Retinotopic organization is a fundamental feature of visual cortex thought to play a vital role in encoding spatial information. One important aspect of normal retinotopy is the representation of the right and left hemifields in contralateral visual cortex. However, in human albinism, many temporal retinal afferents decussate aberrantly at the optic chiasm resulting in partially superimposed representations of opposite hemifields in each hemisphere of visual cortex. Previous functional magnetic resonance imaging (fMRI) studies in human albinism suggest that the right and left hemifield representations are superimposed in a mirror-symmetric manner. This should produce imaging voxels which respond to two separate locations mirrored across the vertical meridian. However, it is not yet clear how retino-cortical miswiring in albinism manifests at the level of single voxel population receptive fields (pRFs). Here, we used pRF modeling to fit both single and dual pRF models to the visual responses of voxels in visual areas V1 to V3 of five subjects with albinism. We found that subjects with albinism (but not controls) have sizable clusters of voxels with unequivocal dual pRFs consistently corresponding to, but not fully coextensive with, regions of hemifield overlap. These dual pRFs were typically positioned at locations roughly mirrored across the vertical meridian and were uniquely clustered within a portion of the visual field for each subject.

^{3}. The SPGR scans were subsequently resampled to 1.0 mm

^{3}. A sync pulse from the scanner at the beginning of each run triggered the onset of visual stimuli.

_{0}and y

_{0}) is the center position, σ is the standard deviation of the distribution, and A is an amplitude scaling factor. Dual pRFs were modeled as the sum of two independent 2D Gaussians:

_{0}and b

_{0}) and (x

_{0}and y

_{0}) are the centers of the two Gaussians, σ

_{1}and σ

_{2}are the standard deviations of the two Gaussians, and A

_{1}and A

_{2}are the amplitude scaling factors.

_{0}and b

_{0}) and (x

_{0}and y

_{0}) starting positions limited to the voxel's preferred eccentricity as determined by the full field expanding ring experiment. Using the starting positions established by the coarse search, polar angle, σ, and amplitude (sensitivity) were refined in a two-pass, coarse-to-fine fashion using an unconstrained nonlinear optimization algorithm (Matlab's fminsearch) to obtain an optimal fit to the empirical rotating wedge fMRI waveform.

*F*-test to determine whether the additional Gaussian parameters optimized in the dual pRF model resulted in a significant increase in variance explained. For each voxel, a partial

*F*-static was computed using the following equation:

_{S}is the

*RSS*error from fitting the single pRF model, RSS

_{D}is the

*RSS*error from fitting the dual pRF model,

*df*is the number of degrees of freedom in the single pRF model, and df

_{S}_{D}is the number of degrees of freedom in the dual pRF model.

*Df*and

_{S}*df*included the number of free parameters optimized in the single and dual pRF models respectively plus the number of degrees of freedom used in the baseline motion/linear trends regression model projected out of the time series data during pre-processing. Voxels were considered to have dual pRFs if this partial

_{D}*F*-statistic fell above the 99% confidence interval in the appropriate

*F*-distribution. Voxels were considered to have single pRFs if their

*F*-statistic fell below the 1% confidence interval. (In other words, only voxels from either extreme of the distribution were selected, thereby excluding voxels with ambiguous signals.) Our selection criteria were intentionally conservative and were intended to select the most unequivocal dual and single pRF voxels for use in subsequent analyses. We note that the

*F*-test assumes that all measurements are fully independent, which is not necessarily true for adjacent fMRI time points (Raz, Zheng, Ombao, & Turetsky, 2003). However, we also assess the model fits in these same voxels using cross-validation (described below), which does not rely on this assumption. The percent variance explained (%

*VE*) by each respective model was also computed on a voxel-wise basis using the following equation:

*TSS*is the total sum of squares for the voxel time course and

*RSS*is the RSS error from the model fitting. Finally, the difference in variance explained (

*∆VE*) by the dual and single pRF models was computed for each voxel in the following manner:

*VE*and %

_{dual}*VE*are the percent variance explained by the dual and single pRF models, respectively.

_{single}*VE*for each model was computed for each run in each subject. These were then averaged across subjects to compute the mean %

*VE*for each model during training. During the validation stage, we tested the models trained on each individual run by separately computing %

*VE*in each of the other two runs. The mean %

*VE*for each model was computed for all six train-test combinations (using the 3 runs) in each subject. These were then averaged across subjects to compute mean %

*VE*for the single and dual pRF models during validation. If the additional parameters in the dual model were simply over-fitting noise in the training data, they should provide no benefit over the single pRF model in the validation stage. However, if the dual pRF model explains significantly more stimulus-driven variance than the single model, it should do so in both stages.

*θ*) between the two pRF centers with respect to the HM:

*x*and

_{1}*y*) are the Cartesian coordinates for pRF center 1 and (

_{1}*x*and

_{2}*y*) are the coordinates for pRF center two. The absolute values of these angles can range from 0 degrees to 90 degrees. Because the dual pRF components were constrained to be at the same eccentricity, an angle of 0 degrees indicated symmetry across the VM, and 90 degrees indicated symmetry across the HM. Histograms of dual pRF angles were generated for each subject.

_{2}^{2}, the threshold previously used to define normal steady fixation (Fujii, De Juan, Humayun, Sunness, Chang, & Rossi, 2003; Woertz et al., 2020).

*cc*> 0.45) to both a right and left hemifield stimulus regardless of the eye stimulated. Within individual subjects, the percent overlap tended to be similar across V1 to V3. However, there were considerable differences in the mean percent overlap across subjects with values ranging from 39% to 70%.

*∆VE*= 32.42%) by the two models (77.25% vs. 39.83%,

*F =*11.4,

*p <*0.001). However, for the voxel time course shown in the right panels, there are only five peaks to fit. Consequently, the additional Gaussian parameters of the dual model provide less benefit, and there is little difference in variance explained (

*∆VE*= 0.06%) by the two models (70.04% vs. 69.97%,

*F =*0.02,

*p >*0.999). (Note: Because the two pRF components are fit independently, they tend to be forced into the same visual field locations when the empirical waveform has only five peaks, thus the two models tend to explain the same amount of variance.)

*F*-test from the comparison of the two models (see Methods) for the same subject. In this analysis, “dual pRF voxels” (red) must have an

*F*-value above the 99% confidence interval, and single pRF voxels (purple) must have an

*F*-value below the 1% confidence interval. It is notable that dual pRF voxels occurred in large contiguous clusters often, although not always, near the HM representations in V1 and on the V2 to V3 boundary, as indicated by the yellow arrows in Figure 2C. In contrast, the single pRF voxels tended to cluster near the VM representations.

*p*< 0.001) but no significant visual area effect (

*p =*0.592) or interaction. Post hoc tests showed that subjects with albinism had higher incidences of dual pRF voxels than controls in every visual area (2-tailed, independent

*t*-tests,

*p*values for V1, V2, and V3, respectively:

*p =*0.006,

*p =*0.006, and

*p =*0.013). On average, dual pRF voxels comprised 8.6% of all visually responsive voxels in the albinism group but ranged from 4% to 15% for different subjects with albinism. This underscores the fact that dual pRF voxel incidence varies across subjects. In contrast, the mean incidence of dual pRFs in control subjects was 1.8% with individual values ranging from of 1% to 3%. Furthermore, visual inspection of many “dual pRF” voxels in controls revealed that they tend to be voxels sampling across the midline or cases in which clear five peaked fMRI responses were overfit by the dual model.

*F*-test analysis. The “Training” column (left) in Figure 4 shows the mean %

*VE*for the dual and single pRF models combined across all subjects’ individual runs in the training stage. The validation column (right) displays the corresponding results averaged across all train-test combinations in the validation stage. For single pRF voxels (see Figure 4, top row), there was no significant difference in the mean %

*VE*for the single and dual pRF models in either the training or validation stages (2 tailed, independent sample

*t*-tests:

*p*= 0.78 and

*p*= 0.57). However, for dual pRF voxels (see Figure 4, bottom row), the dual pRF model significantly outperformed the single pRF model in both the training and validation stages by comparable margins of 10.4% and 7.9%, respectively (2 tailed, independent sample

*t*-tests:

*p*= 0.01 and

*p*= 0.03). This result confirms that the extra Gaussian parameters were not over-fitting noise but instead captured additional BOLD response features (extra peaks) that were not explained by the single pRF model. Note that the overall variance explained by both the single and dual pRF models was lower in this analysis than in those reported above for averaged data because signals from individual fMRI runs have lower signal to noise ratio (SNR; are noisier) than averaged signals. The slight reduction in variance explained by both models in the validation versus the training stage was likely caused by random differences in noise and signal quality across the individual runs.

*SEM*), whereas the individual subject histograms plot the total number of dual pRF voxels at each respective angle. As displayed in Figure 5, the albinism group's dual pRF angle data formed a clear peak centered on 0 degrees. This pattern is also observable in the data of individual subjects with albinism (3, 4, and 5). The dual pRF angle histogram of subject 1 was more sparsely populated and subject 2 showed a wider distribution of angles indicating heterogeneity in the range of angles represented in the albinism group. In contrast, the control dual pRF angle distributions showed no common central tendency. The group histograms in Figure 5 clearly show that the largest group differences and the vast majority of dual pRFs in albinism occur between bins -30 degrees and 30 degrees with the greatest difference occurring at the 0 degree bin. Consequently, we focused all further analyses only on dual pRF voxels with angles between −30 degrees and 30 degrees.

*t*-test

*p =*0.0004). This was not the case for single pRF voxels (60% vs. 40%,

*p =*0.09).

*SEM*). The first five plots in each row display individual subjects’ pRF size scaling data, and the sixth plot in each row displays linear trendlines fit to the data pooled across subjects for each group. The final graphs in Figures 9A and 9B show the same data but are constrained to pRFs falling within 30 degrees polar angle of the HM. The pRFs in this zone are unaffected by the issue mentioned earlier of merged dual pRF's near the VM. This was not included in Figure 9C, as this issue only pertains to single pRF sizes. A color key in the upper left indicates color of symbols representing each visual area. Figures 9A and 9B display single pRF size scaling data for the control and albinism groups respectively, and Figure 9C displays dual pRF component size scaling for the albinism group. A three factor, univariate ANOVA (dependent variable: pRF size, factors: component laterality, visual area, and eccentricity) revealed no significant effect of component laterality (ipsi- versus contralateral) on dual pRF component size (

*p =*0.08). Therefore, we have only displayed the contralateral component sizes. These analyses were also limited to eccentricities of 1 to 10 degrees for the same reasons stated above. Linear trendlines were fit for each visual area in the group data plots to portray size scaling slopes. Colored dotted lines in the group plots demark the 95% confidence interval of the group mean at each eccentricity.

*p <*0.001 for each) and a significant interaction between eccentricity and visual area (

*p <*0.001). These results reflect the consistent scaling of single pRF size with eccentricity and visual area across the control and albinism groups clearly visible in Figures 9A and 9B. Subsequent two-factor ANOVAs run on the control and albinism groups separately revealed that the main effects of eccentricity and visual area were significant for both groups individually (all

*p*values < 0.001) but that the interaction between eccentricity and visual area was only significant for the control group (

*p <*0.001). Post hoc Tukey HSD multiple comparison tests comparing mean single pRF size in each visual area confirmed that the mean pRF size in visual areas V1 to V3 were all significantly different from one another in both the albinism and control groups with single pRF sizes increasing with visual area.

*p <*0.001) with larger single pRFs in albinism. However, we believe this group difference was influenced by the dual pRF modeling limitations near the VM in albinism noted above and illustrated in Figure 7. As dual pRFs in albinism are roughly symmetrical across the VM, the two components and their respective BOLD responses will tend to merge near the VM and be fit equally well by a large single pRF model. Indeed, we confirm in Supplementary Figure S2 that single pRF sizes and scaling are comparable for the two groups on the HM, but that single pRFs are larger in albinism on the VM. Enhancement of single pRF size on the VM in albinism was also clearly exacerbated by increasing eccentricity and visual area where dual pRF components are larger and more likely to merge. To correct for this, we plot single pRF size scaling exclusively for pRFs falling within 30 degrees polar angle of the HM in the rightmost plots of Figures 9A and 9B as these voxels are not affected by this issue. This constraint has little effect on control's single pRF slopes in Figure 9A, but reduces the size scaling slopes for the albinism group in Figure 9B.

*p <*0.001 for each), but also revealed a significant effect of pRF type (

*p <*0.001) and significant interactions of eccentricity*visual area, eccentricity*type, and visual area*type (

*p =*0.028,

*p <*0.001, and

*p <*0.001, respectively). These effects are evident when visually comparing the dual pRF component scaling data to the single pRFs in Figure 9 as the dual components are consistently smaller than their single pRF counterparts and also scale less steeply with eccentricity and visual area. The “merging” issue at the VM discussed earlier likely contributed to this result; however, dual pRF component size scaling is still clearly reduced even when compared to single pRFs restricted to the HM (Figure 9B far right versus 9C far right). Although the ipsi- and contralateral dual pRF component sizes differed for some individual voxels, as mentioned previously, a subsequent ANOVA comparing the size of contralateral and ipsilateral dual pRF components across eccentricity showed no significant effect of component laterality (

*p*= 0.08). However, there were, significant main effects of eccentricity (

*p <*0.001), visual area (

*p =*0.027), and an interaction between eccentricity and laterality (

*p =*0.002). Nonetheless, our post hoc Tukey test comparing dual pRF component sizes of each visual area failed to show significant differences in dual pRF component size between any pair of visual areas. This is consistent with the reduced dual pRF size scaling across visual areas evident in Figure 9C.

*θ*) fell between -30 degrees and 30 degrees. Percent dual pRF voxel incidence correlated significantly with subjects’ mean 95% BCEA measurements (

*R*= 0.978,

*p =*0.004).

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