MEG

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Definition. The quantitative analysis of scalp MEG and EEG data continues to generate a very diverse ..... neuroimaging: a primer with examples. Hum Brain ...
Forward/Inverse Problems of EEG/MEG Encyclopedia of Computational Neuroscience · Article ID: 348594 · Chapter ID: 529 5P06-TBXL-OY2J-R



Definition The quantitative analysis of scalp MEG and EEG data continues to generate a very diverse body of research based on the characterization of time-resolved brain activity. Some questions however require a more direct assessment of the anatomical substrate of cerebral dynamics. In many cases, modeling the neural generators of scalp MEG/EEG data is the method of choice. From a methodological standpoint, MEG/EEG source modeling is an “inverse problem”: a ubiquitous concept in many fields, from medical imaging to geophysics (Tarantola, 2004). It builds a framework that helps conceptualize and formalize the fact that, in experimental sciences, models are confronted with observations to test a set of hypotheses and/or to estimate some parameters that were originally unknown. Parameters are quantities that can be changed without fundamentally invalidating the theoretical model. Predicting observations from a model with a given set of parameters is called forward modeling. The reciprocal situation where observations are used to estimate some unknown model parameters is an inverse modeling problem. The neural sources of scalp signals can be defined in terms of their anatomical location and current-flow orientation in space and amplitude variations in time, which represent the primary unknown parameters of MEG/EEG inverse modeling. To enable the latter, it is crucial to capture how these parameters contribute to scalp data. Hence, MEG/EEG forward modeling consists in predicting the electromagnetic fields and potentials generated by any arbitrary source model, for any possible location, orientation and amplitude parameter values. In practice, MEG/EEG forward modeling assumes that some additional key parameters are known, hence are not estimated from the data: the geometry of the head, conductivity of tissues, sensor locations, etc. (Figure 1). Importantly, it is established that the general inverse problem consisting in characterizing the sources that generate electromagnetic fields outside a volume conductor has an infinite number of solutions. This issue of non-uniqueness is not specific to MEG/EEG: astrophysicists for instance face the same issue to determine the distribution of mass inside a planet by measuring its external gravity field. Hence in principle, an infinite number of source models can fit equivalently any set of MEG and EEG observations. Fortunately, this conundrum is addressed with the powerful mathematics of regularization, which formalizes how additional contextual information can complement a basic theoretical model and guarantee uniqueness and robustness in the determination of its unknown parameters. A considerable amount of research has been produced on practical methods for MEG/EEG forward and inverse modeling. They actually reduce to a handful of classes of approaches, which are now well identified and contribute to making MEG and EEG source imaging a major neuroimaging technique (Salmelin & Baillet, 2009).

MEG and EEG forward modeling Model of neural sources MEG and EEG forward modeling requires two basic models that are bound to work together in a complementary manner: a physical model of neural sources, and a model that predicts how these sources generate electromagnetic fields outside the head. The canonical source model of the net primary intracellular currents within a neural assembly is the electric current dipole. The adequacy of a simple, equivalent current dipole (ECD) model as a building block of cortical current distributions was originally motivated by the topography of scalp MEG/EEG data, which typically consists of dipolar patterns of inward/outward

magnetic fields and positive/negative electrical potentials. From a historical standpoint, the ECD model derived from extensive previous work in electrocardiography, where dipolar field patterns are also omnipresent (Geselowitz, 1964). However, the complexity of the spatio-temporal dynamics of brain activity is more challenging than that of ‘cardiac traces’. Therefore, more generic alternatives to the ECD model have been proposed either as higherorder multipoles (Jerbi et al., 2004) also derived from cardiographic research, or densely distributed source models (Wang et al., 1992). In the latter case, a large number of ECDs are distributed in the entire brain volume or spatially-constrained to the cortical surface, thereby forming a dense grid of elementary sites of activity, the amplitudes/intensities of which are determined from the data. This latter approach is now the prominent technique of choice for MEG and EEG source imaging (Salmelin & Baillet, 2009). To understand how these elementary source models generate signals that are measurable using external sensors, further forward modeling of the sensor array and of the electromagnetic properties of head tissues is required.

MEG/EEG sensor arrays Sensor locations can be readily measured with state-of- the-art MEG devices and ancillary EEG equipment. Sensor locations may also be approximated from montage templates; however this is sub-optimal   with respect to the accuracy of subsequent source estimation (Schwartz et al., 1996). This is critical with MEG, as the subject is relatively free to position his/her head within a sensor array of fixed size and shape, although movement compensation techniques have been proposed. Further detailed modeling of the sensor geometry, pick-up technology and online noise-attenuation processes is state-of-the-art in the field.

Head modeling Quasistatic assumptions MEG and EEG forward modeling also consists in identifying the influence of the head geometry and of the electromagnetic properties of its tissues on data formation. This last step is popularly coined “head modeling”. For a given model of elementary neural currents and sensor array, the physics of MEG and EEG are ruled by the theory of electrodynamics (Feynman, 1964), which reduces in MEG to Maxwell's equations, and to Ohm's law in EEG, under quasistatic assumptions. These latter consider that the propagation delay of the electromagnetic waves from brain sources to the MEG/EEG sensors is negligible. The reason is the relative proximity of MEG/EEG sensors to the brain with respect to the expected frequency range of neural sources (up to 1 kHz) (Hämäläinen et al., 1993). This is a very important, simplifying assumption, which has immediate consequences on the computational aspects of MEG/EEG head modeling. Indeed, the equations of electro- and magneto-statics determine that there exist analytical, closed-form solutions to MEG/EEG head modeling when the head geometry is considered spherical. Hence, the simplest, and consequently by far most popular model of head geometry in MEG/EEG consists of concentric spherical layers: with one sphere per major category of head tissue (scalp, skull, cerebrospinal fluid and brain).

Simplified, spherical head geometry The spherical head geometry has further attractive properties for MEG in particular. Quite remarkably indeed, spherical MEG head models are insensitive to the number of shells and their respective conductivity: a current source within a single homogeneous sphere generates the same MEG fields as when located inside a multilayered set of concentric spheres with different conductivities. The reason for this is that conductivity only influences the distribution of secondary volume currents that circulate within the head volume and which are impressed by the original primary neural currents. The analytic formulation of Maxwell's equations in the spherical geometry demonstrates that these secondary currents

do not generate any magnetic field outside the volume conductor (Sarvas, 1987). Consequently, the conductivity and radius of each spherical layer have no influence on the measured MEG fields, which is not the case in EEG. This is considered a major advantage for MEG. Geometrical registration to individual MRI anatomical data improves the adjustment of the best-fitting sphere geometry to the participant’s head. A spherical head model can be optimally adjusted to the head geometry, or restricted to regions of interest e.g., posterior regions for visual studies. Another remarkable consequence of the spherical symmetry is that radially oriented brain currents produce no magnetic field outside a spherically symmetric volume conductor. For this reason, MEG signals from currents generated at the gyral crests or sulcal depths are attenuated, with respect to those generated by currents flowing perpendicularly to the sulcal walls. The difference in sensitivity to source orientation is another important contrast between MEG and EEG (Hillebrand and Barnes, 2002). Finally, the amplitude of magnetic fields decreases faster than electrical potentials' with the distance from the generators to the sensors. Hence it has been argued that MEG is less sensitive to mesial and subcortical brain structures than EEG. An increasing body of experimental evidence and modeling efforts has shown however that MEG can detect neural activity from deeper brain regions (Attal and Schwartz, 2013).

Realistic head geometry Although spherical head models are convenient, they are poor approximations of the human head shape, which impacts the accuracy of MEG/EEG source estimation (Stenroos et al., 2012). The use of more realistic head geometries, which requires solving Maxwell’s equations with numerical methods, has also been proposed. Boundary Element (BEM) and Finite Element (FEM) methods are generic numerical approaches to the resolution of continuous equations over discrete space. In MEG/EEG, geometric tessellations of the different envelopes forming the head tissues need to be extracted from the individual MRI volume data to yield a realistic approximation of their geometry. In BEM, the conductivity of tissues is supposed to be homogeneous and isotropic within each envelope. Therefore, each tissue envelope is delimited using surface boundaries defined over a triangulation of each of the segmented envelopes obtained from MRI. FEM assumes that tissue conductivity may be anisotropic (such as the skull bone and the white matter); therefore the primary geometric building block needs to be an elementary volume, such as a tetrahedron (Marin et al., 1998). In practice, the FEM’s tessellation step remains tedious and complicated. However, efficient implementations of BEM have been derived recently (e.g., Kybic et al, 2005) and are now readily available in several commercial and academic software applications (Baillet et al., 2011). An important caveat, especially for EEG: Realistic head modeling also depends on the correct estimation of tissue conductivity values, which are difficult to access in vivo. Methods of impedance tomography have been proposed with MRI (Tuch et al., 2001) and EEG (Goncalves et al., 2003) but have had limited practical impact so far. Hence, conductivity values from ex-vivo models are conventionally considered in MEG and EEG head models (Geddes and Baker, 1967).

MEG/EEG source modeling Source localization vs. source imaging The localization approach to MEG/EEG source estimation assumes that brain activity at any time instant is generated by a relatively small number (a handful, at most) of brain regions. An elementary model, such as an ECD accounting for the local distribution of neural

currents is typically used for each source. The resulting elementary sources can then be overlaid to the subject's MRI volume, for visualization and interpretation. Source localization models are therefore compact representation of distributed physiological currents. Although extremely popular in the 1980’s and 90’s, this type of approach is now being gradually abandoned in favor of image-based, distributed inverse models, which are better adapted to streamlined analyses of cohorts of subjects. MEG/EEG source imaging techniques were inspired from the methods and algorithms contributed by research in image restoration and reconstruction in other fields (early digital imaging, geophysics, and other biomedical imaging techniques). Synoptic images of neural current density are obtained using a dense grid of current dipoles distributed as voxel-like current intensity elements over the entire brain volume, or limited to the cortical gray matter surface. The imaging procedure is an estimation of the time-resolved amplitude changes of all elementary currents. Hence, in contrast to the localization model, there is no intrinsic sense of distinct, active source regions per se. Explicit identification of activity issued from discrete brain regions usually necessitates subsequent analysis, such as empirical or inference-driven amplitude thresholding. In that respect, MEG/EEG source images are essentially similar to the maps obtained in fMRI, with the benefit of time resolution however (Figure 2).

Beamforming The first alternatives to source localization were inspired by radar and sonar data analysis, which also concerns array signal processing. MEG and EEG source scanning techniques have therefore emerged: they proceed by systematically sifting through the brain volume to evaluate how a predetermined elementary source model would fit the data at every voxel, while “blocking” contributions from other sources elsewhere in the brain. Hence, these techniques are also known as spatial-filters and beamformers (the simile is a virtual beam directed towards a specific brain region) (Spencer et al., 1992; Hillebrand et al., 2005). In theory, beamformers cannot properly detect signal sources that are strictly synchronized; they are also dependent on the prior characterization of second-order noise vs. signal statistics, which may be challenging in practice (Wax and Anu, 1996). Nevertheless, beamforming has now become a popular technique for MEG and EEG source imaging.

Distributed source imaging Imaging source models consist of distributions of elementary dipole sources in the brain volume or on the cortex, generally with fixed locations and orientations (Dale and Sereno, 1993; Lin et al., 2006). The parameters to estimate through inverse modeling at each time point are therefore the amplitudes of thousands of elementary dipoles given only tens or a few hundreds of data points. Consequently, this image reconstruction inverse problem is dramatically underdetermined and needs to be supplemented by a priori information. The addition of relevant priors can be properly formulated with the mathematics of regularization, reviewed in the context of MEG and EEG (Baillet et al., 2001). The priors on the source image models may take multiple faces e.g., in order to promote current distributions with spatial and temporal smoothness, or to penalize solutions with current amplitudes that are not physiologically plausible, or to fit with fMRI activation maps, or to prefer source image models made of piecewise homogeneous active regions, etc. Wellchosen priors may also guarantee the uniqueness of the optimal solution to the imaging inverse problem. Overall, weighted minimum-norm image models are the most popular in the field: they are relatively robust to noise and head model approximations, generate centimeter resolution images and are fast to compute. Note that the selection of a particular set of image priors is quite subjective. However, methods have emerged to approach model selection in principle manner (Daunizeau et al., 2006).

Appraisal of MEG/EEG source models

Modeling implies dealing with uncertainty and MEG/EEG forward and inverse modeling has uncertainty everywhere: data are complex and contaminated with various nuisances, source models are simplistic, head models are obtained from approximated geometries and conductivity properties. The determination of error bounds and confidence intervals on the estimated parameter values is an important issue, which has been researched with either parametric methods (Mosher et al., 1993) or non-parametric resampling techniques (Baryshnikov et al., 2004; Darvas et al., 2005). Statistical inference and hypothesis testing are other important aspects of the appraisal of MEG and EEG source models; they directly address the neuroscience question that has motivated data acquisition and the experimental design (Guilford and Fruchter, 1978). In the context of MEG/EEG, the population samples supporting inference are either trials or subjects, for hypothesis testing at the individual and group levels, respectively. Here too, parametric (Kiebel et al., 2005) and non-parametric (Nichols and Holmes, 2002; Pantazis et al., 2005) approaches to statistical inference have been proposed, with the important concern of multiple-hypothesis testing (e.g., at thousands of source locations) kept under control (Figure 3).

Conclusion The convergence of forward and inverse modeling techniques with statistical inference solutions has brought electromagnetic source imaging to a considerable degree of maturity that is comparable to other neuroimaging techniques (Lopes da Silva, 2013). The practical impact of methods for EEG/MEG data analysis has been considerably multiplied by the emergence of major commercial and academic software applications (Baillet et al., 2011). The complexity in the data obtained with MEG/EEG source analysis is a direct reflection of the intricate brain dynamics that unfold over multiple spatial and temporal scales. For this reason, MEG and EEG imaging has become a truly unique neuroimaging technique for revealing the fundamental large-scale mechanisms that underlie brain function and dysfunctions.

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Principles of MEG/EEG forward and inverse modeling: a) some unknown brain activity generates magnetic fields and electric potentials at the surface of the scalp. This is illustrated by black traces of time series representing measurements on a subset of sensors. b) A naive illustration of forward and inverse models is shown, with colored arrowheads as ECDs inside a spherical head geometry. The free parameters of each ECD e.g., location, orientation and amplitude, are adjusted by inverse modeling. c) Residuals i.e., the difference between the original data and the measures predicted by a source model are minimized by the inverse modeling procedure.

Inverse modeling: source localization (a) vs. imaging (b) approaches. Source localization consists in decomposing the MEG/EEG generators using a handful of equivalent current dipoles (ECD). This is illustrated here from experimental data testing the somatotopic organization of primary cortical representations of hand fingers. The parameters of the single ECD have been adjusted on the [20, 40] ms time window following electrical stimulus onset at a given hand finger. The ECD was found to localize along the contralateral central sulcus as revealed from the 3D rendering, after registration to the individual anatomy obtained with MRI. In the imaging approach, the source model is spatially distributed using a large number of ECD's. Here, a surface model of MEG generators was constrained to the individual brain surface extracted from T1-weighted MR images. Elementary source amplitudes are interpolated onto the cortex, which yields an image-like distribution of the amplitudes of cortical currents.

MEG source imaging of the response to the presentation of a visual target in a rapid serial visual presentation paradigm. The images show a slightly smoothed version of one participant's cortical surface. Colors encode the contrast of MEG source amplitudes between responses to target versus control faces. Salient visual responses are detected in the target condition by 120ms and rapidly propagate anteriorly. By 250 ms and thereafter, strong posterior cingulate responses are detected. These latter are the main contributors of the brain response to target presentation.