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Smith-Kettlewell Eye Research Institute, San Francisco, California
Submitted 11 September 2006; accepted in final form 1 March 2007
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ABSTRACT |
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INTRODUCTION |
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). This mapping was first studied in human by Inouye (1909), and subsequent work (Holmes 1917; Horton and Hoyt 1991a,b) refined our understanding of the way that the visual world is represented in early visual cortex. Early behavioral studies (Wertheim 1894) and anatomical examination of the retina (Curcio et al. 1990; Østerberg 1935) showed that the fovea is a region of high spatial acuity. It is therefore not unexpected that the visual system allocates proportionally more gray matter to the representation of the fovea. This "cortical magnification" was quantified in the monkey by Talbot and Marshall (1941) and in more detail by Daniel and Whitteridge (1961). Subsequent studies (Dow et al. 1981 1985; Levi et al. 1984; Van Essen et al. 1984; Yap et al. 1987) have improved our estimates of this function, which describes the way that linear magnification changes with eccentricity. Magnification (in mm displacement on the cortex / ° in the visual field) is often described by the function:
 | (1) |
; Tusa et al. 1978): the area of cortical surface (in mm2) per area of visual field (in deg2). The distinctions among these metrics are important in this paper and we provide a detailed analysis in the APPENDIX.
In 1977, Schwartz proposed an analytical expression to describe the 2D mapping between retinal and cortical coordinates. By using complex numbers to represent both retinal eccentricity and angular deviation from the horizontal meridian simultaneously, he showed that the map of visual space in area V1 symbolized by w(E,P), was approximated by a complex-log transform of the retinal image (Schwartz 1977).
 | (2) |
This simple and convenient complex-log transform (or variants thereof—Balasubramanian et al. 2002; Polemeni et al. 2006; Schwartz 1983) captures several important features of the V1 map—most notably the roughly logarithmic scaling of cortical area with eccentricity. It has become the de facto standard for describing the shape of human V1. However, the complex-log representation makes some counterintuitive predictions about the way that space is represented in visual cortex (see APPENDIX). Specifically, the log-polar transform of Eq. 2 requires that the amount of cortex dedicated to a particular area of visual space varies with polar angle as well as eccentricity. There is no a priori reason for believing that this is the case, and in fact, behavioral studies show that psychophysical properties that might be expected to depend on local cortical area (for example Vernier acuity or contrast sensitivity) are essentially independent of angular position in the visual field (Rovamo and Virsu 1984; Toet and Levi 1992).
We term a change in cortical magnification with polar position but constant eccentricity meridional anisotropy todistinguish it from local anisotropies (see APPENDIX), following previous work (Buchanan-Smith and Heeley 1993; Mitchell et al. 1967; Westheimer 2003). It has also been termed "radial bias" (Fiorani et al. 1989; Van Essen et al. 1984).
Since the mid-1990s, researchers have used functional magnetic resonance imaging (fMRI) to measure the mapping of the visual field in the cerebral cortex. Early studies (DeYoe et al. 1996; Engel 1995; Sereno et al. 1995) demonstrated that it was possible to distinguish the "first tier" retinotopic visual areas (V1–V3). More recent work has focused on measuring the linear (1-dimensional) cortical magnification function (Duncan and Boynton 2003; Sereno 1995) or the area of cortex associated with different eccentricity ranges (Dougherty et al. 2003). However, none of these studies has quantified the 2-dimensional shape or the dependence on polar position.
Measuring the shape of V1 in the fovea is critical for several reasons. First the fovea is represented by a disproportionately large amount of cortex. Although the visual field extends over almost 180° of visual angle, approximately 30% of V1 is devoted to the central 3°. Second, the fovea supports a large proportion of functional vision: it is the location where visual acuity and color discrimination are highest, and many visual functions such as object and face identification, stereopsis and reading are essentially limited to the central visual field (Battista et al. 2005; Zegarra-Moran and Geiger 1993). Finally, as we show in this study, different models of retino-cortical mapping can make similar predictions in the periphery but very different predictions near the fovea depending on the strategy adopted to avoid the singularity at zero degrees eccentricity. If a simple analytical expression for the mapping between visual space and cortical location can be found, it is most strongly constrained (and most effectively tested) in the fovea. Conversely, incorrect models produce the most extreme errors in this region.
In the present study, we used high-resolution fMRI to measure the shape and morphometry of V1 and V2 to within 0.5° of the center of the visual field. This allowed us to model the way in which the amount of cortical area dedicated to a particular region of the visual field changes with both polar angle and eccentricity. We used an atlas fitting procedure to determine the retinotopic map in V1–V3 semi-automatically and made detailed local measurements of all the intrinsic distances between arrays of points in the V1 and V2 maps. We then averaged the metric data from eight observers to generate a "mean" V1 morphometry that strongly constrained our subsequent modeling. Our results indicate that the standard complex-log transform does not adequately describe the way in which visual space is mapped to cortex. In particular, we find that this model overestimates the dependence of MA on polar angle near the fovea. We show that a modified version of the complex-log transform provides a good fit to the measured data, and we suggest an analytic mapping function as a useful way to describe the transformation between 2D visual space and cortex. Our new model also provides a good approximation of human V2 and suggests detailed hypotheses about differences between V1 and V2.
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METHODS |
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Functional measurements
Functional magnetic resonance data were acquired on a GE 3T Signa LX scanner (GE Medical Systems, Milwaukee, WI) using a custom-built high-gain head coil. Functional acquisition was taken with a resolution of either 2 x 2 x 3 or 2.5 x 2.5 x 3 mm, using a spiral K-space sampling sequence of the T2*-weighted (TE: 30 ms, 2 interleaves, 70° flip angle) activation (Noll et al. 1995) with 21 or 30 slices perpendicular to the calcarine sulcus, and a volume repetition time (TR) of 3 s. Subjects’ heads were fixed throughout the measurement period by means of snug-fitting pads and tape across the chin and forehead.
Anatomical measurements
Several high-resolution T1 datasets (1 x 1 x 1 mm) were acquired for each subject using a fast three-dimensional (3D) SPGR sequence. These datasets were corrected for inhomogeneity, aligned, and averaged using the FMRIB software library (http:/www.fmrib.ox.ac.uk/fsl) and SPM (http://www.fil.ion.ucl.ac.uk/spm/) to generate low-noise, spatially homogeneous datasets. Segmentation was performed using mrGray software package (http://white.stanford.edu/
brian/mri/segmentUnfold.htm), and each white-matter segmentation was carefully hand-edited to ensure an accurate segmentation and surface reconstruction.
Stimulus generation and presentation
Rotating wedges and expanding rings containing 4-Hz contrast-reversing checkerboard stimuli were generated using custom software and routines from the Psychophysics Toolbox (Brainard 1997; Pelli 1997). To ensure good fixation, an extended fixation grid was present throughout the stimulus presentations (Tyler et al. 2005). The fixation grid was one pixel wide, light gray on dark gray background and consisted of a diagonal cross extending from corner to corner and four concentric circles at 0.5, 5, 10, and 15° eccentricity. All subjects were experienced psychophysical observers and eye-tracker measurements indicated that the SD of fixation position during fMRI experiments was less than 1/4°. Subjects viewed stimuli displayed on a small translucent screen mounted on the head coil. The stimuli were projected using a LCD-projector placed in a shielded box at the foot of the scanner bed. This display system subtended ±16° vertically and ±20° horizontally. The display, calibrated by a PR-650 PhotoResearch Spectroradiometer, had a mean luminance of 30 cd/m2.
Data analysis
The functional data were analyzed using the Stanford VISTA-Tools and an improved version of the ATLAS-toolbox (Dougherty et al. 2003). The ATLAS tools estimate area boundaries by fitting a canonical scheme for V1–V3 simultaneously to retinotopic data for both the eccentricity and meridian sweeps (Fig. 1). It is semi-automated because the operator has to generate a starting scheme and control the results of the fit, which may terminate in local minima of the optimization. Repeated fits were therefore made with different starting points to confirm stable convergence to the V1-V3 region identified by eye at the first pass. The matrix inversion algorithms for this optimization are described in Fischer and Modernitzi (2004). This method estimates the boundaries between the visual areas and also the position of each isopolar or isoeccentricity line. This fit provided an objective method for identifying these key features, providing the basis for the analysis of this paper.
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To specify the shape of cortical projection areas as accurately as possible, we estimated 3-dimensional distances along the cortical manifold using the Dijkstra algorithm in an overcomplete mesh of the boundary between white and gray matter. Measuring distances along a grid (in either 2 or 3 dimension) is subject to errors of the "city-block" type. The mesh was overcomplete in the sense of including all diagonals of the city-block connections because this reduces the city-block error significantly. We estimated the extent of this using a spherical model of the cortex and a Monte Carlo simulation. Irrespective of the radius of the sphere the error distribution was approximately normally distributed with a SD of only ±2.1%. The method also results in a mean overestimation of distances by 7.5%, but we corrected this overestimation by dividing the measures by a factor of 1.075, leaving only the residual error of ±2.1% (see supplementary materials1 ). All distance measurements presented in this paper are measured using this method within the 3D manifold of the undistorted cortex.
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RESULTS |
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We performed retinotopic mapping on 8 subjects and analyzed areas V1 and V2 in the right hemispheres using the semi-automated atlas fitting procedure (Dougherty et al. 2003). Briefly this method fits a schematic model of the early visual areas (V1–V3) to the measured polar angle and eccentricity maps simultaneously (see METHODS). Its purpose is to determine the borders between these early visual areas and to fit a smooth representation of both polar angle and eccentricity within these borders. We restricted our analysis to V1 and V2. For a reliable fit of V3, one would need a clear specification of the organization of its neighboring visual areas (V3A, V3B, hV4), which are still under dispute. Figure 1 shows two examples of our atlas fits.
Dependence of MA on eccentricity
The first parameter we measured was the classical eccentricity magnification function. Each subject's V1 was segmented into annular regions of ±15% eccentricity at several eccentricities (e.g., 0.6–0.8° for the 0.7° eccentricity and 6.8–9.2° for the 8° eccentricity). We then measured the size of each of these subregions to estimate the areal magnification MA (in mm2 of cortex/deg2 of visual field) for these eccentricity ranges, averaged across the range of polar positions (from –
/2 to /2 in radians or –90 to +90 in degrees). The square root of this number (M
) is a linear measurement (the geometric mean of the magnification in the eccentricity and polar angle directions) and can be fitted by Eq. 1. It is these numbers (M) that we show in for V1 and V2 in Fig. 2.
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, there is considerable variability in the linear magnification function across subjects—particularly at eccentricities <3°. However, we find that these differences can be modeled as changes in the scaling parameter. After accounting for differences in the overall size of area V1, the form of the cortical magnification function is remarkably constant (Figs. 2B and S1).
The cortical magnification function has been described previously in several primate species and for macaque monkey by several authors using different procedures and frameworks. In Fig. 3A, we plot several of these to allow a comparison between different species. For illustration, we multiplied the areal magnification functions MA provided by these authors and multiplied them with the values depicted in Fig. 3B. We then took the square root of these numbers to obtain linear magnification (M). Moreover, the concept of inverse magnification has been found useful for characterizing the foveal singularity, such as Duncan and Boynton (2003), Levi et al. (1984), McKee and Nakayama (1984), Rovamo and Virsu (1979), Stensaas et al. (2001), and Tyler (2001). A plot of the average inverse magnification function for our V1 data (Fig. 2C), shows no significant deviation from a straight line with a negative intercept of –0.95.
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To obtain an average 2D map of human V1, we identified the cortical locations for 10 meridians and 11 isoeccentricity rings (110 locations) in each subject's V1. The locations are specified as the voxels lying most closely within the cortical gray matter, defined for the zone 2 mm above the gray/white matter segmentation boundary (in the direction normal to this boundary). We then measured the distance between each pair of these 110 positions (6,050 distances per subject) along the 3D cortical manifold, resulting in a "distance grid" for each subject's cortical map. We averaged these distance grids to obtain an average distance grid, uncontaminated by any flattening distortions. This array constitutes the raw data for the specification of the shape of V1. This dataset is too large for publication in paper form, but we provide the data online because it represents an undistorted reference for the distances between the projections of these 110 visual field positions.
To fit our data to the available mapping models and present it in a digestible form for reference purposes, we projected the 3D data to a 2D surface. We reconstructed a 2D average V1 using a multi-dimensional scaling algorithm (metric MDS; Torgerson 1952). We flattened only once for the mean dataset rather than averaging several individually flattened cortices. The data for all the subjects were averaged in terms of the undistorted distances through the cortical manifold. This approach has several key advantages: the reconstruction is optimized to minimize the error within V1 only; the optimization is performed on a mean dataset with less noise than individual data; and the MDS algorithm gives a direct measurement of the amount of error introduced by the flattening. For our fits, the 2D reconstruction accounts for 95% of the variance in our data. This atlas for human V1 (shown in Fig. 4) provides a ready reference for the distance on the cortex between any two locations in the visual field.
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Schwartz's complex-log models predict (as derived in the APPENDIX) that the area representing the vertical meridian is greater than the area representing the horizontal meridian (see Fig. 5, A and B). This meridional anisotropy reaches the theoretical maximum of a factor of 2 at a = 0.77° eccentricity. We therefore made detailed measurements of this area in our dataset. Specifically, we measured the area of cortex in V1 representing the central 0.5 to 1.5° of eccentricity at nine different polar angles (Fig. 5C). To avoid distortion, we performed all measurements in the 3D surface reconstruction of the cortical manifold (see METHODS). The average cortical area of the areal bins was 52 mm2. There is no significant variation of areal magnification (MA) with polar position at the eccentricity where the complex-log model predicts a maximum variation.
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To clarify the amount of meridional anisotropy inherent in the Schwartz model, we estimated the ratio of the area representing a small square at the horizontal and the vertical meridians (see Fig. 6A, red digits, see also Fig. 5). The meridional anisotropy is 1 (or none) at the fovea, increasing up to a factor of 2 (in area) at 0.77° (at E = a), then decreasing again toward the periphery. In peripheral regions, the predicted anisotropy increases again (as E approaches b). In short, the Schwartz model predicts that there is up to twice as much visual cortex associated with the vertical meridian than the horizontal meridian and does so in a critical part of the visual field: the fovea.
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are consistent with these observations. We therefore suggest a new model that would preserve meridional isotropy of the area (MA) at the expense of a small degree of local anisotropy (linear, ratio of ME/MP).
In principle, there are several ways to achieve equal-area meridional mapping. A mathematically simple approach is to ensure that the imaginary part (azimuth or polar position) of the complex-log transform stays small. In practice, our suggestion is to first divide the polar position value by a large number (the larger the better, in practice 5 is sufficient) and subsequently multiply the real part of w(P,E) by the same value (to form the divisory model).
 | (3) |
For = , the derivative of this function is Eq. 1. Although Eq. 3 solves the meridional anisotropy, it does so at the expense of a strong compression of the local angles in the central fovea (Fig. 6B). To avoid this compression, we introduce a second method that takes an iterative approach to solving the problem of meridional anisotropy by a parametric shear of the polar angle magnification (parametric-2D-shear model). This model calculates the projected area for a given eccentricity and various polar angle ranges starting from the horizontal meridian. The increase of projected area further away from the horizontal meridian is then corrected iteratively by the application of successive local rotational shears in the polar direction. This operation results in the 2D shear map presented in Fig. 6C resulting in an equal-area model (Fig. 6D), note that the amount of shear varies with eccentricity and polar position. This model has near-zero meridional anisotropy (<0.1% after 10 min of iteration) at the cost of introducing a local anisotropy (indicated by the coloration in Fig. 6D). Because this iteration process is time consuming, we approximated the numerical fit with a 2D shear function that provides the same map within 0.3% of the iterative result.
Accordingly we suggest the mapping function
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1 suggested by Balasubramanian et al. (2002), but may not be necessary. We fitted these three models to our V1 data and found that both the divisory and the parametric-2D-shear model describe our data better than the models proposed by Schwartz and its variations proposed by Balasubramanian et al. (2002). (We provide additional descriptions, fits and errors, including Matlab code demonstrating the models, in supplementary materials).
In 2002, the Schwartz group proposed an intriguing extension of their model, the wedge-dipole model (Balasubramanian et al. 2002). This model presents a simple way to integrate V1–V3 into one single mapping function, potentially providing the possibility of predicting the detailed properties of these visual areas, by introducing a single polar shear/scaling parameter 1, 2, 3 for each retinotopic area. Unfortunately, the wedge-dipole model suffers even more from the meridional anisotropy flaw of the original Schwartz model (Fig. 7A). Our equal-area models fix this problem not only for V1 but also for V2 and V3, making the approach suggested by Balasubramanian et al. (2002) feasible. The correction is shown for the divisory and the parametric-2D-shear models in Fig. 7, B and C.
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The widths of area V1 and its neighboring maps (V2, V3) are important properties of the visual cortex mapping. However, these widths have never been measured systematically. All current models of V1 assume that at the representation of the center of the visual field, the width of V1 is 0. The width of V1 then increases with eccentricity until it reaches a maximum at 12–16°. It then decreases again at the peripheral end of the representation (see V1 diagram from Horton and Hoyt 1991a; their Fig. 3). Imaging data for human visual areas are usually presented as flatmaps, computationally unfolded and flattened approximations of the cortical surface (DeYoe et al. 1996; Press et al. 2001; Sereno et al. 1995; Tootell et al. 1996; Van Essen et al. 2001). These illustrations are useful for visualization purposes, but the computational flattening distorts the true shape of V1. We therefore measured the widths of V1 and V2 along isoeccentricity lines through the 3D cortical manifold. Our measurements show that the widths of both areas increases steadily from 0.5 to 8° eccentricity, a key property of human V1 and V2 that has not been previously quantified. The sketch of human V1 provided by Horton and Hoyt (1991a); for example, suggests that human V1 reaches its maximum width as early as 2.5° eccentricity (Fig. 8).
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All the retino-cortical projection functions described so far share one effect of polar position: the representation of the vertical meridian is longer than that of the horizontal meridian. If the cortical manifold can be flattened without distortion, it follows that the true lengths for the meridians, as measured on the 3D cortical sheet, must differ as well. To test if this particular effect of polar position is present in our data, we measured the lengths of lines of constant polar angle within cortex. For each line, we plotted eccentricity in visual space against cortical distance. Crucially, these measurements were made in the undistorted 3D cortical manifold and they provide a critical test of whether V1 can be described with 2D models.
We found that in V1 the average vertical meridian representation (n = 8) is longer than the average horizontal meridian representation by a factor of 1.26 ± 0.06 over the range 0.5–12° eccentricity (Fig. 9) very close to the predicted factor of 1.27. The lengths of these isopolar meridians increase gradually with polar angle, and there was no difference between the upper and lower visual fields (see supplementary material). These results match the predictions from our models (see supplementary materials). This match between the measured lengths and the planar model with no intrinsic curvature suggests that there is little intrinsic curvature in V1. The consequence of the difference in length between the two meridian lines is that MP depends on polar position: it is larger for the vertical meridian than for the horizontal meridian. Because MA is independent of P (Fig. 5), ME must vary with P as well. And because MA = ME*MP, the local anisotropy (ME/MP) must vary with polar position as well.
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DISCUSSION |
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These constraints set up competing requirements among global isotropy, local isotropy, and local conformality of the V1 map and of the two segments of the V2 map lying adjacent to it. In principle, these requirements could all be satisfied if the maps had appropriate intrinsic curvature, but violations of one or other of these principles are necessary if the maps are intrinsically flat or if the curvature is constrained by other factors. Our detailed measurements assess both the intrinsic curvature of the mappings and the particular solution that has evolved in the human brain among the competing mapping principles.
We first consider the foveal measurements in the present study and how they constrain the layout of the foveal confluence. We then discuss the assumptions and implications of the three-dimensional shape of visual cortex, and why we think that flat models for V1 are useful. Finally, we discuss inter-subject variability and show that it consists of predominantly a scaling factor and with little variation in the other mapping parameters.
Measuring the fovea
One of the most important contributions of this paper is to provide detailed data down to 0.5° of eccentricity. Retinotopic mapping in the fovea is technically challenging, and consequently data from this region are not usually analyzed in studies of human cortical magnification. One of confounding factors for foveal measurements is the occurrence of spontaneous eye movements. Assuming that subjects fixated most of the time and that eye movements are random, sufficient averaging should reveal the mapping up to the centermost part of the visual field with gradually increasing noise toward the center of the fovea. Although all subjects are instructed to maintain fixation at the center of the fixation cross, some eye movements are inevitable. However, the use of an invariant fixation grid while viewing the cycling retinotopic stimulus provided substantial improvement in the fixation accuracy during the scans and repeated averaging reduces the effects of momentary eye movements. In separate experiments, we made high-precision measurements of our subjects’ ability to fixate by presenting the retinotopy stimuli while simultaneously imaging the fovea using a Rodenstock scanning laser ophthalmoscope. On average, our subjects fixated during retinotopic stimulation with an SD of only 11 arc min. Our measurements of subject fixation in the MR scanner are consistent with this value although our MR eyetracker can only measure with a precision of 20 min of arc.
Another problem with measuring at the foveal representation is the limited spatial resolution of fMRI. Based on the analysis of this paper, the average human foveal projection (0–0.5° of eccentricity) of V1 covers an area of 142 mm2 (14 voxels) and is 17 mm (5 voxels) wide at 0.5° while being only 9 mm (3 voxels) wide at 0.25° eccentricity. This calculation illustrates how difficult it would be to measure the widths of V1 and V2 at 0.25° eccentricity, where each half of V2 would be only 4 mm wide (i.e., the V2/V3 border is little more than a voxel away from the V1/V2 border). To image the last 0.5° of the fovea, one would therefore need higher-resolution scans. We estimate that using 1.53 mm voxels would allow substantially better detail, allowing one to map down to 0.2° eccentricity.
Layout of the foveal confluence
The maps of the early visual areas V1, V2 (dorsal and ventral), and V3 (dorsal and ventral) converge toward a "foveal confluence" near the occipital pole in human brains (Brewer et al. 2005; Dougherty et al. 2003; Press et al. 2001; Sereno et al. 1995; Tootell et al. 1996). Unfortunately, the layout of early visual areas at the very center of the fovea remains unclear. All current theoretical models based on the Balasubramanian wedge-dipole layout propose that V1, the two halves of V2, and even the two halves of V3 meet at a single point. Current anatomical reviews in monkeys, however (Rosa and Tweedale 2005 for marmoset; Gattass et al. 2005 for cebus and macaque), suggest that V2 does not come down to a point but rather forms a strip with significant width. Our measurements indicate that if V2 forms a strip at the fovea, at least in humans this strip is not more than 7 mm wide at 0.5° (see Fig. 8B). Gattass et al. (2005) on the other hand suggests a rather wide strip of V2 from 0 
2° eccentricity, even proposing a larger representation of foveal V2 than foveal V1. This is in clear disagreement with the width data we present here (see Fig. 8; V2 is narrower than V1 at every eccentricity). The results of Rosa and Tweedale (2002) on the other hand indicate that this strip narrows down <1 mm in the tiny brain of the marmoset monkey.
Flatness of human V1 or the "rugby ball" concept
All published models of the properties of human V1 would be invalid if human V1 contained significant intrinsic curvature. One could imagine that human V1 was intrinsically ellipsoidal, having the 3D shape of the surface a rugby ball (or a part of it), where the poles represent the fovea and the far periphery. Each seam on the rugby ball would represent a different polar angle, and each one could thus have precisely the same geometry, allowing a log-polar projection function with ideal properties: local and global isotropy as well as conformality of the angular mappings. Any 2D projection of this ideal mapping would, however, introduce systematic distortion ruining these perfect properties. A similar argument was presented by Rovamo and Virsu (1984), arguing that macaque operculum is not flat. Although we know that human V1 is a convoluted manifold in 3D space, it appears that the V1 geometry is intrinsically flat and only folded within the calcarine sulcus (see following paragraphs). Human V1 may, thus, be more folded but appears to be flatter than for example squirrel monkey V1, which incorporates substantial intrinsic curvature (D. L. Adams, personal communication). Kaplow and Schwartz (1986) investigated the flatness of macaque V1 and came to the conclusion that the error due to flattening is small (5%) and therefore intrinsic curvature is negligible. However, the fact that the average distortion is small means neither that it is not systematic nor that is small in every location.
The strongest argument for human V1 being intrinsically flat is the increasing length of the isopolar lines with polar angle from the horizontal meridian (see Fig. 9, and supplementary materials). While a rugby-ball-shaped V1 would imply that the lines should have equal length, any flat V1 model consistent with our data would predict that these lengths increase with polar angle. Tootell et al. (1988) reported such a result in macaque. However, these measurements were made in a flattened macaque cortex, and the difference might possibly have been due to some flattening distortion. Here, we measure the same effect directly in the folded and unflattened cortex, providing strong evidence that human V1 is at least not a perfectly symmetric rugby ball. Further, the strength and layout of the anisotropy matches the predictions from our flat model (supplementary Fig. 3), suggesting that V1 is indeed intrinsically flat.
For V2 the situation is different, however. Although the models we propose in Fig. 7 assume that both V2 and V1 lie within a homogeneous flat plane, this may not be the case. To our knowledge, the flatness of V2 has not been investigated so far. The equal-area maps of Fig. 7B and C, illustrate that an equal-area planar map of V2 must violate conformality. Maintaining conformality in a flat V2 would require a huge degree of meridional anisotropy (see Figs. 7A and 8B, red lines). Both problems could be solved if the V2 mapping were intrinsically ellipsoidal (the rugby-ball model). In this context, it is interesting to note that—in contrast to V1—all measured V2-meridian representations from horizontal through vertical, have the same length within the precision of our measurement. This suggests that V2 is probably best described with the rugby-ball model rather than the intrinsically flat and only folded variants of the Balasubramanian model. We do not have sufficient material to test this hypothesis further, but we present and illustrate it in supplementary materials.
Reconstructed average map of human V1
Because we did not find systematic curvature in V1, we reconstructed a 2D reconstruction of human V1 in Fig. 4A. This map allows a ready reference for the average distance between the representations of visual field locations on human primary visual cortex. An interesting feature of this reconstruction is that the isoeccentricity lines beyond 8° appear convex, with their representation around the horizontal meridian closer to the fovea than the ends of the isoeccentricity representations. This is an interesting feature that is strongly present in the data from Tootell et al. (1988) where it is clearly evident as early as 3° of eccentricity. Although the dipole model proposed by Schwartz (1983) predicts such convex isoeccentricity lines, the strength and the onset at small eccentricities would suggest a very small value for b. A very small b, however, would put several other constraints on the model (see supplementary materials).
The map of local deviation of the reconstruction (Fig. 4B) with the originally measured grid shows that there is more local error in the perifoveal part of the reconstruction and more error in the ventral half than in the dorsal half. We did not find any other significant difference between the ventral and the dorsal half of V1. The higher error in the foveal part of the reconstruction could be explained by the fact that smaller distances are more difficult to measure (see supplementary materials).
Intersubject variability
Dougherty et al. (2003) investigated the representation of the visual field in V1–V3 of six human subjects using a similar methodology. The Dougherty et al. study illustrated the high degree of intersubject variability in V1 area (up 2.4 times; see also Stensaas et al. 1974). We confirm these results: the largest V1 in our sample has 2.3 times the area of the smallest. However, as we illustrate, these extreme cases basically differ in overall size and have no significant difference in the shape of their V1s (see supplementary materials). This invariance is consistent with the finding by Tootell et al. (1988) that magnification correlates with the size of striate cortex.
It could be argued that due to the substantial variation between different subjects’ V1s, there is little point in making a detailed average. If one is not in a position to measure the V1 morphometry for individual subjects, however, a good average is certainly the best estimate of its parameters. This is also true if we wish to relate cortical magnification to psychophysical data averaged across subjects. Besides, it is worth noting than a maximum variation of 2.3 in area reflects a factor of only 1.5 in linear size and that the SD of our k parameter is only ±15%.
Is visual cortex conformal?
One of the central questions to distinguish the different variants of models proposed here is the degree to which they exhibit conformality of the angular mapping. The original Schwartz model is perfectly conformal for V1, such that right angles in the visual field project to right angles on cortex, for example. However, subsequent refinements to this model introduce a shear in the polar angle direction (Balasubramanian et al. 2002), and, as the authors note, this eliminates conformality. Unfortunately there are no good estimates as to how conformal the organization of visual cortex is, and even the benefits of a conformal visual cortex are unclear. This issue is even less clear when considering V2 and V3. In the fovea, there is theoretically space only for a conformal V1 and V2 (leaving no room for V3), but this conformality can be achieved only at the cost of a strong meridional anisotropy between the horizontal and vertical meridian representations. That is, conformality would produce a small horizontal meridian and large vertical meridian representations in V1, with the opposite arrangement in V2). However, this does not match our data (Figs. 9 and S3).
Different forms of magnification and anisotropy
Our results agree with current estimates of cortical magnification in humans (Dougherty et al. 2003; Levi et al. 1984; Slotnick et al. 2001), but cortical magnification must be considered as a 2D property of visual cortex, depending on both eccentricity and polar position. When investigating cortical magnification, for example using psychophysical methods, it is essential to be clear about what kind of magnification one is investigating (linear or areal), and it is also important to map it in both dimensions of the visual field. The models that we have developed (Figs. 6 and 7) enforce equal-area magnification within each eccentricity range. This uniformity comes at the cost of a systematic pattern of local anisotropy. In particular, we predict large differences in local anisotropy on the horizontal and the vertical meridian representations at 0.77° eccentricity.
Most psychophysical studies relating performance to (presumed) cortical area have used relatively eccentric stimuli. We suggest that investigation of such properties at the critical eccentricities <1° would allow a test of the predictions we make here, possibly improving the fit of the parameters a, b, and k. Such studies could also evaluate the intersubject variability of the parafoveal mapping in the manner used by Duncan and Boynton (2003) for analyzing mapping functions beyond 1.5° eccentricity
Ocular dominance columns and anisotropy
One of the most prominent anatomical features of human and macaque V1 are the ocular dominance columns (Horton and Hedley-Whyte 1984; LeVay et al. 1975). These and other substructures of visual maps are not part of any current retino-cortical projection model. There is strong evidence that each of these substructures contain continuous representations of most parts of the visual fields, and that these maps are interdigitated (Blasdel and Fitzpatrick 1984; Shipp and Zeki 2002; Tootell et al. 1988) into the overall map of a given area. However, in V1 these interdigitated maps are shown in the monocular input layers (Blasdel and Fitzpatrick 1984; Tootell et al. 1988) and may not extend throughout the binocular layers.
In V2 Shipp and Zeki (2002) have termed the organization of these submaps a "ratchet model" and demonstrated that the mean of each of the 1 mm wide stripes follows the continuous macro map, a concept that could easily be integrated into the current models. With the current resolution of fMRI being 3 mm, however, these micro deviations are invisible. Tootell et al. (1988) found local anisotropy in macaque V1 but no overrepresentation (in area, as estimated by width) of the vertical versus the horizontal meridian. They suggest that the pattern of local anisotropy is associated with the pattern of ocular dominance columns. To be precise, they suggest a higher magnification perpendicular to the ocular dominance columns than along them. We would like to point out that this is precisely the same pattern suggested by our parametric-2D-shear model. Based on this model, we would suggest a magnification of roughly 1.4:1 (perpendicular/parallel) across pairs of ocular dominance columns (or in layer 2/3 and 6 or in the macro map), implying a ratio of 0.7:1 within one ocular dominance column (or within the micro map).
The pattern of local anisotropy suggested by our model matches the pattern observed in several monkey species (Gattass et al. 1987 for cebus monkey; Tootell et al. 1988 for macaque monkey; Fritsches and Rosa 1996 for marmoset monkey, Adams and Horton 2003 for squirrel monkey). In short: sectors along the horizontal meridian are compressed along eccentricity axis, whereas sectors along the vertical meridians are elongated (see Fig. 6D).
For V2, we suggest (and measure) an overall anisotropy of 2:1, that is V2 is elongated along the eccentricity direction or parallel to the stripe structure. Assuming that the complete eccentricity range is represented in each of the three stripe types, this would result in a local anisotropy of 0.66:1 within a stripe.
Conclusions
Cortical magnification is a 2D property of visual cortex, depending on both eccentricity and polar position. It can be described in terms of either linear magnification (in various directions) or areal magnification. When investigating cortical magnification, it is important to express the magnification function for both axes of the maps in terms of both the local and global anisotropies of the mapping. We provide high-density mapping data from many subjects over an extended range of eccentricities and derive an accurate mapping formalism (Eq. 5) that accounts for the cortical morphometry of the first-tier retinotopic areas within a geometric error of 0.3%.
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APPENDIX |
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The concept of cortical magnification was introduced by Talbot and Marshall (1941) and described as a function by Daniel and Whitteridge (1961), who defined it as the displacement of two cells in millimeters on the cortex divided by the displacement of their receptive field centers in degrees (in units of mm/°). This is a 1-dimensional measurement and accordingly magnification is often specified as:
 | (A1) |
).
First, the cortical magnification is the same in both the angular and radial directions (MP = ME). For clarity, we will call this local isotropy (or anisotropy, where violated). However, it is important to note that later studies indicate that V1 may show only approximate local isotropy. For example, Adams and Horton (2003) find a pattern of systematic local anisotropy in V1 of the squirrel monkey.
Second, the cortical magnification for a given eccentricity is the same at different angular positions. This means that the cortical magnification changes with E, but not with P, and is therefore identical along the horizontal and the vertical meridians for a given eccentricity. We will call this property meridional isotropy (but see Van Essen et al. 1984 for a report of meridional anisotropy in macaque monkey).
It is important not to confuse these two properties of the magnification parameters. The degree of local (an)isotropy says little about any meridional difference between the horizontal and the vertical meridians (or intermediate meridians), and the degree of meridional (an)isotropy says nothing about the local ratio between MP and ME at a given point of the visual field. In this paper, we show that V1 exhibits local anisotropy, but a reasonable degree of meridional isotropy is found for the areal magnification MA.
Areal magnification (Myerson et al. 1977; Tusa et al. 1978) is a 2D metric: the area of cortex per area of visual field [MA(E,P), in mm2/deg2], where (P,E) are the coordinates of the visual field. Areal and linear magnifications are closely related. Specifically,
 | (A2) |
. In short, areal magnification, as well as the linear magnifications in any direction, are 2D functions of both visual field dimensions. The analysis of any of these functions in terms of one of its dimensions has to be parameterized by the other, orthogonal, dimension. In particular, any magnification as a function of eccentricity may be specified separately for the horizontal meridian by M(E,0) and at the vertical meridian by M(E,p/2), respectively (see Fig. 6).
Schwartz (1994) pointed out that the linear magnification is a partial derivative of the overall retino-cortical projection function w(E,P). To be precise, there are three interesting derivatives of this function: we can consider the partial derivative with respect to eccentricity which yields the conventional linear eccentricity magnification function, along lines of constant angular position:
 | (A3) |
 | (A4) |
 | (A5) |
Retino-cortical mapping function
It is clear that the various forms of the cortical magnification function depend on the form of the retino-cortical mapping function w(E,P). Schwartz (1977) proposed that this function can be approximated by:
 | (A6) |
 | (A7) |
 | (A8a) |
 | (A8b) |
Estimates of the magnification function based on mapping theory (Schwartz 1977, 1994), on anatomical measurements (Horton and Hoyt 1991a), or on psychophysical estimates (Levi et al. 1984; Yap et al. 1987) suggest a value of 0.75° for a. [Dow et al. (1981, 1985) give a lower value, but their data were reinterpreted by Levi et al. (1984) to be compatible with the 0.75° value]. The Schwartz model therefore predicts the strongest global anisotropy around the perifoveal eccentricity of 0.75°.
The original Schwartz model only describes the singularity at the foveal portion of the projection function. In a figure caption, Schwartz (1983) extended the complex-log transform to a dipole model to account for the peripheral mapping behavior. This model (Fig. 6A) introduces a third parameter b, creating a second singularity in the periphery:
 | (A9) |
 | (A10) |
 | (A11) |
into their power series expression for their magnification function. The dipole model thus provides a simpler approach to the same concept. However, just as estimating the parameter a requires precise measurements in the fovea, estimating the parameter b requires good peripheral data. This dipole model has received considerably less attention than the original model and there are no good estimates of b. Magnification data including peripheral data are therefore usually fit to a power function (Adams and Horton 2003; van Essen et al. 1984) rather than to Eq. A11. Those studies that have attempted to estimate it give a value for b between 90 and 180° (Balasubramanian et al. 2002). In conclusion, there has been no previous publication of the exact form of the 2D magnification function implied by the complex-log transform, which is widely regarded as an adequate characterization of the cortical mapping function. Our analysis reveals that the exact form of this is meridionally anisotropic and that no mapping onto a planar metric can simultaneously preserve the meridional isotropy, local isotropy, and conformality of the resultant map.
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FOOTNOTES |
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1 The online version of this article contains supplemental data. 
Address for reprint requests and other correspondence: M. M. Schira, The Smith-Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco, CA 94115 (E-mail: mark{at}ski.org)
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ABSTRACT
INTRODUCTION
