Directional light-capture efficiency of the foveal and parafoveal photoreceptors at different luminance levels: an experimental and analytical study.

A gradual drop in visibility with obliquely incident light on retinal photoreceptors is namely described by the Stiles-Crawford effect of the first kind and characterized by a directionality parameter. Using a digital micromirror device in a uniaxial flicker system, here we report on variations of this effect with luminance levels, wavelengths within the visible and near-infrared spectrum and retinal regions ranging from the fovea to 7.5° parafoveal. Results show a consistent directionality in mesopic and photopic conditions. Higher directionality is measured for longer wavelengths, and a decrease with retinal eccentricity is observed. Results are discussed in relation to an absorption model for the visual pigments taking the outer-segment packing and thickness of the neural retina into account. Good correspondence is found without enforcing photoreceptor waveguiding.


Introduction
The efficiency of light to trigger vision when obliquely incident onto the retina is described by the psychophysical Stiles-Crawford effect of the first kind (SCE-I) [1]. The SCE-I is characterized monocularly using a subjective visibility comparison of retinal fields produced by two Maxwellian sources of light projected onto the eye's pupil: one near its center (reference) and another (test) at various locations across the pupil plane [1,2]. Two different methods have traditionally been used: (i) flickering of overlapping fields sequentially with incremental brightness [1,[3][4][5][6][7][8][9][10][11] and (ii) simultaneous comparison of adjacent or concentric fields [12][13][14][15]. In either case, the intensity of the test (I test ) and/or the reference (I ref ) field is adjusted until the brightness perception of the two is matched and an effective visibility, η, is determined as the intensity ratio at each pupil point of entry, r . This visibility function is fitted to a Gaussian distribution in the pupil plane [2,12]: where 0 r is the pupil position of maximum visibility and, ( ) ρ λ is the directionality parameter that quantifies the drop-off rate with increased obliqueness at a given wavelength, λ. In photopic conditions, the directionality measured at the fovea is typically ∼0.05 mm −2 but it decreases for eyes with large axial lengths [10,13] and varies slightly with wavelength being typically smallest in the green spectral range for foveal vision [12,14,15]. The SCE-I is mainly a cone effect being largely absent in scotopic conditions where rod vision dominates [5,7]. The peak location of the sensitivity curve determines the pointing direction of cones [2,8] that often has a nasal bias which is more pronounced for myopic eyes [13]. The transition from Maxwellian (point) to Newtonian (normal) view is nontrivial and underestimates the role of the SCE-I in normal vision by up to an order of magnitude as we have recently shown using a mechanical flickering pupil with diameter in the range of 1.4 to 7.4 mm diameter [16] and followingly confirmed by others using a small flickering aperture on a spatial light modulator [17].
Most SCE-I studies have been performed with foveal vision. Yet, the microscopic structure of the retina and the density of photoreceptors changes markedly towards the periphery. Westheimer [3] proposed to relate the variation in shape of the cones across the retina (i.e., rod-shaped at the fovea versus cone-shaped in the periphery) to an increase in directionality at eccentricities up to 3.75° in the temporal retina as determined with deep-red light (Wratten filter 29). Enoch and Hope [4] found similar results up to 10° using light-red light (Wratten filter 23A) but subsequently Bedell and Enoch [6], still using light-red light, reported a reduction in directionality to approximately foveal values at 35° eccentricity notwithstanding the very different shape of the parafoveal cones. In turn, scotopic vision shows a lack of directionality [5,7] despite of the fact that rods are similar in shape to foveal cones. Together, these findings show that the shape of individual photoreceptors does not provide a satisfactory explanation of the SCE-I neither in photopic nor scotopic conditions, but rather the density of visual pigments is a possible cause [16,18]. Also, it has been suggested that dynamical phototropism differences between rods and cones might be involved [19].
The SCE-I is accompanied by a related hue shift, viz., the Stiles-Crawford effect of the second kind (SCE-II), observed after brightness matching of the SCE-I [12]. Whilst the SCE-I has typically been explained by angular-dependent waveguide coupling [2,20], self-screening has detailed the SCE-II [21,22]. This incompatibility seems to suggest that waveguiding may play less of a role in vision than commonly assumed [16,18] although it may well be a relevant factor in high-resolution photoreceptor imaging [23].
The changes in SCE-I with retinal eccentricity is of relevance to understand whether it may play a role for emmetropization. Using light-red light, Choi et al. [13] found a gradual increase in directionality towards 15° eccentricity for emmetropes ( ± 0.75 D), moderate myopes (−3 to −5 D) and high myopes (<-5 D). Retinal thinning and elongation of outer segments have been reported for myopic eyes [24,25]. Thus, microscopic structural changes may plausible relate to changes in the SCE-I directionality and the density of visual pigments at different retinal locations.
Here, we report on SCE-I measurements from the fovea to the parafovea at up to 7.5° eccentricity using a uniaxial flicker system with a digital micromirror device (DMD) [10] at three different wavelengths within the visible spectrum and, to our knowledge, for the first time in the near-infrared. The dependence on retinal luminance is studied from scotopic to photopic conditions. All findings are discussed in relation to an absorption model for the visual pigments using a ray-optics model valid in the limit of weak absorption [16].
The paper is structured as follows: a thorough introduction to the volumetric absorption model is covered in section 2, followed by the experimental procedure performed to characterize the SCE-I at the fovea and parafovea with different brightness conditions and wavelengths, in section 3. Section 4 includes the directionality parameter results of the psychophysical SCE-I and its comparison to the modeled SCE-I plots. Discussion and conclusions of this study are found in sections 5 and 6, respectively.

Volumetric SCE-I modeling
The SCE-I is commonly attributed to waveguide properties of the photoreceptors resembling optical fibers without account for nonguided light [20]. However, discrepancies are inevitable due to low contrast of refractive indices within the cone-photoreceptors, the short length of inner segments (IS) and outer segments (OS), cellular irregularities, and the dense packing of photoreceptors that breaks the ideal conditions of isolated waveguides. Thus, the waveguide model cannot account for the highly diminished SCE-I in rods or for the appearance of the SCE-II. A geometrical-optics model of light absorbance based on the volumetric overlap between a beam of incident light and photoreceptor OS, was recently proposed by the authors [16]. This model accounts for the fragment of light that lies upon the OS, stimulating the visual sensation. The fraction of absorption for a single ray of light can be expressed in terms of Beer-Lambert's law and is proportional to the effective visibility [16]: where L is the total propagation distance across the OS with absorption coefficient , N is the molecular density, and abs σ is the molecular absorption cross section. To a first approximation Eq. (2) becomes linear, i.e., eff L η α ≈ . Thus, the effective visibility is determined as the intersecting volume of the OS and the incident light beam. As Maxwellian view is typically used to characterize the SCE-I, here, a collimated planewave illumination is modeled as a cylindrical beam of parallel light rays, and its intersection volume with an array of cone-photoreceptor OS is calculated for different angles of incidence rotating about the entrance to the OS (rotating about the OS midpoint, as representative of the average location of incident light, results in almost identical results [16]). The diameter of the light beam is chosen to be inversely proportional to the density of cones of a certain kind, as neighboring cones of the same type will have a screening effect on that being analyzed. A foveal cone density of σ = 160,000/mm 2 provides a hexagonal center-to-center spacing of 2.7 μm [26], to which a typical correspondence of 5% S-cones, 30% M-cones and 65% L-cones where broad absorption peaks centered at 420 nm, 534 nm and 564 nm respectively, are assumed [22]. Thus, lower parafoveal densities of cone OS are modeled with a wider beam of light. The length of the cone OS varies with retinal eccentricity with significant variations between reported values [20,27,28]. For the calculation of the OS and light intersection volume, we assume cylindrical OS length of 45 μm for the foveal cones and a 2 μm diameter [28]. In turn, for the parafoveal cones we assume an OS length of 35 μm with a slightly tapered conical reduction in diameter from 3 μm to 2 μm, and center-to-center spacing of 7.6 μm. A slight taper of the OS may partially account for the conical taper of the ellipsoid. For a schematic eye of axial length f AL = 22.2 mm, the angle of incidence of the light on the retina is given by , where r is the pupil entry location measured from the pupil center. The incident beam is angled between 10 ±° with respect to the axis of the OS, limited by a maximum pupil size of 8 mm. Figure 1 includes the normalized intersecting volume of the light beam with (a) a single OS and (b) a group of 3 identical OS in a hexagonal grouping for foveal cones. Figure 2 shows the same calculation for cylindrical and tapered conical OS of parafoveal cones with an assumed density of σ = 20,000/mm 2 .   the SCE-I wit ed at 550 nm h exit pupil of the logarithmic Figure 5 grap -I for the dif r coded scotop ny significant rease rapidly on is observed for y appears to pl ρ = 0.05 mm − na to that at ers [3,4] Secondly, fovea with wa subject. On av not vary signi higher directi for green ligh and 5). The g Thus, both M grouping of c reduction of t of the cone op (2) or a shorte This also resu green between also the diffe variations.
For the pa eccentricities Fig. 6 6. Plot of (a) the m subjects and (b) th -section of the pup correspond to ± 1 S the SCE-I For blue light nal increase is ght (Fig. 7(b)), wer retina, foll tionality param measured at the ality for rods, t w directionalit ed dependence hown in Fig. 6 Fig. 7(a)) seq observed, albe , a distinct 24% lowed by cons meters and their fovea for each the large group ty [16] similar e of the SCE-I (a) for the righ rameter at the f th blue or gree and near-IR lig 2] is only seen n the absorptio o the sensation on. As seen in At wavelength which may be n compared to t n of directiona due to differe r bars in Fig.   8 subjects were o 7.5° nasal in ectionality par is noted over t asal and tempor eimer's findin° eccentricity i to the foveal v noch [6], who f nality parameters a variation with retin . The asterisk (*) was measure quential directi eit with higher % increase is ob secutive 23 an r respective pe h wavelength i pings of adjace to the grouping directionality ht eye of 5 emm fovea is 0.044 en light. Howe ght. The comm n for 3 of the 6 on maxima of n produced cor n Fig. 1, group hs away from modeled by e the wavelength ality. The direc ent M-and L-c . 6(a) may ac e measured usi n steps of 2.5°. rameter decre the first step, ral, as well as ngs [3]. There increases, resp value. This dro found a lower d at the fovea agains nal eccentricity alo denotes myopic p ed both at 4 ionality reduct r variability be bserved again nd 24% reducti ercentages of re is included in T ent rhodopsin i g effect for con y parameter, ρ( metropes and ± 0.001 mm −2 ever, on averag mon dip in dire 6 subjects (sub M-and L-con rresponding to ping of OS res the absorption either a lower h of highest ab ctionality diffe cone densities count for som  Third, bas the experime centered, have angle of incid but, for simp modeled as cy from 3 μm to assuming a c according to t cones have be neighboring c

Discussio
The system u the fovea in dependence w this effect is photoreceptor in adjacent ro prediction as diminished. A and rod-photo the SCE-I ma region.
The wavel lower half of reported by L cones drive t distance from 8. SCE-I chara imental data and t receptors at a) th oveal retina with rs AC and BV, and 650 nm have be spond to ± 1 S.D. on used to charact good corresp with luminance virtually imp rs in scotopic c od OS [16]. T the density o A drastic increa oreceptors prod aintains a con length is seen t f the visible sp terize the psyc pondence to e at the pupil p perceptible, co conditions [5,7 The lower ρ-v of rods increa ase in direction duce the visual nstant contribut to not have a g pectrum but its . [14], and stab nsation produc causing the d parison between ots based on the vo μm length cylindr pered outer segme 0 nm wavelengths placed 0.5 and chophysical SC those reported plane (Fig. 5)  appreciation of the effect is seen when measured using green light ( Fig. 6(b)) and in terms of the volumetric absorption model this is explained by an increased contribution from both Mand L-cone opsins to the net absorption. In turn, when only a single cone type contributes to the sensation produced the directionality is expected to be higher due to a reduced screening by adjacent photoreceptors of a different cone class. The geometrical-optics volumetric absorption model [16] has shown to fit well to this data. On one hand, as the density of cones lessens at higher eccentricities, the diameter of the modeled cylindrical incident light gets larger, as explained in section 2. This causes the modeled SCE-I curves to widen resulting in a lower directionality. On the other hand, photoreceptor outer segments located in the fovea tend to have a longer OS structure, which translates into a steeper curve provided by the model. Both these contributions are in good agreement with the experimental data seen in Fig. 6(b) and Fig. 7, where lower directionality is found at higher eccentricities and complies with results reported by Morris et al. [9]. However, consistent differences between the modeled SCE-I curve and the experimental data noted nearer the edge of the pupil in Fig. 8, could be caused by light propagating from the modeled OS into its neighboring cones of a different kind due to the overlap of the absorption spectra of each type of photopsin. Inclusion of one more term in the series expansion of Eq.
(2) will reduce the predictions at the periphery of the modeled results further. For example, if measuring with 28 nm bandwidth red light centered at 650 nm, M-cones neighboring an Lcone can absorb a fraction of light according to the overlap of its absorption and incident light spectra. To best fit the model to experimental data, the addition of the percentages of each cone type exceeds 100%, which can also be explained by spectral overlaps between the photopsins absorbance and that of the incoming light. Finally, this effect may also account for the hue shift described by the SCE-II, as more oblique incidence will result in more light leakage reaching neighboring cones with a different spectral sensitivity. Thus, the SCE-II hue shift is absent when incident on axis of the photoreceptors but increases with obliqueness and towards both ends of the visible spectrum in agreement with measurements and expectations [12,22]. Taking the volumetric model one step further to account for partial absorption by other photoreceptors will be carried out in future studies.

Conclusions
Light capture efficiency with obliquely incident light on the retina is analyzed with a uniaxial SCE-I flicker system using a DMD to scan a 7 mm pupil with Maxwellian light and allow brightness matching via the duty cycle of the micromirrors. The SCE-I has been tested under luminance variations, confirming a minimum effect in scotopic vision, but maximized throughout the mesopic and scotopic luminance regions. A distinct variation with wavelength is reported, with similar SCE-I curves observed within the lower-half of the visible spectrum, and 22% higher directionality parameters for the upper-half and the near-IR region. The SCE-I has a highest impact at the fovea when measured with green light, where visibility is maximized, and a decrease with parafoveal eccentricity by up to 50% at 7.5° nasal. A similar variation is reported on average with blue light albeit higher variability between subjects. Slightly less of a drop in the directionality parameter with retinal eccentricity is seen with red light.
A volumetric absorption model [16] based on the geometrical overlap between incident light and the cone-photoreceptor outer-segments has proven to be adequate for the SCE-I visibility curves at foveal and parafoveal retinal regions, while accounting for conephotoreceptor density and OS lengths. Ultimately, though, we expect that a full electromagnetic model will provide a more detailed understanding [18,29].