In accordance with institutional guidelines, retinas were obtained from deeply anesthetized monkeys (Macaca mulatta
, Macaca fascicularis
) being euthanized for other experimental procedures, as described previously [15
Eyes were enucleated and hemisected, and the vitreous was removed in room light. Retinas remained attached to the pigment epithelium and were incubated in the dark for at least 30 min prior to recording. For recording, patches of peripheral retina 3–5 mm in diameter and 6–12 mm from the fovea were isolated from the pigment epithelium and held flat against a planar array of 512 extracellular recording electrodes. The preparation was perfused with oxygenated and bicarbonate-buffered AMES medium (Sigma; [pH 7.4] 32–35 °C). The visual stimulus was produced using the optically reduced image of a computer display focused on the photoreceptors. Voltage data for each electrode were digitized at 20 kHz. Offline, spike waveforms were sorted into clusters in a multistep procedure [15
], and clusters with a minimum refractory period between spikes were identified as single neurons. All analyses were performed using custom software.
RFs were mapped by computing the spike-triggered average (STA) stimulus obtained in the presence of a white noise stimulus. Features of the STA were parameterized using a separable model consisting of a two-dimensional difference of Gaussians spatial profile, a biphasic time course, and a spectral profile. For analysis of RF shape, the spatial component of the STA was extracted using singular value decomposition (SVD) across time (see below). Light sensitivity was normalized across neurons by regressing the RF against an elliptical single Gaussian fit with a peak of 1. RFs were low-pass filtered by convolving with a two-dimensional Gaussian function. The standard deviation (SD) of the Gaussian was typically 0.3 to 0.9 pixels. After spatial smoothing, contour lines were linearly interpolated in each RF. For each cell type examined, a common contour level was applied to all cells and adjusted to maximize the UI. The UI was equal to the proportion of space covered by exactly one contour (excluding both gaps and overlaps). It was computed only within the area in which all cells appeared to have been recorded, as defined by an automated procedure (see below). The contour level that maximized the UI tended to produce RF contours that just touched their neighbors, providing the greatest amount of information about whether RF shapes were complementary. If a contour level significantly higher or lower had been used, the area covered by exactly one cell would have been very small, yielding little or no information about the coordination of RF shapes. For analysis of mirrored or rotated RFs, the contour level used for the UI calculation was reoptimized after transforming the RFs.
Simulated RFs (E) were defined in spatial bins the same size as the pixels of the measured RFs, and independent Gaussian noise was added to each bin to match the noise in measured RFs. For reshuffling of measured RFs, each RF was translated to its new location without additional noise. The UI of simulated RFs was computed using the same procedure as for the measured RFs. In particular, the contour level used for the UI calculation was reoptimized after RFs were altered.
Within each pixel of the flickering checkerboard stimulus, the red, green, and blue monitor primaries were modulated based on random draws from a binary distribution, chosen independently in space and time. Pixel sidelengths ranged from 18 to 60 μm (on the retina) and the color changed every 8.33 to 50 ms. For each neuron, the spatiotemporal RF was estimated by computing the STA stimulus over the 250 ms preceding a spike. This RF included the contribution of both center and surround. Although the surround time course was delayed compared to the center time course, the stimulus update temporal period was sufficiently long that this delay typically did not significantly influence the spatial RF estimate.
Some RFs analyzed here were originally recorded for use in other studies, but could also be used to study RF shape coordination. A recording was used only if it satisfied two criteria: (1) the pixels were small enough to resolve the fine shape of each RF (generally at least four to five pixels per RF diameter), and (2) the measurements of RFs exhibited low noise. These criteria were met by stimuli of various spatial and temporal scales, producing a large range of stimulus parameters in the dataset.
The color value of each pixel was chosen from a binary distribution rather than a Gaussian distribution to more rapidly characterize RF shape. The small pixel sizes used produced a relatively low effective contrast, and thus responses were approximately linear. In the linear regime, a binary distribution produces an unbiased estimate of RF shape [41
Each space–time STA described both the RF shape and kinetics. To extract only the spatial component, the STA was approximated by a space–time separable function. First, the STA was put into a matrix M in which each row was the time course of a single pixel. The singular value decomposition (SVD) was performed, yielding a standard decomposition M = UDVt, where U and V are orthogonal matrices and D is diagonal. The first column of U contained the primary spatial component of the STA. Visual inspection showed that this spatial component resembled individual frames of the STA movie in which the spatial structure of the RF was most clear.
The uniformity index.
The UI was computed for the mosaic formed by each cell type in two steps. The first step identified regions in which all cells appeared to have been recorded. The second step revealed the contour level that maximized the area covered by exactly one cell. The details of these steps and justifications are described below.
In some preparations, a complete lattice of RFs was recorded (e.g., A), whereas in other preparations, many RFs appeared to be missing (e.g., D). This variability was likely due to mechanical factors, such as contact with the electrode array. A lattice of RFs was only included in the analysis if sufficiently many RFs appeared to have been recorded, usually about 50%; and in each preparation, only regions with contiguous RFs were used for quantitative analysis. If RFs were precisely uniformly spaced, contiguous regions would be easy to identify. However, RFs exhibited somewhat variable spacing, and thus a multi-step algorithm was required to exclude regions of space not covered by recorded cells.
First, the entire recorded region was subdivided into triangular areas using the Delaunay triangulation [15
] of the collection of Gaussian fit center points. The area within a triangle was considered to be covered by contiguous RFs only if the cells at its vertices were sufficiently close together, with the cutoff distance equal to 1.9 times the median nearest neighbor spacing. The parameters of this algorithm were chosen based on the statistics of RF mosaics, as described below.
The median was used to estimate the typical nearest-neighbor spacing because it is relatively robust to outlying points. In a complete mosaic, nearest-neighbor distances follow an approximately Gaussian distribution. In an incomplete mosaic, however, nearest-neighbor distances will form a distribution composed of a Gaussian plus a long tail that corresponds to cells whose nearest neighbors were not recorded. Thus the median was the most robust estimate of nearest-neighbor spacing.
The scale factor, 1.9, was chosen empirically to match the observed variability of cell spacing. demonstrates the procedure that was used to arrive at this number. To simulate an observed mosaic with missing cells, a complete mosaic (A) was randomly subsampled. For each subsampled mosaic, a range of scale factors was used in the algorithm to identify regions of contiguous cells. D–F shows an 85% subsampled mosaic in which scale factors of 1.5, 1.9, and 2.3 were used, respectively. The areas identified as containing contiguous cells are colored in blue. For a low scale factor (D), the colored region is clearly too small; several groups of contiguous cells were missed. For a high scale factor (F), the colored region is too large, and includes several gaps where cells were not recorded. An intermediate scale factor of 1.9 appears to describe the region of contiguous cells most accurately (E).
Algorithm for Selecting Regions of Contiguous Cells
Quantifying this trend revealed that a scale factor of 1.9 was optimal. For each subsampled mosaic, the region of truly contiguous cells was identified by testing which cells were missing from the original, complete mosaic. As an example, G shows the region of truly contiguous cells for the mosaic shown in A. For the subsampled mosaic shown in D–F, the region of truly contiguous cells is shown in H in red and purple. For each scale factor, the estimated region of contiguous cells was compared to the actual region, thus creating three areas: the area correctly identified as containing contiguous cells, the area of false negatives, and the area of false positives. H shows these three areas for the scale factor value 1.9. The relative sizes of the three areas were summarized by taking the correct area minus the sum of the error areas. I shows this summary value for several scale factors, ranging from 1.5 to 2.3. Data were pooled from mosaics subsampled at 100%, 85%, 70%, and 55%. Higher numbers represent more accurate identification of contiguous regions, and the curve shows that scale factors of 1.8, 1.9, and 2.0 are approximately equally effective. Thus a value of 1.9 was chosen to robustly identify regions of contiguous cells.
Only regions of contiguous cells were used for subsequent analysis. The degree of RF interlocking, and thus the uniformity of RF coverage, was measured by considering how precisely RF shapes fit together. To efficiently describe the RF shapes, each RF was represented by a single contour level at which neighboring cells just touched. To avoid bias, this was done automatically by finding a single contour level for all cells that maximized the total area covered by exactly one cell.
To visualize why this was the most informative contour level, shows a collection of on parasol RFs at a variety of contour levels. For low contours (left column, upper rows), each RF contour was large, and the contours overlapped so much that RF shape interactions were difficult to distinguish. For high contours (left column, lower rows), each RF was too small for neighbor relationships to be revealed. The contour level that provided the most information about RF shape interlocking was the level at which RFs on average just touched their neighbors (0.36, left column, center row), and equivalently, the level that maximized the area covered by exactly one cell (scatter plot).
Analysis of RF Coordination Was Insensitive to the Contour Level
The optimal contour level was different for each preparation and varied from 0.16 to 0.40 (median 0.24). Note that the absolute contour value reflects several factors, including the amount of noise in the measurement (and thus the duration of the recording), the degree of blurring that was applied, and the degree of RF overlap.
The observation of RF coordination was not sensitive to the particular contour level chosen, as illustrated in . Over a broad range of contour levels, the area covered by one cell always declined when RFs were mirrored or rotated (bottom plot). At extreme contour levels, however, there was no effect of mirroring or rotation, because very high or low contours do not reveal the interlocking of RF shapes (upper and lower rows).
Error bars for the measured UI were produced as follows. Within each Delaunay triangle, a local UI was computed as the area covered by exactly one cell within the triangle. The overall reported UI was computed across the area occupied by all triangles. Error bars were equal to the standard error of the mean (SEM) of a subset of the local UIs. The subset was chosen so that no two triangles shared an edge, ensuring that local correlations in the UI did not artificially reduce the SEM. The SEM was validated using a bootstrap simulation. The simulation concluded that the SEM was a conservative estimate, typically overestimating the standard deviation of the UI by 30% to 60%.
Rotation plot (A–D).
Each RF was rotated about its Gaussian fit center point by the same angle; and for each such rotation, the contour level was chosen to maximize the UI. After generating rotated contours, the following procedure was applied to each cell type. First, data were pooled from all preparations in which sufficiently many cells of that type were recorded. A subset of the Delaunay triangles was chosen in each preparation as described above, and the local UI values of these triangles were averaged from all datasets to determine the mean UI at each rotation angle (including 0°), with error bars equal to the SEM. For each nonzero rotation angle, a one-tailed two-sample t-test was used to determine whether the UI was significantly lower (p < 0.01) than the UI of the unrotated RFs.
Generation of simulated RFs.
Mirror test. Each RF was mirrored about an axis passing through the Gaussian fit center point. The angle of the mirror axis was chosen using a procedure that ensured it was both arbitrary and unique. For visualization (A), the axis was parallel to the short edge of the boundary of the region shown. For quantitative analysis (B), the axis was parallel to the short edge of the recording array.
Interlocking polygons. Voronoi domains were computed based on the center points of measured RFs. Each RF was shaped like the Voronoi domain with amplitude following a Gaussian taper matched to the taper of the observed RFs.
Gaussians on a hexagonal lattice. Lattice spacing was equal to the median nearest-neighbor spacing of the measured RFs. Each RF was a circular difference of Gaussians function with center radius equal to 0.5 times the lattice spacing, surround radius equal to twice the center radius, and surround amplitude equal to 0.2 times the center amplitude.
RFs on a hexagonal lattice. Cell centers were located on a regular hexagonal lattice, and the cell at each location was a randomly chosen RF from the original population.
Average RF. Each RF was replaced by the average RF with noise added to match the noise in observed RFs.
Scrambled RFs. Each RF was replaced by a randomly chosen RF from the same preparation. The lattice of RF center locations was held constant.