Improved method for retinotopy constrained source estimation of visual‐evoked responses

Abstract

Retinotopy constrained source estimation (RCSE) is a method for noninvasively measuring the time courses of activation in early visual areas using magnetoencephalography (MEG) or electroencephalography (EEG). Unlike conventional equivalent current dipole or distributed source models, the use of multiple, retinotopically mapped stimulus locations to simultaneously constrain the solutions allows for the estimation of independent waveforms for visual areas V1, V2, and V3, despite their close proximity to each other. We describe modifications that improve the reliability and efficiency of this method. First, we find that increasing the number and size of visual stimuli results in source estimates that are less susceptible to noise. Second, to create a more accurate forward solution, we have explicitly modeled the cortical point spread of individual visual stimuli. Dipoles are represented as extended patches on the cortical surface, which take into account the estimated receptive field size at each location in V1, V2, and V3 as well as the contributions from contralateral, ipsilateral, dorsal, and ventral portions of the visual areas. Third, we implemented a map fitting procedure to deform a template to match individual subject retinotopic maps derived from functional magnetic resonance imaging (fMRI). This improves the efficiency of the overall method by allowing automated dipole selection, and it makes the results less sensitive to physiological noise in fMRI retinotopy data. Finally, the iteratively reweighted least squares (IRLS) method was used to reduce the contribution from stimulus locations with high residual error for robust estimation of visual evoked responses. Hum Brain Mapp, 2013. © 2011 Wiley Periodicals, Inc.

INTRODUCTION

The ability to noninvasively measure activity with high temporal resolution in individual visual cortical areas would allow for better understanding of the properties of those visual areas and of their interactions with higher‐level areas, for example, related to the allocation of attention. With commonly used brain imaging methods, it is not, however, possible to measure activity in areas such as V1, V2, and V3 with adequate spatial or temporal precision. Functional magnetic resonance imaging (fMRI) lacks the necessary temporal resolution, and magnetoencephalography (MEG) and electroencephalography (EEG) lack spatial precision.

Although it is possible to accurately localize early visual evoked responses measured with MEG or EEG to primary visual cortex [Ahlfors et al., 1992; Aine et al., 1996; Baker et al., 2006; Im et al., 2005; Moradi et al., 2003; Sharon et al., 2007], it is generally very difficult to generate distinct, independent source estimates for V1, V2, and V3. This is due to the proximity of these visual areas, resulting in significant crosstalk between them, and therefore ambiguity in the source estimates [Liu et al., 1998; Sereno, 1998]. Source estimation with a few equivalent current dipoles is problematic, usually requiring that multiple visual areas be modeled by a single dipole, even when fMRI and MRI data are used to determine dipole locations or orientations [Ahlfors et al., 1999; Auranen et al., 2009; Di Russo et al., 2005; Vanni et al., 2004]. Distributed source methods are usually no better, because the waveform for any given dipole will be quite similar to neighboring dipoles within about 20 mm [Bonmassar et al., 2001; Dale et al., 2000; Hagler et al., 2009; Kajihara et al., 2004; Liu et al., 2002; Moradi et al., 2003]. Recently, heirachical Bayesian source estimation methods, using fMRI data as a prior, have been shown to improve the localization and separability of the multiple, extended sources involved in the visual evoked response, but limitations related to crosstalk and separation of sources remain [Auranen et al., 2009; Yoshioka et al., 2008].

We recently developed a retinotopy‐constrained source estimation (RCSE) method for visual evoked responses measured with MEG or EEG that is intended to provide independent source estimates for V1, V2, and V3 [Hagler et al., 2009]. FMRI‐derived retinotopic maps are used to identify the cortical locations in V1, V2, and V3 expected to respond to stimuli at various visual field locations. Source models are constructed and time courses are estimated for each visual area. A key feature of this method is that multiple stimulus locations are used to simultaneously constrain the solutions [Ales et al., 2010; Hagler et al., 2009; Slotnick et al., 1999]. As the stimulus location varies, the dipole orientations change in ways that can be predicted by structural and functional MRI data. The pattern of dipole orientations versus stimulus location provides a unique signature for each area, greatly reducing ambiguity and providing more independent source estimates than can be obtained with conventional equivalent current dipole or distributed source estimation methods. Unlike distributed source estimation methods, which have many more sources than sensors, RCSE has only a few sources and potentially thousands of sensor data points, such that the solution is highly over‐determined. A similar implementation of this principle has recently been described to estimate source waveforms for V1 and V2 from multifocal EEG data [Ales et al., 2010].

A limitation of the earlier implementations of this method was that dipoles for a given stimulus location were modeled as individual vertices on a cortical surface mesh [Ales et al., 2010; Hagler et al., 2009], without taking into account the cortical point spread function of a stimulus, and how this can vary between visual areas. Receptive field sizes determine how large a patch of cortex is activated in response to a stimulus, and these are known to be larger for V2 and V3, relative to V1 [Dumoulin and Wandell, 2008; Zeki, 1978]. Given the complicated folding pattern of the cortex, if the cortical point spread of a stimulus is large, cancellation becomes more likely, thus reducing the measurable field relative to an unfolded cortical patch of similar area. A slight misspecification of the cortical location can have a large effect on the expected orientation and amplitude of the dipole. Accurately modeling the spatial extent of the cortical activation should provide greater stability of the estimates because they will be less sensitive to these problems.

For our initial application of the RCSE method, dipoles were selected manually, which is undesirable for three reasons. First, it is extremely time consuming, making studies with more than a few subjects impractical. Second, dipole orientations vary substantially between nearest neighbor locations on the cortical surface. Because inaccurate dipole orientations will result in inaccurate source estimates, the choice between several plausible locations becomes a critical question. Third, fMRI retinotopy mapping data is sometimes noisy, making manual judgements difficult, arbitrary, and often wrong. For these reasons, we have developed a method for automatically selecting extended cortical surface patches using a template map fitted to fMRI retinotopy data.

Despite these improvements, inaccurately specified dipole locations and orientations remain a concern. Errors specific to one or more particular stimulus locations, for example because of local errors in the retinotopic map fitting or cortical surface reconstruction, may contaminate the source estimates, even when many stimulus locations are used to constrain the solution. One way to address this problem is to perform a multidimensional, nonlinear search for better fitting dipole locations or orientations [Ales et al., 2010]. The problem with this type of solution is that the over‐dertermined nature of the RCSE method is eroded by the introduction of many free parameters (e.g., one or more per stimulus location). The approach we have taken is to selectively reduce the contribution of stimulus locations with the largest residual error, through the use of iteratively reweighted least squares (IRLS) [Holland and Welsch, 1977; Huber 1981].

METHODS

Participants

Two adult females were included in this study (ages 25 and 26). Two additional subjects (one male, one female) were excluded because fMRI retinotopy data were extremely noisy and therefore unusable. Subjects were right handed and had normal or corrected to normal vision. The experimental protocol was approved by the UCSD institutional review board, and informed consent was obtained from all participants.

Stimulus Presentation and Behavioral Monitoring

For fMRI sessions, stimuli were presented via a mirror reflection of a plastic screen placed inside the bore of the magnet, and a standard video projector with a custom zoom lens was used to project images onto this screen from a distance. For MEG sessions, visual stimuli were presented with a three mirror DLP projector. An MRI‐compatible fiber‐optical button box was used for fMRI experiments; for MEG sessions, a finger lifting response was used with a laser and light sensor. Dental impression bitebars to immobilize the top row of teeth were used for fMRI sessions. Subjects' heads were supported and surrounded by foam padding. To relate the eccentricity of stimuli between fMRI and MEG sessions, the distances between the nasion and the screen was measured for each session and 1 cm added to account for the approximate distance to the eyes from the nasion. The maximum visual angles for fMRI sessions ranged from 26° to 29° (top to bottom of displayable area), and were fixed at 25°for MEG sessions.

MEG Measurement

MEG signals were measured with an Elekta/Neuromag Vectorview 306 channel whole head neuromagnetometer, which comprises 2 planar gradiometers and one magnetometer at each of 102 locations. Electrooculogram electrodes were used to monitor eye blinks and movements. The sampling frequency for the MEG recording was 1,000 Hz with an antialiasing low‐pass filter of 333 Hz. Data were band‐pass filtered offline between 0.2 and 200 Hz with a notch filter at 60 Hz. The locations of the nasion, preauricular points, and additional locations on the scalp were measured using a FastTrack 3D digitizer (Polhemus, Colchester, VT). Very noisy or flat channels were excluded from analysis. After rejecting trials containing artifacts such as eye blinks and movements, data from remaining trials for a given stimulus location were used to calculate average time series time‐locked to stimulus onset.

Stimuli For MEG Sessions

Visual stimuli were portions of a black and white dartboard pattern presented for 100 ms at maximum contrast on a gray background. The size of stimuli and number of stimulus locations are described in Results and depicted in Figures 4 and 6. To ensure that subjects maintained a stable level of alertness and maintained central fixation, subjects performed a simple task in which they made a finger lift response upon rare dimming of the central fixation cross (approximately once every 5–10 s). Stimulus trials within 500 ms before or after a button press were excluded. The interval between successive stimulus onsets was fixed at 117 ms. Some stimulus events were “null” events in which no stimulus was presented (four times as often as an ordinary stimulus). The average of these null events reflects the ongoing activity from a given trial that overlaps with the response to successive trials. This overlap was removed by subtracting the averaged null event from the other stimulus conditions. In a single MEG session with up to 45 min of stimulus presentation (separated into 2.5 min blocks with rest periods of 30 s or more), up to ˜16,000 total trials were acquired, divided approximately equally across all stimulus locations.

Figure 4.

Variation between sets of twelve stimuli and full set of 36. A, D, and G. Subsets of twelve stimuli used to constrain RCSE. Red outlines, which were not visible during the actual MEG session, show the locations of all 36 stimuli that were presented one at time. B, E, and H. RCSE source waveforms constrained by 12 stimuli. C, F, and I. Normalized variance of data, fit, and residual error. J–L: Stimuli used, RCSE source waveforms, and normalized variance for full set of 36 stimuli. Number of trials per location was limited so that total number of trials was approximately equal (∼1,600) for all estimates in Figures 4 and 5. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Figure 6.

Effect of stimulus size on sensor waveforms and source estimates. A: Comparison of small (0.5° visual angle long, 5° polar angle wide) and large (1.2° visual angle long, 22° polar angle wide) stimuli used in separate MEG sessions with Subject 1. B: Sensor waveforms for all gradiometers for small stimulus shown in A. C: Sensor waveforms for large stimulus shown in A. D: Locations of all 100 small stimuli outlined in red. E: RCSE source estimates constrained by 100 small stimuli. F: Normalized variance for data, fit, and residual error. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

MRI and fMRI Image Collection

Subjects were scanned with a GE 3T scanner using a GE 8‐channel phased array head coil. High‐resolution T ₁‐weighted images were acquired to generate cortical surface models (TR = 10.5 ms, flip angle = 15°, bandwidth = 20.83 kHz, 256 × 256 matrix, 180 sagittal slices, 1 × 1 × 1 mm³ voxels). Echo‐planar imaging (EPI) was used to obtain T ₂ ^*‐weighted functional images in the axial plane with 3.5 mm isotropic resolution (TR = 2,500 ms, TE = 30 ms, flip angle = 90°, bandwidth = 62.5 kHz, 41 axial slices, 64 × 64 matrix, FOV = 224 mm).

Correction of Distortion in Structural and Functional MRI Images

In‐plane and through‐plane gradient warping in structural and function MRI images was corrected by applying a predefined, scanner specific nonlinear transformation [Jovicich et al., 2006]. B0‐inhomogeneity distortions in fMRI data were corrected using the reversing gradient method in which a pair of test images with opposite phase‐encode polarity was acquired to estimate a displacement field [Chang and Fitzpatrick, 1992; Holland et al., 2010; Morgan et al., 2004]. The resulting displacement field was then applied to full‐length EPI fMRI images with identical slice prescriptions as the test images.

Cortical Surface Modeling

The FreeSurfer software package version 3.0.5 (http://surfer.nmr.mgh.harvard.edu/) was used to create cortical surface models from T ₁‐weighted MRI images. The processing includes creation of a brain mask, intensity normalization, segmentation of subcortical white matter and deep gray matter structures, tessellation of the boundary between gray and white matter, topology correction, and surface deformation following intensity gradients to optimally place the outer white matter and pial surfaces [Dale et al., 1999; Dale and Sereno, 1993; Fischl et al., 2001; Fischl et al., 2002; Fischl et al., 1999; Segonne et al., 2004; Segonne et al., 2007]. The resulting surfaces were thoroughly checked for errors in occipital cortex, and manual editing of the white matter segmentation and brain mask was performed to correct local defects. Gray‐mid surfaces, roughly half‐way between the white and pial surfaces were generated by expanding the white matter surface a distance equal to half the estimated cortical thickness.

MEG Forward Solution

Forward solutions were calculated using the boundary element method (BEM) with the linear collocation algorithm [Mosher et al., 1999; Oostendorp and van Oosterom, 1989]. Software used for these calculations originated from the BrainStorm software package version 2.0 (http://neuroimage.usc.edu/brainstorm), and were modified further by M.X. Huang. The FreeSurfer watershed program was used to create surface tessellations from high‐resolution anatomical MRI data for the following three boundary layers: inner skull, outer skull, and outer scalp [Yvert et al., 1995; Zanow and Peters, 1995]. Note that unlike EEG, MEG signals are less sensitive to the conductivity profile of the head because of the low conductivity of the skull that confines almost all the current within it, and thus only the inner skull boundary was used for the MEG forward solution. Brain conductivity was assumed to be 0.3 S/m. Gain matrices, specifying the predicted sensor amplitudes for a set of cortical surface locations, were calculated with three vector components for each location, along the three cardinal axes. A normal vector gain matrix was calculated from this by taking the dot product of each gain vector with the normal vector for each vertex (perpendicular to the cortical surface). To determine the rigid body transformation between MRI and MEG reference frames, 100 or more digitized locations on the scalp were manually aligned to the surface representation of the outer scalp surface using a graphical interface written with MATLAB (The Mathworks, Natick, MA).

Phase‐Encoded Retinotopic Mapping

Procedures for the acquisition and analysis of phase‐encoded fMRI data have been described in detail previously [Hagler et al., 2007; Hagler and Sereno, 2006; Sereno et al., 1995]. Briefly, retinotopic maps of polar angle were measured using a black and white checkerboard wedge revolving around a central fixation cross. Eccentricity was mapped using an expanding or contracting ring. To ensure that subjects maintained a stable level of alertness and maintained central fixation, subjects performed a simple task in which they pressed a button upon rare dimming of the central fixation cross (approximately once every 5–10 s). For each subject, there were equal numbers of scans with counterclockwise or clockwise stimulus revolutions. Similarly for eccentricity mapping, expansion, and contraction scans were counterbalanced. Fourier transforms of the fMRI time series were computed to estimate the amplitude and phase of periodic signals at the stimulus frequency (10 cycles per 320‐s scan), with phase corresponding to the preferred stimulus location for a given voxel. For Subject 1, fMRI retinotopy data were collected in three separate sessions (Session 1: 6 polar angle scans; Session 2: 4 ecentricity scans, Session 3: 4 polar angle and 2 eccentricity). For Subject 2, fMRI retinotopy data were collected in a single session (4 polar angle and 2 eccentricity).

Retinotopic Map Fitting

Nonlinear optimization methods were used to fit a template map including V1, V2, and V3 to polar angle and eccentricity mapping data derived from fMRI. This procedure is similar in principle to work by Dougherty et al. [2003], but different in implementation. The template maps were initialized as rectangular grids, and each grid node was assigned a preferred polar angle and eccentricity and a unique area code, corresponding to the lower or upper field portions of V1, V2, and V3 (Fig 1). V3 is often treated as two separate areas, V3 and VP, but for simplicity, we refer to them as the lower and upper field portions of V3. To align the template map with the cortical surface, regions of interest (ROIs) were first manually drawn for each cortical hemisphere of each subject to encompass all of V1, V2, and V3, up to the maximum eccentricity measured with fMRI; a buffer zone was included, extending to the middle field representations of V3A and V4 (Fig. 1). A two part fitting procedure was then performed. First, a coarse fitting step with 9 or more parameters determined the overall shape and location of the template map that best fit the data (Table I). Second, a fine‐scale fitting step smoothly deformed the template to better match the data.

Figure 1.

Map fitting. A: fMRI retinotopy data with manually drawn region of interest around V1, V2, and V3 compared to fitted template maps for left hemisphere of Subject 1. B: Left hemisphere data and map fits. C: Examples of coarse fitting templates with rotation and radial bending. D: Fine scale fitting deforms the template by small displacements of each node. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Table I.

Parameters for coarse map fitting

Fit parameter	Description	Min valuea	Max valuea
u scale	Horizontal stretching of template	0.5	0.8
v scale	Vertical stretching of template	0.3	0.8
Rotation	Rotation of template relative to fMRI data	−10°	10°
u shift	Displacement along “u” axis	−0.1	0.1
v shift	Displacement along “v” axis	−0.1	0.1
radial wedge factor	Bend template into radial wedge	0.1	0.4
radial offset	Eccentricity offset of radial wedge	1	4
V2 width	Width of V2 relative to V1	0.6	1
V3 width	Width of V3 relative to V1	0.5	1
min eccentricityb	Minimum eccentricity of template	1	2
max eccentricityb	Maximum eccentricity of template	10	12
V1− lengthb	Length of lower field half of V1	0.8	1.2
V1+ lengthb	Length of upper field half of V1	0.8	1.2
V2− lengthb	Length of lower field half of V2	0.8	1.2
V2+ lengthb	Length of upper field half of V2	0.8	1.2
V3− lengthb	Length of lower field half of V3	0.8	1.2
V3+ lengthb	Length of upper field half of V3	0.8	1.2

^aTypical values; can be varied across subjects if needed.

^bUsually held constant; only allowed to vary when needed.

As part of the cortical reconstruction performed by FreeSurfer, the folded cortical hemisphere surfaces were inflated into unit spheres [Fischl et al., 1999]. To simplify map fitting, we transformed the spherical coordinates of the surface vertices into two‐dimensional Cartesian coordinates. The spherical reference frame was first rotated about the y and z axes such that the center of mass of the manually defined ROI was centered at the prime meridian and equator of the sphere. Additional rotation about the x axis was then applied so that the horizontal meridian of V1 was roughly parallel to the prime meridian of the sphere. The x axis rotation was chosen manually for each subject, but was generally +65° for the right hemisphere and −65° for the left hemisphere. The theta and phi components of the spherical coordinates were renamed u and v, rescaled, and shifted so that both u and v ranged from 0 to 1. The data were then interpolated onto a uniform grid to make comparisons between the template map and data more efficient. After the fitting procedure (described below), the deformed template was interpolated onto the uniform grid and then resampled back to the original cortical surface vertices, providing an area code for each vertex within the ROI, and a preferred polar angle and eccentricity for each vertex with a nonzero area code.

The coarse fitting step was a nonlinear search for a set of nine or more parameters defining the overall shape and location of the template map relative to the ROI (Table I). The location, scaling, and rotation of the template map were determined by five of these parameters. Two parameters were used to curve a rectangular template into a radial wedge (Fig. 1C). Two parameters specified the width of areas V2 and V3 relative to V1. Additional parameters can be used for some subjects, for example, specifying the length of lower and upper field portions of V1, V2, and V3, to accommodate more unusual map layouts. For each parameter, a range of allowed values was specified, and MATLAB's fmincon function was used to search for the combination of parameters that minimized a cost function (Eq. 1). To avoid local minima, the fitting procedure was repeated 200 times, each time randomly selecting a set of starting parameters from the specified ranges. For each subject, the fitting results were inspected for potential mismatches with the retinotopy data. For example, noisy retinotopy data could cause the fitted map to exclude V3+ or V3−. In such cases, the coarse fit parameter ranges can be manually adjusted—most commonly, the u scale, v scale, radial wedge factor, and radial offset parameters (see Table I)—and the fitting procedure repeated to force the template to better match the entire V1‐V2‐V3 complex. The coarse fitting procedure generally took several hours to run.

The fine‐scale fitting step further deformed the template map by allowing small displacements for each node (Fig. 1D). Iterative gradient descent was used to calculate the displacements. The gradients of the cost function with respect to the u and v coordinates were calculated analytically (see below). A minimum cost would typically be achieved within a few hundred iterations. To avoid local minima, random perturbations were made for each subsequent iteration to slightly scale or shift the map template. If the cost was further reduced, gradient descent was resumed until a new minimum was found. A total of 500 iterations were allowed for the fine‐scale fitting step, which generally completed within a few minutes.

The cost function was defined according to:

(1)

where C is the overall cost, k is the index for each of n template nodes, E _k, R _k, S _k, and F _k are components of the cost function, and λ_E, λ_R, λ_S, and λ_F are scaling factors controlling the relative weighting of each cost function component.

E _k is the residual error component of the cost function and was calculated as:

(2)

where θr _k and θi _k are the real and imaginary components of the polar angle retinotopy data (normalized to amplitude = 1) at the current location of each map node k, and are the fitted polar angle values, ϕr _k and ϕi _k are the eccentricity retinotopy data, and and are the fitted eccentricity values. σ² _θr,σ² _θi,σ² _ϕr, and σ² _ϕi, are the variances across all ROI locations calculated for each of the data components. n is the total number of map nodes. For the gradient descent method used for the fine scale fitting, the gradients of E _k were calculated by applying the chain rule and using the discrete gradients of the polar angle and eccentricity mapping data:

(3a)

(3b)

R _k is a penalty for nodes located outside the manually defined ROI (Fig. 1A,B). To obtain this value for each template node, an outside‐ROI penalty map was created on the uniform grid used to represent the retinotopy data. Grid points inside the ROI were set equal to 1 and the grid values were iteratively smoothed (sigma = 0.2 grid units, 10 iterations), resetting the inside‐ROI grid points to 1 after each smoothing iteration. For a given node,

(4)

with m representing the ROI mask values, which range from 0 to 1. The cost gradient was calculated from the discrete gradient of the outside‐ROI penalty map R.

S _k imposes a smoothness constraint on the deformation related to between‐neighbor differences in the displacements:

(5)

Δu _k and Δv _k are the displacements for node k relative to the starting locations obtained from the coarse fit. is the discrete gradient of the u displacements along the polar angle grid axis, and is the gradient along the eccentricity grid axis. Gradients of S _k were calculated as:

(6a)

(6b)

F _k is an additional constraint to prevent folding (e.g. neighboring nodes switching places):

(7)

F _k is the negative of the Jacobian determinant of the displacements. It is doubled because it is repeated for both u and v parameters for each node. Positive values indicate grid points whose neighbors have switched places and are thus penalized. Negative values of F, which indicate the absence of folding, were set to 0 so they did not contribute to the cost function. Gradients of F _k were calculated as:

(8a)

(8b)

As in shown in Equation 1, the components of the cost function were variously weighted by setting λ_E, λ_R, λ_S, and λ_F. λ_E was set to 1 and λ_R was 100. For the coarse fitting procedure, λ_S and λ_F were set to 0. For fine fitting, λ_S and λ_F were set to 0.5 and 7, respectively; in the case of noisy fMRI retinotopy data, λ_S and λ_F can be increased (e.g., 3 and 15) to avoid distortions in the map.

Construction of Retinotopic Models of the Generators of Evoked Visual Responses

Dipolar models of the cortical sources of evoked visual responses, limited to visual areas V1, V2, and V3, were generated for each subject. Dipole orientations were assumed to be perpendicular to the cortical surface mesh derived from the structural MRI data (∼0.8 mm intervertex distance). Dipole locations were determined by selecting a patch on the cortical surface, with vertex weights corresponding to the expected strength of activation by a stimulus at a given visual field location. Generally, dipole locations in ipsilateral cortex were allowed (e.g., near vertical meridians) as was crossover between the upper and lower field subareas (e.g., near horizontal meridians). For comparison, source estimates were also calculated with dipole locations restricted to a single subarea (e.g., left dorsal V2 for a right lower field stimulus).

To select cortical patches, weighting factors for each cortical surface vertex in V1, V2, and V3 were calculated for each stimulus presented during a given MEG session. These weighting factors were based on the preferred stimulus location for each vertex derived from the fMRI retinotopy template fit. For each stimulus, a binary image of the visual field was created with 100 rows and columns. For each pixel within the boundary of the stimulus, weights were calculated based on the difference between a visual field location and the fitted map data. The weight for a given vertex i, was calculated as:

(9)

where p is the number of visual field pixels for a given stimulus, x _j and y _j are the Cartesian coordinates for pixel j, and ŷ _i are Cartesian coordinates for the fitted preferred location for vertex i, and σ_i is the estimated receptive field size for vertex i (see below for receptive field size estimates). Those weights were then normalized so that the sum across visual field locations equaled one, and thresholded at 0.01. The mapping between each pixel of the visual field and each vertex in V1, V2, and V3 was precalculated and represented as a sparse matrix, and then multiplied by the binary stimulus images to calculate vertex weights for each stimulus.

Retinotopy Constrained Source Estimation

Retinotopy constrained forward and inverse matrices were calculated as described previously [Hagler et al., 2009], except that the dipole models were derived from retinotopic map fits rather than manually selected locations. The retinotopy constrained forward solution is described by:

(10)

where y, for each time point t, is a vector of measurements (all available MEG sensors and multiple stimulus locations). F is a matrix of forward solution sensor amplitudes for each visual area. The size of F is the number of measurements multiplied by the number of sources (visual areas). s is a vector of amplitudes for each source. n is a sensor noise vector. y, s, and n, are each functions of time t, although F is time invariant. F was constructed according to:

(11)

where G is the normal vector gain matrix (see MEG forward solution), and w _j,V1 is a vector of weights for every vertex in the retinotopy map fit (Equation 9) for stimulus location j (of m total locations) and visual area V1. As G is a matrix of sensors by sources, the result of multiplying G and w _j,V1 is a gain vector of amplitudes for each sensor. Note that for a given visual area, the gain vectors for all stimulus locations were arranged into a single column. This model configuration assumes that a given visual area has the same evoked response regardless of stimulus location [Ales et al., 2010; Hagler et al., 2009; Slotnick et al., 1999].

The following defines the inverse operator used to estimate source amplitudes at each time point:

(12)

where F ^T is the transpose of forward matrix F, C is the noise covariance matrix, R is the source covariance matrix, and λ² is a regularization parameter equal to the mean of the diagonal elements of FRF ^T divided by the mean of the diagonal elements of C, divided by the square of the assumed signal‐to‐noise ratio (SNR; a value of 1 was used). C was a diagonal matrix with variance values for each sensor calculated from the average baseline periods, averaged across all stimulus locations. R was a 3 by 3 identity matrix. To ensure numerical stability, y, F, and C were scaled to remove the orders of magnitude difference between gradiometers and magnetometers (10¹³ and 10¹⁵, respectively). The inverse operator W, was applied to the data as follows:

(13)

where ŝ is the estimated source vector and y is the measurement vector, each functions of time t. The size of W is the number of sources, which is 3, multiplied by the number of measurements, which depends on the number of stimulus locations; e.g., 36 locations and 306 sensors provides 11,016 measurements. The residual error was calculated as:

(14)

Normalized variance of residual error was calculated as the ratio between the across‐sensor variance of the residual error and the maximum variance of the data over time for a given sensor type, and then averaged across sensor types.

Receptive Field Sizes

Receptive field sizes are known to increase for areas higher in the visual hierarchy [Zeki, 1978]. The use of extended cortical surface patches to define dipoles allows us to incorporate information about receptive field sizes and potentially create more accurate dipole models. Dumoulin and Wandell recently published receptive field sizes for V1, V2, and V3 estimated from human fMRI data [Dumoulin and Wandell, 2008]. At an eccentriciy of ∼6° visual angle, they found that average (n = 6) receptive field sizes were ∼0.66, 1.03, and 1.88 for V1, V2, and V3, respectively (measured from Dumoulin and Wandell's Fig. 9). Receptive field sizes increase as a function of eccentricity, and the slopes of this function were ∼0.06, 0.10, and 0.15.

Robust Estimation Using Iteratively Reweighted Least Squares

The absolute value of the residual error, that is, the difference between sensor data and predicted data, was summed across all time points and sensors, to provide residual error r for each stimulus location (Fig. 8C). The minimum of these error values was subtracted from all values, the result of which was then normalized by the median absolute deviation (MAD), a robust estimator of the standard deviation.

(15)

(16)

Thus, r′ is the offset residual error normalized by a multiple of the MAD. The sensitivity of outlier detection is tunable by adjusting κ, for which a value of 2 was used. Smaller values of κ impose a greater penalty for deviation from the minimum error than larger values and consequently penalize more stimulus locations. If κ is too large, the result will be minimally different from the ordinary least squares solution. If too small, the result may rely too heavily on only a few stimulus locations, potentially resulting in noisier source estimates. Weighting factors were calculated from r′ using Tukey's bisquare function [Tukey, 1960].

(17)

These weights were used to scale both the sensor data and retinotopy‐constrained forward matrix before calculating the inverse operator and source estimates. Predicted sensor waveforms were calculated as in Eq. (14) using the updated source estimates and the unweighted forward matrix. Stimulus locations that were outliers continued to have errors larger than most other locations. As their influence was reduced, their error increased, whereas error for the remaining locations decreased. This process was repeated for at most 100 iterations or until the solution converged (i.e., source estimates change less than 10^‐7), which typically occurred within 10–20 iterations.

Figure 8.

Robust estimation using iteratively reweighted least squares. A: RCSE source waveforms before (thick lines) and after (thin lines) IRLS. B: Normalized variance of data, fit, and residual error before (thick) and after (thin) IRLS. C: Map of residual error summed across sensors and time for each of 36 stimulus locations, normalized to twice the median absolute deviation. D: Histogram of normalized residual error. E: IRLS weights as function of normalized residual error for each stimulus location. F: Map of complement of IRLS weights for each stimulus location. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

RESULTS

Comparison of Source Estimation Methods

To illustrate the difficulty of estimating visual evoked responses for individual visual areas, we used dynamic statistical parametric mapping (dSPM), a cortically constrained, minimum‐norm, distributed source estimation method [Dale et al., 2000]. Noise normalized source amplitudes, with and without orientation constraints, were calculated for each stimulus location for ∼5,000 dipoles, spaced ∼7 mm apart, and average time courses were calculated for V1, V2, and V3 ROIs created from the fMRI map fitting procedure. As expected, this method was not able to provide meaningful separation of the signals in the three visual areas, with similar time courses for each ROI (Fig. 2). When dipoles were constrained to be perpendicular to the cortical surface, the timing of activation was again similar, but with apparently arbitrary polarity of waveform peaks.

Figure 2.

Comparison of source estimation methods. A–D are source estimates from Subject 1, and E–H are source estimates from Subject 2. A and E, Average dSPM source waveforms for upper and lower field V1, V2, and V3 ROIs, with unconstrained dipole orientations. B and F, dSPM ROI source waveforms with dipole orientations perpendicular to the cortical surface. C and G, RCSE source waveforms for V1, V2, and V3 with independent solutions for each stimulus location. D and H, RCSE source waveforms with a common solution for all stimulus locations. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

An alternative to distributed source estimation methods is to model the evoked response with a few equivalent current dipoles. We used fMRI retinotopy to determine the cortical locations in V1, V2, and V3 expected to be activated by each MEG stimulus, and dipole orientations were made to be perpendicular to the cortical surface. When source estimates were calculated independently for each stimulus location, waveforms varied considerably between stimulus locations, displaying polarity inversions or large shifts in latency (Fig. 2). Variation of these estimates between the two subjects was similarly pronounced. When source waveforms were forced to be identical for all stimulus locations, the resultant waveforms are still somewhat different for the two subjects, but there are also clear similarities. An initial, negative peak for V1 occurs ∼70 to 80 ms post‐stimulus, followed by a positive peak at ˜150 ms, and a second negative peak at ∼180 ms. V2 and V3 waveforms have similar shapes, but with slightly later peak latencies.

Extended Cortical Patches Improve Fit

In our earlier study, dipolar sources for a given stimulus location were modeled with a single cortical surface vertex in each contralateral visual area [Hagler et al., 2009]. For a more accurate forward model, we now model each source as an extended cortical patch. These cortical patches are derived from the template‐fitted fMRI retinotopic map and take into account the size of the stimuli as well as the estimated receptive field (RF) sizes for each visual area at a given eccentricity (see Methods). This also allows us to model the contribution of neurons in both contralateral and ipsilateral hemispheres—e.g., for a stimulus close to the vertical meridian—and in both dorsal and ventral subareas—e.g., for a stimulus close to the horizontal meridian. We find that the explained variance is increased slightly (and residual error reduced) when the sources are modeled as extended patches rather than single surface vertices (Fig. 3). Including the contribution of contralateral, ipsilateral, dorsal, and ventral portions of the visual areas further improved the fit relative to a model with sources restricted, for example, to the dorsal right hemisphere for a lower left visual field stimulus (Fig. 3). The source waveforms themselves—particularly the relative amplitudes of the different visual areas—were also affected by these manipulations. Results for Subject 2 showed a similar pattern of increased explained variance and alteration of source estimates with the more realistic dipole models (Supporting Information Fig. 1). To isolate the effect of RF sizes on these changes, we compared source estimates calculated with RF sizes informed by a prior study [Dumoulin and Wandell, 2008], RF sizes estimated through nonlinear optimization, and very small RF sizes (0.01). We found small changes in the estimated waveforms, slight improvement in explained variance with optimized RF sizes, and a slight reduction in explained variance with very small RF sizes (Supporting Information Fig. 2).

Figure 3.

Modeling dipoles as extended cortical surface patches. A: RCSE dipole modeled as single vertex on cortical surface mesh, in left hemisphere V3 corresponding to a lower field stimulus near the horizontal meridian. B: Source waveforms with single vertex RCSE dipoles. C: Normalized variance of data, fit, and residual error for single vertex RCSE dipoles. D–F: Extended cortical surface patch and corresponding source estimates and normalized variance. G–I: Extended cortical surface patch, source estimates, and normalized variance, allowing contribution from both upper and lower field, contralateral and ipsilateral subareas. For A–C, dipoles were chosen as the single vertices closest to the weighted center of mass of the extended patches used for D–F. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Sensitivity to Location, Number, and Size of Stimuli

Consistent with the variability between independent source estimates for individual stimulus locations, we also found that RCSE waveforms are sensitive to the location, number, and size of stimuli presented to subjects and used to simultaneously constrain the solutions. For our original applications of the method, we used 16 stimulus locations arranged in a single ring. The stimulus layout that we chose for this study has 36 total stimulus locations, divided between 3 eccentricities (3.6, 5.3, 8.2° visual angle, with sizes 1.2, 2.2, 3.6° visual angle, respectively) and 12 polar angles (22° polar angle wide, contiguous, nonoverlapping portions of the visual field, excluding 24° polar angle centered on each horizontal or vertical meridian).

To investigate the sensitivity to stimulus location and number, we generated RCSE waveforms with subsets of the 36 stimulus locations for Subject 1′s data. For comparing across different numbers of stimulus locations, the total number of trials was kept constant. For 36, 12, and 4 locations, the number of trials per location was limited to 44, 133, and 400 respectively, for a total of ∼1,600 trials. Comparing sets of twelve stimuli defined by the stimuli at each of three eccentricities, there were large differences in peak amplitude, ranging from ∼10 to 25 nA m for V1, but relatively little variation in peak latency. Source waveforms were most similar for the inner and middle rings of stimuli, both of which had lower residual error than the outer ring of stimuli. Using all 36 locations provided a consensus solution, different from merely calculating the average of three separate estimates. The residual error is relatively high when using all 36 locations, comparable with the outer ring of stimuli, which had the highest residual error (Fig. 4I,L). Note also that the variance during the prestimulus baseline period was higher when all 36 locations were used, presumably related to the small number of trials per location (44) used for these estimates (Fig. 4L). When all available trials were used, the baseline variance and the residual error during the evoked response were smaller (Fig. 3I), and the source estimates themselves were less noisy (Fig. 3H).

When only four stimulus locations were used, the variation between estimates from the different sets of stimuli was greater. We selected three sets of four stimulus locations—one in each visual field quadrant—keeping eccentricity constant and varying polar angle. Between these sets of stimuli, the RCSE peak amplitudes varied substantially, similar to the differences between the sets of 12, but there was additional variation in peak latency, particularly for V2 and V3 (Fig. 5B,E,H). As with the sets of twelve, one of the three sets of four stimuli had relatively high residual error compared to the others (Fig. 5C,F,I).

Figure 5.

Variation between sets of four stimuli. A, D, and G. Subsets of four stimuli used to constrain RCSE. Red outlines, which were not visible during the actual MEG session, show the locations of all 36 stimuli that were presented one at time. B, E, and H. RCSE source waveforms constrained by four stimuli. C, F, and I. Normalized variance of data, fit, and residual error. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

To test the effect of stimulus size on the RCSE estimates, we collected MEG data in an additional session for Subject 1, using smaller stimuli (0.5° visual angle long and 5° polar angle wide) presented at 100 different locations in the visual field (25 locations in each quarterfield, 0.5° visual angle and 5° polar angle between centers of stimuli, with each group of 25 centered at 5° visual angle and 45°, 135°, 225°, or 315° polar angle). A comparison of the sensor waveforms from two similarly located stimuli shows that the smaller stimulus evoked a much smaller response (Fig. 6A–C). Summed across all stimulus locations and comparing the squared amplitudes between 70 and 120 ms poststimulus to those between −70 and −20 ms, the SNR was 16.4 for the 36 large stimuli and 2.9 for the 100 small stimuli. Reflecting this lower SNR, the resulting RCSE waveforms were quite noisy, particularly for V3 (Fig. 6E). Despite this, waveform shapes were similar to those with the large stimuli, with comparable peak latencies.

Within‐Subject Variability of Source Estimates

To assess the reliability of RCSE, we compared source estimates calculated for a single subject as we manipulated three important sources of potential variability. First, there is intrinsic variability in the measurement of neuronal activation using MEG. Between two different MEG sessions, levels of physiological and measurement noise will vary, as will head position relative to sensors, resulting in differences in the measured sensor waveforms. Second, fMRI retinotopic mapping data is contaminated by physiological noise that will vary between sessions. As dipole selection relies on the fMRI retinotopy data, variability in fMRI measurements will result in variable source estimates. Third, cortical surface reconstructions on two different images of the same brain will be slightly different, introducing differences in dipole orientations and map fitting on the cortical surface.

Source estimates were calculated for two separate MEG sessions with Subject 1 (Fig. 7A), each time using the 36 stimulus layout described above and depicted in Figure 4J. Differences in peak latencies were minimal, but peak amplitudes did vary substantially, by ∼25%. We observed similar differences between estimates for two blocks of trials within a single MEG session, and these differences reflect similar variability of sensor waveforms (Supporting Information Fig. 3). Functional and structural MRIs were also collected in multiple sessions for this subject, and RCSE waveforms were calculated using two alternate fMRI retinotopy data sets (Fig. 7B) and two alternate cortical surface reconstructions (Fig. 7C). For the other source estimates for this subject (e.g., Fig. 7A), averaged fMRI retinotopy and averaged structural MRIs were used. As with the separate MEG sessions, there were substantial differences in the amplitudes of the RCSE waveforms, but minimal differences in the peak latencies. The difference between source estimates for the separate cortical surface reconstructions was smaller than for separate MEG or fMRI sessions.

Figure 7.

Within‐subject variability of RCSE estimates. A: RCSE source waveforms from two separate MEG sessions for Subject 1. For both sets of estimates, fMRI retinotopy data, map fits, and cortical surface reconstructions were held constant. B: RCSE source waveforms estimated using two different sets of fMRI retinotopy data and map fits, holding MEG data and cortical surface reconstructions constant. C: RCSE source waveforms estimated using cortical surface reconstructions from two different structural MRI scans with a separate map fit for each, holding MEG and fMRI retinotopy data constant. For A–C, one of the two sets of source estimates is plotted with thinner lines. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Robust Estimation Using Iteratively Reweighted Least Squares

One explanation for the apparent disagreement between source estimates derived from different sets of stimulus locations (Figs. 2, 4, and 5) is that the forward model is less accurate for some stimulus locations than others. For example, noisy fMRI retinotopy data could introduce errors in the map fitting that affect some portions of the visual field more than others. The resulting displacement along the cortical surface would likely result in incorrectly modeled dipole orientations, particularly in regions of high cortical curvature. Such regions may also be more susceptible to cancellation of the evoked fields, increasing the difficulty in accurate forward modeling. Some stimulus locations do indeed exhibit larger residual error than others (Fig. 8C,D). We applied the IRLS algorithm (see Methods) to automatically detect potential outliers and reduce or eliminate the influence they had on the source estimates. The derived weights range from 0 to 1, with smaller weights given to stimulus locations with the largest residual error (Fig. 8E,F). With a small number of outliers, use of IRLS results in an increase in the explained variance with slight changes to the RCSE waveforms (Fig. 8A,B). If more stimulus locations are identified as outliers, the derived weights could be used to diagnose problems, such as a poor map fit, that should be resolved with manually adjusted parameter constraints.

DISCUSSION

Source estimation for MEG or EEG is typically an ill‐posed problem given the large number of possible current sources in the brain and the relatively small number of sensors. Distributed source methods generally estimate amplitudes for thousands of cortical locations, and regularization—a mathematical technique that amounts to a smoothness constraint—is used to make the solution well‐posed, if not entirely accurate. With or without orientation constraints, it is very difficult to meaningfully separate signals from cortical locations as closely spaced as V1, V2, and V3. With equivalent current dipole (ECD) methods that use nonlinear searches for dipole locations and orientations, a large number of potential locations are sampled, and with multiple dipoles, there are many, equally likely combinations. Furthermore, dipoles must be prevented from being too close to each other or else cancellation can result in nonsensical estimates. Even when functional and structural MRI data is used to specify locations and orientations of predicted current dipoles, small inaccuracies can, because of cortical folding, result in relatively large changes in the expected dipole orientation, making the estimated responses for a single stimulus location unreliable. Furthermore, for some stimulus locations, an activated cortical patch may produce little MEG signal because it could be quite distant from the sensors, produce a radial dipole, or have a folding pattern that results in nearly complete cancellation. The source estimates for such stimulus locations will be particularly noisy. Another problematic scenario is when the dipoles for a stimulus location are nearly parallel for two visual areas. In that case, separating signals from those sources is more difficult. These difficulties are illustrated by the implausible variability of waveforms estimated independently for individual stimulus locations (Fig. 2C,G).

Improved Modeling of Visual‐Evoked Responses

With RCSE, the number of sources is reduced to as few as one, and the number of measurements is equal to the number of sensors multiplied by the number of stimulus locations (e.g., 11,016 for 306 sensors and 36 locations), so the problem becomes well‐posed by virtue of being massively over‐constrained. Confidence in the source estimates is increased by simultaneously constraining the solutions with many stimulus locations. Crosstalk, a measure of the ambiguity between two sources, is greatly reduced when they are defined as a collection of multiple, variously oriented dipoles [Hagler et al., 2009]. It may be possible to obtain plausible source estimates with relatively low residual error using only a few stimulus locations (see Fig. 4). An unfortunate choice of stimulus locations, may, however, yield improbable source estimates—for example, polarity inversions or peaks with abnormal latencies or amplitudes—due to incorrectly specified dipole locations and orientations. Using stimuli from all parts of the visual field avoids localized errors and provides the consensus estimate of the visual evoked response. There is no precise number of stimulus locations required to reliably obtain good‐quality source estimates, but there are practical considerations that place upper limits on that choice, including computation time and memory requirements, data collection time, and the SNR of sensor waveforms.

Despite the reduced sensitivity to forward modeling errors granted by simultaneously constraining the solution with many stimulus locations, if dipole locations or orientations are specified incorrectly, source estimates will be inaccurate to some degree and contribute to greater residual error. RCSE results are naturally influenced by the quality of the structural and functional MRI data used to generate the forward models. Source estimates calculated using different cortical surface reconstructions and different fMRI sessions showed that these sources of variability are comparable with that from the MEG data itself, but such variability could have been even larger if certain precautions were not taken. First, we corrected for the nonlinear distortions found in both structural and functional MRI that would otherwise prevent accurate registration between fMRI retinotopy data, cortical surface reconstructions, and MEG sensors. Second, careful manual editing of white matter and brain segmentations was done to ensure accurate cortical surface reconstruction, which is important because of the direct influence on modeled dipole orientations. Third, retinotopic map fitting was used to reduce the influence of physiological noise in fMRI retinotopy data and to automate dipole selection.

Additional improvements to RCSE were achieved through the use of larger stimuli and modeling the extended nature of the cortical activations. Larger visual stimuli activate larger cortical surface patches; so measured MEG signals were naturally larger, resulting in source estimates with higher SNR. The use of larger stimuli also likely reduces the sensitivity to small displacements of the modeled dipole locations along the cortical surface. Explicitly modeling the extent of cortical surface activated by a stimulus effectively smooths over rapidly changing cortical surface orientations in regions of high curvature, particularly with large stimuli. A cortical patch is a collection of surface mesh vertices, each with its own normal vector, oriented perpendicularly to the cortical surface. The forward solution for a given patch is the weighted sum of the forward solutions for each vertex within the patch. Modeling sources this way accounts for potential cancellation due to cortical folding, enables the use of realistic estimates of receptive field sizes, and allows for modeling the contribution from opposite hemifield sources (e.g., ipsilateral or upper vs. lower field). Without resorting to nonlinear optimization or otherwise introducing free parameters, using extended cortical surface patches reduced residual error, presumably providing more accurate and reliable source estimates. Despite this improvement, it is difficult, if not impossible, to be certain that we have achieved the best possible fit to the data. Since the amplitude of the source estimates appears to increase along with the goodness of fit, it is difficult to have high confidence in the nominal units of the waveforms. It should be noted that despite large differences in waveform amplitudes for different models with varying goodness of fit, the peak latencies were quite consistent (Fig. 3). Furthermore, for parametric manipulations, for example varying luminance contrast, issues related to scaling can be avoided through normalization.

Residual Error

Compared with other multidipole modeling methods that use a single stimulus location, the overall residual error for these RCSE solutions can be relative large. The inherent noise in the MEG sensor waveforms accounts for at least some of the residual error. Because fewer trials are possible with a large number of stimulus locations—for example, 400 trials per 36 conditions versus 14,400 trials for a single condition—the sensor noise may be relatively large. Noise in the average MEG sensor waveforms is assumed to be independent for different stimulus locations, such that it is excluded from the source estimates and results in larger residual error (Supporting Information Fig. 4). If dipole patch locations and extents are specified incorrectly for some stimulus locations more so than others, this will also contribute to greater residual error when all stimulus locations are included, approximately equal to the error for the worst subset of stimulus locations (see Figs. 4 and 5). We have attempted to reduce the residual error by improving the accuracy of the forward model as much as possible, but more comprehensive or sophisticated models are certainly possible.

In the future, additional improvement may be possible by modeling the more complicated center‐surround structure of receptive fields. Another potential improvement to RCSE would be to model additional retinotopic visual areas; for example V4, MT/V5, or V3A. The utility of this would depend on how reliably these areas can be distinguished from their neighbors and retinotopically mapped with fMRI. There are several examples in the literature of successful mapping of these areas [Amano et al., 2009; Brewer et al., 2005; DeYoe et al., 1996; Gardner et al., 2005; Huk et al., 2002; Sereno et al., 1995; Swisher et al., 2007; Tootell et al., 1997; Wade et al., 2002; Wandell et al., 2007]; however, their relatively small sizes, poorly defined borders, and weak eccentricity tuning will likely make this technically challenging. In any event, while it is desirable to explain as much of the VER as possible, and including V3 does modestly increase the explained variance, we found that omitting V3 from the model has small or negligible effects on the source estimates for V1 and V2 (Supporting Information Fig. 5). This finding is indicative of the very low crosstalk between the multilocation dipole models used for RCSE. Similarly, a simulation we performed in our previous study demonstrated that V1, V2, and V3 source estimates are not affected by the inclusion or exclusion of dipoles to model activity in MT/V5 [Hagler et al., 2009].

Nonlinear Optimization

As discussed above, larger residual error may occur for particular stimulus locations due to local map fitting errors, subtle surface variations, or noise in the original MEG sensor data. If such errors cannot be eliminated through more accurate map fitting or cortical reconstruction, correcting those errors through additional nonlinear fitting is an attractive idea. For example, if higher residual error is caused by a displacement along the cortical surface of the modeled dipole relative to the correct cortical location, one could perform a nonlinear search for displacements for each stimulus location that provide a better fit to the data. Alternatively, incorrect dipole orientations could be adjusted by first linearly estimating source waveforms and then using them to linearly estimate better fitting orientations, an example of bilinear estimation. Similarly, one could estimate source waveforms and then search for the single vertex within a defined cortical neighborhood for each stimulus location that provides the best fit [Ales et al., 2010].

Doing so, however, introduces many free parameters and potentially results in inaccurate source estimates. Even when the solutions appear reasonable, one cannot be sure that the optimization procedure did not manage to reorient some of the poorer fitting dipoles of V1, V2, or V3 to align with some other, unmodeled visual area, thus reducing the residual error, but contaminating the source estimates. Furthermore, since the actual response to a stimulus can be evoked simultaneously in contralateral and ipsilateral sub‐areas or dorsal and ventral sub‐areas, modeling that with a single vertex means that the resulting dipole orientation is basically arbitrary, forced to be the linear combination of the four subareas. Finally, if noisy MEG data is the source of error, fitting that by displacement or reorientation of dipoles is not physically meaningful and thus arbitrary as well. Reductions in residual error do not necessarily indicate more accurate source estimates, particularly if obtained through nonlinear optimization. Despite these reservations, an approach for future study will be to use group average or group‐constrained source estimates as a priori information to guide optimization of dipole locations in individual subjects.

The approach we have taken presently is to selectively reduce the contribution of stimulus locations with the largest residual error, through the use of iteratively reweighted least squares (IRLS) [Holland and Welsch, 1977; Huber, 1981]. IRLS is a method for robust estimation that has found wide‐spread application in a number of fields, including neuroimaging [Gorodnitsky et al., 1995; Mangin et al., 2002; Wager et al., 2005]. This method allows for the detection of outliers and provides a way to identify map fitting or cortical surface reconstruction errors that may be correctable through manual interventions such as adjusting map fit parameter ranges, redefining occipital regions of interest, or segmentation editing.

Potential Applications

Retinotopy‐constrained source estimation is a promising method for measuring activity in individual visual areas, enabling non‐invasive studies of the temporal dynamics of visual processing. Potential applications include studying the modulation of visual evoked responses by various stimulus properties such as contrast, color, spatial frequency, motion, etc. It would also be interesting to calculate separate source estimates for upper and lower field stimuli, as there are known differences in psychophysical and electrophysiological measures [Lehmann and Skrandies, 1979; Portin et al., 1999; Previc, 1990; Skrandies, 1987], and the upper and lower field portions of V3 may be distinct visual areas V3 and VP [Burkhalter and Van Essen, 1986; Felleman et al., 1997; Zeki, 2003]. As we found that different sets of stimuli inevitably give rise to some differences in source estimates, group analysis with several subjects would be required to determine whether there are indeed systematic differences between upper and lower field responses. Bottom‐up and top‐down interactions between individual visual areas and higher level cortical areas, several of which are known to also contain maps of visual space [Hagler et al., 2007; Hagler and Sereno, 2006; Kastner et al., 2007; Schluppeck et al., 2005; Sereno et al., 2001; Silver et al., 2005; Swisher et al., 2007], is another potential area of future research.

Supporting information

Additional Supporting Information may be found in the online version of this article.

Supporting Information Figure 1.

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