Packing Geometry of Human Cone Photoreceptors: variations with eccentricity and evidence for local anisotropy. K.R. Sloan and C.A.

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1 Packing Geometry of Human Cone Photoreceptors: variations with eccentricity and evidence for local anisotropy K.R. loan and C.A. Curcio University of Alabama at Birmingham Birmingham, AL, UA ABTRACT Disorder in the packing geometry of the human cone mosaic is believed to help alleviate spatial aliasing effects. In order to characterize cone packing geometry we gathered positions of cone inner segments at 7 locations along 4 primary and 2 oblique meridians in an adult human retina. We generated statistical descriptors based on the distribution of distances and angles to Voronoi neighbors. Parameters of a compressed-jittered model were fit to the actual mosaic. Local anisotropies were investigated using correlograms. We find that: 1) median distance between Voronoi neighbors increases with eccentricity, but the minimum distance is constant (6 8um) across peripheral retina; 2) the cone mosaic is most orderly at the edge of the foveal rod-free zone; 3) in periphery, cone spacing is 1 15% less in one direction than in the orthogonal direction; 4) cone spacing is minimal perpendicular to meridians emanating from the foveal center. The nearly constant minimum distance implies that high spatial frequencies may be sampled even in peripheral retina. Local anisotropy of the cone mosaic is qualitatively consistent with the meridional resolution effect previously described for the discrimination of gratings. 1 Introduction An image on the retina of a human eye enters the visual system through an array of photoreceptors that sets the upper limit on the spatial detail available for the rest of the visual system. The period of the highest spatial frequency that can be unambiguously reconstructed by the visual system (the Nyquist frequency) is half the angular subtense of the row-to-row spacing of cones packed in a perfect triangular lattice [22]. A grating whose highest spatial frequency exceeds the Nyquist frequency is misinterpreted as a low frequency pattern, a phenomenon known as aliasing. However, the cone mosaic is not a perfect triangular lattice (see below); there has been ample speculation about the consequences for vision of this sampling disorder. For example, an early study suggested that if position of cones in a perfect lattice were jittered randomly, and the nervous system attributed each receptor's quantum catch to the nominal rather than to the actual position, then image contrast is reduced and noise is increased, especially at high spatial frequencies [9]. On the other hand, disordered sampling might in fact be advantageous, because spatial frequencies above the Nyquist frequency established by local mean cone spacing are scattered into broadband noise, minimizing coherent aliasing [26]. 124 / PIE Vol Human Vision, Visual Processing, and Digital Display 11(1991) /91 /$4.

2 Despite the interest in the significance of sampling disorder from a theoretical point of view, anatomical data on cone packing geometry are sparse. The cone mosaic of the primate fovea has an approximately triangular lattice of tightly packed inner segments [1, 11, 12, 17], and variability in distances and angles to near neighbor cones (short range disorder) increases with eccentricity [1, 12]. All that is known about cone packing in peripheral retina is that it is not random [24]. Information about short range disorder in cone packing over a wide range of eccentricities is currently lacking. There is also medium range disorder in the packing of cones, such as the curvature of rows of cones and abrupt changes in the orientation of adjacent rows in the fovea [1, 6, 7, 15]; the significance of these phenomena for vision is unknown. Our goal in this study was to characterize short range disorder in cone packing at many retinal locations. We examined the distribution of distances and angles between neighboring cones by comparing the actual mosaic with artificial lattices generated according to a new model [21]. We also wanted to look for evidence of anisotropy in cone spacing, because laser interferometry has suggested that foveal cones are spaced more closely in the horizontal direction than in the vertical direction [25]. We gathered positions of cone inner segments at numerous locations in an adult human retina whose overall distribution of photoreceptors [6, 5] and foveal cone spacing [1] has been described previously. 2 Materials and Methods A retina was obtained from a 35 yr old male corneal transplant donor and fixed by immersion in mixed aldehydes. A retinal whole mount was prepared and cleared with dimethyl sulfoxide [4]. Tissue was viewed with Nomarski differential interference contrast (NDIC) microscopy and video at a focal plane where individual photoreceptor inner segments are just visible, a level slightly vitread to the ellipsoid-myoid junction. Only tissue sites in which the external limiting membrane was intact were analyzed, in order to ensure that cone packing geometry had not been disrupted by fixation or processing artifacts. At 7 locations ranging from the foveal center to far periphery along 4 primary and 2 oblique meridians, we used the stylus of a digitizing tablet and a video overlay to mark the centers of cone inner segments on NDIC-video images [6]. The size of the video image was scaled using a calibrated slide viewed in horizontal and vertical orientations, and adjustments in the camera's internal size controls were made as necessary. Cone positions were assigned using a 1X objective within the fovea and 4X objective when cones were completely surrounded by a ring of rods, at distances 1 mm or more from the foveal center. Cone positions were collected within slightly overlapping adjacent windows within a square grid. A computer program discarded cones that were counted more than once in the overlap zones. At each site, the sample contained 4 or more cones, and the statistics described below were based on 1 cones in a square centered within the sample. We generated statistical descriptors based on the distribution of distances (L) and angles (A) to Voronoi neighbors, as described by [2]. At each site, we defined a square window containing 1 cones. This central window was then expanded by a linear factor of 2 the larger windows contained approximately 4 cone positions. A Voronoi diagram (and Delaunay triangulation) was computed from all of the points, but statistics were gathered from the central 1 points at each site. This effectively eliminates edge effects. Parameters of a compressed-jittered model [21] were found which best fit the L and A statistics of the actual mosaic. In the compressed-jittered model, a regular, square lattice is compressed along one PIE Vol Human Vision, Visual Processing, and Digital Display 11(1991) / 125

3 diagonal and expanded along the other diagonal by a scale-factor c. The points are then jittered. By varying the compression, this model can produce a wide range of regular lattices, including both the pure square (c=1.o) and pure triangular (c=o.76) lattices as special cases. The amount of jitter is specified by the standard deviation of a Normal distribution. Local anisotropies were investigated by using correlograms, in which each cone was placed at the origin of a scatter plot and all of the other (nearby) cones were superimposed. Moments were calculated for this cloud of points, and these moments were used to characterize the local orientation and spacing of cones. We calculated the direction in which cones were furthest apart (the orientation) and the ratio between the average cone spacing in that direction and the spacing in the orthogonal direction (the axial ratio). 3 Results Figure 1 shows that the mean distance between Voronoi neighbors increases with eccentricity, as would be expected from the fact that cone density decreases with eccentricity [6]. Mean distance between neighbors is larger in temporal retina than in nasal retina, where cone density is 4% higher. The minimum distance between neighbors is similar to the dimensions of cone inner segment diameter [2]. Minimum spacing is roughly 6 8km across peripheral retina, suggesting that some cones may be directly adjacent to each other. By reexamining the tissue, we confirmed that at each site a small number of cone inner segments were actually adjacent. By plotting cone positions at the same density in order to remove the influence of scale, qualitative differences in packing geometry are apparent. Figure 2 shows the positions of cones in the center of two sites with distinctly different cone packing, a site with orderly triangular packing at the edge of the rodfree zone (Figure 2A) and the other with more disorderly packing at 5 mm from the fovea (Figure 2B). These differences were quantified by finding the best fitting parameters of a compressed-jitter model for each site. Figure 3 shows that in this specimen, both compression and jitter are minimal at eccentricities of.2.35mm, indicating that the cone mosaic resembles a perfect triangular lattice most closely near the edge of the foveal rod-free zone, rather than at the foveal center, where both parameters overlap the range of values observed for peripheral retina. Compression increases at greater eccentricities and levels off between.88 and.92 by about 3 mm, indicating that cone packing is neither purely square nor purely triangular over most of the retina. Jitter also increases with eccentricity to about 1 mm, and most of the values for peripheral retina fall between.12 and.18. Correlograms for the sites shown in Figure 2 are shown in Figure 4. The strong six-fold symmetry of the foveal cone mosaic is apparent in Figure 4A, where the reference cone is surrounded by 6 equally spaced clouds of neighbors. In contrast, at the peripheral site the first ring of neighbors is more indistinct (Figure 4B). The symmetry of the distribution of neighbors around each reference cone was quantified by using moments to determine axial ratio and orientation. At the edge of the rod-free zone, where cone packing is most regular (Figure 4A), spacing between cones was approximately the same in all directions, i.e., the axial ratio was less than 1.5. Over the rest of the retina, however, the spacing between cones is 1 15% greater in one direction than in the orthogonal direction. Regional differences in the orientation of the minimal spacings are demonstrated by Figure 5, in which orientations are plotted. Orientations for the horizontal and the vertical meridians fell within a small 126 / PIE Vol Human Vision, Visual Processing, and Digital Display 11(1991)

4 and non-overlapping range. For the nasal and temporal horizontal meridian, orientations were between _3 and 3 (mean = 6.7 ), or roughly horizontal. For the superior and inferior vertical meridians, orientations were between 9 and 125 (mean = 12.5 ), or roughly vertical. Orientations for oblique meridians (not shown) fall between those for horizontal and vertical meridians and are orthogonal to each other. Thus, cone spacing is greater in the radial direction (along meridians) than in the tangential direction (along lines of isoeccentricity) for all meridians examined. 4.1 Cone spacing 4 Discussion The packing geometry of the human cone mosaic, as assessed by finding the best fitting artificial lattice, becomes increasingly disordered with eccentricity up to about 3 mm, in qualitative agreement with results from a more limited range of eccentricities in monkey [12]. Of the two parameters of our artificial lattice, compression is roughly constant across eccentricities from 3 17 mm, and jitter is roughly constant across eccentricities from 1 17mm. However, we find that the cone mosaic is more disordered at the foveal center than at the edge of the rod-free zone. This finding is in agreement with a previous study using different methods to analyze cone spacing in the fovea of retina 114 [1] and contrasts with data from macaque showing that the cone mosaic is most orderly at the foveal center [12]. We stress that our results in retina H4 may not apply universally. This retina was unusual in that the peak density of cones was confined to a very small area [6], so that the central field used for statistical analysis contained a gradient of densities and therefore a wider range of spacings. A different retina with a larger area of peak density { 6] would have orderly packing of cones sustained over a larger area. Furthermore, measurements of cone spacing in vivo using laser interferometry [25] have suggested that disorder in cone packing geometry increases monotonically with eccentricity, because only within 1.75 of the foveal center is the cone mosaic sufficently ordered to allow the analysis of its packing geometry by aliasing phenomena. We find that although mean spacing between neighboring cones increases with eccentricity, the minimum spacing is roughly constant at 6 8,am across much of the retina. Minimum spacing is approximately equal to the diameter of the inner segment [2], because a small number of cones (3% or less by our observations) may be directly adjacent to their neighbors even where most cones are surrounded by one or more rings of rods. The Nyquist frequency for cones 6-8,um apart is 2 27 c/deg, suggesting that high spatial frequencies may be accurately sampled even in peripheral retina. In another study [3], we have found that the closest neighboring cones tend to be those of unlike spectral sensitivity, i.e., blue and red/green, so any information about high frequencies is likely to be avaliable to post-receptoral cells in luminance rather than chromatic channels. 4.2 Anisotropy Across the retina, the cone mosaic is locally anisotropic in that cones are 1 15% farther apart in the radial direction than in the tangential direction. This finding is not explained by either a systematic processing-related stretching of the tissue or by a non-linearity in our video camera. The dendritic fields of ganglion cells in cat [13] and human [18] retina are also radially oriented, and these cells exhibit an PIE Vol Human Vision, Visual Processing, and Digital Display II (1991) / 127

5 orientation bias for high spatial frequencies [14]. Laser interferometry [25] has revealed that in some observers, aliasing is perceived for horizontal gratings at spatial frequencies that are 14% higher than for vertical gratings, a result interpreted to mean that cone spacing is slightly greater horizontally than vertically. However, this phenomenon occured along all meridians within the fovea, in contrast to our observation that horizontal distances between neighboring cones are largest only along the horizontal meridian, and thus, the relationship between the anatomically and psychophysically measured anisotropies is unclear. It is possible that a larger anatomical sample may have revealed results more similar to the psychophysics. A meridional resolution effect has been described for the discrimination of gratings, in which the lowest thresholds for resolution [19] and the best seen gratings [23] are those that are parallel to meridians of the visual field. However, this effect was seen only at eccentricities greater than 2 (about 5.6 mm) whereas our effect was seen at all eccentricities except for at the edge of the rod-free zone ( ). Furthermore, the difference between optimal and orthogonal orientations was much greater (a factor of 2) than can be explained on the basis of either cone spacing or the dendritic field sizes of retinal ganglion cells [18]. The anisotropy in cone spacing across the retina is probably not related to the differences in magnification of retinal images in the radial and tangential directions that occurs only at eccentricities exceeding 6 [8]. Because the magnification of retinal images is smaller radially than tangentially, a symmetric image such as a circle of light would occupy fewer mm on the retina in the radial direction than the tangential direction. Thus, to maintain equal resolution in both directions, one would like to have cones more closely spaced radially than tangentially, rather than the other way around. It is also not clear how the anisotropy in cone spacing relates to the dominance of radial over tangential growth during development of the primate retina [16], because radial growth is much larger than the observed anisotropy, and it is greatest in the far periphery. Acknowledgments upported by N.E.I. grant EY619. Kimberly Allen provided excellent technical assistance. References [ 1] P.K. Ahnelt and D. Pum. A low frequency component in the foveal cone mosaic. Invest. Ophthalrrt. Vis. ci., 28 (uppl):262, [ 2] C.A. Curcio. Diameters of presumed cone apertures in human retina. J. Opt. oc. Am. A, 4:83, [3] C.A. Curcio, K. Allen, K.R. loan, C. Lerea, J. Hurley, I. Klock, and A. Milam. Distribution and morphology of human cone photoreceptors stained with anti-blue opsin. submitted, [4] C.A. Curcio,. Packer, and R.E. Kalina. A whole mount technique for sequential analysis of photoreceptor and ganglion cell topography in a single retina. Vision Research, 27:9 15, [5] C.A. Curcio, K.R. loan, Jr.,. Packer, A.E. Hendrickson, and R.E. Kalina. Distribution of cones in human and monkey retina: Individual variability and radial asymmetry. cience, 236: , May / PIE Vol Human Vision, Visual Processing, arid Digital Display 11(1991)

6 [6] Christine A. Curcio, Kenneth R. loan, Robert E. Kalina, and Anita E. Hendrickson. Human photoreceptor topography. J. Cornp. Neuro., 292: , 199. [7] F.M. de Monasterio, E.P. McCrane, J.K. Newlander, and.j. chein. Density profile of blue-sensitive cones along the horizontal meridian of macaque retina. Invest. Ophthalrnol. Vis. ci., 26:289 32, [8] N. Drasdo and C.W. Fowler. Non-linear projection of the retinal image in a wide-angle schematic eye. Br. J. Ophthalm., 58:79 714, [9] A.. French, A.W. nyder, and D.G. tavenga. Image degradation by an irregular retinal mosaic. Biol. Cybernet., 27: , [1] J. Hirsch and C.A. Curcio. The spatial resolution capacity of the human fovea. Vision Research, 29: , [11] J. Hirsch and R. Hylton. Quality of the primate photoreceptor lattice and limits of spatial vision. Vision Research, 24: , [12] J. Hirsch and W.H. Miller. Does cones positional disorder limit resolution? J. Opt. oc. Am. A, 4: , [13] A.G. Leventhal and J.D. chall. tructural basis of orientation sensitivity of cat retinal ganglion cells. J. Cornp. Nenol., 22: , [14] W.R. Levick and L.N. Thibos. Analysis of orientation bias in cat retina. J. Physiol. (Lond.), 329: , [15]. Packer, A.E. Hendrickson, and C.A. Curcio. Photoreceptor topography of the adult pigtail macaque (rnacaca nernestriria) retina. J. Cornp. Neurol., 288: , [16]. Packer, A.E. Hendrickson, and C.A. Curcio. Developmental redistribution of photoreceptors across the rnacaca nernestrina (pigtail macaque) retina. J. Cornp. Neurol, 298: , 199. [17].L. Polyak. The Retina. University of Chicago, [18] R.W. Rodieck, K.F. Binmoeller, and J. Dineen. Parasol and midget ganglion cells of the human retina. J. Comp. Neurol., 233: , [19] J. Rovamo, V. Virsu, P. Laurinen, and Hyvärinen. Resolution of gratings oriented along and across meridians in the visual fields. Invest. Ophthalrn., 23:666 67, [2] M.B. hapiro,.j. chein, and F.M. de Monasterio. Regularity and structure of the spatial pattern of blue cones of macaque retina. J. Am. tat. Assoc., 8:83 812, [21] K. R. loan, C. A. Curcio, and D. Meyers. Models for cone packing geometry in human retina. Invest. Ophthalm. Vis. ci, 3 (suppl):347, [22] A.W. nyder and W.H. Miller. Photoreceptor diameter and spacing for highest resolving power. J. Opt. oc. Am., 67: , [23] L.A. Temme, L. Malcus, and W.K. Noell. Peripheral visual field is radially organized. Am. J. Opt. Physiol. Optics, 62: , PIE Vol Human Vision, Visual Processing, arid Digital Display 11(1991)1 129

7 [24] H. Wässle and H.J. Reimann. The mosaic of nerve cells in the retina. Proc. R. oc. Lorzd. B, 2: , [25] D.R. Williams. Topography of the foveal cone mosaic in the living human eye. Vision Research, 28: , [26] J.I. Yellott, Jr. pectral analysis of spatial sampling by photoreceptor topological disorder prevents aliasing. Vision Research, 22: , E 4 a) U C (I) ) a) z Eccentricity, mm Eccentricity, mm Figure 1. Distribution of distances between near neighbor cones along the temporal and nasal horizontal meridian, including mean, maximum, minimum, and 25th and 75th quartiles. 13 / PIE Vol Human Vision. Visual Processing, and Digital Display II (1991)

8 a.. ;. Figure 2. Positions of cones at 2 sites with distinctly different packing geometry. tatistical tests were performed on the 1 cones shown. A surrounding frame (not shown) served as a guard zone to prevent edge effects. Dots representing inner segments are not to scale. A:.35 mm inferior to the foveal center, near the edge of the rod-free zone, where packing is nearly triangular and least jittered; B: 5 mm inferior, where packing is neither purely square nor purely triangular and jitter is characteristic of peripheral retina _ - :'- ; '.::'.. ::-;.. ;' - ;-i I -' '.'; ':: C.' ;..,&jp,lr,, ;Jt.. ' r " ;; rr t :4 r:ir. tg 4t >' 1,..'..'.,,r...d %.:f_. : :..' ":'.#., t'.. '. t.. ' : :'.'t... % ,.. J. I..,...% F s,._ :i.i. '.-,%%t ':.: :.:q.. 4 f3( -F 't :,... ',- ' 1' :::' :T1 1 Figure 4. Correlograms for the sites shown in Figure 2. Note the strong hexagonal symmetry of the inner ring of neighbors in A, a site with nearly triangular packing, and the less distinct and somewhat elongated inner ring of neighbors in B, a site with the more disorderly packing characteristic of peripheral retina. PIE Vol Human Vision, Visual Processing, and Digital Display II (1991) / 131

9 1. C.92 A. E.i;; o U) 84 FC OT ON V N, T Eccentricity, mm V U - Q).15 a o.:; FC.1 P 1 ON V N, T Eccentricity, mm Figure 3. Parameters of best-fitting artificial lattice as a function of eccentricity. FC: foveal center; T: temporal; N: nasal; : superior; I: inferior; T: superior-temporal; N: superior-nasal. A: Compression ranges from.76 (purely triangular) to 1. (purely square). B: Jitter is expressed as the standard deviation of a Gaussian distribution. Each point in the artificial lattice was jittered in a random direction by a distance with the specified standard deviation. 132 / PIE Vol Human Vision, Visual Processing, and Digital Display II (1991)

10 I Aup Inf otemp Nas A. A : 3 o C A. r - '_) =::, : Eccentricity, mm Figure 5. Ellipses based on moments calculated from the correlograms, demonstrating quadrant-specific differences in local anisotropy of cone spacing. Ellipses with the major axis parallel to the horizontal meridian have an orientation of ; those parallel to the vertical meridian have an orientation of 9. The orientation indicates the direction along which cones are more widely spaced. Cones are more narrowly spaced in the orthogonal direction. Along all meridians, cones are more widely spaced in the radial direction than they are in the tangential direction. PIE Vol Human Vision, Visual Processing, and Digital Display 11(1991) / 133