Previous studies have demonstrated an inverse relation between the size of the complete spatial summation area and ganglion cell density. We hypothesized that if this relation is dynamic, the spatial summation area at 6° nasal would expand to compensate for age-related losses of retinal ganglion cells but not in the fovea where age-related loss in ganglion cell density is not significant. This hypothesis was tested by measuring contrast thresholds with a series of Gabor patches varying in size. The spatial summation area was defined by the intersection of the segments of a two-branched, piece-wise linear function fitted to the data with slopes of −0.5 and 0 on a plot of log threshold vs. log area. Results demonstrate a 31% increase in the parafoveal spatial summation area in older observers with no significant age-related change in the fovea. The average foveal data show a significant increase in thresholds with age. Contrary to the foveal data, age comparisons of the parafoveal peak contrast thresholds display no significant difference above the summation area. Nevertheless, as expected from the increase in summation area, expressing the parafoveal thresholds as contrast energy reveals a significant difference for stimuli that are smaller than the maximal summation area.

*t*-tests) of the seven observers who completed all measurements revealed no significant differences (

*p*> 0.05) from those reported for the complete samples.

*σ*= stimulus size / 4) in sine-wave phase. The sinusoidal grating was modulated in counterphase (light–dark reversal) at a reversal rate of 0.5 cycle/s. All stimuli were achromatic with a chromaticity of CIE Illuminant C (CIE

*x, y*= 0.310, 0.316). The stimuli were presented on a background of the same mean luminance and chromaticity. Stimulus contrast was defined as the Michelson contrast of the sine-wave component, (

*L*

_{max}−

*L*

_{min}) / (

*L*

_{max}+

*L*

_{min}), where

*L*

_{max}is the maximal luminance and

*L*

_{min}is the minimal luminance in the sine wave. We manipulated the stimulus size, varying it in area from 0.24 to 28.26 deg

^{2}(diameters: 0.55 to 6.00 deg), while keeping the spatial frequency constant. Stimulus duration was 750 ms with 500-ms interstimulus intervals. Each stimulus size was tested in separate blocks, and observers were aware of which stimulus was being used as a target. A temporal two-alternative forced-choice (2-AFC) task, controlled by an adaptive staircase procedure, QUEST (Watson & Pelli, 1983), was used to obtain contrast detection thresholds. For each test session, data were collapsed across the two randomly interleaved staircases. The QUEST procedure terminated if the standard deviation of the threshold estimate dropped below 0.05 log unit of contrast after a minimum of 45 trials (per staircase) or if both staircases reached 100 trials. The observer's task was to detect the interval containing the stimulus and respond by pushing a button. The order of the signal and blank intervals was randomized. An auditory signal denoted the beginning of each stimulus interval. Observers were instructed to fixate the center of the test screen. A dark fixation point appeared on the screen before the start of each trial. It was displayed on the screen continuously during parafoveal stimulus presentations.

Model | df | Log likelihood | χ ^{2} | Δdf | Pr(>χ ^{2}) |
---|---|---|---|---|---|

Area | 14 | 378.45 | |||

Area + Age | 15 | 385.56 | 14.21 | 1 | 2.00e − 04 |

Area + Age + Age:Area | 25 | 393.81 | 16.5 | 10 | 0.086 |

Model | df | Log likelihood | χ ^{2} | Δdf | Pr(>χ ^{2}) |
---|---|---|---|---|---|

Area | 14 | 415.08 | |||

Area + Age | 15 | 417.44 | 4.72 | 1 | 0.03 |

Area + Age + Age:Area | 25 | 433.41 | 31.94 | 10 | 4.00e − 04 |

*A*is the stimulus area, and

*k*and

*A*

_{max}are estimated parameters. The first segment has a slope of −0.5 and the second a slope of 0 at an ordinate value given by

*k*. The two segments intersect at the abscissa value of

*A*

_{max}, which corresponds to the maximal area showing square root summation. The two parameters

*k*and

*A*

_{max}control, respectively, the vertical and horizontal positions of the function. While each segment is linear, the problem requires a nonlinear regression because parameter

*A*

_{max}must also be estimated. Equation 1 can be parameterized in a single equation as follows:

*β*= 0.25 yields the same behavior as Equation 1. Equation 2 was fit to each observer's data initially with 3 parameters (

*β*,

*A*

_{max},

*k*) and subsequently with only two parameters with

*β*= 0.25. Parameter estimation was based on a least-squares criterion using a Gauss–Newton algorithm to search for the best fit. All calculations and statistical tests were done within R (R Development Core Team, 2010).

*p*> 0.05 (18 for

*p*> 0.01), and this number increased to 18 after adjusting for multiple testing using Bonferroni's correction. The parameter estimates and summary information of the fits of the two-parameter model for each observer are shown in Table 3. The approximate standard errors for each parameter estimate (columns 3 and 5) are based on the square root of the diagonal of the variance–covariance matrix. The residual standard error of the fit (column 7) is the square root of the sum of squared residuals (column 6) divided by the square root of degrees of freedom (the number of points minus the number of estimated parameters). The fits of the two-parameter model to individual observer's data are shown as the solid lines in Figure 2 for representative observers, and fits based on the average parameters for 10 younger and 10 older observers are shown in Figure 3. Arrows point to the spatial summation areas defined by the intersection of the two-segmented function. In units of diameter, these points correspond to 3.16 and 2.67 deg, in the younger and older groups, respectively.

Observer ID number | log_{10} A _{max} | Standard error (log_{10} A _{max}) | k | Standard error (k) | Residual SSE | Residual standard error | Age (years) |
---|---|---|---|---|---|---|---|

S02 | 0.748 | 0.073 | −1.878 | 0.027 | 0.071 | 0.06 | 21 |

S05 | 0.682 | 0.098 | −1.793 | 0.042 | 0.038 | 0.052 | 23 |

S07 | 0.764 | 0.073 | −1.57 | 0.027 | 0.17 | 0.092 | 21 |

S08 | 0.893 | 0.075 | −1.615 | 0.03 | 0.17 | 0.092 | 23 |

S10 | 0.982 | 0.075 | −1.901 | 0.03 | 0.09 | 0.067 | 21 |

S15 | 1.367 | 0.126 | −1.937 | 0.06 | 0.117 | 0.076 | 21 |

S16 | 0.923 | 0.083 | −1.983 | 0.035 | 0.088 | 0.07 | 21 |

S17 | 0.408 | 0.087 | −1.428 | 0.035 | 0.195 | 0.118 | 21 |

S19 | 1.363 | 0.126 | −1.905 | 0.06 | 0.238 | 0.109 | 19 |

S23 | 0.817 | 0.085 | −1.825 | 0.035 | 0.174 | 0.104 | 20 |

S03 | 0.742 | 0.077 | −1.443 | 0.03 | 0.079 | 0.066 | 74 |

S06 | 0.666 | 0.073 | −1.575 | 0.027 | 0.123 | 0.078 | 67 |

S09 | 0.313 | 0.087 | −0.958 | 0.035 | 0.238 | 0.13 | 70 |

S11 | 1.234 | 0.094 | −1.625 | 0.042 | 0.146 | 0.085 | 67 |

S12 | 0.655 | 0.077 | −1.191 | 0.03 | 0.086 | 0.069 | 83 |

S13 | 0.766 | 0.077 | −1.782 | 0.03 | 0.205 | 0.107 | 68 |

S14 | 0.955 | 0.075 | −1.331 | 0.03 | 0.123 | 0.079 | 77 |

S20 | 0.758 | 0.073 | −1.207 | 0.027 | 0.127 | 0.08 | 80 |

S21 | 0.569 | 0.073 | −1.538 | 0.027 | 0.043 | 0.046 | 83 |

S22 | 0.813 | 0.073 | −1.246 | 0.027 | 0.133 | 0.081 | 72 |

*t*-tests. As suggested by the mixed-effects models in the previous section, the difference in summation area, log

_{10}

*A*

_{max}, is not significant (

*t*= 1.228,

*df*= 18,

*p*= 0.24), but the difference in heights of the curves as indexed by parameter

*k*is significant (

*t*= −4.043,

*df*= 18,

*p*= 8e−4). Parameter

*k*, however, corresponds to the thresholds in the data above the areal summation limit,

*A*

_{max}, while most of the data are for stimulus sizes below this value.

*A*

_{max}would appear flat if expressed in contrast energy units and the region to the right would rise with slope of 0.5. By transforming to contrast energy, then, we are able to test for differences in the region to the left of the summation limit. We transform the contrast thresholds to deciBarlows,

^{1}which are proportional to the stimulus detection efficiency (Watson, 2000), using the following equation:

*t*= −3.958,

*df*= 18,

*p*= 9e − 4). The average contrast energy values obtained (and 95% confidence intervals) were 17.94 (16.01, 19.94) for the younger observers and 24.34 (21.29, 27.47) for the older observers. As expected from previous investigations (Brown, Peierken, Bowman, & Crassini, 1989; Latham, Whitaker, & Wild, 1994; Werner et al., 2000), photopic increment thresholds are significantly elevated in older observers for foveal stimuli, but there is no significant change in spatial summation area with age.

^{2}

*p*= 0.05 (16 for

*p*= 0.01) by a likelihood ratio test and 18 after adjusting for multiple testing by Bonferroni's correction. The parameter estimate and summary information from the fits of the 2-parameter model for each observer are shown in Table 4.

Observer ID number | log_{10} A _{max} | Standard error (log_{10} A _{max}) | k | Standard error (k) | Residual SSE | Residual Standard error | Age (years) |
---|---|---|---|---|---|---|---|

S01 | 0.895 | 0.076 | −1.005 | 0.03 | 0.379 | 0.138 | 21 |

S02 | 1.063 | 0.076 | −1.44 | 0.03 | 0.056 | 0.053 | 21 |

S04 | 1.057 | 0.076 | −1.586 | 0.03 | 0.069 | 0.059 | 21 |

S07 | 1.033 | 0.076 | −1.454 | 0.03 | 0.361 | 0.134 | 21 |

S08 | 1.179 | 0.082 | −1.282 | 0.035 | 0.3 | 0.122 | 23 |

S10 | 0.909 | 0.076 | −1.376 | 0.03 | 0.14 | 0.084 | 21 |

S15 | 1.047 | 0.076 | −1.552 | 0.03 | 0.062 | 0.056 | 21 |

S18 | 0.787 | 0.073 | −1.384 | 0.027 | 0.097 | 0.07 | 21 |

S19 | 1.08 | 0.076 | −1.328 | 0.03 | 0.195 | 0.099 | 10 |

S23 | 0.815 | 0.073 | −1.563 | 0.027 | 0.065 | 0.057 | 20 |

S03 | 1.061 | 0.076 | −1.194 | 0.03 | 0.077 | 0.062 | 74 |

S06 | 1.293 | 0.127 | −1.384 | 0.06 | 0.14 | 0.084 | 67 |

S09 | 1.16 | 0.082 | −1.304 | 0.035 | 0.099 | 0.07 | 70 |

S11 | 1.032 | 0.076 | −1.415 | 0.03 | 0.089 | 0.067 | 67 |

S12 | 1.29 | 0.081 | −0.956 | 0.035 | 0.094 | 0.065 | 83 |

S13 | 1.21 | 0.095 | −1.513 | 0.043 | 0.161 | 0.09 | 68 |

S20 | 1.083 | 0.076 | −1.224 | 0.03 | 0.095 | 0.069 | 80 |

S21 | 1.398 | 0.094 | −1.624 | 0.043 | 0.16 | 0.085 | 83 |

S22 | 1.428 | 0.127 | −1.319 | 0.06 | 0.168 | 0.092 | 72 |

*t*-tests indicate a significant difference in

*A*

_{max}(

*t*= −3.69,

*df*= 17,

*p*= 0.002) but not in

*k*(

*t*= −0.849,

*df*= 17,

*p*= 0.41). These results demonstrate an age-related change in the area of maximal summation in the parafoveal data but not a change in sensitivity for the branch of the summation curve that shows independence. To test the branch below this region, we transformed the data to contrast energy. The results indicate a significant difference (

*t*= −2.189,

*df*= 17,

*p*= 0.043) if we take significance as

*p*< 0.05. The average contrast energy values (and 95% confidence intervals) are 26.59 (24.06, 29.19) for the younger observers and 30.31 (27.40, 33.31) for the older observers. The confidence intervals are based on the individual means and overlap, in apparent contradiction to the results from the

*t*-test. The

*t*-test, however, uses a pooled estimate of error, based on both samples, yielding a more sensitive indicator of the differences than that provided by the individual confidence intervals.

*A*is the stimulus area,

*A*

_{ca}is the critical summation area, which marks the cessation of Piper's law (intersection of branches with slopes of −0.5 and −0.25), and

*A*

_{max}is the maximal summation area, corresponding to the intersection of lines with slopes of −0.25 and 0. Because the tested stimulus areas are out of the range of the classically defined Ricco's area, we fitted the three-branched model to our data without an initial part (with slope = −1) of the summation curve. The two-branched model is nested within the three-branched model, so we could compare the fits by a likelihood ratio test For the foveal data, the difference in fits of the two models was not significant for 16/20 observers (20/20 after adjustment by Bonferroni's correction). For the parafoveal data, 15/19 of the data sets (17/19 after adjustment) showed no significant difference for the two models. Figure 6 shows a comparison of fits with two-branched and three-branched piece-wise linear models.

*A*

_{max}* (10

^{ k })

^{2}. The leading coefficient is due to the reduction of contrast energy that results from counterphase modulation of the stimulus. If we combine these two equations and incorporate the spatial and temporal extent of the stimulus (presentation duration and the standard deviation of the Gaussian envelope) and a correction factor for the sinusoidal waveform, we obtain Equation 3 in the text.