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Glaucoma  |   February 2004
Perimetric Defects and Ganglion Cell Damage: Interpreting Linear Relations Using a Two-Stage Neural Model
Author Affiliations
  • William H. Swanson
    From the Glaucoma Institute, SUNY State College of Optometry, New York, New York; and the
    Retina Foundation of the Southwest, Dallas, Texas.
  • Joost Felius
    Retina Foundation of the Southwest, Dallas, Texas.
  • Fei Pan
    From the Glaucoma Institute, SUNY State College of Optometry, New York, New York; and the
Investigative Ophthalmology & Visual Science February 2004, Vol.45, 466-472. doi:10.1167/iovs.03-0374
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      William H. Swanson, Joost Felius, Fei Pan; Perimetric Defects and Ganglion Cell Damage: Interpreting Linear Relations Using a Two-Stage Neural Model. Invest. Ophthalmol. Vis. Sci. 2004;45(2):466-472. doi: 10.1167/iovs.03-0374.

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      © 2015 Association for Research in Vision and Ophthalmology.

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purpose. To better understand the relations between glaucomatous perimetric defects and ganglion cell damage, a neural model was developed to interpret empiric findings on linear relations between perimetric defects and measures of ganglion cell loss.

methods. A two-stage model computed responses of ganglion cell mosaics (first stage), then computed perimetric sensitivity in terms of processing by spatial filters (second stage) that pool the ganglion cell responses. Cell death and dysfunction were introduced in a local patch of the first-stage ganglion cell mosaic, and perimetric defect depth was computed for the corresponding region of the visual field. Calculations were performed for both sparse and dense ganglion cell mosaics and for spatial filters with peak frequencies from 0.5 to 4.0 cyc/deg.

results. The model yielded nonlinear functions for perimetric defect depth in decibel versus the percentage of ganglion cell damage, but functions for lower spatial frequencies became linear when perimetric defect was expressed as a percentage of normal. The relations between perimetric defects and percentage of ganglion cell loss were determined primarily by spatial tuning of the second-stage spatial filters. For averaging sensitivities across different visual field locations, linear units (arithmetic mean) can more closely approximate mean ganglion cell loss than decibel units (geometric mean). Fits to data from experimental glaucoma required ganglion cell dysfunction in addition to ganglion cell loss.

conclusions. Pooling by second-stage spatial filters can account for empiric findings of linear relations between perimetric defects and measures of ganglion cell loss.

Standard automated perimetry is one of the most frequently performed psychophysical tests, because patients being treated for glaucoma routinely undergo perimetric testing one or more times per year as long as they retain functional vision. 1 Perimetry is used clinically to assess functional consequences of retinal ganglion cell damage, yet the relations between perimetric defects and ganglion cell damage are not well understood. This is in part because standard perimetric stimuli were developed in an empiric framework for spatial vision from the first half of the 20th century. The literature on perimetry has not yet fully integrated the linear systems approach for spatial vision that has been widely used over the past half century in both psychophysical and electrophysiological studies of spatial vision. A systematic application of the linear systems approach found that effects of eccentricity on perimetric sensitivity in normal eyes could be accounted for by magnification of spatial filters with eccentricity, 2 but this finding has not yet been applied to analysis of effects of ganglion cell loss. The present study provides a quantitative application of the linear systems approach to estimate the potential contributions of ganglion cell death and dysfunction to glaucomatous perimetric defects. 
Highly nonlinear correlations between depth of perimetric defect and extent of ganglion cell loss have been obtained by histologic studies of humans 3 4 and monkeys, 5 when perimetric defect is presented in decibels (1.0 dB = 0.1 log unit) and measures of ganglion cell loss are presented in linear units (e.g., percentage of normal cells). Depth of perimetric defect in decibels shows minimal change until cell density decreases to half of normal, and then increases dramatically as cell loss proceeds. A possible explanation for this was provided by Hood et al., 6 7 who found that the amplitude of the multifocal visual evoked potential (mVEP) was linearly related with perimetric sensitivity when perimetric sensitivities were averaged in linear units. They assumed that the mVEP signal amplitude reflects a linear sum of ganglion cell responses and hypothesized that, when perimetric sensitivity is expressed in linear units, the relation between local perimetric defect and local ganglion cell loss is nearly linear. 
Clinical studies have obtained similar results when using structural measures (area of neuroretinal rim, nerve fiber layer thickness) as indices of ganglion cell loss. When mean perimetric sensitivity is computed in decibel units, there are highly nonlinear correlations with indices of structural loss—that is, depth of perimetric defect is minimal for mild structural loss and then increases dramatically when structural indices fall below half of mean normal. 8 9 10 11 When perimetric sensitivities are averaged in linear units the relation with rim area becomes more linear. 12 The present study presents a theoretical interpretation of the empirically observed relations between clinical measures of structural abnormalities and functional defects. 
Nerve fiber layer defects and notches in the neuroretinal rim are consistent with localized retinal regions of extensive cell death, but the pattern of cell damage is not clear within these regions. Ganglion cells can die by apoptosis, 13 and it has been argued that cell death could be the sole cause of clinically significant glaucomatous defects. 14 However, clinical reports of improvement in visual function after treatment 15 16 have suggested that dysfunction of ganglion cells, in addition to cell death, also contributes to depth of perimetric defects. 
The purpose of the present study was to develop a quantitative model for perimetric sensitivity that incorporates key features of psychophysical models of spatial vision as a way to improve understanding of the relation between perimetric defects and ganglion cell damage. The basic parameters of the model are varied to cover the ranges of values found in the literature on ganglion cells and on perimetric spatial summation and to establish conditions for which linearity would be expected. 
Methods
A two-stage model was developed for visual processing of perimetric stimuli: the first stage represents a mosaic of retinal ganglion cells responding to presentation of a perimetric stimulus, and the second stage represents psychophysical spatial filters that pool the responses of the ganglion cell mosaic. Effects of ganglion cell death and cell dysfunction were simulated by altering ganglion cell sensitivities in the first-stage mosaic. The model was implemented on computer (Igor Pro 4.01 software; Wavemetrics, Inc., Lake Oswego, OR). 
The model used two different types of ganglion cell mosaics and six different peak spatial frequencies for the spatial filters, giving a total of 12 different sets of calculations for each degree of ganglion cell loss. The two types of first-stage ganglion cell mosaics were based on the literature on macaque and human ganglion cells, and the six types of second-stage filters were based on analysis of the literature on spatial summation of perimetry in normal eyes. Robustness of the results was evaluated by comparisons among these 12 sets of calculations, and by secondary calculations (see the Appendix). 
First Stage: Ganglion Cell Mosaics
The first stage of the model computes the responses of an array of ganglion cells to the perimetric stimulus: a circular luminance increment with a diameter of 0.43° on a uniform background, typically referred to as stimulus size III, as defined by Goldmann. 17 This stage of the model requires five parameters to define the mosaic: diameters and relative sensitivities of ganglion cell receptive field centers and surrounds, and ganglion cell spacing. Figure 1 shows diameters of receptive field centers of macaque parvocellular and magnocellular ganglion cells as a function of eccentricity from studies using gratings 18 19 or circular achromatic increments. 20 Across the different ganglion cell types and studies and across eccentricities from 10° to 30°, there is a fourfold range of center diameters. To evaluate the robustness of the model in the face of uncertainty about receptive field diameters, we used two sets of parameters reported for ganglion cells from the parvocellular and magnocellular pathways from the only one of these studies that also provided diameter and weight for the inhibitory surround. 19 These parameter sets span much of the range of reported receptive field diameters from 10° to 30° across both ganglion cell types and are shown as large open triangles in Figure 1
Implementation of the first stage of the model also requires estimates of ganglion cell spacing. For eccentricities from 10° to 30°, the center–center separation between human ganglion cells is estimated to average from 0.033° to 0.067° for all cells and from 0.10° to 0.20° if only magnocellular cells are considered. 21 Hexagonal spacing was used to create local patches of a mosaic. The dense mosaic patch had a center–center separation of 0.036°, and the sparse mosaic patch had a center–center separation of 0.16°. A mosaic patch was constructed that spanned an area whose diameter equaled the diameter of the stimulus plus six times the space constant of a single cell’s receptive field surround. Cells falling outside this region would have minimal response to presentation of the stimulus. 
Progressive ganglion cell damage in glaucomatous defects was simulated by generating a series of degraded mosaics, in which cells were successively removed on a random basis to produce mosaics retaining only 75%, 50%, 25%, 12.5%, 6.3%, 3.1%, and 1.6% of normal ganglion cell density. To evaluate effects of cell death, normal responsiveness was retained by the remaining cells. To evaluate the effects of cell dysfunction, responsiveness of surviving cells was decreased using both homogeneous and heterogeneous patterns of dysfunction. For heterogeneous dysfunction, sensitivity of a fraction of ganglion cells was reduced to 10% of normal, whereas the rest of the ganglion cells retained normal sensitivity. For homogeneous dysfunction, sensitivity of all ganglion cells was reduced by the same amount. 
Second Stage: Psychophysical Spatial Filters
Key Features of Spatial Filters.
Over the past half century, application of linear systems theory has allowed rapid measurement of spatial properties of receptive fields of retinal ganglion cells and visual cortical cells 22 and has enabled psychophysicists 23 to select experimental stimuli that characterize luminance profiles for psychophysical spatial filters. The sensitivity of an individual psychophysical spatial filter to a given stimulus can be estimated by a linear computation (i.e., the convolution of the filter’s receptive field by the luminance profile of the stimulus). Just as there can be thousands of cells in the input layer of visual cortex responding to a perimetric stimulus, there can be thousands of linear spatial filters that contribute to perimetric sensitivity. Spatial filters responding to a stimulus can be distinguished in terms of offset from stimulus center, peak spatial frequency, spatial phase, spatial bandwidth, and orientation tuning. The combination of sensitivities of all the spatial filters into a single psychophysical sensitivity requires a nonlinear process referred to as probability summation. The probability of detecting a stimulus is equal to the probability that enough spatial filters are sufficiently responsive to the stimulus. Probability summation can be well-approximated by vector summation of filter sensitivities with an exponent of 4.0. 24 25 26  
Most of the effects of eccentricity on sensitivity to perimetric stimuli can be accounted for by magnification of effective stimulus size with eccentricity, 2 which compensates for corresponding decreases in retinal ganglion cell density and area of visual cortex. Magnification of the stimulus to obtain constant sensitivity across eccentricities is consistent with spatial vision models for which the second-stage spatial filters are magnified with eccentricity. 27 28 29 We have shown 30 that perimetric spatial summation data can be accounted for by a simple two-parameter function for which the primary parameter is the critical area, a measure of magnification; the sensitivity parameter shows little or no variation with eccentricity. 31 For the two-parameter function, sensitivity increases with stimulus area with a slope of 10 dB/log unit below the critical diameter and 2.5 dB/log unit above the critical diameter. 
To evaluate the extent to which magnification of the second-stage spatial filters affects the relation between perimetric sensitivity and ganglion cell density in normal eyes, we reanalyzed data from a recent comparison 32 of normative data on human ganglion cell density 33 and perimetric sensitivity. 34 We tested the hypothesis that magnification of the second-stage spatial filters was the only factor affecting perimetric sensitivity and that the magnification factor was related to ganglion cell density. For the standard stimulus size III, perimetric sensitivity would be determined by its size relative to the critical area at each location: When the critical area is smaller than the stimulus, sensitivity decreases at a rate of −10 dB per log unit difference in area. When the critical area is larger than the stimulus, sensitivity increases at a rate of 2.5 dB per log unit difference in area. 
Figure 2 shows perimetric sensitivity versus log ganglion cell number. Our two-parameter spatial summation function 30 accounted for 82% of the variance in the data. This confirms that much of the effect of eccentricity on perimetric sensitivity can be explained in terms of magnification of spatial filters. The magnification factor obtained by fitting the data was equivalent to 31 ganglion cells filling the critical area at each retinal location. Based on the anatomic data used for ganglion cell density, 33 at eccentricities greater than 15° the size III stimulus would cover a retinal region that contains fewer than 31 cells. Most of the visual field locations used by the empiric model were for eccentricities greater than 15° (left side of Fig. 2 , ganglion cell number <1.5 log unit), and thus, in these locations, perimetric sensitivity would be linearly related to ganglion cell density. In the remaining locations within ±15° (right side of Fig. 2 , ganglion cell number >1.5 log unit), a nonlinear relation would hold in which sensitivity increases much more slowly with ganglion cell number. This is qualitatively consistent with the empiric model, 32 which used ganglion cell pooling factors that were highly nonlinear in the macula and became more linear as eccentricity increased. 
Features of the Model.
To compute psychophysical sensitivity, the responses of the first-stage ganglion cell mosaics were combined by spatially tuned psychophysical spatial filters, then the responses of the second-stage filters were combined by probability summation. 
The range of peak spatial frequencies used for the spatial filters was determined by fitting the two-parameter function 30 to published perimetric spatial summation data from normal eyes 2 tested at a range of eccentricities along four different meridians. Figure 3 shows equivalent peak spatial frequency (one half cycle = critical diameter) as a function of eccentricity. Because equivalent peak spatial frequency varied from 0.6 to 4.0 cyc/deg, the model used filters with peak spatial frequencies in the range of 0.5 to 4.0 cyc/deg. The forms used for the spatial filters are described in detail in the Appendix
In areas with deep perimetric defects, dramatic changes in perimetric sensitivity can be caused by minor changes in stimulus location. 35 For perimetric subjects with good fixation, normal microsaccades and drift can cause stimulus location to vary by as much as ±1° of the nominal stimulus center. 36 For the model, psychophysical sensitivity for a damaged patch of ganglion cell mosaics was computed for 13 different locations within ±1° of nominal stimulus center, using a hexagonal grid with 0.5° separation. The mean of these 13 sensitivities was used as the perimetric sensitivity. 
Results
Figure 4 shows depth of defect versus percentage of ganglion cells remaining. The left graph shows perimetric defects in decibel units, for which all functions are nonlinear. Defects did not reach −3 dB until the percentage of ganglion cells remaining dropped to 50%, and they deepened dramatically when the remaining cells declined below 10%. The right graph shows perimetric defects in linear units (percentage of normal sensitivity). The functions are linear when the peak spatial frequency of the second-stage spatial filters is sufficiently low and become nonlinear as spatial frequency increases. At lower spatial frequencies (0.5–2.0 cyc/deg for the dense mosaic, 0.5–1.0 cyc/deg for the sparse mosaic), the difference between perimetric defect and ganglion cell loss was less than 6% at all stages of loss. At higher spatial frequencies (2.0–4.0 cyc/deg for the sparse mosaic, 4.0 cyc/deg for the dense mosaic), perimetric sensitivity remained within 50% of normal until the percentage of remaining ganglion cells decreased below 25% of normal, and perimetric sensitivity declined dramatically with further ganglion cell loss. 
To estimate potential roles for ganglion cell dysfunction, we analyzed published data from a study of perimetric sensitivity versus ganglion cell number in monkeys with experimental glaucoma. 37 These data are plotted as filled circles in Figure 5 (top) with both perimetric defect and ganglion cell loss as percent of normal. Perfect linearity is shown as a thick straight line, and the open circles show an example of highly linear results from the model (dense mosaic, 1.0 cyc/deg). The data from experimental glaucoma deviate substantially from linearity, consistent with dysfunction preceding anatomic cell death. When the second-stage spatial filters are tuned to low enough spatial frequencies to produce highly linear results, homogeneous and heterogeneous dysfunction have identical effects. The lower panel shows the amount of dysfunction necessary to fit the data from experimental glaucoma. When ganglion cell loss was minimal, dysfunction reduced sensitivity (for homogeneous dysfunction) or fraction of normal cells (for heterogeneous dysfunction) to approximately 20% to 30% of normal. When the percentage of ganglion cells remaining decreased below 50%, the effects of dysfunction became even more pronounced. 
Discussion
Our two-stage neural model of perimetric sensitivity found that effects of ganglion cell death yielded nonlinear functions for perimetric defect in decibels versus ganglion cell loss in percentages, for both sparse and dense ganglion cell mosaics and for spatial filters tuned to a wide range of spatial frequencies. The functions are consistent with results of studies comparing perimetric sensitivity to human ganglion cell counts 3 4 and to human neuroretinal rim area. 9 Perimetric defects do not become deeper than −5 dB until ganglion cell density is reduced to below half of normal density, and then, at greater amounts of cell death, perimetric defects rapidly become deeper. 
Perimetric defect expressed in linear units (percentage of normal) was linearly related to ganglion cell loss for spatial filters tuned to relatively low spatial frequencies. This is consistent with the findings of recent studies that compared perimetric sensitivity with mVEP signal amplitude, 6 7 neuroretinal rim area and pattern electroretinogram amplitude. 12 The reason for the linear relation in the model is that spatial filters mediating detection linearly sum the weighted responses of the ganglion cells, and therefore their sensitivity tends to decrease linearly as the number of ganglion cells declines. As long as each spatial filter pools responses of a large number of ganglion cells, then cell loss affects all filters similarly. However, if the filters are tuned to a high enough spatial frequency that they sum responses of only a few ganglion cells, ganglion cell loss causes some filters to have greatly reduced sensitivity whereas others retain near-normal sensitivity. At higher spatial frequencies, probability summation across spatial filters results in perimetric sensitivity remaining near normal until the degree of loss becomes so great that all filters under the stimulus are affected. This is a quantitative expression of an idea presented qualitatively in prior studies. 35 37  
Cell death alone cannot account for the finding in experimental glaucoma that perimetric defects can occur in the absence of ganglion cell loss. 5 37 For the model to account for this finding, it must include dysfunction in surviving ganglion cells, with the degree of dysfunction increasing as the number of cells remaining declines below 50% (Fig. 5) . This type of dysfunction would introduce nonlinearities in the relations between perimetric defect and cell loss for spatial filters tuned to lower spatial frequencies, and could make the relations more linear for spatial filters tuned to higher spatial frequencies. Various combinations of cell death and dysfunction could account for the finding of linear relations between perimetric defect (in percentage) and neuroretinal rim area. 
Structural measures are for larger retinal areas than standard perimetric measures; thus, for structure and function studies, it is common to average perimetric sensitivities across a number of locations. The traditional methods for averaging perimetric sensitivities, such as the mean deviation (MD), use a geometric mean (average in decibel units). However, to the extent that perimetric sensitivity is linearly related to ganglion cell loss, geometric means tend to overestimate average loss. For instance, if one location has normal ganglion cell density, and a second location has only 1% of ganglion cells remaining, the average number of ganglion cells would be 50.5%, whereas the geometric mean would be approximately −10 dB, or 10%. For averaging sensitivities across regions with inhomogeneous damage, the arithmetic mean (average in linear units) may provide a better estimate of the average amount of loss. 6 7  
In a recent study, arithmetic means yielded more linear relations than geometric means for comparisons of perimetric defects with neuroretinal rim area and with pattern ERG amplitude. 12 That study restricted visual field location to a central square (±10°), which is the region of the visual field for which critical areas yield the highest equivalent spatial frequencies (Fig. 3) . Nonlinearities introduced by higher-spatial frequency filters are more likely for this region than for the rest of the visual field, and therefore it is possible that the linear relation they obtained reflects a combination of effects of ganglion cell death combined with ganglion cell dysfunction. Our analysis of the effects of eccentricity on normal ganglion cell density found that, at eccentricities beyond 15°, the relation between perimetric defect and ganglion cell loss should be linear (Fig. 2) . To understand better the finding of linear relations between perimetric sensitivity and clinical measures of ganglion cell loss, it would be useful to assess linearity in structure and function studies in retinal regions outside the central ±15°. 
Conclusions
A quantitative two-stage neural model for perimetric sensitivity provides a theoretical framework that complements and extends the empiric framework used in the historical development of perimetric methods. The model demonstrates how ganglion cell death and dysfunction can account for empiric findings about relations between perimetric and structural measures of glaucomatous damage, and provides theoretical support for the use of linear units (arithmetic mean) for averaging perimetric sensitivities. The key role of second-stage spatial filters in providing linearity illustrates the potential value of using perimetric stimuli for which sensitivity is mediated by spatial filters that pool the responses of a large number of ganglion cells. 38  
Appendix
Spatial Filters
The primary factor in the model affecting linearity of relations between perimetric defect and ganglion cell defect is the peak spatial frequency of the second-stage spatial filters that pool ganglion cell responses. Secondary calculations were performed to determine the extent to which the conclusions of the model are not affected by varying the form (orientation tuning, spatial phase, and spatial bandwidth) used for the spatial filters. 
The primary calculations, for which results are shown in Figure 4 , used spatial filters with a spatial bandwidth (at half-height) of one octave, an orientation bandwidth of 54°, sine phase, and no DC component (i.e., no change in mean luminance). Each filter was produced by multiplying a one-dimensional fifth-derivative-of-Gaussian (D5) function 39 with the desired peak spatial frequency by an orthogonal Gaussian whose space constant was equal to the square root of three divided by π times the spatial frequency. 40 There were eight filters centered at each filter location—identical except that orientation tuning varied from 0° to 315° in steps of 45°. 
The effects of phase of the spatial filters were explored by comparing results for sine phase and cosine phase. The sine phase filters were the D5 filters used in the primary calculations, and the cosine phase filters were sixth-derivative-of-Gaussian (D6) filters with the same orientation tuning. The effects of orientation tuning of the spatial filters were explored by using nonoriented (circularly-symmetric) filters as well as oriented filters with much narrower orientation bandwidths than the filters used in the primary calculations. The nonoriented circularly symmetric spatial filters were two-dimensional difference-of-Gaussians functions (DoGs) with no DC component (the surround had sensitivity one quarter that of the center and space constant twice as large, giving a spatial bandwidth at half-height of two octaves). The strongly oriented spatial filters had a space constant for the orthogonal Gaussian that was four times larger than for the primary orientations, yielding an orientation bandwidth of 14° (orientation of different filters was varied in steps of 15°). 
The only effect of varying the form of the spatial filters was for the cosine phase D6 filters when the percentage of ganglion cells remaining was moderate (25%–75%). At higher spatial frequencies (4.0 cyc/deg for the dense mosaic, 2.0 and 4.0 cyc/deg for the sparse mosaic) sensitivity became as much as 4 dB better than normal. This effect was minimal or absent with the other types of filters. Because the D6 filters are in cosine phase, there are two inhibitory flanks with similar magnitude, and for the most sensitive filters the stimulus falls on only one of the two inhibitory flanks. The increased sensitivity to moderate amounts of ganglion cell loss reflects the effects of loss of ganglion cells providing input to the inhibitory flanks of spatial filters near the inside edge of the stimulus. For the D5 filters used in the primary calculations, there was only one major inhibitory flank, and therefore the most sensitive filters were those with the inhibitory flank outside the stimulus. For these filters, there was little inhibition produced by the stimulus, and loss of ganglion cell input to the flank therefore had little or no effect on sensitivity. The circular DoG filters have a circular inhibitory surround, so the stimulus must fall on at least a portion of the inhibitory region of filters near the edge of the stimulus and therefore must produce some inhibition. However, the surround of the circular DoG filters is weaker than the flanks for the D6 filters, and thus loss of cells providing input to the surround produces a smaller increase in sensitivity. The prediction of increased sensitivity does not affect the conclusions that for higher spatial frequencies there is relatively little loss in perimetric sensitivity until the percentage of ganglion cells remaining decreases below 25% and that perimetric sensitivity decreases dramatically when the percentage of ganglion cells remaining declines below 10%. In summary, varying the form used for the spatial filters did not alter the conclusions of the model. 
Spacing of Ganglion Cells and Spatial Filters
In the primary calculations, the spacing between the ganglion cells in the mosaics varied with the size of the receptive field centers. The primary difference between the sparse and dense mosaics was that sensitivity-versus-cell-loss functions became nonlinear at higher spatial frequencies for the dense mosaic than for the sparse mosaic (Fig. 4) . In secondary calculations that explored the effects of ganglion cell spacing, the small receptive field size of the dense mosaic was used with the ganglion cell spacing of the sparse mosaic. The results were similar to results for the sparse mosaic with large receptive fields: functions were linear for 1.0 cyc/deg and became nonlinear for 2.0 and 4.0 cyc/deg. The main effect of smaller ganglion cell receptive fields was to make the nonlinearity at 2.0 cyc/deg slightly less severe for the smaller receptive fields than for the larger receptive fields (i.e., defects were as much as 3 dB deeper). Density of ganglion cells, rather than receptive field size, accounted for the primary differences between the two mosaics. 
For the primary calculations the spacing between spatial filters was set equal to the spacing between the ganglion cells. 41 The diameter of the array of locations used for the filters equaled the sum of the stimulus diameter plus the width of four cycles of the filter’s peak spatial frequency; filters more distant than this would have minimal response to the stimulus. The number of filter locations ranged from a minimum of 120 for the sparse mosaic sampled by filters with a peak spatial frequency of 4.0 cyc/deg (cyc/deg) to more than 60,000 for the dense mosaic sampled by filters with a peak spatial frequency of 0.5 cyc/deg. For the dense mosaic with spatial filters tuned to 0.5 cyc/deg, calculation on a computer (466 MHz Macintosh G4; Apple Computer, Cupertino, CA) of perimetric sensitivity for the eight different patterns of loss required over 30 days. 
In secondary calculations, the spacing of the spatial filters was varied independently of the spacing of the ganglion cells. Results were similar to those for the primary calculations, except that the nonlinearity at 4.0 cyc/deg became more pronounced for the sparse mosaic sampled with spatial filters at the density of the dense mosaic (i.e., defect depth was not as deep as in the primary calculations). Variation in spacing of the spatial filters had no effect on the basic result that for higher spatial frequencies there is relatively little loss in perimetric sensitivity until the percentage of ganglion cells remaining decreases to below 25% and then increases dramatically as the number of ganglion cells declines below 10%. 
Dysfunction of Ganglion Cells
Effects of heterogeneous ganglion cell dysfunction were simulated in secondary calculations for the sparse mosaic with second-stage filters tuned to 1.0 cyc/deg. A binary distribution was used in which half of the cells (randomly selected) retained normal sensitivity and half had sensitivity reduced to 10% of normal. This gave the same result as an equivalent loss of ganglion cells. In general, for low-frequency filters, heterogeneous dysfunction has results similar to the effects of cell loss, in that dysfunctional cells make little contribution to perimetric sensitivity. 
Effects of homogeneous dysfunction were simulated by reducing responsiveness of all ganglion cells in a mosaic by a fixed amount. For all degrees of ganglion cell loss, homogeneous dysfunction decreased perimetric sensitivity by the same amount. For low-spatial-frequency filters, perimetric sensitivity is linearly related to percentage of ganglion cells remaining, and thus homogeneous and heterogeneous dysfunction are equivalent. 
Cortical Reorganization in Response to Ganglion Cell Loss
The model assumes that the relative weights assigned by the second-stage spatial filters to the ganglion cell inputs remains unchanged as the ganglion cells die or become dysfunctional. This assumption may not be valid, because a lesion study found that, when local regions of retinal cells are destroyed, the corresponding cortical cells can undergo reorganization of their receptive fields. 42 The lesion study, in which all cells from a given region are damaged, is not directly comparable to the heterogeneous damage used in the current model. However, it is possible that there is a reorganization of the weighting functions of the spatial filters. If the filters lost orientation tuning or expanded in size, there would be little impact on the predictions of the model for filters with peak frequencies of 1.0 cyc/deg or lower. If filters tuned to higher spatial frequencies expanded in size, then the perimetric defect would be more linearly related to ganglion cell loss (as seen with lower spatial frequencies). If the filters shifted the location in visual space where they have peak sensitivity, then they would be similar to filters in the model already at that location and would have little effect on the predictions of the model. Reorganization of second-stage filters may affect the precise shape of the functions in Figure 4 , but would have little impact on the general conclusions drawn from these functions, other than possibly to extend linearity to higher spatial frequencies. 
Figure 1.
 
Average receptive field center diameters of macaque parvocellular (PC) and magnocellular (MC) cells reported by studies using gratings1819 or circular achromatic increments.20Large open triangles: correspond to center diameters for the two sets of parameters used for the dense and sparse mosaics in the primary calculations of the two-stage neural model. Horizontal dashed line: diameter of the standard size III perimetric stimulus.
Figure 1.
 
Average receptive field center diameters of macaque parvocellular (PC) and magnocellular (MC) cells reported by studies using gratings1819 or circular achromatic increments.20Large open triangles: correspond to center diameters for the two sets of parameters used for the dense and sparse mosaics in the primary calculations of the two-stage neural model. Horizontal dashed line: diameter of the standard size III perimetric stimulus.
Figure 2.
 
Number of ganglion cells versus perimetric sensitivity for the size III stimulus from a recent empiric study,32 fit by the two-parameter spatial summation function30 discussed in the text. The function is consistent with critical area increasing with eccentricity to compensate for decrease in ganglion cell density. In locations at eccentricities greater than 15° (left side of figure, ganglion cell number <1.5 log unit), perimetric defect is linearly related to the number of ganglion cells.
Figure 2.
 
Number of ganglion cells versus perimetric sensitivity for the size III stimulus from a recent empiric study,32 fit by the two-parameter spatial summation function30 discussed in the text. The function is consistent with critical area increasing with eccentricity to compensate for decrease in ganglion cell density. In locations at eccentricities greater than 15° (left side of figure, ganglion cell number <1.5 log unit), perimetric defect is linearly related to the number of ganglion cells.
Figure 3.
 
Equivalent spatial frequency versus eccentricity derived by fitting normative spatial summation data gathered along four different meridians.2 To span this range of values, the peak spatial frequencies for the second-stage filters varied from 0.5 to 4.0 cyc/deg.
Figure 3.
 
Equivalent spatial frequency versus eccentricity derived by fitting normative spatial summation data gathered along four different meridians.2 To span this range of values, the peak spatial frequencies for the second-stage filters varied from 0.5 to 4.0 cyc/deg.
Figure 4.
 
Effects of ganglion cell loss in the first stage of the model, with defect depth shown as a function of the percentage of ganglion cells remaining. Left: depth of defect in dB; right: depth of defect as a percentage of normal; thick gray line: perfect linearity.
Figure 4.
 
Effects of ganglion cell loss in the first stage of the model, with defect depth shown as a function of the percentage of ganglion cells remaining. Left: depth of defect in dB; right: depth of defect as a percentage of normal; thick gray line: perfect linearity.
Figure 5.
 
Use of ganglion cell dysfunction to fit data from experimental glaucoma, with both perimetric defect and ganglion cell loss in percentage of normal. Top: filled circles show perimetric defect versus ganglion cell loss in experimental glaucoma37 ; Bottom: The amounts of dysfunction used for the fits shown (top).
Figure 5.
 
Use of ganglion cell dysfunction to fit data from experimental glaucoma, with both perimetric defect and ganglion cell loss in percentage of normal. Top: filled circles show perimetric defect versus ganglion cell loss in experimental glaucoma37 ; Bottom: The amounts of dysfunction used for the fits shown (top).
 
The authors thank Mitchell W. Dul for helpful comments on a draft of the manuscript. 
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