated from equiluminance settings independently from LPOD and MPOD, when equiluminance is determined using three primary (matched to a fourth, standard) lights (Webster, personal communication, June 2021). Our results apply to the more common case where there are only three primary lights, and specifically to the case where L- and M-cones are modulated while attempting to keep the S-cones unmodulated. Under these circumstances, our results indicate that changes in cone weights cannot be distinguished from variations in MPOD, LPOD, PPOD, and L-cone λ

_{max}.

_{1}/k

_{2}ratio, despite there being variation in L:M cone numerosity. That constancy would likely result from a long-term, high-level adaptation (see Webster (2015) for a review) to the relative number of L- and M-cones in an individual's eye. Testing that idea would require estimating the low-level factors that determine the cone fundamentals.

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**p**(each of which varies between −1 and +1), and the vector of three cone contrasts

**c**(Eskew et al., 1999). For observer

*i,*

*N*maps the RGB stimuli to cone contrasts. It consists of the product of two 3 × 3 matrices,

_{i}**A**and

^{−1}**D**, corresponding to the denominator and numerator of the cone contrasts:

**A**represent the adaptation states of the three cones; these are plotted in the bottom row of Figure 11. When

**A**is inverted it produces the three denominators of the cone contrasts. Thus, the transformation in Equation (A2) is accomplished by having

**A**are set to 1.0, the same computation creates cone excitations (in arbitrary units) instead of cone contrasts. If instead these three diagonal elements were set to L, M, and S threshold values, the computation would be made in threshold units. Neither of these changes would have altered any of the conclusions of this study.

*L*(λ) is the quantal Stockman-Sharpe L-cone fundamental

_{i}*(L*, and similarly for the M- and S-cone fundamentals (

_{SS}(λ))*M*(λ)

_{SS}*and S*(λ)). For the other model observers, L-cone functions (as well as the corresponding M- and S-cone functions) were recalculated, starting from the Stockman-Sharpe absorbance curves and adding in the effects of L-cone λ

_{SS}_{max}shift, photopigment peak density, macular pigment density, and lens optical density (Brainard & Stockman, 2010). The changes in the physiological factors, therefore, produce changes in cone fundamentals. Each set of fundamentals renders a different

*N*for converting between RGB vectors and LMS cone contrast vectors.

_{i}*p*for the

*ith*observer, we have from Equation (A1)

*L*/

*L*)

_{SS}and (Δ

*M*/

*M*)

_{SS},

_{1}/k

_{2}= 2 (Equation 2), so the standard observer selects the RGB in that plane that produces a −2:1 ratio of M:L contrasts when calculated via the standard cone fundamentals. That computation is described next, for a nonstandard observer (but it is the same calculation that is used for the standard observer).

_{1}/k

_{2}= 2 (Equation 2) for

*all*observers (the neural equiluminance constancy assumption), and thus for each observer the ratio of M-to-L contrasts at equiluminance is −2 in terms of their own cone fundamentals. For a nonstandard observer Z (with cone fundamentals shown in Figure 2),

*L*/

*L*)

_{Z}and (Δ

*S*/

*S*)

_{Z}

_{max}shift, whose cone fundamentals are plotted in Figure 2 and whose results are plotted in blue in Figure 9.

*S*/

*S*= 0 plane for the standard observer, and the plane defined by (Δ

*M*/

*M*)/(Δ

*L*/

*L*) = −2 for a nonstandard observer.

*r*, and substituting that into Equation A10. The point at which that line meets the gamut limits (where

*max*(|

*r*|, |

*g*|, |

*b*|) = 1) represents the peak of the HFP setting at maximum contrast for observer

*Z*: that is the stimulus

**p**

_{Z}from those used in the experiment that produces an individual M:L contrast ratio of −2 (of course this would also be true had the blue plane been calculated using the standard cone fundamentals, using N

^{−1}

_{SS}in Equation A9). Calculating

**N**

_{Z}·

**p**

_{Z}(from Equation A9) provides the actual S-cone contrast—using individual cone fundamentals—for the nonstandard observer (as reported in the

*S-cone modulation*section of Results and Discussion). That RGB is then transformed into Stockman-Sharpe cone contrasts

*c*must have (Δ

_{i}*S*/

*S*)

_{SS}= 0, but it will not, in general, have a −2:1 M:L ratio; the analysis of the obtained ratios (and their corresponding angles in the Stockman-Sharpe LM plane) forms the major result of the study.