{"title": "Subject-Independent Magnetoencephalographic Source Localization by a Multilayer Perceptron", "book": "Advances in Neural Information Processing Systems", "page_first": 741, "page_last": 748, "abstract": "", "full_text": "Subject-Independent Magnetoencephalographic\nSource Localization by a Multilayer Perceptron\n\nSung C. Jun\n\nBiological and Quantum Physics Group\n\nMS-D454, Los Alamos National Laboratory\n\nLos Alamos, NM 87545, USA\n\nBarak A. Pearlmutter\n\nHamilton Institute\n\nNUI Maynooth\n\nMaynooth, Co. Kildare, Ireland\n\njschan@lanl.gov\n\nbarak@cs.may.ie\n\nAbstract\n\nWe describe a system that localizes a single dipole to reasonable accu-\nracy from noisy magnetoencephalographic (MEG) measurements in real\ntime. At its core is a multilayer perceptron (MLP) trained to map sen-\nsor signals and head position to dipole location. Including head position\novercomes the previous need to retrain the MLP for each subject and ses-\nsion. The training dataset was generated by mapping randomly chosen\ndipoles and head positions through an analytic model and adding noise\nfrom real MEG recordings. After training, a localization took 0.7 ms with\nan average error of 0.90 cm. A few iterations of a Levenberg-Marquardt\nroutine using the MLP\u2019s output as its initial guess took 15 ms and im-\nproved the accuracy to 0.53 cm, only slightly above the statistical limits\non accuracy imposed by the noise. We applied these methods to localize\nsingle dipole sources from MEG components isolated by blind source\nseparation and compared the estimated locations to those generated by\nstandard manually-assisted commercial software.\n\n1\n\nIntroduction\n\nThe goal of MEG/EEG localization is to identify and measure the signals emitted by elec-\ntrically active brain regions. A number of methods are in widespread use, most assuming\ndipolar sources (H\u00a8am\u00a8al\u00a8ainen et al., 1993). Recently MLPs (Rumelhart et al., 1986) have\nbecome popular for building fast dipole localizers (Abeyratne et al., 1991; Kinouchi et al.,\n1996). Since it is easy to use a forward model to create synthetic data consisting of dipole\nlocations and corresponding sensor signals, one can train a MLP on the inverse problem.\nHoey et al. (2000) took EEG measurements for both spherical and realistic head models\nand trained MLPs on randomly generated noise-free datasets. Integrated approaches to the\nEEG/MEG dipole source localization, in which the trained MLPs are used as initializers\nfor iterative methods, have also been studied (Jun et al., 2002) along with distributed output\nrepresentations (Jun et al., 2003). Interestingly, all work to date trained with a \ufb01xed head\nmodel. However, for MEG, head movement relative to the \ufb01xed sensor array is very dif-\n\ufb01cult to avoid, and even with heroic measures (bite bars) the position of the head relative\nto the sensor array varies from subject to subject and session to session. This either results\nin signi\ufb01cant localization error (Kwon et al., 2002), or requires laborious retraining and\n\n\frevalidation of the system.\n\nWe propose an augmented system which takes head position into account, yet remains\nable to localize a single dipole to reasonable accuracy within a fraction of a millisecond\non a standard PC, even when the signals are contaminated by considerable noise. The\nsystem uses a MLP trained on random dipoles and random head positions, which takes\nas inputs both the coordinates of the center of a sphere \ufb01tted to the head and the sensor\nmeasurements, uses two hidden layers, and generates the source location (in Cartesian\ncoordinates) as its output. Adding head position as an extra input overcomes the primary\npractical limitation of previous MLP-based MEG localization systems: the need to retrain\nthe network for each new head position.\n\nWe use an analytical model of quasi-static\nelectromagnetic propagation through a\nspherical head to map randomly cho-\nsen dipoles and head positions to super-\nconducting quantum interference device\n(SQUID) sensor activities according to the\nsensor geometry of a 4D Neuroimaging\nNeuromag-122 MEG system, and trained\na MLP to invert this mapping in the pres-\nence of real brain noise. To improve the lo-\ncalization accuracy we use a hybrid MLP-\nstart-LM method,\nin which the MLP\u2019s\noutput provides the starting point for a\nLevenberg-Marquardt (LM) optimization\n(Press et al., 1988). We use the MLP and\nMLP-start-LM methods to localize single-\ndipole sources from actual MEG signal\ncomponents isolated by a blind source sep-\naration (BSS) algorithm (Vig\u00b4ario et al.,\n2000; Tang et al., 2002) and compare the\nresults with the output of standard interac-\ntive commercial localization software.\n\nSensor surface\n\n11.343 cm\n\n10.851cm\n\n10.851 cm\n\n13.594 cm\n\n3.605 cm\n\nz\n\nx\n\nCoronal View \n\nSaggital View\n\nz\n\ny\n\n10.5 cm\n\n7.5 cm\n\nHead Model B\n\nHead Model A\n\n7.5 cm\n\nB\n\nA\n\n3 cm\n\n4 cm\n\nTraining Region\n\nTraining Region and various Head Models \n\nFigure 1: Sensor surface and training region.\nThe center of the spherical head model was\nvaried within the given region. Diamonds de-\nnote sensors.\n\nSection 2 describes our synthetic data, the\nforward model, the noise used to additively contaminate the training data, and the MLP\nstructure. Section 3 presents the localization performance of both the MLP and MLP-start-\nLM, and compares them with various conventional LM methods. In Section 3.2, compar-\native localization results for our proposed methods and standard Neuromag commercial\nsoftware on actual BSS-separated MEG signals are presented.\n\n2 Data and MLP structure\n\nWe constructed noisy data using the procedure of Jun et al. (2002), except that an additional\ninput was associated with each exemplar, namely the (x, y, z) coordinates of the center of\na sphere \ufb01tted to the head, and the forward model was modi\ufb01ed to account for this offset.\nEach exemplar thus consisted of the (x, y, z) coordinates of the center of a sphere \ufb01tted to\nthe head, sensor activations generated by a forward model, and the target dipole location.1\nWe made two datasets: one for training and another for testing. Centers of spherical head\n\n1Given the sensor activations and a dipole location, the minimum error dipole moment can be\ncalculated analytically (H\u00a8am\u00a8al\u00a8ainen et al., 1993). Therefore, although the dipoles used in generating\nthe dataset had both location and moment, the moments were not included in the datasets used for\ntraining or testing.\n\n\fmodels in the training set were drawn from a ball of radius 3 cm centered 4 cm above the\nbottom of the training region,2 as shown in Figure 1. The dipoles in the training set were\ndrawn uniformly from a spherical region centered at the corresponding center, with a radius\nof 7.5 cm, and truncated at the bottom. Their moments were drawn uniformly from vectors\nof strength \u2264200 nAm. The corresponding sensor activations were calculated by adding\nthe results of a forward model and a noise model. To check the performance of the network\nduring training, a test set was generated in the same fashion as the training set. We used the\nsensor geometry of a 4D Neuroimaging Neuromag-122 whole-head gradiometer (Ahonen\net al., 1993) and a standard analytic model of quasistatic electromagnetic propagation in a\nspherical head (Jun et al., 2002).\n\nThis work could be easily extended to a more realistic head model. In that case the integral\nequations are solved by the boundary element method (BEM) or the \ufb01nite element method\n(FEM) numerically (H\u00a8am\u00a8al\u00a8ainen et al., 1993). The human skull phantom study in Leahy\net al. (1998) shows that the \ufb01tted spherical head model for MEG localization is slightly\ninferior in accuracy to the realistic head model numerically calculated by BEM. In forward\ncalculation, a spherical head model has some advantages: it is more easily implemented and\nis much faster. Despite its inferiority in terms of localization accuracy, we use a spherical\nhead model in this work.\n\nIn order to properly compare the performance of various localizers, we need a dataset for\nwhich we know the ground truth, but which contains the sorts of noise encountered in\nactual MEG recordings. To this end, we measured real brain noise and used it to additively\ncontaminate synthetic sensor readings (Jun et al., 2002). This noise was taken, unaveraged,\nfrom MEG recordings during periods in which the brain region of interest in the experiment\nwas quiescent, and therefore included all sources of noise present in actual data: brain\nnoise, external noise, sensor noise, etc. This had a RMS (square root of mean square)\nmagnitude of roughly P n = 50\u2013200 fT/cm, where we measure the SNR of a dataset using\nthe ratios of the powers in the signal and noise, SNR (in dB) = 20 log10 P s/P n, where P s\nand P n are the RMS sensor readings from the dipole and noise, respectively. The datasets\nused for training and testing were made by adding the noise to synthetic sensor activations\ngenerated by the forward model, and exemplars whose resulting SNR was below \u22124 dB\nwere rejected.\n\nThe MLP charged with approximating the inverse mapping had an input layer of 125 units\nconsisting of the three Cartesian coordinate of the center of the sphere \ufb01tted to the head, and\nthe 122 sensor activations. It had two hidden layers with 320 and 30 units respectively, and\nan output layer of three units representing the Cartesian coordinates of the \ufb01tted dipole. The\noutput units had linear activation functions, while the hidden unit had hyperbolic tangent\nactivation functions. Adjacent layers were fully connected, with no cut-through connec-\ntions. The 122 sensor activation inputs were scaled to an RMS value of 0.5, and the target\noutputs were scaled into [\u22121, +1]. The network weights were initialized with uniformly\ndistributed random values between \u00b10.1, and online stochastic gradient decent with no mo-\nmentum and an empirically chosen constant of proportionality was used for optimization.\n\n2Fitted spheres from twelve subjects performing various tasks on a 4D Neuroimaging Neuromag-\n122 MEG system were collected, and this distribution of head positions was chosen to include all\ntwelve cases. Just as the position of the center of the head varies from session to session and subject to\nsubject, so does head orientation and radius. Because a sphere is rotationally symmetric, our forward\nmodel is insensitive to orientation, and similarly the external magnetic \ufb01eld caused by a dipole in a\nhomogeneous sphere is invariant to the sphere\u2019s radius. On the other hand, the noise process would\nnot be invariant to orientation or radius, so we might expect a slight increase in performance if the\nnetwork had orientation and radius available as inputs, rather than just the position of the center.\n\n\fTop\n\n0.58\n\n0.59\n\n0.85\n\n0.85\n\n0.58\n\n0.6\n\nLeft\n\n0.79\n\n0.9\n\nRight\n\n1.78\n\n0.7\n\n1.07\n\n1.08\n\n0.81\n\n14\n\n12\n\n10\n\n8\n\n6\n\n4\n\n2\n\n0\n\n\u22122\n\n\u22124\n\n\u22126\n\n14\n\n12\n\n10\n\n8\n\n6\n\n4\n\n2\n\n0\n\n\u22122\n\n\u22124\n\n\u22126\n\nTop\n\n0.92\n\n0.82\n\n0.87\n\n1.04\n\n0.81\n\n1.37\n\nBack\n\n0.89\n\n1.52\n\nFront\n\n1.71\n\n0.92\n\n1.1\n\n2.42\n\n1.67\n\n\u221215\n\n\u221210\n\n\u22125\n\n0\n\n5\n\n10\n\n15\n\n\u221215\n\n\u221210\n\n\u22125\n\n0\n\n5\n\n10\n\n15\n\nFigure 2: Mean localization errors of the trained MLP as a function of correct dipole loca-\ntion, binned into regions. All units are in cm. Left: Coronal cross section. Right: Sagittal\ncross section.\n\n3 Results and discussion\n\n3.1 Training and localization results\n\nDatasets of 100,000 (training) and 25,000\n(testing) patterns, all contaminated by real\nbrain noise, were constructed. As is typ-\nical, the incremental gains per epoch de-\ncrease exponentially with training. From\nthe training curves (not shown)\nis\nevident\ntraining would\nhave further decreased the error, but we\nnonetheless stopped after 1000 epochs,\nwhich took about three days on 2.8 GHz\nIntel Xeon CPU.\n\nthat additional\n\nit\n\n)\n\nm\nc\n(\n \nr\no\nr\nr\nE\n \nn\no\ni\nt\na\nz\ni\nl\na\nc\no\nL\n \nn\na\ne\n\nM\n\n 2.5\n\n 2\n\n 1.5\n\n 1\n\n 0.5\n\n 0\n\nfixed\u22124\u2212start\u2212LM\nMLP\nMLP\u2212start\u2212LM\noptimal\u2212start\u2212LM\n\n 0\n\n 5\nS/N (dB)\n\n 10\n\n 15\n\nWe investigated localization error distribu-\ntions over various regions of interest. We\nconsidered two cross sections (coronal and\nsagittal views) with width of 2 cm, and\neach of these was divided into 19 regions,\nas shown in Figure 2. We extracted the noisy signals and the corresponding dipoles from\ntesting datasets. For each region 49\u2013500 patterns were collected. A dipole localization was\nperformed using the trained MLP, and the average localization error for each region was\ncalculated. Figure 2 shows the localization error distribution over two cross sections. In\ngeneral, dipoles closer to the sensor surface were better localized.\n\nFigure 3: Mean localization error vs. SNR.\nMLP, MLP-start-LM, and optimal-start-LM\nwere tested on signals from 25,000 random\ndipoles, contaminated by real brain noise.\n\nWe compared various automatic localization methods, most of which consist of LM used\nin different ways:\n\n\u2022 MLP-start-LM\n\nLM was started with the trained MLP\u2019s output.\n\n\u2022 \ufb01xed-4-start-LM\n\nLM was tuned for good performance using restarts at the four \ufb01xed initial points\n(0, 0, 6), (\u22125, 2, \u22121), (5, 2, \u22121), and (0, \u22125, \u22121), in units of cm relative to the\ncenter of the spherical head model. The best result among four results was chosen.\n\n\fTable 1: Comparison of performance on real brain noise test set of Levenberg-Marquardt\nsource localizers with three LM restarts strategies, the trained MLP, and a hybrid system.\nEach number is an average over 25,000 localizations, so the error bars are negligible.\n\nAlgorithm\n\n\ufb01xed-4-start-LM\nrandom-20-start-LM\noptimal-start-LM\nMLP\nMLP-start-LM\n\nComputation\n\ntime (ms)\n\nLocalization\nerror (cm)\n\n120\n663\n14\n0.7\n15\n\n0.83\n0.54\n0.49\n0.90\n0.53\n\n\u2022 random-n-start-LM\n\nLM was restarted with n random (uniformly distributed) points within the spheri-\ncal head model. We checked how many restarts were needed to match the accuracy\nof the MLP-start-LM, yielding n = 20, which is the same as in Jun et al. (2002).\n\n\u2022 optimal-start-LM\n\nLM was started with the known exact dipole source location.\n\nFigure 3 shows the localization performance as a function of SNR for \ufb01xed-4-start-LM,\noptimal-start-LM, the trained MLP, and MLP-start-LM. Optimal-start-LM shows the best\nlocalization performance across the whole range of SNRs, but the hybrid system shows\nalmost the same performance as optimal-start-LM except at very high SNRs, while the\ntrained MLP is more robust to noise than \ufb01xed-4-start-LM. In this experiment, most of the\nsources with very high SNR were super\ufb01cial, located around the upper neck or back of the\nhead. These sorts of sources are often very hard to localize well, as it is easy to become\ntrapped in a local minimum (Jun et al., 2002). It is expected that, under these conditions,\na better initial guess than the MLP output (which are 0.7 cm on average from the exact\nsource) would be required to obtain near-optimal performance from LM.\n\nA grand summary, averaged across various SNR conditions, is shown in Table 1. The\ntrained MLP is fastest, and its hybrid system is about 40\u00d7 faster than random-20-start-LM,\nwhile the hybrid system is about 9\u00d7 faster, yet more accurate than, \ufb01xed-4-start-LM. This\nmeans that MLP-start-LM was about two times faster than might be naively expected.\n\n3.2 Localization on real MEG signals and comparison with commercial software\n\nThe sensors in MEG systems have poor signal-to-noise ratios (SNRs) for single-trial data,\nsince MEG data is strongly contaminated by various noises. Blind source separation of\nMEG data segregates noise from signal (Vig\u00b4ario et al., 2000; Tang et al., 2000a; Sander\net al., 2002), raising the SNR suf\ufb01ciently to allow single-trial analysis (Tang et al., 2000b).\nEven though the sensor attenuation vectors of the BSS-separated components can be well\nlocalized to equivalent current dipoles (Vig\u00b4ario et al., 2000; Tang et al., 2002), the recov-\nered \ufb01eld maps can be quite noisy. We applied the MLP and MLP-start-LM to localize\nsingle dipolar sources from various actual BSS-separated MEG signals.3 The x\ufb01t program\n\n3Continuous 300 Hz MEG data for four right-handed subjects was collected using a cognitive\nprotocol developed by Michael P. Weisend, band-pass \ufb01ltered at 0.03\u2013100 Hz, separated using sec-\nond order blind identi\ufb01cation algorithm (SOBI), and scanned for neuronal sources of interest. The\nfollowing four visual reaction time tasks were performed by each subject: stimulus pre-exposure\ntask, trump card task, elemental discrimination task, and transverse patterning task. For each subject,\nall four experiments were performed on the same day, but each in a separate session. Subjects were\npermitted to move their heads between experiments.\n\n\f 15\n\n 10\n\n 5\n\n 0\n\n\u22125\n\n\u221210\n\n\u221215\n\n\u221215\n\nMLP\u2212start\u2212LM\n\nMLP\nxfit\n\nSV\n\nPV\n\n 20\n\n 15\n\n 10\n\n 5\n\n 0\n\n\u22125\n\nMLP\u2212start\u2212LM\n\nMLP\nxfit\n\nPV\n\nSV\n\n 20\n\n 15\n\n 10\n\n 5\n\n 0\n\n\u22125\n\nMLP\u2212start\u2212LM\n\nMLP\nxfit\n\nPV\n\nSV\n\n\u221210\n\n\u22125\n\n 0\n\n 5\n\n 10\n\n\u221210\n\n\u221215\n\n 15\n\n\u221210\n\n\u22125\n\n 0\n\n 5\n\n 10\n\n\u221210\n\n\u221215\n\n 15\n\n\u221210\n\n\u22125\n\n 0\n\n 5\n\n 10\n\n 15\n\nFigure 4: Dipole source localization results of Neuromag software (x\ufb01t), our MLP, MLP-\nstart-LM for four BSS-separated primary visual and four secondary visual MEG signal\ncomponents of S01, over four sorts of tasks. PV and SVdenote primary visual source and\nsecondary visual source, respectively. Left: Axial view. Center: Coronal view. Right:\nSagittal view. The outer surface denotes the sensor surface, and diamonds on this surface\ndenote sensors. The inner surface denotes a spherical head model \ufb01t to the subject.\n\n(standard commercial software bundled with the 4D Neuroimaging Neuromag-122 MEG\nsystem) is compared with the methods developed here.\n\nA \ufb01eld map of each component was scaled to an RMS of 0.5 and inputed to the trained\nMLP. Their MLP\u2019s outputs were scaled back to their dipole location vectors and were used\nfor initializing LM. Figure 4 shows the dipole locations estimated by the MLP, MLP-start-\nLM, and Neuromag\u2019s x\ufb01t software, for two sorts of sensory sources: primary visual sources\nand secondary visual sources, respectively, over four tasks in subject S01. In Figure 5, the\nestimated dipole locations are shown for somatosensory sources over three different sub-\njects. Each \ufb01gure consists of three viewpoints: axial (x-y plane), coronal (x-z plane), and\nsagittal (y-z plane). The center of a \ufb01tted spherical head model (S01: trump card task) is\n(0.335, 0.698, 3.157). All units are in cm. All dipole locations estimated by the MLP and\nMLP-start-LM are clustered within about 3 cm, and about 0.7 cm, of x\ufb01t\u2019s results, respec-\ntively. We see that the primary visual sources are more consistently localized, across all\nfour tasks, than the secondary visual sources. The secondary sources also had more vari-\nable stimulus-locked average time courses (Tang and Pearlmutter, 2003). It is noticeable\nthat somatosensory sources on the right hemisphere are localized poorly by the MLP, but\nwell localized by the hybrid method. Even though the auditory sources are the weakest (not\nshown here), i.e. have the lowest SNRs, they are reasonably well localized.\n\nWhile the MLP-estimated location is about 1.16 cm (|dx| \u2248 0.90, |dy| \u2248 0.57, |dz| \u2248\n0.46) on average (N = 14) from those of x\ufb01t, the hybrid method\u2019s result is about 0.35 cm\n(|dx| \u2248 0.20, |dy| \u2248 0.22, |dz| \u2248 0.10) from x\ufb01t\u2019s estimated location. Considering that\nx\ufb01t had extra information, namely the identity of a subset of the sensors to use, this hybrid\nmethod result is believed to be almost as good as the x\ufb01t result. The trained MLP and the\nhybrid method are applicable to actual MEG signals, and seem to offer comparable and\nperhaps superior localization relative to x\ufb01t, with clear advantages in both speed and in the\nlack of required human interaction or subjective human input.\n\nSOBI was performed on continuous 122-channel data collected during the entire period of the\nexperiment. It generated 122 components, each a one-dimensional time series with an associated \ufb01eld\nmap. Event triggered averages were calculated from their continuous single-trial time series for all\n122 separated components. A dipole \ufb01tting method was applied to the identi\ufb01ed neural components.\nThe input to the dipole \ufb01tting algorithm of x\ufb01t was the \ufb01eld map and the output was the location of\nECDs. From all separated components for four subjects and four sorts of tasks taken as in Tang et al.\n(2002). only fourteen components were localized and compared. For further experimental details and\na detailed SOBI algorithm, see Tang et al. (2002).\n\n\f 15\n\n 10\n\n 5\n\n 0\n\n\u22125\n\n\u221210\n\n\u221215\n\n\u221215\n\nMLP\u2212start\u2212LM\nMLP\nxfit\n\ny\n\n 20\n\n 15\n\n 10\n\n 5\n\n 0\n\n\u22125\n\nMLP\u2212start\u2212LM\n\nMLP\n\nxfit\n\nz\n\n 20\n\n 15\n\n 10\n\n 5\n\n 0\n\n\u22125\n\n\u221210\n\n\u22125\n\n 0\n\n 5\n\n 10\n\nx\n\n\u221210\n\n\u221215\n\n 15\n\n\u221210\n\n\u22125\n\n 0\n\n 5\n\n 10\n\nx\n\n\u221210\n\n\u221215\n\n 15\n\nMLP\u2212start\u2212LM\n\nMLP\nxfit\n\nz\n\ny\n\n\u221210\n\n\u22125\n\n 0\n\n 5\n\n 10\n\n 15\n\nFigure 5: Dipole source localization results of Neuromag software (x\ufb01t), our MLP, MLP-\nstart-LM for three real BSS-separated somatosensory MEG signal components from the\ntransverse patterning task over three different subjects (S01, S02, S03). Even the center of\na \ufb01tted spherical head model is varied over three subjects, the only \ufb01tted sphere of subject\nS01 transverse patterning task, centered at (0.373, 0.642, 3.205), is depicted. Left: Axial\nview. Center: Coronal view. Right: Sagittal view. The outer surface denotes the sensor\nsurface, and diamonds on this surface denote sensors. The inner surface denotes a spherical\nhead model \ufb01t to the subject.\n\n4 Conclusion\n\nWe propose the inclusion of a head position input for MLP-based MEG dipole localizers.\nThis overcomes the limitation of previous MLP-based MEG localization systems, namely\nthe need to retrain the network for each session or subject. Experiments showed that the\ntrained MLP was far faster, albeit slightly less accurate, than \ufb01xed-4-start-LM. This mo-\ntivated us to construct a hybrid system, MLP-start-LM, which improves the localization\naccuracy while reducing the computational burden to less than one ninth than that of \ufb01xed-\n4-start-LM. This hybrid method was comparable in accuracy to random-20-start-LM, at\n1/40-th the computation burden, which is about two times faster than might be naively\nexpected. Over the whole range of SNRs, the hybrid system showed almost as good per-\nformance in accuracy and computation time as the hypothetical optimal-start-LM.\n\nWe applied the MLP and MLP-start-LM to localize single dipolar sources from actual BSS-\nseparated MEG signals, and compared these with the results of the commercial Neuromag\nprogram x\ufb01t. The MLP yielded dipole locations close to those of x\ufb01t, and MLP-start-LM\ngave locations that were even closer to those of x\ufb01t.\n\nIn conclusion, our MLP can itself serve as a reasonably accurate real-time MEG dipole\nlocalizer, even when the head position changes regularly. This MLP also constitutes an\nexcellent dipole guessor for LM. Because this MLP receives a head position input, the\nneed to retrain for various subjects or sessions has been eliminated without sacri\ufb01cing the\nmany advantages of the universal approximator direct inverse approach to localization.\n\nAcknowledgements\n\nThis work was supported by NSF CAREER award 97-02-311, the Mental Illness and Neu-\nroscience Discovery Institute, a gift from the NEC Research Institute, NIH grant 2 R01\nEB000310-05, and Science Foundation Ireland grant 00/PI.1/C067. We would like to thank\nGuido Nolte for help with the forward model, Michael Weisend for allowing us to use his\ndata, and Michael Weisend, Akaysha Tang, and Natalie Malaszenko for providing experi-\nmental details.\n\n\fReferences\n\nAbeyratne, U. R., Kinouchi, Y., Oki, H., Okada, J., Shichijo, F., and Matsumoto, K. (1991).\nArti\ufb01cial neural networks for source localization in the human brain. Brain Topography,\n4:3\u201321.\n\nAhonen, A. I., H\u00a8am\u00a8al\u00a8ainen, M. 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