In pilot analyses of the fMRI data, we noticed a high propensity for local minima in model solutions. To reduce inaccuracies and biases due to local minima, we designed the following fitting approach. We first performed a grid fit in which a range of parameter combinations were evaluated. We densely sampled parameter space using 25 nonlinearly spaced eccentricity values between 0° and 16° (0°, 0.04°, 0.09°, 0.16°, 0.22°, 0.33°, 0.43°, 0.58°, 0.73°, 0.95°, 1.2°, 1.5°, 1.8°, 2.2°, 2.7°, 3.3°, 3.9°, 4.7°, 5.6°, 6.8°, 8.0°, 9.6°, 11.3°, 13.7°, and 16°), 32 angle values between 0 and 360 degrees (0°, 11.25°, 22.5°, ..., and 348.75°), and 13 size values on a log scale between 1 and 128 pixels (equivalent to 0.08°, 0.11°, 0.16°, 0.23°, 0.32°, 0.45°, 0.64°, 0.91°, 1.3°, 1.8°, 2.6°, 3.6°, 5.1°, 7.2°, and 10.2°), yielding 25 × 32 × 13 = 10,400 parameter combinations. The combination yielding the optimal fit (in a least-squares sense) to the data was identified. We then used this parameter combination as the initial seed in a nonlinear optimization procedure (MATLAB Optimization Toolbox, Levenberg-Marquardt algorithm). For this initial seed, the gain parameter was changed to 75% of its discovered (optimal) value to allow room for adjustment in the optimization. Also, the gain parameter was restricted to be nonnegative to constrain the space of fits to solutions that predict positive BOLD responses to stimulation. Note that no spatial constraints are incorporated into the model fitting process (e.g., smoothing, priors on expected parameter values, etc.). Thus, parameter estimates for grayordinates are computed independently, thereby maximizing resolution and minimizing bias.
We fit the pRF model not only to the data from each subject (S1–S181), but also to the data from three group-average pseudosubjects, which were constructed by averaging time series data across subjects. One group average is the result of averaging all 181 subjects (S184); the second group average is the result of averaging a randomly chosen half of the subjects (S182); and the third group average is the result of averaging the other half of the subjects (S183). For each individual subject and each group average, we performed three separate model fits: One fit uses all six runs, a second fit uses only the first half of each of the six runs, and the third fit uses only the second half of each of the six runs. The rationale for these fits is that the first fit provides the best estimate of model parameters, whereas the second and third fits can be used to assess the reliability of parameter estimates.
Each fit produces six quantities of interest: pRF angle, pRF eccentricity, pRF size (calculated as
σ/√
n), pRF gain, percentage of variance explained, and mean signal intensity (calculated as the mean of all time points). The dimensions of the final results are (181+3) subjects × 91,282 grayordinates × 3 model fits × 6 quantities (see
Figure 2).