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Adabella Frierdich
Several domains of neuroscience offer map-like models that link location on the cortical surface to properties of sensory representation. Within cortical visual areas V1, V2, and V3, algebraic transformations can relate position in the visual field to the retinotopic representation on the flattened cortical sheet. A limit to the practical application of this structure-function model is that the cortex, while topologically a two-dimensional surface, is curved. Flattening of the curved surface to a plane unavoidably introduces local geometric distortions that are not accounted for in idealized models. Here, we show that this limitation is overcome by correcting the geometric distortion induced by cortical flattening. We use a mass-spring-damper simulation to create a registration between functional MRI retinotopic mapping data of visual areas V1, V2, and V3 and an algebraic model of retinotopy. This registration is then applied to the flattened cortical surface anatomy to create an anatomical template that is linked to the algebraic retinotopic model. This registered cortical template can be used to accurately predict the location and retinotopic organization of these early visual areas from cortical anatomy alone. Moreover, we show that prediction accuracy remains when extrapolating beyond the range of data used to inform the model, indicating that the registration reflects the retinotopic organization of visual cortex. We provide code for the mass-spring-damper technique, which has general utility for the registration of cortical structure and function beyond the visual cortex.
A two-dimensional projection of the visual world, termed a retinotopic map, is spread across the striate and extra-striate areas of the human brain. The organization of retinotopic maps has been described with algebraic functions that map position in the visual field to points on the cortical surface. These functions represent the cortical surface as a flat sheet. In fact, the surface of the brain is intrinsically curved. Flattening the cortical surface thus introduces geometric distortions of the cortical sheet that limit the fitting of algebraic functions to actual brain imaging data. We present a technique to fix the problem of geometric distortions. We collected retinotopic mapping data using functional MRI from a group of people. We treated the cortical surface as a mass-spring-damper system and corrected the topology of the cortical surface to register the functional imaging data to an algebraic model of retinotopic organization. From this registration we construct a template that is able to predict the retinotopic organization of cortical visual areas V1, V2, and V3 using only the brain anatomy of a subject. The accuracy of this prediction is comparable to that of functional measurement itself.
Copyright: 2014 Benson et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
The human occipital cortex contains multiple representations of the visual field, starting with primary visual cortex (V1; also called striate cortex). V1 lies primarily within the calcarine sulcus and represents the contralateral visual hemifield. The cortical surface dorsal and ventral to V1 contains the neighboring extrastriate regions V2 and V3, each of which represents a complete visual hemifield that is split into the upper visual quarterfield, ventral to V1, and the lower visual quarterfield, dorsal to V1. These three distinct retinotopic maps are organized on the cortical surface by distance from the fovea (eccentricity) and angle from the vertical meridian (polar angle) [1]. Polar angle sweeps dorsally down and ventrally up from the horizontal meridian in V1 (lying along the calcarine sulcus) around the foveal confluence then reverses direction at the V1/V2 and V2/V3 borders (Fig. 1A). Eccentricity radiates uniformly outward from the foveal confluence in all three visual field maps (Fig. 1B).
The result of the MSD simulation maps individual vertices within the cortical surface atlas to a specific visual area and visual field position. We show that this mapping may be used to accurately predict the retinotopic organization of extrastriate cortex in a novel subject who's brain anatomy is brought into register with the cortical surface atlas. Further, because the algebraic model is continuous, we find that the mapping may be used to accurately predict retinotopic data collected from beyond the eccentricity range of data used in the aggregate to derive the mapping.
By simulating the system until a low PE is achieved, we allow the constraints imposed by both the cortical anatomy and the functional model to relax into a solution that respects both kinds of information. Because the simulation incorporates KE, a nonlocal energy minimum may be found; it is therefore beneficial to use simulated annealing. Four simulations of 10 s (5,000 steps) each were performed such that the final arrangement of vertices in each simulation was used as the starting arrangement for the next simulation; spring ideal lengths were not recalculated, however, and the velocities were re-randomized such that the KE of the system was 10 rad2 g/s2 at the beginning of each simulation. The final arrangement of the four simulations with the lowest PE was chosen as the arrangement of the corrected topology. Vertices were assigned model polar angle and eccentricity values from their positions in the corrected topology by inverting the algebraic model of retinotopy. In other words, if the algebraic model of retinotopy predicts that a point (θ, ρ) in the visual field should lie at position (x, y) on the cortical surface, then a vertex with position (x, y) in the corrected topology would be assigned a polar angle of θ and an eccentricity of ρ.
Retinotopic mapping data was obtained from 19 subjects to an eccentricity of 10 of visual angle (dataset D10) (Fig. 1A, 1B). The brain anatomy from each subject was registered to an atlas of cortical surface topology (fsaverage_sym), and the across-subject, confidence-weighted mean aggregate of polar angle and eccentricity obtained. The reversals of polar angle that mark the boundaries of visual areas, and the regular progression of eccentricity from the occipital pole, is readily seen in the aggregate data. The goal of our work is to register the measured retinotopy in the volume with an algebraic model on the flattened cortical surface (Fig. 1C). The registration is performed within a flattened patch of the cortical surface atlas (Fig. 1D).
Registration via MSD simulation brought the aggregate retinotopic mapping values into alignment with the algebraic model by warping the cortex. The magnitude and direction of warping induced by this registration (i.e., the distance and angle between each vertex position in the flattened fsaverage_sym atlas space and its position in the corrected topology) is shown in Figs. 2A and 2B. Notably, the greatest displacement of vertices is found around the occipital pole. We presume that the warping of the cortex in our registration is correcting the geometric distortions created during flattening of this region of high curvature. The sulcal folding pattern of the original cortical surface atlas and the corrected topology following MSD simulation are shown in Figs. 2C and 2D respectively, along with the regional assignment (V1, V2, or V3) predicted by applying the algebraic model of retinotopic organization to vertices in the corrected topology.
(A) The magnitude of the warping of each vertex on the flattened cortical surface. The Hinds et al. V1 border [7] is marked by the dashed black line (throughout). (B) The direction of warping induced by MSD simulation upon each vertex from the original cortical surface atlas space (fsaverage_sym). Vertices with distortion magnitudes below 0.01 rad are plotted in unsaturated colors. (C) The flattened patch of the cortical surface atlas (fsaverage_sym) with sulcal curvature shown in light and dark gray. Regions V1, V2, and V3, as predicted by our method in the corrected topology then projected back to surface atlas space, are tinted red, green, and blue respectively. (D) The flattened cortical surface of the corrected topology with sulcal curvature shown in light and dark gray. A line plot of the algebraic model (Fig. 1C) to which the MSD simulation registered the functional data is shown. Regions V1, V2, and V3 are tinted as in panel C.
When aggregated within the cortical surface atlas, polar angle organization is largely consistent across subjects. A flattened aggregate map of the confidence-weighted mean polar angle of the 19 subjects in our 10 eccentricity dataset D10 is shown in Fig. 3A. Although regional boundaries in the aggregate map are apparent, the iso-angular curves in this organization do not resemble the smooth curves found in the algebraic model of retinotopic organization (Fig. 1C), suggesting an opportunity for registration via simulation to improve the predictive accuracy of the algebraic model. The polar angle organization following the MSD simulation is shown in Fig. 3B. As would be expected, minimization of energy in the MSD simulation has warped the cortex to bring the aggregate polar angle data into better alignment with the algebraic model of retinotopic organization. The algebraic model of retinotopy can then be projected back to the original cortical surface atlas (Fig. 3C). This smooth, continuous map of polar angle organization should resemble the measured polar angle functional data of any subject following the registration of their brain anatomy to the cortical surface atlas. We therefore refer to this representation as an anatomical template of retinotopy.
(A) The mean weighted aggregate polar angle map of all subjects in dataset D10 shown in the cortical surface atlas space. (B) The mean weighted aggregate polar angle map from panel A shown in the corrected topology following MSD warping. A line plot of the algebraic model to which the MSD simulation registered the functional data is shown over the functional data. (C) The polar angle template plotted on the fsaverage_sym pial surface. This template was calculated by converting the prediction of polar angle from the idealized model, as applied to vertices in the corrected topology, back to the fsaverage_sym atlas. (D) Median absolute leave-one-out polar angle error for all vertices with predicted eccentricties between 1.25 and 8.75 shown in the fsaverage_sym atlas space. This error was calculated by comparing the predicted polar angle generated from each subset of 18 of the 19 subjects in the 10 dataset to the observed polar angle of the remaining subject. The median absolute overall leave-one-out error is 10.93 (Tab. 1). The highest errors occur near the foveal confluence and at the dorsal border of V3. (E) Absolute leave-one-out error of the polar angle prediction across all regions (V1, V2, and V3), plotted according to the predicted polar angle value. The thin gray line represents the median error while the thick black line shows a best-fit 5th order polynomial to the median error. The dashed lines demarcate similar fits to the upper and lower error quartiles. Error plots for individual regions are given in Fig. S1.
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