Contemporary models of perimetric sensitivity assume probability summation of retinal ganglion cell sensitivities, ignoring cortical processing. To assess the role of cortical processing in perimetric spatial summation, we used a common form of multiple-mechanism spatial vision model in which the stimulus is sampled by receptive fields analogous to those of simple cells in primary visual cortex. Psychophysical threshold was computed by probability summation across the receptive fields. When the receptive fields were nonoriented (like ganglion cells), the spatial summation function had a large nonmonotonic transitional region that was inconsistent with perimetric spatial summation data. When the receptive fields were orientation tuned (like cortical cells), the model was able to give good fits to perimetric spatial summation data. The predictions of the model were evaluated with a masking study, in which noise masks either enlarged the critical area or changed the shape of the spatial summation functions. We conclude that cortical pooling by multiple spatial mechanisms can account for perimetric spatial summation, whereas probability summation across ganglion cells cannot.

*k,*which varies with eccentricity: Sensitivity =

*cG*

^{k}

*,*where

*G*is the number of ganglion cell bodies in the region being tested and

*c*and

*k*are free parameters. The other approach (Inui, Mimura, & Kani, 1981; Wilson, 1970) uses Ricco's law for small stimuli (threshold is inversely related to stimulus area) and characterizes the effects of eccentricity in terms of increase in critical diameter (the largest stimulus for which Ricco's law holds). For the first approach, the empirical parameters have no straightforward theoretical interpretation and can vary dramatically depending on how the data are analyzed. For the second approach, there is no standard way of describing the effects of stimulus size for stimuli larger than the critical diameter. Both approaches assume that detection is mediated by ganglion cells, with little role for cortical processing (Gardiner et al., 2006; Glezer, 1965).

*spatial filters*that are each characterized by their receptive field structure. A single filter is an array of

*filter-elements*that all have the same receptive field structure (the “filter kernel”) and are centered at different locations in visual space. The model simulation was for retinal regions outside the fovea; thus, it was assumed that nearby filter-elements have similar spatial and temporal features. Sensitivity of a

*spatial mechanism*was then computed with probability summation across spatial filters that are tuned to different orientations but to the same spatial frequency. In a degenerate form of spatial vision model, a single spatial mechanism with circularly symmetric filters was used to represent a ganglion-cell-based model.

^{2}(0.025° to 5.64° in diameter) in steps of 0.1 log unit.

Spatial filters | Spatial bandwidth (octaves) | Orientation bandwidth (deg) | Space constant of the orthogonal Gaussian (deg) | Log critical area (log deg^{2})/critical
diameter (deg) | Extended slope |
---|---|---|---|---|---|

D1 long | 2.6 | 34 | 0.11 | −1.6/0.18 | 0.15 |

D1 short | 2.6 | 120 | 0.45 | −1.6/0.18 | 0.13 |

D2 long | 1.8 | 24 | 0.16 | −1.5/0.19 | 0.19 |

D2 short | 1.8 | 90 | 0.64 | −1.6/0.18 | 0.13 |

D6 long | 1.0 | 14 | 0.28 | −1.4/0.23 | 0.24 |

D6 short | 1.0 | 54 | 1.10 | −1.5/0.20 | 0.15 |

DoG 4× | 1.9 | – | – | −1.6/0.18 | 0.13 |

DoG 2× | 2.4 | – | – | −1.7/0.16 | 0.11 |

DoG 1× | 3.2 | – | – | −1.7/0.16 | 0.11 |

Gaussian no surround | Low pass | – | – | −1.6/0.18 | 0.27 |

*Differences of Gaussians*(

*DoG*) filters. The width of the inhibitory surround of the DoG filters was varied to produce filters with four different spatial bandwidths ranging from low pass (no inhibitory surround) to 1.9 octaves. Cortical filters were represented by

*DN filters*:

*N*th derivatives of Gaussian windowed by an orthogonal Gaussian, which provide both sine-phase and cosine-phase filters that integrate to zero and have a small number of zero crossings (and, therefore, only a few excitatory and inhibitory regions). The DN filters were orientation selective. More details on the choice of spatial filters can be found in Swanson, Felius, and Pan (2004). Quantitative expression for DN filter kernels is given in Swanson, Wilson, and Giese (1984).

^{2}. Multiplicative noise from eye movements should be minimal because the stimuli are briefly flashed.

*critical area*was similar for many different filters having the same peak spatial frequency, varying from −1.6 to −1.7 log deg

^{2}for the DoG filters and from −1.4 to −1.6 log deg

^{2}for the DN filters. Critical area increased systematically as peak spatial frequency was decreased, as shown in Figure 2c. The critical area increased by 1.8 log unit as peak spatial frequency decreased from 4.0 to 0.5 cpd, corresponding to a linear relation between peak spatial frequency and square root of the critical area. The

*extended slope*varied from 0.11 to 0.13 for DoG filters and from 0.13 to 0.24 for DN filters. Critical area and extended slope were greatest for the mechanism whose filter-elements had the narrowest orientation and spatial frequency bandwidths (long D6 receptive fields).

*V*(

*λ*) functions used by the spectroradiometer (CIE 1931) and by Smith and Pokorny cone fundamentals (Judd, revised 1951).

^{2}along the luminance axis to give a luminance contrast of 48%. For the chromatic stimuli, the squares of the mask were modulated along the equiluminant L–M axis. The chromaticity of each square was set to either (0.59, 2.08) or (0.70, 2.08) in the Boynton–Kambe cone excitation space (computed without macular pigment) to give a mean equal to the EEW (0.64, 2.08) and an L-cone contrast of 17%. The squares of the chromatic mask were 6 arcmin on a side because smaller squares produced minimal threshold elevation; this is consistent with the L–M chromatic mechanisms being less sensitive to higher spatial frequencies (Anderson, Mullen, & Hess, 1991; Kelly, 1983; Losada & Mullen, 1994). In all cases, the mean chromaticity was EEW and the mean luminance was 50 cd/m

^{2}.

*SEM*s of reversals were always smaller than 0.06 log unit, and symbol size for all figures is greater than ±1

*SEM*.

^{2}and were composed of fine square pixels so that the mask would primarily stimulate filter-elements tuned to high spatial frequencies. A filter-element with a peak spatial frequency of 0.5 cpd would have a receptive field as wide as at least 60 squares of the luminance mask, which is a sampling density sufficient enough that each filter-element would give a response similar to that for a uniform field. In contrast, a filter-element with a peak spatial frequency of 5 cpd could have a width of as few as 6 squares, which is a low-enough sampling that many of the filter-elements would have responses substantially different from that for a uniform field. Because substantial masking can still occur across spatial frequencies (Meese & Hess, 2004), we tested the simple hypothesis that masking effects would decrease with stimulus size to avoid any artifact from a generalized masking effect.

*p*value of .316 for any individual regression, which corresponds to a

*z*score of −0.48. The secondary prediction was that some of the data sets would have negative slopes, whereas others would not, which could occur if not all masks had the same effect. Including a Bonferonni correction for four tests, we required a significance of

*p*< .0125 for each slope.

*z*score greater than 0.48) and the secondary prediction was that only some of the data sets would show increased critical areas (at

*p*< .0125).

*z*< −1.3,

*p*< .0001), and for secondary predictions, slopes for two of the data sets reached statistical significance (chromatic data for subject W.S. and luminance data for subject P.F.).

*SD*of the parameter estimate when the mask was present, confirming the primary prediction. The amount of increase ranged from 0.15 log unit to 0.38 log unit, and for secondary predictions (increase by more than 2.24

*SD*), the increase in critical area only reached statistical significance for the luminance data (both subjects).

W.S., critical area (log deg ^{2})/diameter (deg) | P.F., critical area (log deg ^{2})/diameter (deg) | |
---|---|---|

Luminance | −1.21/0.28 | −1.24/0.27 |

Luminance with mask | −0.85/0.42 | −1.00/0.35 |

Difference | 0.36/0.14 | 0.24/0.08 |

Chromatic | −0.53/0.61 | −0.43/0.69 |

Chromatic with mask | −0.24/0.86 | −0.28/0.81 |

Difference | 0.29/0.25 | 0.15/0.12 |

*summation coefficient,*reported as the slope of a line fit to data (on log–log axes) for a subset of stimulus sizes (Goldmann, 1999; Sloan, 1961). When the summation coefficient is derived from sensitivities for stimulus diameters near the diameter of the standard perimetric stimulus (Goldman Size III, 0.43° diameter), its value increases from 0.2–0.4 in the fovea to 0.6–0.9 at 50° eccentricity (Garway-Heath et al., 2000). Figure 10 shows the summation coefficient derived as the derivatives of the spatial summation functions fit to the data from the classic study of Sloan (1961). For stimulus areas equal to the standard Size III stimulus (vertical dashed line), the summation coefficients in Figure 10 increased from 0.28 at 10° eccentricity to 0.82 at 40° eccentricity. This illustrates that the effect of eccentricity on the empirical summation coefficient can be interpreted as reflecting the use of a fixed stimulus size in the presence of changes in spatial scale of the mechanisms mediating detection.

*N*filter-elements tuned to different orientations would yield an increase in log sensitivity by log(

*N*)/4, where

*N*is the number of orientations used.

*N*different orientations, we would expect log sensitivity to be increased by log(

*N*)/4.

*N*= 4; that is, there were four filters with 45° orientation steps. The test was performed for the whole range of stimulus sizes for the 2 cpd spatial mechanism. For the weakly orientation-tuned D1 and D4 filters, the increase in sensitivity was within 0.007 log unit of the predicted amount of increase. For the strongly orientation-tuned D1 and D4 filters, the increase in sensitivity was within 0.003 log unit. These results validate the use of a single orientation in the model.