A Two-Stage Cascade Model of BOLD Responses in Human Visual Cortex

<div><p>Visual neuroscientists have discovered fundamental properties of neural representation through careful analysis of responses to controlled stimuli. Typically, different properties are studied and modeled separately. To integrate our knowledge, it is necessary to build general models that begin with an input image and predict responses to a wide range of stimuli. In this study, we develop a model that accepts an arbitrary band-pass grayscale image as input and predicts blood oxygenation level dependent (BOLD) responses in early visual cortex as output. The model has a cascade architecture, consisting of two stages of linear and nonlinear operations. The first stage involves well-established computations—local oriented filters and divisive normalization—whereas the second stage involves novel computations—compressive spatial summation (a form of normalization) and a variance-like nonlinearity that generates selectivity for second-order contrast. The parameters of the model, which are estimated from BOLD data, vary systematically across visual field maps: compared to primary visual cortex, extrastriate maps generally have larger receptive field size, stronger levels of normalization, and increased selectivity for second-order contrast. Our results provide insight into how stimuli are encoded and transformed in successive stages of visual processing.</p></div

Rotation-invariant statistics based on the SFM.

<p>The SFM leads to two rotation-invariant statistics that are calculated in every voxel, and shown here in axial sections at the height of the Centrum Semiovale (A, B; compare to Figs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0123272#pone.0123272.g004" target="_blank">4B</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0123272#pone.0123272.g007" target="_blank">7B</a>) and the Optic Radiation (C, D; compare to Figs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0123272#pone.0123272.g004" target="_blank">4C</a> and <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0123272#pone.0123272.g007" target="_blank">7C</a>). The Fiber Anisotropy (FA; A, C) is an indication of the total fiber fraction, relative to W0. The Dispersion Index (DI; B, D) is an indication of the degree to which different fascicles cross each other within each voxel.</p

DTM model-accuracy is high (low rRMSE) in a large portion of the white matter, but systematically deviates in two locations (3% of the voxels): the centrum semiovale (A, top row) and the optic radiation (A, bottom row).

<p>The columns show data obtained at three different b-values (b = 1000, 2000, 4000) in one individual, and this pattern is observed in a second individual as well. The color overlay measures the rRMSE. Poor cross-validation (rRMSE > 1) is denoted by the yellow-red colors. (B, C) The color overlays are rRMSE maps calculated at b = 4000. The two images illustrate the poor fits in the optic radiation (B) and the centrum semiovale (C).</p

The SFM fits the data better than the DTM.

<p>(A) Image histograms comparing the rRMSE of the SFM and DTM in each white matter voxel. (B) Median rRMSE of the DTM and SFM +/- 95% confidence interval estimated with a bootstrapping procedure.</p

A simulation study of parameter-validity and parameter-reliability of fiber ODF estimates.

<p>(A) Parameter-reliability of the DTM and SFM estimates is defined as the angular difference of the PDD between model parameters in two simulations of the same fascicle configuration with different noise. (B) Parameter-validity is estimated by examining the angular difference between the peaks of the estimated and true fODF entered in the simulation. (C) Summary of parameter-reliability and parameter-validity. The black lines represent the true simulation fascicle directions and colored lines represent the difference between the estimated and the true fODF peak (parameter-validity). The shaded region represents parameter-reliability in estimating the peak of the fODF with different noise samples. The DTM PDD is an invalid estimate of the fiber directions over a wide range of crossing angles and unreliable when crossing angles are near 90 degrees. The SFM provides a valid estimate of fiber directions, and is reliable throughout.</p

The effects of number of measurement directions on model-accuracy.

<p>Subsamples from the 150 direction measurements were used to fit the DTM (bright grey) and SFM (dark grey) and to estimate rRMSE. Median in one participant is presented. Similar results are found in a second participant, and in 6 participants in b-value of 2000 s/mm².</p

RMSE and SNR of diffusion MRI measurements.

<p>Error bars delineate the 95% interquantile range. RMSE does not change across b-values, but SNR changes substantially, with the median decreasing from approximately 7 (b = 1000 s/mm²) to approximately 2 (b = 2000 s/mm²).</p

The diffusion tensor model cross-validates to an independent data set better than the data cross-validate.

<p>(A) The relative diffusion-attenuated signals S(θ, b) in a single voxel in two measurements are compared. Each point in the scatter-plot represents the repeated measurement in one of 150 diffusion directions. (B) The signal measured in the one data set is compared to the predicted signal from fitting a tensor model to the other data set. (C) The distribution of rRMSE values in the white matter for the diffusion tensor model (DTM). The rRMSE is calculated for each voxel as ratio of the RMSE in (B) (model prediction vs. data) divided by the RMSE in (A) (test-retest reliability). When values of rRMSE are smaller than 1 (right dashed line), the DTM better predicts a subsequent data set than repeated measurement. An optimal model will have an rRMSE distribution centered on </p><p></p><p></p><p></p><p><mn>1</mn></p><p></p><p><mn>2</mn></p><p></p><p></p><p></p><p></p> (left dashed line). Different curves show measurements at different b-values.<p></p

The sparse fascicle model (SFM) predicts the data better than data-to-data repeatability in almost all voxels in the white matter.

<p>For higher b-values (2000, 4000) more than 99.9% of the white matter voxels have rRMSE<1.</p

Local extrema in the diffusion signal attenuation do not cross-validate well.

<p>The two middle columns are independent measurements of the same voxel from the centrum semiovale. The three rows show measurements of this voxel obtained at b = 1000, 2000, and 4000. Notice that local minima and maxima differ between replications (arrows). The DTM (left column) and SFM (right column) predictions generally cross-validate well and are much smoother than the data. This particular voxel was chosen to illustrate a case where there are likely to be crossing fascicles. At this location and at b = 4000, the rRMSE of the DTM is greater than 1, while the rRMSE of the SFM is less than 1.</p