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.

^{ 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.

^{ 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.

^{ 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.

^{ 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.

^{Appendix}).

^{ 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 .

^{ 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.

^{ 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 }

^{ 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.

^{ 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.

^{ 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.

^{ 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}.

^{ 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.

^{ 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.

^{ 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.

^{ 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 }

^{ 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.

^{ 6 }

^{ 7 }

^{ 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°.

^{ 38 }

^{ 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°.

^{ 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.

^{ 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.

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