The spectral sensitivities of the cones

Let there be three types of cone , which are most sensitive to long, medium, and short wavelengths respectively (they are sometimes called red, green, and blue cones). They have tuning curves , such that the average cone absorption of a single cone to a monochromatic light of intensity at wavelength is . If cones of type are excited by a uniform patch of light, then the essential quantities for determining input color, regardless of the spatial shape of the input patch, are the total responses from each of the three cone types. For the task of color discrimination, it is equivalent to view the cones of type collectively as a single giant cone with sensitivity , for this giant cone's sensitivity provides a sufficient statistic for the task (i.e., this sensitivity provides all the information relevant to the task) such that viewing individual cones separately does not provide any additional useful information for the task. The all-important ratios depend on both the relative densities and the relative sensitivities of the different cone types.

According to various experimental data on the responses from and light absorption by cones [10, 11, 12], for different cones should peak to the same peak value, if one ignores the pre-receptor absorption by the ocular media. We denote this normalized spectral sensitivity as , and will call it the cone fundamental. However, pre-receptor absorption of the input lights by the ocular media makes where is the pre-receptor absorption factor. Let , where is the wavelength where peaks; then should correspond to the behaviorally measured (normalized) cone fundamental, and for notation simplicity we still denote it as and thus . Meanwhile, assuming that does not change as quickly as with near , then where is the optical density of the pre-receptor ocular media at wavelength .

In our analysis later, we will include the cone density factor and use the notation . Furthermore, we normalize such that Max. Given these normalizations, the total photon absorptions of the cones will also scale with the size of the input light field (which determines the total number of cones for each cone type) and the effective input integration time by the viewing of the observers. These scale factors will be absorbed into the input intensity parameter , which also scales with the input radiance. We will see later that, given , the shape of the curve relating the discrimination threshold to wavelength is completely determined by the optimal decoding, and the parameter merely scales the threshold.

As our illustrative starting point, we approximate and . These numerical values arise from the following considerations. Firstly, various sources suggest that S cones are almost absent within 0.3 deg from the center of fovea but their contribution to the total cone density rises and peaks to 15% around 1 deg from the center[13] and approaches 7–10% in the periphery[13], [ 14]. Meanwhile, the Pokorny and Smith data[1] were from experiments using a centrally viewed disc containing the bipartite field of color inputs. We combine this information to assume that the S cones contribute 10% to all cones excited by the Pokorny and Smith stimuli. Secondly, various sources suggest that L cones are about twice as numerous as the M cones[14], we hence assume that L and M cones contribute 60% and 30%, respectively, of all the excited cones by the stimuli. This gives us . Thirdly, the optical density of the pre-receptor ocular media is almost constant in the medium and long wavelength region, giving , but rises with decreasing by 0.7 log units when nm[14], giving . Additionally, although the cone fundamentals from various literature sources are similar, we use those from Smith and Pokorny[15] (obtained from the CVRL website (http://www.cvrl.org) by Andrew Stockman), since we will be fitting their wavelength discrimination data[1]. Combining the considerations above gives as shown in Fig. 1. It turns out that these 's are not far from those by Vos and Walraven[16], who made where is the luminous efficiency function, a measure of the visual effectiveness of lights at different wavelengths for luminosity, normalized such that the maximum value of is 1, i.e., Max. The biggest discrepancy between the two sets of 's is that the S cone contribution is weaker in Vos and Walraven's composition[16] than in ours. This is not too surprising, as although the relative contributions by different cone types to luminosity perception are not necessarily the same as their relative contributions to color perception, they should be related or quite close to each other, except that S cones may contribute to the luminosity perception less than suggested by their density[17]. Our analysis and conclusions do not depend sensitively on our actual approximation for . We will later explore how our results vary quantitatively when we use other choices for the ratio . This ratio depends on cone densities and the optical density of the pre-receptor ocular media, which both vary substantially between observers (e.g., by up to one log unit in optical density[14]). This ratio also depends on the cone spectral sensitivities, which do not vary as substantially between observers but different literature sources provide slightly different quantitative values for them.

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Figure 1. Illustrations of noisy encoding of monochromatic inputs by the cone responses.

On the left is the cone spectral sensitivity (with , where s are derived from the Smith and Pokorny cone fundamentals[15], the cone density ratio is , the pre-receptor light transmission factors , and Max). A monochromatic input of wavelength evokes response from the three cones, L, M, and S. Due to input noise, there is a range of possible responses from this input. If the mean response to a monochromatic input of nearby wavelength is one of the typical responses within this range of responses to input , then it will be difficult to perceptually distinguish the input from input .

https://doi.org/10.1371/journal.pone.0019248.g001

Stochastic cone absorptions in response to monochromatic light

In this paper, we only consider monochromatic inputs. Hence, we describe our input stimulus by , a vector of two parameters, and , for the wavelength and intensity of the input light. The actual cone absorption for cone is stochastic following a Poisson distribution with a mean (1)

Sometimes we also call the response of the cone to the input light. The population response has the probability(2)

Fig. 1 shows how an input of particular wavelength could give rise to many possible responses in the three dimensional space near the mean response .

Maximum likelihood decoding

Given the responses , one can decode the input stimulus from the conditional probability (of given ) by finding the that makes maximum or large. So the most likely input to evoke is the one that maximizes . By Bayes's formula, we have where is the prior probability of input and . When the prior probability is constant so that it does not favour one over another, then varies with only through , i.e.,(3)

Therefore, the input for responses can be found by maximizing . As is also called the likelihood of given , decoding by maximizing is called maximum likelihood decoding. We will use this method to understand wavelength discrimination.

Decoding for input wavelength when input intensity is known and fixed

When input intensity is known and fixed, knowing the response enables us to estimate the input wavelength using maximum likelihood decoding. We call this the simple model of optimal input wavelength estimation, in the sense that we are not considering the variation of (as in experimental procedure of Pokorny and Smith[1]) in decoding. With a flat prior expectation that could be any value (within the visible light spectrum), the best estimate for the input is the one that maximizes the probability or equivalently its natural logarithm, ,(4)which we call the log likelihood.

The best estimate is the value of satisfying(5)

In a special case, if under input for all three cones (i.e., the response of each cone type is exactly equal to the mean absorption), then is the value satisfying the above equation. In general, there is no to make exactly for all three cones simultaneously, but one can still find a to satisfy the equation above. In any case, given an input wavelength , different responses will lead to different estimates ; most of them will be near to but not equal to the actual input wavelength . So if two different input wavelengths and are similar enough, the estimated wavelengths and may appear to be drawn from the same probability distribution. In such a case, these two input wavelengths would appear perceptually indiscriminable, or within the discrimination threshold; see Fig. 1.

With strong enough responses (effectively responses collected from enough cones and sufficiently many captured photons), it is known that the variance of these maximum likelihood decoded for a given input should approach[18](6)

where is the Fisher information defined as(7)

where denotes average of over . Since(8)

and , we have(9)

As , a larger Fisher information gives a smaller estimation error . This estimation error can be expressed as(10)

in which does not depend on intensity .

The estimation error is identified here as the discrimination threshold, as it characterizes the uncertainty of the perceived wavelength. Fig. 2 shows this threshold as a function of , together with the experimentally observed threshold from Pokorny and Smith[1]. Let and be the mean and the standard deviation of the wavelength discrimination thresholds of the four observers in Pokorny and Smith[1]. The input intensity in Fig. 2 is chosen as the one that minimizes the average square difference:(11)

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Figure 2. Wavelength discrimination assuming input intensity is fixed and known during color matching.

It is by maximum likelihood decoding of the cone responses using the simple model. The solid curve plots the discrimination threshold as a function of from the model. The data points with error bars are the mean and the standard deviation of the discrimination thresholds of the four observers of Pokorny and Smith[1]. In fitting the model to the data, is chosen such that the quantity is minimized.

https://doi.org/10.1371/journal.pone.0019248.g002

The that minimizes is the one that gives , leading to (since )(12)

One can see that the model prediction greatly underestimates the threshold for long wavelengths nm. Also, the peak location near 550 nm is not quite right. This best fit gives , indiciating that for most data points, the model predicts a threshold which departs from the data by more than a standard deviation of the data point.

The poor fit of the simple model arises because of the following. In Pokorny and Smith's experiment, observers adjusted the intensity of the comparison input field with wavelength to make it look as perceptually indistinguishable as possible from the standard input field which has input wavelength . This adjustment makes the comparison and standard input fields look indistinguishable until is too large, and the wavelength discrimination threshold is defined as the when this matching between the two fields starts to become impossible, so the comparison field is perceptually discriminable from the standard field no matter how observers adjust the intensity . If the observers somehow had the full knowledge of the intensities in both fields, they should in principle still be able to decode and thus discriminate the wavelength to roughly the same accuracy as predicted by the simple model when the intensity is held fixed and identical in the two fields. The reason the predictions overestimate the human accuracy is because one should not assume that the observers know the intensities , which also have to be decoded from the same sensory stimuli used to decode the wavelength. To explain the experimental data, our model should let be unknown and changeable rather than known and fixed. We call this the full model (rather than the simple model) of optimal wavelength estimation, and this model is explained next.

Sensory discrimination under perceptual confound – wavelength discrimination when input intensity is not fixed

Wavelength discrimination when input intensity is not fixed is just one example of a general problem of sensory discrimination under perceptual confound: sensory discrimination along one sensory feature dimension when neural responses are also affected by feature changes in another feature dimension. In the wavelength discrimination case, the two feature dimensions are input light wavelength and input intensity . Here, we formulate this problem in general, and it will be clear that our result in equation (20) is general and not specific to our example of monochromatic wavelength discrimination. Meanwhile, we will use our wavelength discrimination problem as an example to illustrate this general result.

Let the sensory input be , where and are feature values in the two feature dimensions, e.g., and . Let be the neural responses evoked by with probability . The maximum likelihood estimation of from can be arrived at by finding the solution to(13)(14)

The estimation error is(15)

This error depends on the specific response in each trial. Over many trials, these two dimensional errors have a covariance, generalizing from the simple 1-dimensional case above, given by(16)

where is the matrix inverse of the Fisher information matrix(17)

We note that, when is in our example, the matrix element is exactly the Fisher information we had in our simple model of wavelength discrimination.

Let the be the probability of obtaining the maximum likelihood estimate when the true input is . Since the estimation error has the covariance structure in equation (16), we can approximate as(18)

Note that this approximation makes the error have zero mean and gives the correct error covariance.

Now the threshold to discriminate while is not fixed is the largest value that can be obtained to maintain , i.e., to give(19)

Applying the above to the example of wavelength discrimination, the threshold for wavelength discrimination while is not fixed is the largest value that can be obtained to maintain , a particular example of equation (19). This can be illustrated in Fig. 3. This figure shows the contour plot of the posterior probability . This probability peaks at the origin of the coordinates in this plot. As deviation of from increases, the probability decreases, as indicated by the contours of probabilities, with larger, darker, contours indicating smaller probabilities. When , the largest to make is , the color discrimination threshold in the simple model and indicated by in Fig. 3. If , then the largest wavelength deviation on the contour should be larger, as indicated in the figure. This condition of means that the decoding system assumes that can be different from the default , i.e., the intensity of the comparison field can be different from the intensity of the standard field in the color matching.

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Figure 3. Illustration of 2D decoding in the full model.

Given the true input , is the estimated input parameters. This plot illustrates the conditional probability , since a given may evoke different responses leading to different . The wavelength discrimination threshold when is allowed to deviate from is larger than otherwise.

https://doi.org/10.1371/journal.pone.0019248.g003

We can show (detailed derivation in the next subsection after equation (28)) that the discrimination threshold for feature when input feature is not fixed (e.g., wavelength discrimination threshold at wavelength when input intensity is not fixed) is(20)

In particular, in our wavelength discrimination problem is(21)

Since we have(22)(23)

and(24)(25)(26)

then, given , we have(27)

Plugging the above into equation (20) we have(28)

Again, this threshold can be writen as . This predicts precisely how wavelength discrimination threshold should vary with wavelength , and that it should scale with as in the simple model. Like the simple model, the full model only has one free parameter, .

Mathematical proof of equation (20)

For matrix , let us denote its normalized eigenvectors as and , with corresponding eigenvalues and . Note that the two eigenvectors and are orthogonal to each other, since is a symmetric matrix, so any 2 dimensional vector (where the superscript denotes transpose) can be expanded in their basis as with coefficients and respectively. Then due to the invariance of this quantity to the bases used. Note that since is positive definite, and . Defining , we have(29)

Analogous to the 1-d case, we find the discrimination threshold by looking at the vs. curve such that , and find the largest on this curve, and this largest should be the discrmination threshold.

One can always find a parameter (see Fig. 3), such that the eigenvectors are(30)

One notes that the dot product . Then we have(31)

From these we can solve for in terms of and as(32)

The values of and on the curve can be described by a parameter such that(33)

Hence, we can write as a function of as(34)

The largest is when , giving(35)

The above is satisfied when(36)and(37)and since , and ; then we have(38)

Plug equation (36) to equation (34), writing for this extreme (the discrimination threshold) when , we have(39)or(40)(41)(42)

Noting that, as properties of eigenvectors , and ,(43)we have, equating on the right hand side of the equation to that in the left hand side(44)

Also noting that , the determinant of the matrix, we have(45)

The wavelength-intensity confound and the divergence of threshold near the red and blue ends of the spectrum

The thresholds predicted by the full and simple models differ most towards the red and blue ends of the spectrum. This is because only one cone type can be substantially activated at the spectrum ends, making the system practically color blind, just like in scotopic vision when only the rods are active. For example, in the red end of the spectrum when the M and S cones are almost silent, an increase in , i.e., , weakens the L cone response , i.e, . The simple model uses for wavelength discrimination by attributing it to with the relationship . The full model however sees this as equally attributable to a reduced input , i.e., , with , making it hard to distinguish whether the input gets redder or darker. This wavelength-intensity confound for the same makes wavelength discrimination difficult. In the procedure of the Pokorny and Smith experiment[1], it means that an increase in can be easily compensated by an increase in , making the threshold large.

The wavelength-intensity confound is present generally even when all cone types are substantially activated. Let , , and , with , be the preferred wavelengths of the L, M, and S cones respectively. This confound is stronger when or , when the predictions from the simple and full models differ most (see Fig. 4. In these wavelength regions, a change causes response change , which either simultaneously increases or simultaneously decreases the responses from all cone types, just like the response change caused by an intensity change . Although a slightly changes the ratio while a does not, the difference between the caused by and the caused by could be submerged under noise such that the two causes are perceptually indistinguishable.

This confound is weaker when , when a wavelength change will raise responses from some cone types while lowering responses from other cone types. In this case, a cannot be easily compensated for by an , which raises or lowers the responses from all cone types simultaneously. Hence, the simple and full model predict similar thresholds, particularly when is in between the preferred wavelengths of the two most numerous cone types, L and M. For nm, the S cones are still insensitive, while both the L and M cones prefer larger , and the confound is again significant, causing a substantial difference in the predicted thresholds from the simple and the full models. This is because a increases or decreases the responses from the L and M cones simultaneously (while affecting the S cone response relatively little), and can be easily compensated for by a .

Implications of the wavelength-intensity confound on the experimental procedures and on the stability of the threshold measurements

The wavelength-intensity confound, especially when , means that there can be problems with some experimental methods used to measure wavelength discrimination threshold. In many such experiments (e.g., [19], [ 20]), the procedure requires adjusting the intensity of the comparison field such that the brightnesses of the two fields match. The confound means that, when observers see a difference between the two fields, it is not easy to tell whether it is a brightness difference or a hue difference. This is a known difficulty noted in the accompanying discussions of Wright and Pitt's paper by fellow color vision scientists (pages 469–473 of [20]). Supposedly, the threshold is the smallest wavelength difference between the two fields when observers deem the two fields to differ in hue but not in brightness. However, whether the observers judge some perceptual difference to be a brightness or hue difference is likely to be dependent on the following factors: observers' internal criteria based on their expectations or biases, specific task instructions given by the experimenters, and perhaps even the visual environment around the experimental set up. These factors cannot be predicted straightforwardly from our optimal decoding theory, and could also cause variabilities between data from different observers and from different laboratories.

The procedure used by Pokorny and Smith[1] differs from the procedure above. They ask the observers to adjust the intensity of the comparison field until the two fields match in both hue and brightness, and the threshold is the smallest wavelength difference when this match is impossible by any intensity adjustment. This procedure does not require observers to decide whether any perceptual difference is due to brightness or hue, as they simply need to judge whether the two fields differ or not. This makes the threshold data more stable. Therefore, we do not intend to compare our theoretical prediction with data other than those by Pokorny and Smith[1].

The effect of the cone densities and pre-receptor light transmission on wavelength discrimination

It is clear from the analysis that the discrimination threshold depends on the relative sensitivities for different cone types . Since our scales with the relative cone density and the relative pre-receptor transmission factor for each cone , and should affect discrimination. We remind ourselves that the cone fundamental for all cones have the same peak value Max, and we have the normalization Max. Let us denote by , which could be understood as the effective cone density for cone type . We can rewrite the threshold in equation (28) as(46)

So at any particular wavelength scales roughly with for the cone type that dominates at . For example, increasing the fraction of the L cones among all cones would relatively lower the discrimination threshold near the red end of the spectrum, and increasing light absorption by the pre-receptor ocular media near the short wavelength region would decrease and thus raise the threshold near the blue end of the spectrum.

Fig. 5A–C shows the predictions using Smith and Pokorny cone fundamentals[15] but with different settings for . Fig. 5A is a replot of Fig. 4 with different scales on the axes. Its arises from our estimated and from experimental data[13], [ 14]. We note that its worst predictions are near wavelength nm, which is in the region where S cones' sensitivity has large slopes and hence a high sensitivity to wavelength changes. Fig. 5B has which could be seen as a situation when all cones have the same density and pre-receptor optical transmission. It raises the relative density for the S cones way over the physiological reality, and slightly raises the relative density of the M cones over the L cones. Consequently, it vastly over-estimates the discrimination sensitivities near the region nm, in the domain of the S and M cone contribution. As a result, it gives a that is substantially worse than the in Fig. 5A. Fig. 5C has a ratio that minimizes , such that the predicted thresholds best agree with experimental data. This ratio is obtained by exhaustively searching all integer values of with held fixed. With , almost all the data points are within a standard deviation from the predicted values. Compared with Fig. 5A, Fig. 5C raises the weights for the S cones (and slightly for the M cones), but not as dramatically as Fig. 5B does. Hence, it corrects the worst predictions in Fig. 5A near the nm region without overshooting the correction.

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Figure 5. Variations of the model predictions due to variations in the cone fundamentals, cone densities, and pre-receptor transmission.

The is normalized to the same peak value Max, the cone factor combines the cone density and pre-receptor transmission factor , to determine the cone sensitivity , with normalizations Max. Each plot is like Fig. 4A, having a full model predicted threshold with an optimal . Each is labeled with the literature source for and the used. A–C have the Smith and Pokorny cone fundamentals[15] with different . A is a modified plot of Fig. 4A. C–F show the best predictions (the that minimizes ) for four different cone fundamentals. Only integer values of , , and are used ( in all cases).

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Fig. 5D–F show the best predicted thresholds like Fig. 5C by three other cone fundamentals obtained from different sources in the literature: [21], [16], and [22] (see Andrew Stockman's webpage http://www.cvrl.org/). Compared with the predictions when using the Smith and Pokorny cone fundamentals[15] in Fig. 5C, their best predicted ratios are similar, and so are their goodness of fit , , and , which are only slightly worse than in Fig. 5C. This finding is not so surprising, as the cone fundamentals from different literature sources are similar to each other. Meanwhile, it may not be a coincidence that the cone fundamentals of Smith and Pokorny[15] best fitted the wavelength discrimination data obtained by them. It is likely that different researchers have different research styles and experimental procedures and hence different sets of experimental data obtained by the same style are more likely to be consistent with each other.

The importance of the S cone minority

Experimental data for wavelength discrimination for nm are scarse and very variable. These may be caused by many factors, including the large inter-subject variabilities (e.g., in cone densities and optical density of the ocular media) in that wavelength region, the difficulties of delivering stimulus in the short wavelength region, where light absorption by ocular media is dramatic[14], and, as discussed above, the wavelength-intensity confound makes some experimental procedures problematic in that wavelength region. However, Bedford & Wyszecki[19] reported that, as threshold rises with decreasing below nm, it dips again around nm before rising sharply. Wright and Pitt reported in 1934[20] a much shallower dip at a slightly larger nm. As we argued, a perceptual confound between wavelength and intensity for , the most preferred wavelength by S cones, should make threshold rise continuously with decreasing as all three cone types become less and less sensitive. So it may seem puzzling how this dip could arise from our full model, which shows a continuous rise of the threshold as decreases. Bedford and Wyszecki[19] acknowledged and discussed that the presence of this dip was controversial experimentally. In fact, a dip in the very long wavelength region was also seen by earlier studies and was then invalidated by later studies[20], and is no longer seen in modern day data[19], [ 1].

We suggest that the extra dip near nm may be the side effect of an extra peak in threshold at nm caused by too few blue cones involved in some experiments. We note that Bedford and Wyszecki[19] used input bipartite fields that were or smaller. This is smaller than the input field used by Pokorny and Smith[1]. As the density of S cones drops drastically to zero within from the center of the fovea[14], there are fewer S cones involved if the central viewing color matching fields are smaller than . (Note that observers in Bedford and Wyszecki's experiment[19] used free viewing for their task. We consider such free viewing in this attention demanding task as central viewing since gaze follows attention mandatorily in free viewing[23]). If there are no S cones, wavelength discrimination relies on L and M cones only. A close examination of the L and M cone spectral sensitivities reveals that, in a small region of around nm, with a scale factor that is almost constant within that region. This means, as changes in that region, the responses of the L and M cones co-vary almost completely (except for noise) so that they act together as if a single rather than two different cone types. This makes the L+M dichromatic system almost color blind in that local wavelength region, and consequently the discrimination threshold shoots up. This covariance of the two cone types can be seen in the signature , and we can define a degree of co-variance as(47)Mathematically, the 2x2 Fisher information matrix reduces its rank to 1 when both cones have their scale with each other, and thus the two dimensional wavelength-intensity input space is collapsed into one by the two redundant cone types acting as one. Fig. 6 illustrates how this Degree of Co-variance between the L and M cones shoots up near , thus giving a peak in threshold around that wavelength when there are too few S cones. The exact location of the peak depends slightly on the cone fundamentals used, whether it is the[15] cone fundamentals or other cone fundamentals, but this difference is not big. This rise in threshold around nm can be prevented by having sufficiently many S cones to remove the collapse of dimensionality. The dramatically worse discriminability at nm with smaller color matching field sizes or in tritanopic dichromats (who lack S cones) has been observed in previous studies ([24], [ 25].

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Figure 6. Illustration of how reducing the density of S cones should create a threshold peak near nm.

Because the L and M cones have their spectral sensitivity co-vary with each other as varies near nm, they act as if they are a single cone type around that . As threshold eventually increases when approaches nm, this local threshold peak at 460 nm creates a threshold dip between and nm.

https://doi.org/10.1371/journal.pone.0019248.g006