Packing geometry of human cone photoreceptors: Variation with eccentricity and evidence for local anisotropy

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1 Visual Neuroscience (1992), 9, Printed in the USA. Copyright Cambridge University Press / 9 2 $ Packing geometry of human cone photoreceptors: Variation with eccentricity and evidence for local anisotropy CHRISTINE A. CURCIO 1 AND KENNETH R. SLOAN 2 'Departments of Ophthalmology and Cell Biology and Anatomy, University of Alabama at Birmingham, Birmingham 2 Computer and Information Sciences, University of Alabama at Birmingham, Birmingham (RECEIVED September 2 7, ; ACCEPTED December 3 0, ) Abstract Disorder in the packing geometry of the human cone mosaic is believed to help alleviate spatial aliasing effects. To characterize cone packing geometry, we gathered positions of cone inner segments at seven locations along four primary and two 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 anisotropics were investigated using correlograms. We find that (1) median distance between Voronoi neighbors increases with eccentricity, but the minimum distance is constant (6-8 ^m) across peripheral retina; (2) the cone mosaic is least compressed and jittered at the edge of the foveal rod-free zone; (3) disorder in the foveal center resembles that described by Pum et al. (1990); (4) cone spacing is 10-15% less in one direction than in the orthogonal direction; and (5) cone spacing is greater in the radial direction (along meridians) than in the tangential direction (along lines of isoeccentricity). The nearly constant minimum distance implies that high spatial frequencies may be sampled even in peripheral retina. Local anisotropy of the cone mosaic is discussed in relation to the growth of the primate retina during development and to the orientation biases of retinal ganglion cells. Keywords: Cones, Photoreceptor mosaic, Disorder, Spatial sampling, Anisotropy, Orientation bias 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 of the angular subtense of the row-to-row spacing of cones packed in a perfect triangular lattice (Snyder & Miller, 1977). 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 the positions 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 (French et al., 1977). On the other hand, disordered sampling might in fact be advantageous, because spa- Reprint requests to: Christine A. Curcio, Department of Ophthalmology E F H , University of Alabama at Birmingham, U.A.B. Station, Birmingham, AL , USA. tial frequencies above the Nyquist frequency established by local mean cone spacing are scattered into broadband noise, minimizing coherent aliasing (Yellott, 1982). 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 (Hirsch & Curcio, 1989; Hirsch & Hylton, 1984; Hirsch & Miller, 1987; Polyak, 1941), and variability in distances and angles to near neighbor cones (short range disorder) increases with eccentricity (Hirsch & Curcio, 1989; Hirsch & Miller, 1987). All that is known about cone packing in peripheral retina is that it is not random (Curcio et al., 1991; Wassle & Reimann, 1978). 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 (Pum et al., 1990); 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 (Sloan et al., 1989). We also wanted 169

2 170 C.A. Curdo and K.R. Sloan to look for evidence of anisotropy in cone spacing, because laser interferometry has suggested that foveal cones are spaced more closely in the vertical direction than in the horizontal direction (Williams, 1988). Methods We gathered positions of cone inner segments at numerous locations in an adult human retina whose overall distribution of photoreceptors (Curcio et al., 1990; Curcio et al., 19876) and foveal cone spacing (Hirsch & Curcio, 1989) has been described previously. This retina, obtained from a 35-year-old male corneal transplant donor, was fixed by immersion in mixed aldehydes, mounted scleral side up on a slide, cleared with dimethyl sulfoxide, and coverslipped with glycerol (Curcio et al., 1987o). 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. To ensure that cone packing geometry had not been disrupted by fixation or processing artifacts, we analyzed only those tissue sites in which the external limiting membrane, a structural element that maintains the spatial integrity of the outer retina, was intact, and the mosaic of inner segments was clearly visible. We did not analyze the far periphery where cone inner segments are slightly tilted and optical cross sections are elliptical. At seven locations ranging from the foveal center to far periphery along four primary and two 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 (Curcio et al., 1990). 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 loox objective within the fovea and 40x 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 400 or more cones. We generated statistical descriptors based on the distribution of distances (L) and angles (A) to Voronoi neighbors, as described by Shapiro et al. (1985). At each site, we defined a square window containing 100 cones. This central window was then expanded by a linear factor of 2, resulting in a larger window containing approximately 400 cone positions. A Voronoi diagram (and Delaunay triangulation) was computed from all of the points, but statistics were determined from the central 100 points only, thus effectively eliminating edge effects. Disorder was characterized by comparing the actual cone mosaic to a two-parameter model. We developed this model because it required less computation than rejection models (Ahumada & Poirson, 1987; Shapiro et al., 1985), and preliminary studies indicated that it fit our data at least as well as these models, by criteria described below. In this model, a regular, square lattice (Fig. 1A) is compressed along one diagonal and expanded along the other diagonal (Fig. IB) by a scale factor c. By varying the compression, this model can produce a wide range of regular lattices, including both the pure square (c = 1.0) and pure triangular (c = 0.76) lattices as special cases. The points are then jittered (Fig. 1C), with the amount of jitter specified by the standard deviation of a normal distribution. Compression and jitter were systematically varied over a narrow range to find the best combination for each site. The best combination of compression and jitter for any site was that which minimized the discrepancy in the L and A statistics of the artificial and actual lattices (Curcio et al., 1991; Shapiro et al., 1985). The L statistic (Ripley, 1981) measures distances between all pairs of points. We computed the cumulative distribution function for a restricted range of distances (approximately, between an individual cone and its first three rings of neighbors). Features that influence the L statistic include the distance to nearest neighbors and the extent to which cones are clustered together. The A statistic (Shapiro et al., 1985) measures internal angles in Voronoi regions associated with individual cones. It is very sensitive to local arrangements of neighboring cones. We used the discrepancies in the (restricted) L statistic and the A statistic to measure the similarity between different cone lattices. Local anisotropics were investigated by using correlograms (Fig. 8). In this analysis, each cone is translated to the origin, and then all cone positions are plotted under the same translation. This procedure results in a single point at the origin and a dense cloud of points representing the "average neighborhood." Superficially, the images in Fig. 8 look like the plots of the Fourier transform of the set of points (e.g. Yellott, 1982; Pum et al., 1990), and they carry essentially the same information. However, these correlograms should not be confused with the Fourier transform, because they are entirely in the spatial domain (see also Rodieck, 1991). To characterize the "average neighborhoods," we computed moments around the origin for the cloud of points that were closer than three times the distance to the nearest neighbor. From these moments, we specified the eccentricity (ratio of length to width) and orientation (direction of the long axis) of the point cloud. Fig. 1. We compared cone mosaics to a two-parameter model that produces lattices ranging from square to triangular with varying amounts of jitter. A perfect square lattice (A) is compressed along one diagonal and expanded along the other (B) so that density is constant. Each point is then jittered (C).

3 Cone packing geometry Eccentricity, mm A Eccentricity, mm Fig. 2. Distribution of distances between near-neighbor cones along the temporal (A) and nasal (B) horizontal meridian, including median, maximum, minimum, and 25th and 75th quartiles. Open circles indicate mean inner segment diameter of cones in the same retina, measured by fitting circles at a level just sclerad to that at which photoreceptors are optically separate as determined by NDIC-video imaging (Curcio, 1987). Error bars are not shown for the diameter measurements; standard deviations are about 10% of the mean. B As previously described (Curcio et al., 1990), we converted retinal eccentricity in millimeters to degrees of visual angle using a schematic eye (Drasdo & Fowler, 1974), scaling retinal arc length in millimeters from their estimated retinal radius of mm to the average of mm observed for a sample of young human retinas. This conversion provides radial magnification factors of m m / d e g at the fovea and m m / deg at 17 mm (63 deg). Results Fig. 2 shows that the median distance between Voronoi neighbors increases with eccentricity, as would be expected from the fact that cone density decreases with eccentricity (Curcio et al., 1990). Median distance between neighbors is larger in temporal retina than in nasal retina, where cone density is 40% higher. However, minimum spacing is roughly constant at 6-8 /jm at eccentricities exceeding 1 mm, where cones are completely separated by rods. This distance is similar to the dimensions of cone inner segment diameters, shown in Fig. 2 as open circles (Curcio, 1987), suggesting that cones may be directly adjacent to each other even in peripheral retina. We confirmed that at sites beyond 1-mm eccentricity there were about three pairs of closely spaced cone inner segments per 100 cones. Furthermore, these doublets tend to be of unlike spectral sensitivity (Fig. 3). In the near periphery of two retinas stained with anti-blue opsin (Curcio et al., 1991), the proportion of directly adjacent pairs that were blue-red/green (R/G)* rather than R / G - R / G (62% and 4 9 %, respectively) was much higher than would be predicted by the low density of blue cones ( 7 % of the total). Blue-blue pairs were not found in peripheral retina. By plotting cone positions at the same density in order to remove the influence of scale, qualitative differences in packing This terminology is shorthand for blue-sensitive and red/greensensitive cones, the latter type being currently indistinguishable into red- and green-sensitive subclasses by anatomical methods. Equivalent terminology is the more correct but long-winded short-wavelength sensitive and long- and middle-wavelength sensitive. Fig. 3. NDIC-video image of closely apposed blue and R/G cones in peripheral human retina. Donor retinas were labeled with an antibody to the blue cone opsin (Lerea et al., 1989) using standard peroxidaseanti-peroxidase histochemistry (Curcio et al., 1991). Scale bar = 10 iim. A: Three cone inner segments at 3-mm eccentricity (large arrowheads) form a row without intervening rods. Numerous rods (small arrowheads) are present elsewhere. B: At a more sclerad focal plane, the outer segment of the middle cone in A is labeled (arrowhead) with antibody to blue opsin and the others are not. geometry are apparent. Fig. 4 shows the photoreceptor mosaic (Figs. 4A and 4B) and cone positions (Figs. 4C and 4D) at two sites with distinctly different cone packing, a site with ordered triangular packing at the edge of the rod-free zone (Fig. 4A) and the other with more disordered packing at 5 mm from the fovea (Fig. 4B). These differences were quantified by finding the best-fitting parameters of a compressed-jitter model for each site, examples of which are shown in Figs. 4E and 4F. The fit of the model was assessed by the mean discrepancy in the L and A statistic! between the actual and artificial mosaics, which ranged between x 10~ 3 at all sites. However, qualitatively, even the best-fitting lattices (Figs. 4E and 4F) fail to capture the medium-range disorder of the actual lattice, such as the curving rows of cones seen in Figs. 4C and 4D. Fig. 5 shows that in this specimen, both parameters of our model, compression and jitter, are minimal at eccentricities of tnote that we used both the L (over a restricted range) and A statistics to calculate the discrepancies.

4 172 C.A. Curdo and K.R. Sloan.c. «Fig. 4. Two sites with distinctly different packing geometry. Panels A and B are NDIC-video images of the layer of inner segments; scale bar = 10 /mi and applies to both. Panels C and D show 100 cones at the sites in A and B at normalized density. Statistical tests were performed on these 100 cones, and a surrounding frame containing about 300 cones (not shown) served as a guard zone to prevent edge effects. Panels E and F show examples of artificial lattices that best fit the actual lattices shown in C and D, respectively. Dots representing inner segments are not to scale. A, C, E : 0.2 mm inferior to the foveal center, near the edge of the rod-free zone, where packing is nearly triangular and least jittered. In A, occasional rods are denoted by small arrowheads, and the site of an apparently missing cone (arrow) appears as a gap in the lattice just below and to the right of the center of C. B, D, F : 5 mm inferior, where packing is neither purely square nor purely triangular and both compression and jitter are characteristic of peripheral retina. The cones in B are above the center of panel D mm. This result indicates 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. For reasons discussed in detail below, the foveal center was omitted when we fit curves through the data in Fig. 5. Jitter decreases to the edge of the rod-free zone at about 0.2-mm eccentricity, then increases rapidly to 1 mm and slowly from there throughout the periphery (Figs. 5A and 5B). These data are well fit by a trilinear function, and poorly fit by a single line, bilinear, or exponential functions. Within the fovea, the generally low values for compression indicate that packing is more triangular than it is in the periphery (Fig. 5C). By about 3 mm, compression levels off between 0.88 and 0.92, indicating that packing is neither square nor triangular. Thus, two parameters of our model have different eccentricity dependencies, and their separate analysis is warranted. The nature of the disorder in the foveal center is illustrated in more detail in Fig. 6, which demonstrates that the packing of foveal cones in retina H4 is similar to that described for other human retinas by Pum et al. (1990). An isodensity contour map for this retina (Fig. 6A) reveals high density in the foveal center and a decline with eccentricity. Fig. 6B shows Voronoi regions of approximately 2000 cones in the rod-free zone of H4, color coded with respect to the number of their Voronoi neighbors. The foveal cone mosaic can be subdivided into patches of

5 Cone packing geometry 173 r Temporal I n f e r i o r A >Fov Ctr Nasal 'Superior 0blique Ecce ntricity, mm B Eccentricity, mm Fig. 5. Parameters of best-fitting artificial lattice as a function of eccentricity. A and C show eccentricities from 0-1 mm at an expanded scale. In B and D, eccentricities less than 1 mm are omitted for illustrative clarity. Note that points at 1 mm are represented twice, in the foveal graphs (A,C) and graphs of the entire retina (B,D). Fov Ctr: foveal center; Oblique: superiortemporal and superior-nasal combined. A,B: Jitter is specified as the standard deviation of a normal distribution. Jitter decreases from the foveal center to a minimum around 0.2 mm. It increases steeply from there to 1 mm and slowly from 1 mm to 20 mm. Equations of best-fitting lines (r > 0.80 for all): for mm, y = -0.14* ; for mm,^ = 0.078x ; for 1-20 mm, y = * The foveal center is not included in the linear fits, for reasons given in the text. C,D: As illustrated in Fig. 1, compression ranges from 0.76 (triangular packing) to 1.0 (square packing). With the exception of the foveal center, compression at eccentricities less than 1 mm is generally low, indicating that packing is triangular rather than square. Compression at 3-20 mm ranges from , indicating that packing is neither triangular nor square. Linear, bilinear, and exponential fits to these data were not particularly good (r < 0.5 for all), and therefore, descriptive curves are not shown. six-neighbored, triangularly packed cones (green in Fig. 6B). The patches are separated by irregular lines of cones with four, five, seven, or eight neighbors (other colors in Fig. 6B, see legend). These cones with a non-standard number of neighbors abound in the foveal center, in accordance with the findings of Pum et al. (1990). In Fig. 6C, Voronoi regions of cones with six neighbors are color coded with respect to their orientation. Along rows within a patch, orientations are similar. Adjacent patches differ in orientation, and patches tend to get larger with eccentricity within the rod-free zone. These results are in agreement with the findings of Pum et al. (1990), who defined orientation of rows within patches in the same way we did (see their Figs. 6 and 7). Thus, like other retinas, the foveal cone mosaic of H4 contains discrete, iso-orientation areas. The presence of greater disorder in the foveal center relative to elsewhere in the rod-free zone (Figs. 5A and 5C) is attributable to the fact that our statistical window of 100 cones (see legend to Fig. 6) contains a disproportionately large number of cones with a non-standard number of neighbors and parts of patches with different orientations, when compared to 100-cone windows at slightly higher eccentricities. We then asked if the increase in jitter beyond 0.2-mm eccentricity could be explained by the intrusion of rods into the allcone fovea. This relationship may be obscured by plotting jitter as a function of eccentricity (Fig. 5), because the distributions of both cones and rods are slightly asymmetric around the fovea (Curcio et al., 1990). Therefore, we replotted jitter and compression as a function of local rod/cone ratio in Fig. 7. With the exception of the foveal center, jitter is lowest where rod/ cone ratio is less than 2 (Fig. 7A). Geometric considerations indicate that at this ratio each cone can be completely encircled by rods.* Because jitter is maximal where cones are encircled by rods, it is probably not rod intrusion per se that regulates jitter, but loss of contact between cone inner segments. With the exception of the foveal center, compression is lowest (i.e. packing is most triangular) where the rod/cone ratio is 0.2 or less (Fig. 7C). Compression tends to increase slightly (i.e. packing is more square) with rod/cone ratio (Fig. 7D). Fig. 8 shows correlograms for the two sites in Fig. 4 and demonstrates local anisotropy of cone spacing. The strong sixfold symmetry of the foveal cone mosaic is apparent in Fig. 8A, where the reference cone is surrounded by six equally spaced clouds of neighbors. At the peripheral site (Fig. 8B), there again appears to be an enforced minimal distance between cones, but the first ring of neighbors is less distinct, because there is more variability in the distance to neighbors. The symmetry of the Each cone is considered to have radius c and numerical density C, and each rod, r and R, respectively. The number of rods that can completely encircle a cone is then i ( c + r)/r. At any site, a cone can be adjacent to rods encircling each of (on average) six neighbors. Thus, the number of rods actually encircling any individual cone is 6RT(C + r)/cr. Using rod and cone densities reported in Curcio et al. (1990) and unpublished measurements of rod and cone inner segment diameters in this retina, we determined that cones can be completely encircled by a ring of rods when the local rod/cone ratio is slightly over 2, which occurs at about 0.8-mm eccentricity.

6 174 C.A. Curcio and K.R. Sloan Fig. 6. Voronoi regions of cones in the rod-free zone of retina H4, color coded to reveal isodensity contours (A), the number of neighbors (B), and the orientation of six-sided regions (C) for each. Voronoi regions are defined mathematically (Shapiro et al., 1985), and their shapes should not be confused with the shapes of actual cones. The overall square field is 119 fun on a side. The 100 cones used for statistical analysis lie in a central square 23 jim on a side. Superior is at the top and nasal to the left of all panels. A: Cone density is highest in the foveal center and declines rapidly with eccentricity. The area of each Voronoi region was converted to its reciprocal, producing a density, and averaged with the areas of neighboring Voronoi regions twice. Isodensity contours (x 1000 cones/mm 2 ): violet, <100; cyan, ; green, ; yellow, ; red, ; magenta, ; white, >200. B: Number of Voronoi neighbors: violet, 4; cyan, 5; green, 6; yellow, 7; red, 8. Patches of six-neighbored cones (green) are separated by ragged lines of cones with non-standard number of neighbors (non-green). C: Each cone with six Voronoi neighbors (green in B) is color coded according to its orientation, computed as described by Pum et al. (1990). Cones that do not have six neighbors (non-green cones in B) are gray. To produce this graph, each six-sided cone was assumed to be a perfect hexagon concentric with the actual Voronoi region, with all such perfect hexagons oriented identically. Then, a vector to each neighbor was calculated with respect to the appropriate central (60 deg) angle of the perfect hexagon around each cone center. Directions therefore range from 0-60 deg, here shown in 10-deg bins: violet, 0-10 deg; cyan, deg; green, deg; yellow, deg; red, deg; magenta, deg. This graph shows that within the patches separated by cones with other than six neighbors, the direction to near neighbors is similar and differs discretely from the directions in surrounding patches.

7 Cone packing geometry B r o o OO V 0 o o D D 0 <n Q io 0 o y & D 0 0 O O OO D A O O Fov Ctr Temporal onasal "Inferior 'Superior Oblique 1 ' 1 1 > , Q. E o Rod:cone ratio Rodxone ratio 32 Fig. 7. Jitter (A,B) and compression (C,D) as a function of local rod/cone ratio. Scales for the y axes of all graphs are the same as for Fig. 5. A and C show rod/cone ratios <4 at expanded scale. Note that a point at rod/cone = 4 is shown twice in both pairs of graphs. A: With the exception of the foveal center, jitter is low at rod/cone ratios less than 2 and is virtually constant at greater ratios. B: Compression is lowest, i.e. packing is most triangular, for sites in the rodfree zone (rod/cone ratio = 0) and increases slightly at rod/cone ratios above 0.2. distribution of neighbors around each reference cone was characterized by its axial ratio and orientation, as determined by moments. At the edge of the rod-free zone ( mm eccentricity), where cone packing is most regular, spacing between cones was approximately the same in all directions (Fig. 8A), i.e. the mean axial ratio was 1.05 (range ). Over the rest of the retina, however, the spacing between cones is 10-15% greater in one direction than in the orthogonal direction (axial ratios = ). Regional differences in the orientation of the axis of greatest cone spacing, as determined by moments, are demonstrated by Fig. 9A. In this graph, axes parallel to the horizontal meridian have an orientation of 0 deg and those parallel to the vertical meridian have an orientation of 90 deg. This figure shows that orientations for the horizontal and the vertical meridians fall within small and nonoverlapping ranges. For the nasal and temporal horizontal meridian, orientations were between 30 and 30 deg (mean = -6.7 deg), indicating that cone spacing was Fig. 8. Correlograms for the sites shown in Fig. 4. The point in the center of each plot is the reference cone. A: At 0.2 mm, a site with nearly triangular packing, the inner ring of neighbors has strong hexagonal symmetry. The inner ring is slightly elongated (axial ratio = 1.06) and oriented at 129 deg. Frame width is 25.7 fim. B: At 5 mm, a site with the more disordered packing characteristic of peripheral retina, the inner ring of neighbors is less distinct and more elliptical. The major axis of the ellipse is oriented at 129 deg. Cone spacings in this direction are about 10% longer than spacings in the orthogonal direction. Frame width is 87.1 ^m.

8 176 C.A. Curcio and K.R. Sloan greatest in a roughly horizontal direction. For the superior and inferior vertical meridians, orientations were between 90 and 125 deg (mean = deg), indicating that cone spacing was greatest in a roughly vertical direction. At the foveal center (not shown), the orientation was 107 deg. Orientations for oblique meridians (Fig. 9B) 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. Discussion Minimum cone spacing Although mean spacing between neighboring cones increases with eccentricity, the minimum spacing is roughly constant at 6-8 /tm across much of the retina. Minimum spacing is approximately equal to the diameter of the inner segment (Curcio, 1987), because a small number of cones may be directly adjacent to their neighbors even where most cones are surrounded by one or more rings of rods. A disproportionate number of these doublets are of unlike spectral sensitivity, i.e. blue and A 150 A # B * Superior VM Inferior VM 30 otemporal HM A A ANasal HM 6 A 0- A o O 2 O -30- o A A o Sup-temp Sup-nasal o Horizontal o Vertical Eccentricity, mm Fig. 9. Orientation of axis of greatest cone spacing (Fig. 8) as a function of eccentricity, showing that cones are more widely spaced radially than tangentially. Ellipses with the major axis parallel to the horizontal meridian have an orientation of 0 deg, indicating that cones are more widely spaced horizontally. Those parallel to the vertical meridian have an orientation of 90 deg, indicating that cones are more widely spaced vertically. A: Horizontal and vertical meridians. B: Horizontal and vertical meridians from A are shown as open symbols, and oblique meridians are shown as closed symbols. A A o o R/G. However, many doublets of peripheral cones are R/G- R/G, in contrast to the observation that these closely spaced cones are always of unlike type, as assessed by carbonic anhydrase activity (Nork et al., 1990). Directly adjacent cones are also found in the periphery of cat and echidna but not monkey retina (Wassle & Reimann, 1978; Young & Pettigrew, 1991). Such cells in human should not be confused with the twin (morphologically identical) or double (morphologically dissimilar) cones common in species below placental mammals. The inner segments of twin and double cones are closely apposed along a considerable length and contain underlying subsurface cisternae (Borwein, 1981). In primate retina, small processes with occasional subsurface cisternae connect inner segments of cones to adjacent cones and rods (Cohen, 1989) and the inner segments of rods to adjacent rods (Krebs & Krebs, 1991; Ringvold & Davanger, 1989), but no membrane specializations at the site of contact have been seen. It is more likely that the closely spaced cones of human retina represent the outcome of a developmental superimposition of multiple independent mosaics (Curcio etal., 1991). The Nyquist frequency for cones that are 6-8 /im apart in a triangular array at 10 mm (35 deg) eccentricity is cycle/deg, suggesting that some high spatial frequencies at the correct orientation may be accurately sampled even in peripheral retina. Although the contribution of these frequencies to the overall power spectrum of the sampled image is small, our finding underscores the notion that the common practice of using a mean spacing calculated from spatial density overlooks the potential contribution of closely spaced cones. It also emphasizes the difficulty in calculating a Nyquist frequency for anything other than a regular array, for which it is strictly defined. Jitter in cone spacing Our finding that the jitter component of our two-parameter model is greater at the foveal center of retina H4 than at the edge of the rod-free zone agrees with a previous analysis of foveal cone spacing in the same retina using other techniques (Hirsch & Curcio, 1989). It also agrees well with another study of human retina by Pum et al. (1990), who analyzed four normal retinas that were fixed more rapidly after death or enucleation than our retina and processed in a completely different manner. Like Pum et al. (1990), we find that the disorder is due to the presence of irregular lines of cones with less than or greater than six neighbors separating patches of similarly oriented rows. These iso-orientation patches tend to decrease in area close to the foveal center, so that a statistical sampling window of constant cone number contains multiple patches of differing orientations, or no patches at all. Although similar patches have been observed in macaque retina (Shapiro et al., 1985), the only quantitative data to date indicate that cone packing in the foveal center is less jittered than elsewhere (Hirsch & Miller, 1987). Our results in human retina are not inconsistent with measurements of cone spacing in vivo using laser interferometry (Williams, 1985; Williams, 1988). Only within 1.75 deg (0.5 mm) of the foveal center is the cone mosaic sufficiently ordered to allow the analysis of its packing geometry by the appearance of moire patterns that resemble zebra stripes. However, the smallest retinal area used in these studies appears to be deg in diameter at the point of fixation, an area considerably larger than the entire window in Fig. 6A. Greater disorder in the foveal center would be revealed

9 Cone packing geometry 177 by disruption of just a few zebra stripes at the very center of an overall pattern, and thus, it may be difficult to detect using psychophysical techniques. Because the foveal center is the site of highest interindividual variability in cone density (Curcio et al., 1987; Curcio & Allen, 1990), it is important to ask whether disorder in the foveal center is present in eyes with widely different foveal cone density. In particular, the peak density of cones in retina H4 was confined to a very small area (Curcio et al., 1990), so that the central field used for statistical analysis (Fig. 6B) contained a gradient of densities and therefore a wider range of spacings. For this reason, our compression-jitter model is not appropriate for the foveal center, because the spatial statistics we used assume that point processes are constant over the sample site (Ripley, 1981). However, the results of Pum et al. (1990) were found in eyes whose peak cone density ranged from 141, ,000 cones/mm 2, bracketing the peak density of H4 at 181,800 cones/mm 2. In other material from our laboratory, disorder at the foveal center is qualitatively apparent (Fig. 2 of Curcio et al., 1990) in two other well-preserved retinas with peak cone density of 166,300 cones/mm 2 sustained over a large area and 324,000 cones/mm 2. The presence of this disorder may be related to the finding that the zonula adherences that are formed between photoreceptor inner segments and Miiller cells at the external limiting membrane and serve to maintain the structural stability of the photoreceptor mosaic across the retina are actually discontinuous in the foveal center of humans, nonhuman primates, and birds (Krebs & Krebs, 1989; Snyder & Miller, 1978). Maximum jitter in cone packing is achieved at mm eccentricity and changes little from there to 17 mm. This relationship of jitter with eccentricity is not inconsistent with measurements of cone spacing using laser interferometry (Williams, 1985, 1988), because moire patterns tend to become less distinct beyond 1.25 deg (0.36 mm). Our finding that maximum jitter is achieved when cones are completely encircled by rods (at a rod/cone ratio of 2) contrasts with studies showing that fractional spacing disorder (Standard Deviation + Mean Spacing) is maximal in macaque when the rod/cone ratio is well below 2 (Hirsch & Miller, 1987; Samy & Hirsch, 1989), even though cones and rods are about the same absolute size in monkeys as they are in humans (Samy & Hirsch, 1989; Curcio and Packer, unpublished observations). We suggest that the jitter component of packing disorder is regulated by several factors, on the basis of our results and a more detailed study of the fovea of H4 (Hirsch & Curcio, 1989). Across the retina, the appropriate spacing of cone inner segments may be maintained in part by competitive interactions among the telodendria that connect cone pedicles to the synaptic endings of other cones and rods (Ahnelt et al., 1990; Ahnelt & Pflug, 1986; Boycott et al., 1987; Raviola & Gilula, 1973, 1975). At low eccentricities where rods are present in small numbers, one factor regulating jitter is the degree to which the inner segments of neighboring cones are closely apposed and thereby become subject to the physical constraints of close packing more than they are subject to interactions at the level of the telodendria. Within the rod-free zone, a second factor is the steepness of the gradient of cone density, and concomitantly, cone size, because jitter is greater in a mosaic where these two parameters change rapidly with eccentricity (as they do in H4) than one in which they change slowly (Ahumada & Poirson, 1987). The significance of packing disorder in alleviating aliasing effects is as yet uncertain. The optical quality of the eye is best for foveal vision and declines with distance from the fovea (Jennings & Charman, 1981). Because the Nyquist frequency for mean cone spacing falls off more rapidly with eccentricity than does the cutoff frequency of the ocular optics, most of the human retina (and the entire retina of other species) is potentially exposed to spatial frequencies higher than the resolution afforded by the cone mosaic (Snyder et al., 1986), thus allowing aliasing to occur. The fact that aliasing is not encountered in peripheral vision outside of laboratory conditions was attributed to the protective effects of disorder in the packing of peripheral cones, which serves to smear frequencies above the Nyquist frequency into broadband spatial noise (Yellott, 1982). In fact, grating orientation may be correctly discriminated at spatial frequencies 50% higher than the Nyquist frequency set by local mean spacing of parafoveal cones (Williams & Coletta, 1987). Thus, sampling disorder is not necessarily detrimental, because perfect image reconstruction is not required for pattern discrimination. Additional theoretical analysis indicated that if the jittered positions of cones are preserved in subsequent neural processing, then aliasing effects are eliminated, and information about supra-nyquist frequencies is retained (Yellott, 1987), leading to further speculation about the mechanism and significance of preserving cone positions (Ahumada & Yellott, 1988; Bossomaier & Snyder, 1985; Costaridou et al., 1990; Maloney, 1988). Consistent with this notion is the fact that individual cones are represented in two postreceptoral channels at parafoveal eccentricities (Curcio & Allen, 1990) where supra-nyquist resolution occurs (Williams & Coletta, 1987). However, a simpler explanation for the lack of aliasing in peripheral vision may be that frequencies exceeding the Nyquist frequency of local cone spacing are minimally present in natural scenes, even in edges (Galvin & Williams, 1991). Anisotropy in cone spacing Meridional asymmetries have been reported for a number of visual functions, with the general finding that visual tasks are easier for horizontally and vertically oriented (i.e. non-oblique) stimuli in foveal vision and for radially aligned stimuli in peripheral vision. Visual functions that are better for radially aligned stimuli than for tangentially aligned stimuli include grating resolution (Rovamo et al., 1982), grating visibility (Temme et al., 1985), discrimination of mirror-symmetric gratings (Berardi & Fiorentini, 1991), contrast sensitivity (Pointer & Hess, 1989), phase discrimination (Bennett & Banks, 1991), displacement thresholds (Scobey & van Kan, 1991), and 3-dot bisections (Yap et al., 1987). The meridional effect shows up at eccentricities of 13 deg or more, with the exception of mirror-image discrimination (Berardi & Fiorentini, 1991), and it is not present along all meridians for all tasks. Because these effects are not explained by optical factors (Pointer & Hess, 1989; Rovamo et al., 1982; Temme et al., 1985; Yap et al., 1987), it is commonly thought that the underlying mechanisms are neural and include the elongated receptive fields and orientation biases of individual visual neurons. Encountered less frequently is the idea that sampling density by receptive fields may be higher along isoeccentric contours than along radial contours (e.g. Yap et al., 1987). To our knowledge, our finding that human cones are 10-15% farther apart in the radial direction than in the tangential direction (except at the edge of the rod-free zone) is the first ev-

10 178 C.A. Curcio and K.R. Sloan idence of an anisotropy in retinal cell spacing as opposed to dendritic field shape. This anisotropy is not likely to be explained by a systematic processing-related stretching of the tissue. The method by which this retina was processed produces a 2-12% areal expansion of tissue (Curcio et al., 1987a) that may be nonuniform because of regional differences in the thickness of the retina and its constituent layers. However, even if all of the expansion were systematically oriented in the radial direction, it would be less than the observed radial anisotropy. Nor is this finding explained by a nonlinearity in our video camera, which was frequently calibrated. Anisotropy in cone spacing is not related to differences in retinal image magnification in the far periphery (Drasdo & Fowler, 1974), where magnification is less in the radial direction than in the tangential direction and a symmetric image such as a circle of light occupies a shorter distance on the retina radially than tangentially. These differences occur only at eccentricities greater than 60 deg, about where our measurements ended. The radial anisotropy in spacing is distinct from the meridional anisotropy in cone density, in which cones are more numerous along the horizontal than on the vertical meridian and are more numerous nasally than temporally (Curcio et al., 1990). These two effects might appear to be related, but with the exception of the foveal center, our statistical windows of 100 cones were very small with respect to the eccentricity-dependent density decline, and density may be considered constant within the sample. These two anisotropies occur at different spatial scales, i.e. the radial anisotropy occurs locally, whereas the meridional anisotropy in density occurs over large distances, so that the two phenomena may in fact vary independently. For example, one could envision a model in which cones are more closely spaced vertically across the retina, in the face of a decline in spatial density that is symmetric around the fovea. The difficulty in distinguishing between effects occurring on different spatial scales is seen in the study of Pum et al. (1990), who noted that their Fourier transforms of the foveal cone mosaic were elongated by a factor of 15% horizontally in the frequency domain, suggesting that cone spacing is greater in the vertical direction in the spatial domain. It is not clear, however, that given the large size of their sample (on the order of our Fig. 6), whether they have shown an isotropy in local cone spacing, such as we found in the foveal center, or horizontally oriented isodensity contours; the authors interpreted it both ways. Our original motivation for this analysis was to seek a substrate within the photoreceptor mosaic for the finding by Williams (1988) that aliasing is perceived by some observers for horizontal gratings at spatial frequencies that are 14% higher than for vertical gratings. This result suggested that cone spacing is slightly greater horizontally than vertically along all meridians within the fovea. In contrast, we noted that cone spacings were actually larger vertically in our foveal sample, and that in general horizontal distances between neighboring cones are largest only along the horizontal meridian. However, our data are consistent with another study from Williams's laboratory that used an orientation reversal paradigm to show that parafoveal cones on the nasal horizontal meridian are 7.4% more widely spaced horizontally (Coletta & Williams, 1987). During development of the primate retina from late gestation to adulthood, radial growth exceeds tangential by a factor of throughout most of the periphery (Packer et al., 1990). Thus, barring significant cell death or lateral movement of cones within the photoreceptor mosaic, this manner of growth would be expected to produce cones that are more widely spaced radially than tangentially by a similar factor. However, cones are spaced more widely in the radial direction by only 10-15%, suggesting that an active process may resist growth in the photoreceptor mosaic. Retinal growth has also been proposed to explain the fact that dendritic fields of many ganglion cell types are elongated and radially oriented in the peripheral retina of cat (Leventhal & Schall, 1983), monkey (Schall et al., 1986), and human (Rodieck et al., 1985), but not baboon (Watanabe & Rodieck, 1989). The axial ratio of 90% of monkey ganglion cell dendritic fields exceeds 1.11 (calculated from Schall et al., 1986), and the mean axial ratios of human midget and parasols are 1.3 and 1.33, respectively (calculated from Rodieck et al., 1985), values that are larger than the anisotropy in cone spacing. It should be pointed out that we, Rodieck et al. (1985), and Schall et al. (1986) used different methods to assess elongation, and therefore, quantitative comparisons should be interpreted cautiously. Nevertheless, the differences in the eccentricity range and the extent of elongation between cone spacings and ganglion cell dendritic fields argues against a common developmental mechanism for both phenomena. Retinal ganglion cells exhibit an orientation bias for high spatial frequencies (Levick & Thibos, 1982) that is well explained by an elliptical center mechanism (Soodak et al., 1987). Cells in the lateral geniculate nucleus (LGN) of cat and monkey have orientation biases similar to those found in retinal ganglion cells, as shown by close correspondence of responses for single LGN units and presynaptic S-potentials (Smith et al., 1991; Soodak et al., 1987). The fact that even foveal parvocellular units exhibit such a bias was interpreted by Smith et al. (1991) to mean that the foveal midget ganglion cells projecting to these units receive input (at least 10-15%) from more than one midget bipolar cell. However, recent serial section electron microscopy indicates that a single midget bipolar accounts for 95% or more of the input onto a single foveal midget ganglion cell (Kolb & Dekorver, 1991). Another explanation for the orientation bias of foveal units is that the receptive fields of retinal cells distal to the ganglion cells may be elliptical. Although empirical studies in nonmammalian retinas (Baylor et al., 1971) and theoretical studies in cat (Smith & Sterling, 1990) indicate that the receptive fields of cones are broad, shallow, and have an antagonistic center-surround organization, it is not known if the center mechanism is circular or elliptical. Receptive-field centers of nonmammalian cones receive facilitatory input from neighboring cones by way of telodendria (Baylor et al., 1971; Detwiler & Hodgkin, 1979; Kraft & Burkhardt, 1986; Normann et al., 1984). Primate cones are also interconnected by telodendria (Ahnelt et al., 1990; Ahnelt & Pflug, 1986; Boycott et al., 1987; Raviola & Gilula, 1973, 1975), and if these connect to anisotropically spaced neighboring cones, then cone receptive-field centers may be slightly elliptical. This idea is consistent with findings in turtle retina that some cone telodendria are long and asymmetrically distributed (Ohtsuka & Kawamata, 1990) and that cones exhibit directionally asymmetric responses (Carras & DeVoe, 1991). Thus, orientation biases may begin at the earliest stage of image sampling by the retina and acquire further sharpening by both intraretinal and intracortical processing. Acknowledgments Supported by N.E.I. Grant EY We thank Drs. Joy Hirsch, Christopher Tyler, and two anonymous reviewers for their helpful comments

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