Vision in Observers With Enhanced S-Cone Syndrome: An Excess of S-Cones but Connected Mainly to Conventional S-Cone Pathways

Abstract

Purpose.: The effect of increased numbers of S-cone photoreceptors in enhanced S-cone syndrome (ESCS) was investigated psychophysically in six ESCS observers to understand more about relative cone sensitivities and postreceptoral organization.

Methods.: Measures of temporal sensitivity or delay were made: S- and L-cone temporal acuity (critical flicker fusion, or CFF), S-cone temporal contrast sensitivity, and S-cone delay.

Results.: ESCS observers showed uniform enhancements of S-cone CFF of between 0.85 and 6.25 Hz, but reductions in L-cone CFF. They also showed higher S-cone temporal contrast sensitivities at medium and high S-cone adaptation levels, with sensitivity functions that peaked near 7.5 Hz but fell off at lower and higher frequencies. In contrast, the mean normal function was flat at low frequencies and fell off only at high frequencies. The S-cone signal, as in the normal, is subject to large phase delays.

Conclusions.: We interpret the enhancements in CFF as increases in S-cone number in ESCS of between 1.39 and 11.32 times normal density (with a mean of 3.48). The peaked ESCS contrast-sensitivity functions are consistent with S-cone signal interactions that increase sensitivity at intermediate frequencies through constructive interference but decrease it at lower and higher frequencies through destructive interference. Measurements of S-cone delays relative to L- and M-cone signals show that the predominant S-cone signals in ESCS are negative and delayed as in normal observers, but reveal another faster, positive S-cone signal. This signal is also likely to be the cause of constructive and destructive interference in the contrast-sensitivity data of ESCS observers

Introduction

Enhanced S-cone syndrome (ESCS) is a rare inherited degenerative retinal disease named after the associated unusual gain in function—an increase in short-wavelength–sensitive (S) cone sensitivity. ¹ The syndrome is also characterized by severely reduced rod sensitivity (night blindness), foveal schisis and macular cysts, varying degree of visual acuity loss, and atypical ERGs that show little or no responses to dim rod (scotopic) stimuli, but have large, slow responses to brighter cone (photopic) stimuli. ^1–4 The photopic ERG was originally thought to be of rod origin, ^5–7 but spectral measurements have shown that it is dominated by S-cones with reduced contributions from long- and middle-wavelength–sensitive (L and M) cones. ^2,8,9 Psychophysical studies also show increased short-wavelength and decreased middle- and long-wavelength sensitivities consistent with S-cone enhancement and L- and M-cone impairment; moreover, the increased sensitivities on yellow adapting fields have an S-cone spectral sensitivity. ^2,3,10 Despite the reduction in L- and M-cone sensitivity, color vision in ESCS observers assessed by standard tests is usually normal ^1,3 ; but some deficits have been reported. ¹¹  

The increased S-cone sensitivity has been linked to a larger than normal number of S-cones in the retinae of ESCS patients. ^9,10,12 Hood et al. ⁹ estimated that S-cone ERG a-waves in ESCS affected individuals—which are 4 to 6 times bigger than normal—are consistent with as many as 75 times more S-cones than normal in the affected retinae. A modest improvement in S-cone acuity has also been reported, but only with high adaptation levels of “yellow” background light where the normal S-cone acuity falls but that of ESCS observers does not. ¹⁰ More direct evidence for a relative increase in the number of S-cones was provided by histological examination of a disordered ESCS retina of a 77-year-old woman, ¹² in which twice the normal number of cones was found, 92% of which were S-cones. A more recent study used adaptive optics imaging to attempt to visualize individual cones directly in vivo in three young adults with ESCS. ¹³ The authors found a disordered cone mosaic, but were unable to quantify either cone-cell density or type. 

The excess of S-cones can be related to a molecular defect. The gene NR2E3 codes for a photoreceptor-specific nuclear receptor NR2E3 ^12,14,15 that, acting in concert with CRX and NRL, is thought to promote the differentiation and survival of rod photoreceptors by regulating the transcription of rod- and cone-specific genes and, in particular, repressing the expression of cone-specific genes in rods that would otherwise favor an S-cone fate for the precursor cell. ^12,16–19 More than 30 different mutations of NR2E3 have been linked to ESCS, Goldmann-Favre syndrome, and clumped pigmentary retinal degeneration. ^20–23  

One goal of this work is to characterize the disorder more fully by measuring temporal acuity for S-cone and L-cone-detected flicker in ESCS-affected individuals and by measuring S-cone temporal contrast sensitivity functions. These S-cone measurements may reveal the postreceptoral organization of S-cone signals in ESCS observers and, in particular, whether the postreceptoral organization differs from normal. 

In normal observers, under conditions where L- and M-cone–detected flicker can be resolved up to 50 Hz, ^24,25 S-cone–detected flicker can be perceptually resolved up to only 18 to 28 Hz. ^26–28 (Under the conditions of our experiment, L-cone flicker can be resolved in normal observers up to a frequency of approximately 40 Hz; see Fig. 2.) These sensitivities depend on both receptoral and postreceptoral properties of the S-cone, and on the L- and M-cone pathways. Given that the S-cone photoreceptors are as fast as their L- and M-cone counterparts, ^9,29–31 the perceptual differences between the S- and L-/M-cone sensitivities in normal observers must be due to postreceptoral differences. ^32,33 And, indeed, under most conditions in normal observers, the S-cone signals seem to be confined to sluggish visual pathways with low-pass temporal frequency responses that carry chromatic information with little or no access to the faster pathways that carry luminance or intensity information. ^34–43 However, under some conditions of long-wavelength adaptation often used to isolate the S-cone response (such as those used here), S-cone signals make a delayed, negative contribution to fast luminance pathways, probably by way of some indirect connection, perhaps via horizontal cells. ^28,44–46  

Given the excess of S-cones in the ESCS retinae, it is conceivable that some S-cones may replace the direct L- and M-cone inputs into the luminance pathway. And, because the normal luminance pathway consists of fast, positive contributions from the L- and M- pathways, we might then expect to find evidence in individuals with ESCS for a faster S-cone temporal signal that also makes a positive contribution to luminance. If, instead, the normal postreceptoral organization be preserved, then the S-cone contribution in the ESCS observer would be delayed and negative just as it is in normal observers. Further, S-cone temporal sensitivity measures in ESCS observers might be expected to show modest improvements that are consistent with an increase in the numbers of S-cones, rather than large improvements indicative of a change to a faster pathway. 

Our results suggest that while the predominant S-cone pathway in ESCS corresponds to the same pathway found in normals, evidence for a second, faster pathway can be found in the temporal contrast-sensitivity and phase-delay data. 

Methods

Observers

The experimental group of observers comprised six individuals with ESCS. All had a history of night blindness, maculopathy, and relatively mild peripheral visual field loss. The availability of ESCS observers constrained how many of the experiments each could perform: All six ESCS observers participated in the S-cone critical flicker fusion (CFF) measurements, ES1 through ES4 participated in the L-cone CFF measurements, and only ES1 through ES3 participated in the S-cone temporal contrast-sensitivity and phase-delay measurements. The stimuli were presented to the observers' right eye, except for observers ES1 and ES6, who preferred to use their left eyes. A group of up to 12 adults with normal or corrected to normal visual acuity provided representative control data. All participants, including the ESCS individuals, had normal color vision as assessed by the Farnsworth-Munsell 100-hue test, and by red-green Rayleigh and blue-green Moreland anomaloscope matches. 

This study conformed to the standards set by the Declaration of Helsinki, and the procedures have been approved by local ethics committees at Moorfields Eye Hospital and at University College London. 

The sequence variants identified in NR2E3 in the six ESCS observers, their age at the time of testing, and their right and left eye acuities are given in Table 1. 

Table 1

 

The Age at Testing, Genotype, and Visual Acuities in the Left and Right Eyes for Observers ES1 to ES6

Observer	Age	Genotype	Right Eye, Left Eye Visual Acuity
ES1	37	IVS1-2A > C, p.E341K	6/24 OD, 6/36 OS
ES2	29	p.R311Q, p.L371W	6/9 OD, 6/9 OS
ES3	39	IVS1-2A > C, p.A256E	6/9 OD, 6/9 OS
ES4	32	Unknown	6/12 OD, 6/18 OS
ES5	28	IVS1-2A > C homozygous	6/9 OD, 6/12 OS
ES6	27	IVS1-3A > G homozygous	3/60 OD, 6/60 OS

Apparatus

We used a conventional Maxwellian-view optical system with a 2-mm entrance pupil illuminated by a 900-W Xenon arc lamp. Wavelengths were selected by the use of interference filters with full width at half-maximum bandwidths of between 7 and 11 nm (Ealing, Holliston, MA, or Oriel, Stratford, CT). The radiance of each beam could be controlled by the insertion of fixed neutral density filters (Oriel) or by the rotation of circular, variable neutral-density filters (Rolyn Optics, Covina, CA). Sinusoidal modulation was generated by pulse-width modulation of fast, liquid-crystal light shutters running at 400 Hz with rise and fall times faster than 50 μs (Displaytech, Carlsbad, CA), thus effectively producing rectangular pulses of variable width at a fixed frequency of 400 Hz. The pulse width was varied sinusoidally under computer control using programmable timers (DT2819; Data Translation, Marlborough, MA) to produce the sinusoidal stimuli at the desired visible frequencies and at signal modulations up to 92%. (Frequencies near the 400-Hz rectangular-pulse frequency and above were much too high to be resolved, so that observers saw only the sinusoidally varying stimuli produced by the variation of the pulse width.) 

The position of the observer's head was maintained by a dental wax impression fixed to a milling-machine head that could be moved in three dimensions to align the observer's pupil in the optical system. The system is described in full detail elsewhere. ⁴⁷  

Stimuli

Visual targets were centrally fixated, monochromatic, 4° diameter discs that flickered sinusoidally about a fixed mean radiance, R̄. The flickering waveform, A(t), was thus given by:  

 

where f is the frequency of the flicker (in Hz), t is the time (in seconds), θ is the phase, and the ripple ratio or “modulation,” m, is defined as the conventional Michelson contrast:  

 

 

I_max and I_min are the maximum and minimum radiances of the stimulus, respectively. The maximum target modulation that could be achieved was 92%. Fixation was always central. 

S-Cone Measurements.

A flickering 4° diameter target of 440 nm was presented in the center of a steady, 9° diameter, 620-nm background field. The radiance of the background was fixed at 11.41 log₁₀ quanta s⁻¹ deg⁻². This background selectively desensitizes the M- and L-cones, but has comparatively little direct effect on the S-cones. For normal observers, the background ensures that flicker detection is mediated by the S-cones up to a target radiance of approximately 10.0 log₁₀ quanta s⁻¹ deg⁻². ^28,31,48 Above 10.0 log₁₀ quanta s⁻¹ deg⁻², M-cones may also contribute to flicker detection in normal observers. In the ESCS observers, who have reduced L- and M-cone sensitivities (see Fig. 2), any M-cone contribution will occur at still higher radiances or not at all. 

Three types of S-cone measurement were made, each described in more details below: CFF, temporal contrast–sensitivity measurements, and phase delay measurements. 

L-Cone Measurements.

A flickering 4° diameter target of 650-nm wavelength was presented in the center of a 9°, 481-nm background field. The radiance of the 481-nm background was 8.26 log quanta s⁻¹ deg⁻² (1.39 log₁₀ photopic trolands or 2.53 log₁₀ scotopic trolands). The 650-nm target wavelength was chosen to favor detection by L-cones rather than rods or S-cones. The 480-nm background served to suppress the rods, but also selectively desensitized the M-cones. Consequently, these conditions isolate the L-cone response over most of the 6.5 to 11.5 log₁₀ quanta s⁻¹ deg⁻² range of target intensities. However, at the highest intensities, the M-cones are also likely to contribute to flicker detection; we were not concerned about the possibility of mixed M- and L-cone detection at those levels. Only CFF measurements were made for L-cone–detected flicker. 

Procedures

All observers light adapted to the background and target for 3 minutes before measurements began. They interacted with the computer controlling the experiment by means of an eight-button keypad, and received information and instructions via tones and a computer-controlled voice synthesizer. Each measurement was the average of at least three settings and the experiment was repeated two or three times, usually on separate days. For a few measurements, noted below, only one repeat of the measurements could be made. The visual stimulus, focused in the plane of the pupil, was the only visible light source for the observers in an otherwise dark room. The image of the source in the plane of the observers' pupils was always less than the minimal pupil size so that retinal illumination was not affected by pupil size. The method of adjustment was used in measuring the CFF, temporal contrast–sensitivity functions (TCSFs), and phase delays. 

Calibration

The radiant fluxes of test and background fields were measured at the plane of the observer's entrance pupil with a radiometer (UDT Instruments, San Diego, CA) that had been calibrated against a standard, traceable to the National Bureau of Standards. Neutral density filters, fixed and variable, were calibrated in situ for all test and field wavelengths used. Interference filters were calibrated in situ with a spectroradiometer (Gamma Scientific, San Diego, CA). All radiances are given as time-averaged values. 

Experiment I: Critical Flicker Fusion Measurements

In the first experiments, we measured the CFF for S-cone- and L-cone-detected flicker to gauge the temporal acuity limits and relative sensitivities of ESCS observers compared with normals. 

Methods

All six ESCS observers participated in the S-cone CFF measurements; ES1 through ES4 participated in the L-cone CFF measurements. 

For the S-cone measurements, a 440-nm target was presented in the center of steady, 620-nm background field of 11.41 log₁₀ quanta s⁻¹ deg⁻², and the target radiance was varied from 6.30 to 11.00 log₁₀ quanta s⁻¹ deg⁻² in steps of approximately 0.3 log₁₀ unit. For the L-cone measurements, a 650-nm target was presented in the center of a steady, 481-nm background field of 8.26 log quanta s⁻¹ deg⁻², and the target radiance was varied from 6.50 to 10.50 log₁₀ quanta s⁻¹ deg⁻² in steps of approximately 0.3 log₁₀ unit. 

At each target radiance, observers adjusted the flicker frequency (at the fixed maximum stimulus modulation of 92%) using the method of adjustment to find the frequency at which the flicker just disappeared. Observers were instructed to approach the CFF from both lower and higher frequencies. 

During a single run of the experiment, three settings were made at each radiance and averaged. The experimental runs were repeated on three separate occasions, except for ES5 (one occasion) and ES6 (two occasions). 

Results

S-Cone Critical Flicker Fusion Measurements

The six panels of Figure 1 show the S-cone CFF (temporal acuity) data for the six ESCS observers (ES1 through ES6, colored circles) plotted as a function of log₁₀ target radiance. For comparison, the mean CFF data for 12 normal control subjects are also plotted in each panel (dark-blue squares). The error bars where visible in all figures are ±1 SEM within observers for the ESCS measurements, and between observers for the normal measurements. The small open circles are shifted ESCS data, the significance of which will be discussed later. 

Figure 1

 

S-cone critical flicker fusion frequencies (Hz) measured on a 620-nm background of 11.41 log₁₀ quanta s⁻¹ deg⁻² plotted as a function of the log₁₀ mean radiance of the 440-nm flickering target. Data are plotted for each of the six ESCS observers in separate panels: ES1 (yellow circles), ES2 (red circles), ES3 (green circles), ES4 (orange circles), ES5 (violet circles) and ES6 (light-blue circles). Each panel also shows the mean data for 12 normal observers (dark-blue squares). The small open circles in each panel show the ESCS data below 9.0 log quanta s⁻¹ deg⁻² shifted vertically to align with the normal data using a least-squares fitting criterion. In all figures, the error bars are ±1 SEM within observers for the ESCS measurements, and between observers for the normal measurements. ES5 made only one set of measurements.

Figure 2

 

L-cone critical flicker fusion frequencies measured on a 481-nm background of 8.26 log₁₀ quanta s⁻¹ deg⁻² plotted as a function of the log₁₀ mean radiance of a 650-nm flickering target. Data are plotted for four ESCS observers in separate panels: ES1 (yellow circles), ES2 (red circles), ES3 (green circles), and ES4 (orange circles). Each panel also shows the mean data for 12 normal observers (dark-red squares). The error bars are ±1 SEM within observers for the ESCS measurements, and between observers for the normal measurements.

In the normal observer, S-cone CFF rises steadily from just above a radiance of 6.5 log₁₀ quanta s⁻¹ deg⁻² until approximately 9.0 log₁₀ quanta s⁻¹ deg⁻², after which it reaches a plateau and decreases slightly. This decrease may result partly from saturation of the S-cone signal, ³¹ but is also due, in part, to chromatically opponent interactions with the other cone types. ^31,49,50 The rise in CFF above approximately 9.9 log₁₀ quanta s⁻¹ deg⁻² in normal observers is due to flicker detection by M-cones (see Fig. 4 of Stockman and Plummer ³¹ ). 

Table 2

 

The Best-Fitting Parameters and Their Standard Errors and R² Values for the Model Given by Equations 3 and 4

	ES1	ES2	ES3
High
Delay, ms	37.49 ± 1.46
w	0.37 ± 0.12	0.24 ± 0.12	0.63 ± 0.16
s	0.25 ± 0.04	0.25 ± 0.03	0.51 ± 0.04
R2	0.924
Medium
Delay, ms	44.93 ± 1.18
w	0.36 ± 0.08	0.23 ± 0.07	0.43 ± 0.05
s	0.02 ± 0.02	0.30 ± 0.02	0.27 ± 0.02
R2	0.981
Low
Delay, ms	41.21*	−	−
w	0.38 ± 0.17	0*	0*
s	−0.23 ± 0.05	−0.06 ± 0.05	0.13 ± 0.04
R2	0.913

 Fitted to the S-cone modulation sensitivity data for ES1, ES2 and ES3 at the High, Medium and Low adaptation levels. See text for details.

Table 3

 

The Best-Fitting Parameters and Their Standard Errors and R² Values of the Model Given by Equations 3 and 6

	ES1	ES2	ES3
Delay, ms	66.66 ± 5.38	41.10 ± 3.19	44.60 ± 1.49
w	1.000 ± 0.18	0.00 ± 0.30	0.78 ± 0.15
R2	0.818	0.881	0.970

 Fitted to the S-cone phase delay data for ES1, ES2 and ES3 measured at the Medium adaptation level. See text for details.

Below approximately 9.0 log₁₀ quanta s⁻¹ deg⁻², the ESCS CFF functions have similar slopes to the mean normal function. Above 9.0 log₁₀ quanta s⁻¹ deg⁻², however, the shapes of the ESCS CFF functions show sizable individual differences, which we consider in the Discussion. 

L-Cone Critical Flicker Fusion Measurements

The four panels of Figure 2 show the mean L-cone critical flicker fusion frequencies for four ESCS observers (ES1 through ES4, colored circles), again plotted as a function of log₁₀ target radiance. For comparison, the mean CFF data for 12 normal control subjects are plotted in each panel (dark red squares). The error bars are ±1 SEM within observers for the ESCS measurements, and between observers for the normal measurements. 

In the normal observer, L-cone CFF rises steadily from just above 6.5 log₁₀ quanta s⁻¹ deg⁻² until approximately 9.0 log₁₀ quanta s⁻¹ deg⁻², after which the CFF approaches a plateau near 40 Hz. ^51,52 The L-cone CFF functions for the four ESCS observers (colored symbols) all show losses in CFF of varying degrees relative to that for normal observers. ES1 shows the greatest loss and ES2 the least. The slopes of the CFF functions are also variable. Compared with the slope of the normal CFF, those for ES2 and ES3 are shallower, whereas those for ES1 and (possibly) ES4 are similar to the normal slopes. The L-cone CFF functions for ES1, ES2, ES3, and ES4 reach 29, 35, 30, and 31 Hz, respectively, compared with 40 Hz for the normal function. 

General Discussion

The S- and L-cone CFF data for ESCS observers are broadly consistent with the work described in the Introduction and show both improvements in S-cone sensitivity and reductions in L- and M-cone sensitivities. ^2,3,8–10  

All ESCS observers show some improvements in S-cone CFF relative to the mean normal observer. This is a notable feature of ESCS, where despite the progressive retinal degeneration, there is a specific gain in visual function. Two of the observers (ES1 and ES6) show relatively small improvements in S-cone CFF. Although this might seem at odds with their diagnosis, S-cone sensitivity improvements can be greater at frequencies below the CFF, as the next experiment illustrates. For instance, although ES1 shows only a small improvement in S-cone CFF relative to the mean normal observer (see Fig. 1), ES1 shows clear improvements in contrast sensitivity between 7.5 and 20 Hz at high radiance levels (see Fig. 3). 

Figure 3

 

Log₁₀ S-cone modulation sensitivities measured using a sinusoidally flickering 440-nm target superimposed on a steady 620-nm background of 11.41 log₁₀ quanta s⁻¹ deg⁻² plotted as a function of temporal frequency (logarithmic axis). The top, middle, and bottom rows of panels show data for high, medium, and low 440-nm target radiances of 9.61, 8.73, and 7.40 log₁₀ quanta s⁻¹ deg⁻², respectively. The left-hand, middle, and right-hand columns show data for ES1 (yellow symbols), ES2 (red symbols), and ES3 (green symbols), respectively. The data for the ESCS observers are plotted as circles (high level), triangles (medium level), and inverted triangles (low level). In each panel, the mean normal data are plotted as dark-blue squares. The error bars are ±1 SEM within observers for the ESCS measurements, and between observers for the normal measurements. The red continuous lines show fits of a two-signal S-cone model described in the text.

Figure 4

 

S-cone versus L/M-cone phase delays (degrees, linear scale) measured between a flickering 440-nm target with a mean radiance of 8.73 log₁₀ quanta s⁻¹ deg⁻² and a flickering 610-nm target with a mean radiance of either 10.32 (ES1), 9.95 (ES2), or 10.04 (ES3) log₁₀ quanta s⁻¹ deg⁻². The phase delays (degrees) are plotted linearly as a function of temporal frequency (Hz) for ES1 (yellow circles), ES2 (red circles), and ES3 (green circles). The two flickering targets were superimposed on a 620-nm background of 11.41 log₁₀ quanta s⁻¹ deg⁻². In each panel, the mean normal S-cone phase delays are plotted as dark-blue diamonds. The continuous lines are versions of a model in which it is assumed that the S-cone phase delay can be accounted for by the interaction between a delayed, negative S-cone signal and a fast, positive one (see Equations 3 and 6). The green horizontal lines show the predictions if there were only a fast signal (with the same delay as the L-/M-cone signal). The blue line shows best-fitting versions of the model applied to the normal data in which the fast signal is set to zero (the best-fitting delay relative to L-/M-cone flicker is 39.6 ms). The red lines are best-fitting versions of the model fitted to each ESCS data set (see Table 3 for the best-fitting parameters). Lastly, the fine black lines are phase delay predictions based on the model fitted to the temporal contrast functions for the ESCS observers also measured at the medium level (see middle row of Fig. 3 and Table 2).

The shapes of the ESCS CFF functions show sizable individual differences above 9.0 log₁₀ quanta s⁻¹ deg⁻². Those for ES1, ES2, and ES5 reach a plateau at approximately 30 Hz—well above the 22-Hz plateau of the normal observers—but show none of the subsequent increase in performance attributed in normals to M-cone intrusion. In contrast, the function for ES6 shows a substantial loss of CFF above 9.5 log₁₀ quanta s⁻¹ deg⁻², which is difficult to interpret because of the variability in the measurements of ES6 at these high radiance levels. The data for ES4 and particularly ES3 continue to rise at high target radiances, perhaps because in normal observers, the interactions between S-cones and L- and M-cones that contributes to their 22-Hz plateau is weaker. We assume that the differences in CFF at radiances above 9.0 log₁₀ quanta s⁻¹ deg⁻² both among the ESCS observers and between them and the mean normal observer partially reflect individual differences in the strengths of signals from the L- and M-cones. In this region, the L- and M-cones are implicated in limiting the effectiveness of the S-cone signals ^31,49,50 ; and in normal observers, the M-cones take over flicker detection at the highest radiances. ³¹  

As can be seen in Figure 2, the L- and M-cone loss is greater for some ESCS observers than others. This implies, perhaps, that the influence of the L- and M-cones on the S-cone CFF at the highest levels is likely to be much less for ES1 than for ES2. 

Consistent with earlier measurements, ^2,8,9 all four of the ESCS observers who made L-cone CFF measurements show some loss in CFF sensitivity (see Fig. 3), but the extent and form of the loss varies among observers, making simple interpretation difficult. 

Experiment II: S-Cone Temporal Contrast Sensitivity Measurements

S-cone TCSFs were measured at three mean radiance levels to assess the temporal responses of the ESCS observers at frequencies below CFF, and to compare them with the normal response. 

Methods

Only ES1, ES2, and ES3 were available to participate in these experiments. Measurements were made at three time-averaged 440-nm target radiances: 7.40 (low), 8.73 (medium) and 9.61 (high) log₁₀ quanta s⁻¹ deg⁻². As for the CFF measurements, the 440-nm targets were presented in the center of steady, 620-nm background field with radiance fixed at 11.41 log₁₀ quanta s⁻¹ deg⁻². 

At each target frequency, observers adjusted the flicker contrast using the method of adjustment to find the contrast at which the flicker just disappeared. Observers were instructed to approach the threshold contrast from both below and above threshold. 

During a single run of the experiment at one of the mean target radiances, three threshold settings were made at each flicker frequency and then averaged. The experimental runs were repeated on two or three separate occasions, depending on observer availability. 

Results

The results are shown in the nine panels of Figure 3. Each panel illustrates the mean log₁₀ S-cone contrast threshold plotted as a function of frequency (logarithmic axis). The upper, middle, and bottom rows show data for the high (9.61 quanta s⁻¹ deg⁻²); medium (8.73 quanta s⁻¹ deg⁻²); and low (7.40 log₁₀ quanta s⁻¹ deg⁻²) target radiances, respectively. The left-hand, middle, and right-hand columns show data for ES1 (yellow symbols); ES2 (red symbols); and ES3 (green symbols), respectively. The dark-blue squares in each panel show the mean results of 12 normal observers. The error bars in all figures are ±1 SEM within observers for the ESCS measurements, and between observers for the normal measurements. In many cases, the error bars are smaller than the symbols. 

The mean normal data (dark-blue squares) are low-pass in shape at all radiances—that is they are constant at low temporal frequency and fall only at higher frequencies. In contrast, the data for ES1, ES2, and ES3 are low-pass only at the lowest target radiance; at the medium and high radiances, they are band-pass in shape, peaking in sensitivity near 7.5 Hz, and falling off at both lower and higher frequencies. At the two higher radiance levels, performance of the ESCS observers is as good as—and in 5 out of 6 cases, much better than—that of the normal observers. The model fits shown by the red continuous lines will be described in the General Discussion. 

Discussion

The temporal contrast–sensitivity data shown in Figure 3 provide new details about the ESCS phenotype. At the medium and high 440-nm target level, the contrast sensitivities for the ESCS observers, unlike those for normal observers, are clearly band-pass in shape, peaking at 7.5 Hz. Low-frequency attenuation is usually attributed to a sluggish or delayed surround antagonism, ^53,54 so one possibility is that higher S-cone density in ESCS observers results in greater S-cone surround antagonism between neighboring S-cones perhaps via horizontal, H2 cells, which have mixed inputs from all three cone types. ⁵⁵ The normal density of S-cones reaches approximately 2000 cells mm⁻² (approximately 8% of the cone population) within the area of the target (see Table 1 of Calkins ⁵⁶ ). Consequently, any increase in density will increase the number of S-cones likely to be contacted by each H2 horizontal cell, ^55,57,58 and thus will potentially increase the S-cone surround signal. 

However, the results in Figure 3 show that at the medium and high levels, the high-frequency slope on these double logarithmic coordinates is steeper for the ESCS observers than for the mean normal observer. A very sluggish surround signal is unlikely to cause such a steepening at high frequencies, which suggests that the interaction may be between the normal S-cone signal and a faster S-cone signal that destructively interferes not only at low frequencies, but also at high frequencies. A model, the predictions of which are shown by the red continuous lines in Figure 3, is presented in the General Discussion. 

We note that although the mean temporal contrast sensitivity functions (dark-blue squares, Fig. 3) are low-pass in shape, the shapes of underlying individual functions vary. In particular, two of the normal observers show low-frequency attenuation comparable with that found in the ESCS observers. We speculate that these observers have higher than average S-cone densities, so that their results are more similar to those of ESCS observers. 

Experiment III: S-Cone Phase Delay Measurements

Under intense long-wavelength adaptation, S-cone–detected flicker of the appropriate phase will cancel or null L-/M-cone–detected flicker, suggesting that under those conditions, the signals are transmitted by a common luminance pathway. ^28,46 This type of cancellation is known as “flicker-photometric cancellation.” Because cancellation between S-cone–detected flicker and L-/M-cone–detected flicker occurs, not only can the relative size of the S- and L-/M-cone responses be determined, but also the relative delay and sign of the S-cone signal within the common pathway. 

Methods

Only ES1, ES2, and ES3 were available to participate in these experiments. 

Phase delay measurements were made at the medium S-cone level between S-cone–detected flicker and L-/M-cone–detected flicker. The S-cone–detected flicker was generated by the fixed 440-nm 4° diameter target of 8.73 log₁₀ quanta s⁻¹ deg⁻² presented in the center of the steady, 620-nm background field of 11.41 log₁₀ quanta s⁻¹ deg⁻². The L-/M-cone–detected flicker was generated by a 4° diameter 610-nm target superimposed on the 440-nm target. Its radiance, viewed alone on the 620-nm background, was adjusted by the observer so that 15-Hz flicker at the maximum 92% modulation was just visible (this ensured that the 610-nm target was at the radiance necessary for the experiment, and was always much dimmer than the background). The 610-nm target radiances in log₁₀ quanta s⁻¹ deg⁻² were 10.32 for ES1, 9.95 for ES2, and 10.04 for ES3. 

The S-cone and the L-/M-cone targets were flickered at frequencies between 2.5 and 15 Hz in 2.5-Hz steps. The two targets were initially presented in opposite phase, but their relative phase and strength of modulation could be adjusted by the observer or by the experimenter to find the relative phase and modulation strength at which the flicker appeared nulled or had a clear minimum (i.e., when the summed flickering targets appeared to cancel each other). The initial adjustments of phase for the three ESCS observers were carried out by the experimenter, who was unaware of the absolute phase delay, but aware of the relative changes in phase that could be made by pressing various keys (steps of 2, 10, and 180°). By asking the observer whether he or she could see more flicker in one phase condition versus another, it was possible for the experimenter to quickly determine, within approximately 10°, the relative phase that produced the best null. Fine adjustments could then be made by the observer. This interactive approach proved successful and circumvented the need for much more extensive training, which was not feasible given the limited availability of the ESCS observers. 

A single interactive run of the experiment was carried out. 

Results

The colored circles in each of the three panels of Figure 4 show the S-cone phase delays (in degrees) for ES1, ES2, and ES3 (yellow, red, and green circles, respectively) plotted as a function of frequency (both axes linear). 

For comparison, the mean results for five normal observers (dark-blue diamonds) are plotted in each panel. The mean function for normal observers follows a roughly straight line, the extrapolation of which crosses the y-axis (0 Hz) at −180°. Note that the normal observers' error bars are fairly large, which reflects the variability in the underlying data. The significance of the solid red, green, and blue lines will be described later. 

The phase delay data for the three ESCS observers show some variability. The data for ES1 (yellow circles) lie substantially above the normal mean data, but those for ES2 (red circles) and ES3 (green circles) are closer. The differences between the phase delays for ES2 and the normal observers are relatively small, but the phase delays for ES3 are consistently above the normal function and have a different slope. 

Discussion

The mean normal S-cone phase delays are consistent with a time delay between the L/M-cone signals of Δt ms, and a signal inversion. Thus, the phase delay in Δθ (degrees) as a function of f (frequency in Hz) can be written:  

 

where the addition of 180° represents (and has the same effect as) signal inversion or inhibition. The blue lines fitted to the normal data are the best-fitting version of a pure delay model defined by Equation 3 for which Δt = 39.57 ± 0.59 ms (R² = 0.991). Thus, as expected, the phase delays of the normal observers are consistent with a time delay and sign inversion. ^28,44–46  

The deviations of the ESCS data from the normal data in Figure 4 potentially yield insights into differences between the normal and ESCS in the postreceptoral organization of the S-cone signal. A model, the predictions of which are shown by the red lines in each panel of Figure 4, is developed in the General Discussion that follows next. 

The fact that ESCS observers, like normal observers, can cancel 440- and 610-nm flicker on a 620-nm background suggests that against that background the S-cone and L/M-cone signals for the ESCS observers are also transmitted by a common pathway, the properties of which are consistent with those of the luminance channel. ²⁸  

General Discussion

The psychophysical results for the ESCS observers reveal clear differences between their data and the mean normal data. In this section, we endeavor by analysis and modeling to interpret the differences in terms of changes in cone density or in postreceptoral organization. 

Enhancements of S-Cone CFF and Increases in Cone Density

Up to approximately 9.0 log₁₀ quanta s⁻¹ deg⁻², the enhancements in S-cone CFF for the ESCS observers are approximately consistent with simple vertical shifts of the normal CFF function to higher frequencies (see Fig. 1). We can therefore use these shifts to quantify the relative improvement for each ESCS observer. Accordingly, we determined the vertical shift in CFF that minimized the squared differences between the ESCS and normal CFF functions below 9.0 log₁₀ quanta s⁻¹ deg⁻². The best-fitting shifts and ± the standard errors are 1.14 ± 0.33, 5.30 ± 0.27, 2.58 ± 0.22, 6.25 ± 0.25, 4.77 ± 0.12, and 0.85 ± 0.65 Hz for ES1 to ES6, respectively, with R² values of 0.974, 0.981, 0.989, 0.985, 0.997, and 0.902. The shifted ESCS data are shown in each panel of Figure 1 by the small open circles superimposed on the normal data. The sizes of the standard errors show that significant enhancements in CFF are found for five of the six ESCS observers. In the Appendix of this paper, we develop a metric that enables the shift in CFF to be translated into an estimate of the change in photoreceptor number. 

The analysis on which the metric is based, equates changes in cone number caused by changing target area with changes in cone number caused by photoreceptor gain or loss within a fixed target area (see Equation A4). Using this equation, we can estimate the factor by which the cone number increases that corresponds to the vertical shift in CFF that we find for each observer. The factors we infer are increases by 1.56, 7.83, 2.72, 11.32, 6.37, and 1.39 for ES1, ES2, ES3, ES4, ES5, and ES6, respectively. These factors are much less than the increase of 75 times estimated from ERG measurements, ⁹ but we emphasize that these approximations are dependent on several assumptions (see Appendix). 

S-Cone Temporal Contrast Threshold Measurements and Signal Interactions

The shapes of the S-cone ESCS contrast sensitivities (see Fig. 3) suggest that two S-cone signals interact in a way that lowers sensitivity at both high and low frequencies but raises it at intermediate frequencies. We model this interaction by assuming that there are two S-cone signals: a fast “center” signal, A_c (f), and a second inhibitory “surround” signal, A_s (f), that is delayed by Δt ms. Given a fixed time delay, the phase delay in Δθ (degrees) as a function of f (frequency in Hz) is given by Equation 3 above (but between two S-cone signals rather than between S-cone and L-/M-cone signals). 

The resultant signals produced by adding together A_c (f) and A_s (f) at the angles determined by their relative (frequency dependent) phase differences (Equation 3) is then:  

 

 

Figure 5 illustrates the vector addition of A_c (open arrow) and A_s (grey arrow) separated by a relative phase delay of Δθ(f) (red arc). Their addition yields the resultant vector A_r (black arrow), which has a phase delay of ϕ relative to A_c (green arc). (The component of A_s in the direction of A_c [A_scos(Δθ)] and the component at right angles to A_c [A_ssin(Δθ)] are shown as thin black lines.) 

Figure 5

 

Vector addition: the weighted “center” signal A_c (white arrow) and “surround” signal A_s (grey arrow), are added with a phase shift of Δθ (red arc) to produce the resultant vector A_r (black arrow) with phase shift of ϕ relative to the center signal (green arc). The components of the “surround” signal that are in phase and 90° out-of-phase with the “center” signal are also indicated.

The model defined by Equations 3 and 4 was fitted to the ESCS data of Figure 3. Various schemes were tried for determining the shapes of A_c (f) and A_s (f). Somewhat surprisingly, however, we found that at the medium and high levels we could simplify the model by assuming that both A_c (f) and A_s (f) were similar in shape to the mean normal function (blue squares, Fig. 3)—and that they differed only by a scaling constant (such that A_c [f] = wA_s [f], where w is the relative weight of the center signals, thus, the vector length A_c (f) in Fig. 5 and Equation 4 becomes wA_s [f]). The model fits based on this assumption are shown by the continuous red lines in Figure 3. An additional logarithmic scaling constant, s, that simply shifts the fitted functions up or down was also allowed in the fitting procedure. Since, in preliminary fits, the delay Δt was similar across ESCS observers at the high and medium levels, we constrained it to have the same value for all three observers at these levels. 

The fits at the low level were different. The data for ES2 and ES3 were similar in shape to the mean normal S-cone contrast sensitivity function without an additional second signal. Thus for them, w at the low level was fixed at 0. For ES1, a second signal was required but since the delay Δt was poorly constrained (and tended toward either very small or very high values), we fixed it at the mean of the best-fitting values of ES1 obtained at the high and medium levels. 

Best-fitting versions of the model were obtained using a standard nonlinear, least-squares curve-fitting algorithm (implemented in SigmaPlot, SPSS). The best-fitting parameters and their standard errors are given in Table 2, along with the R² values for the fits at each radiance level. The fits, shown by the red continuous lines in each panel of Figure 3, are good, having R² values of 0.913 or better. Fixed values are noted in Table 2 by the asterisks. The values of s capture the individual differences in frequency-independent overall sensitivity. 

The model and fits are intended to be largely illustrative. Although they clearly demonstrate that the ESCS contrast sensitivity data are consistent with an interaction between two similar S-cone signals of opposite sign, one of which is delayed relative to the other, the details of the model are less secure. For example, the simplification that the two signals have the same dependence on frequency is unlikely to be precisely correct. More sluggish signals are usually subject to some form of filtering that causes more attenuation at high frequencies. ⁵⁹ And, we find that slightly better fits can be obtained at the high and medium levels by filtering the slow signal at these levels with a single-stage low-pass filter with a corner frequency of approximately 6 and 11 Hz, respectively. 

Given that we assume that A_c (f) and A_s (f) are of the same shape, we cannot determine from the fits which of them is bigger since—in principle—because of symmetry, w could be applied either to A_s (f) or to A_c (f). In the model, we assume that A_c (f) is smaller than A_s (f) (so w <1) on the grounds that the S-cone phase delays (see Fig. 4 and the next section) suggest that the sign of the S-cone resultant signal is negative (which implies that when Δθ = 180, at 0 Hz, A_s [f] > A_c [f]). 

S-Cone Phase Delays and Signal Interactions

The analysis in the previous section suggests that two S-cone signals separated by a delay interact in the ESCS observer to produce the band-pass temporal contrast–sensitivity functions of Figure 3 at the medium and high levels. One of these signals is likely to be the S-cone signal also found in normals, which is delayed and inverted in sign (shown by the dark-blue diamonds, Fig. 4). Conceivably, the second signal could be more delayed than the normal signal, in which case it might be a surround signal, or it could be less delayed, in which case it might be a more direct S-cone signal that avoids the normal S-cone pathway. The green lines in Figure 4 illustrate the expected phase delays if there were only a fast signal with the same delay as the L/M-cone signal and this prediction clearly does not fit the data. 

We modeled the S-cone phase delay data by assuming, as before, that there are two S-cone signals, A_c(f) and A_s(f), of opposite sign and separated by a time delay of Δt ms (as in Equation 3). The phase delay of the resultant S-cone signal (see Fig. 5), ϕ(f), is:  

 

Again, if we assume that A_c(f) = wA_s(f), so that w = A_c(f)/A_s(f), Equation 5 simplifies to:  

 

 

The model defined by Equations 3 and 6 was fitted individually to the S-cone phase delays for each ESCS observer and the fits are shown as red lines in Figure 4. The best-fitting parameters and their standard errors are given in Table 3, along with the R² values. 

The fits for ES2 and ES3 are good. Moreover, the delays between the two S-cone signals for the two observers are comparable with the delay of 44.93 ms inferred from the temporal contrast sensitivity data at the same level (see Table 2); the values of w agree less well. The continuous black lines shown in Figure 4 are the phase predictions from the model used to account for the temporal contrast sensitivity functions, which are plausible for ES2 and ES3. The fits for ES1, however, agree much less well. Indeed, the requirement of the phase model—that the two S-cone signals should be equal in size and should therefore cancel each other when they are in opposite phase (at 0 and 15 Hz)—is clearly inconsistent with the temporal contrast sensitivity data of ES1, which show much less attenuation at those frequencies. We do not understand the cause of this discrepancy. Note that when w = 1, as in the best fit for ES1, Equation 6 simplifies to ϕ(f) = Δθ/2 (red line, upper panel, Fig. 4). 

Conclusions

On balance, the phase and contrast-sensitivity models suggest that two S-cone signals generate S-cone flicker in the ESCS observer under long-wavelength adaptation. The larger S-cone signal is comparable to the slow, inverted S-cone signal found in normal observers. The second, smaller S-cone signal reduces the S-cone phase delays (relative to L-/M-cone flicker), which suggests that it is a faster, positive signal more similar to the L-/M-cone signals against which the S-cone phase delays are measured. 

The existence of “normal” S-cone pathways in the ESCS observer is supported by other work. ESCS patients, for instance, are also affected by the phenomenon of transient tritanopia (the loss of S-cone sensitivity following the offset of a yellow field; see Mollon ⁶⁰ ), which is also consistent with a normal, spectrally opponent postreceptoral organization of S-cone signals. ³ Given that in the “normal” S-cone pathways the signals from S-cones are opposed by signals from M- and L-cones (for reviews, see Refs. 61 and 56), improvements in chromatic S-cone sensitivity in ESCS observers could be due in part to a reduction in the number of L- and M-cones. 

This evidence for a faster S-cone signal in ESCS both from the temporal contrast sensitivity measurements at the medium and high levels, and from the S-cone phase delay measurements does not address the issue of how such a fast signal arises. One possibility is that some of the abundant S-cones in ESCS observers displace the direct L- and M-cone inputs into the luminance pathway, thus giving rise to a fast, positive S-cone luminance signal. However, if S-cones do feed prominently into faster L- and M-cone pathways, then a faster response should be evident in the ERG. Yet, although the leading edge of the a-wave is similar to other cone responses, ⁹ the b-wave of the S-cone response is substantially delayed in these observers as it is in normal observers. ³ The normal leading edge of the a-wave yet anomalous b-wave are consistent with the S-cones in the central macula region feeding through the normal S-cone pathways, ⁴ although rod pathways may be involved in the periphery. ⁴ Indeed, one possibility is that the “fast” S-cone signals in our measurements are transmitted via sluggish rod pathways, which would explain why the relative amplitudes of the fast and slow S-cone signals required to model the temporal contrast-sensitivity functions are not more dissimilar. However, it would not explain their phase characteristics that, unlike rod signals, show relatively little delay (see Experiment III). 

We note that comparisons between the postreceptoral organization of the ESCS and normal retina may not be straightforward. Substantial rewiring can occur after photoreceptor degeneration, ^62,63 so that the faster S-cone signal in ESCS may be transmitted through a novel pathway that has no normal counterpart. 

If the S-cones in ESCS observers have access to normal L- and M-cone pathways, or even if there are simply more S-cone pathways to match the increase in S-cone number, then we should expect improvements in S-cone spatial contrast-sensitivity and spatial-acuity measures. Unfortunately, the most relevant evidence from Greenstein et al. ¹⁰ is equivocal at best, since it shows very modest improvements in spatial acuity. The maximum spatial acuities for their ESCS observers P2 and P3 reach approximately 6 cyc/deg, and for their observer P1 approximately 10 cyc/deg (see Fig. 5 of Greenstein et al. ¹⁰ ). By contrast, the maximum normal S-cone acuity reported by Humanski and Wilson ⁶⁴ is 4 cyc/deg, and as yet unpublished spatial contrast-sensitivity measurements from our laboratory suggest S-cone mediated spatial-acuity limits in normal observers can be as high as 7 to 10 cyc/deg. (These values should be contrasted to comparable L- and M-cone spatial acuity limits, which can be as high as 55 cyc/deg. ⁶⁵ ) 

Acknowledgments

The authors thank the ESCS observers who participated in this study. 

Supported by grants from Fight for Sight; the Biotechnology and Biological Sciences Research Council; the Engineering and Physical Sciences Research Council; the National Institute for Health Research Biomedical Research Centre at Moorfields Eye Hospital NHS Foundation Trust; the UCL Institute of Ophthalmology; a Foundation for Fighting Blindness Research Center grant for the Study of Retinal Degenerative Diseases (ARW, ATM); and a Foundation Fighting Blindness Career Development Award (MM). 

Disclosure: C. Ripamonti, None; J. Aboshiha, None; G.B. Henning, None; P.I. Sergouniotis, None; M. Michaelides, None; A.T. Moore, None; A.R. Webster, None; A. Stockman, None 

APPENDIX

The question we address here is how the shifts in S-cone CFF might be usefully related to increases or decreases in photoreceptor number. We start with a useful approximation for targets of between approximately 1° and 5° of visual angle in diameter first noted by Granit and Harper that CFF increases linearly with the logarithm of the target area. ⁶⁶ See Figure 1 of Kugelmass and Landis ⁶⁷ for a graphical summary of relevant results up to 1955. Here, we take advantage of the detailed set of CFF measurements made by Kugelmass and Landis ⁶⁷ as function of both target luminance and area, for which the Granit-Harper law is approximately obeyed for foveally fixated target diameters of between 3° and 7° of visual angle (see Fig. A1, upper panel). These form part of an extensive historical literature on CFF measurements (for reviews, see Refs. 68 and 69) that dominated research on temporal processing before linear systems theory and measurements of temporal contrast sensitivity of sinusoidally flickering stimuli became common in the late 50s and early 60s. ^37,70,71 More recent work that nicely links CFF to linear systems theory has been carried out in several studies by Tyler and Hamer. ^25,72–74  

Figure A1

 

Upper panel: The mean critical flicker fusion frequencies (Hz) for two observers (ES, SK) for luminance levels of 1.24 (blue diamonds), 1.64 (cyan inverted triangles), 2.44 (green squares), 3.04 (yellow triangles), and 3.64 (red circles) log phot. td. replotted from Table 1 of Kugelmass and Landis ⁶⁷ as a function of the logarithm of target area (log deg²) and vertically shifted to align with their mean between 2.98 and 7.10° deg of visual angle in diameter (i.e., aligned between the vertical red line labeled 2.98 and that labeled 7.10 and containing our 4° target diameter). Over this range, the data can be well approximated by a linear function, as shown by the black line, the equation for which is given by Equation A1. The dashed lines show the Equation A1 extended outside the fitted range. The troland values were converted from millilamberts by assuming a 2-mm pupil diameter. See text for other details. Bottom panel: The mean critical flicker fusion data from Kugelmass and Landis ⁶⁷ from 2.98° to 7.10° deg of visual angle in diameter from the upper panel replotted as a function of the logarithm of cone number. Symbols as in the upper panel. The shifted data can be approximated by the linear function (black line) given by Equation A4.

We relate the changes in area to the changes in the number of cones underlying the target by a theoretical analysis of data that links changes in CFF to changes in cone number. A similar approach that linked CFF to ganglion cell number over the whole retina was developed by Rovamo and Raninen. ⁷⁵ The upper panel of Figure A1 shows mean CFF (linear) as a function of the logarithm of target area (log₁₀ deg²) for targets between 1.27° and 14.60° in visual diameter from Table 1 (columns “for the average results of ES and SK”) of Kugelmass and Landis ⁶⁷ and plotted in the upper part of their Figure 3 as open symbols. Their targets were circular, centrally fixated, and the flicker waveform was square-wave. Data are shown for luminance levels of 1.24 (blue diamonds); 1.64 (cyan inverted triangles); 2.44 (green squares); and 3.64 (red circles) log photopic trolands (phot. td). (Converted from mL assuming a 2-mm pupil diameter.) 

The CFF data have been vertically aligned with their mean using the data between target diameters of 2.98° and 7.10° over four luminances: vertical shifts of 10.81, 6.55, −4.13, −10.05, and −13.23 Hz for the CFF data for 1.24, 1.34, 2.44, 3.04, and 3.64 log phot. td levels, respectively. The alignment minimized the squared differences between each data set and the mean. (The vertical red lines indicate target diameters of 2.98° and 7.10° between which the fit was made and 4.00°—the size used in our experiments.) 

The data between 2.98° to 7.10° of visual angle are seen to be differentiated only by a (vertical) shift in CFF so that the shape of the function relating CFF to retinal area is approximately independent of luminance level. The straight line (black line in the upper panel) fitted to the aligned data between 2.98° and 7.10° has the formula:  

 

where A is the area in deg². The R ² for the fit is 0.860. The dashed lines show the fitted line extended outside the fitted range.  

Next, we need to convert target area in deg ² to cone number. To make the conversion, we used the human cone-density estimates for temporal retina made by Curcio et al., ⁷⁶ which we have replotted in Figure A2 as yellow circles. Using a curve discovery program (TableCurve 2D, Jandel Scientific), we generated an arbitrary continuous exponential function that fit the Curcio et al. ⁷⁶ data from 0 to 60° with an R² of 0.998. The function fitted to the Curcio et al. ⁷⁶ data shown by the red line in Figure A2 was generated to provide a continuous function of cone density versus eccentricity. The function is:  

 

where the density is in cones per deg², ε is the eccentricity in deg, a = 9.570, b = 1.866, c = −6.504 × 10⁻⁴, d = 2.295 × 10⁻², f = 3.839 × 10⁻¹, g = −1.343 × 10⁻², and h = 4.464 × 10⁻³.  

Figure A2

 

Cone density (cones per deg²) plotted as a function of retinal eccentricity (degrees) for human temporal retina (yellow circles) taken from Figure 6 of Curcio et al. ⁷⁶ (their open squares). A continuous exponential function (red line, given by Equation A2) was fitted to the densities from 0° to 60° of eccentricity.

We then used this function to calculate the number of cones in successive annular rings with inner and outer diameters differing by 0.01° for inner diameters between 0 and 60° of eccentricity also in steps of 0.01°. The cone numbers for each of the particular targets used by Kugelmass and Landis ⁶⁷ were then calculated using the continuous exponential function by summing the cone numbers in all the annular rings that made up each target. These calculations enabled us to plot the CFF against cone number instead of the retinal area. The lower panel of Figure A1 shows the aligned CFF data from between 2.98 and 7.1° from the upper panel plotted as a function of the logarithm of the number of cones. (Tyler ⁷⁷ accounted for comparable cone density data from Oesterberg ⁷⁸ with a logarithmic slope of −0.667 for eccentricities between 0.1 and 15°. This function can also plausibly account for the Curcio data between 0.1 and 15°, but substantially overestimates cone densities below 0.1°; the densities below 0.1° are needed to estimate cone number for centrally fixated targets.) 

The aligned CFF data plotted in the lower panel can also be approximated by a straight line (black line), the formula for which is:  

 

where N is the number of cone photoreceptors. The R² for the fit is 0.861. Thus, for a change in CFF of ΔCFF Hz, the change in the ratio of the number of cones, r, is given by:  

 

 

Equation A4 is independent of the absolute CFF over the range 17 to 43 Hz and of the luminance level over the range 1.24 to 3.64 log phot. td, and provides an estimate of the relative change in cone density. The absolute cone densities can be calculated using Equation A4 for the target size used in our experiments. Consequently, for ESCS observers, for whom the vertical shifts in CFF increased by between 0.85 and 6.25 Hz (see above) we estimate the increase in S-cone number to be between 1.39 and 11.32 times normal. 

Equation A4 provides a useful indication of the approximate relative increase or decrease in photoreceptor number with CFF. However, we stress that the approximation is inevitably limited most obviously to the range of target diameters from approximately 3 to 7° over which the Granit-Harper law holds (see above). Implicit, too, in the approximation is the assumption that both the effective quantum catch and the temporal characteristics of the cones remains constant as their density varies between eccentricities of 1.59 and 3.55°. This assumption is likely to be only approximate, although the result that the cone photopigment optical density measured psychophysically is roughly constant over this range ⁷⁹ suggests some uniformity. 

Another useful approximation for accounting for CFF results is known as the Ferry-Porter “law,” which holds that for intermediate luminance levels, the growth of CFF is proportional to the logarithm of the luminance (that is, a plot of CFF versus log luminance should have a linear slope). ^80,81 Of relevance in this context are observations that the proportionality constant and thus the CFF increases as a function of eccentricity (under conditions where the same number of cones are stimulated at each eccentricity). ^25,74 This change is evident in the data shown in Figure A1, in which the slope of the underlying CFF versus target luminance functions must be shallower below a target diameter of 2.98° than between diameters of 2.98 and 7.10°. Obedience to the Granit-Harper law between 2.98 and 7.10° suggests, however, that the Ferry-Porter proportionality constant for these targets must be roughly constant. Nevertheless, we acknowledge that the variation of the Ferry-Porter slope with eccentricity may be a confounding factor. 

A complication in relating CFF to changes in cone number in clinical populations is that reductions in CFF can also arise when disease or mutations disrupt normal photoreceptor sensitivity regulation. However, since such photoreceptor impairment is likely to result in a change in the slope of the CFF versus luminance function rather than a simple vertical shift, it may be possible to some extent to disambiguate photoreceptor impairment from loss. Further discussion of photoreceptor sensitivity regulation and temporal sensitivity can be found in our earlier papers. ^82–84