Source Model Estimation

Source localization from EEG and MEG began with educated inspection of surface field topographies. In EEG, a radially oriented current will produce a potential extremum over the source. In MEG, a radial current source produces little externally detectable magnetic field, but the tangential component of a neural source produces a field distribution with extrema that straddle the source. The distance separating the field extrema allows an estimation of the source depth, given assumptions regarding the nature of the current source. In both EEG and MEG, many observed response topographies can be explained by an equivalent current dipole (ECD) source, i.e., an isolated point current with a given location, orientation, and amplitude. Theory suggests that such a model provides a reasonable estimate of the field or potential distributions due to a small cluster of oriented neurons measured at a distance. Even extended patches of activated cortex often produce a dipole-like distribution, although the estimated location and strength of the ECD will contain systematic errors. An extended patch of parallel current elements produces an ECD estimate that is deeper and stronger than the center of mass and integral current estimated from the actual source distribution.

Because field topographies are typically diffuse, source estimation by eye is a rather inexact process. Some investigators have employed image processing techniques to allow for easier detection of features in the field topography. For example, computation ofthe spatial derivative (the Laplacean) of the observed magnetic field or potential topography tends to place maxima over the current sources. Indeed, one form of MEG sensor—a first-order planar gradiometer— effectively implements this transform in the detector coil configuration.

Although inspection of response topographies is a useful starting point for finding NEM sources, modelbased parameter estimation procedures provide a more objective strategy that allows quantitative assessment of the goodness of fit. Nonlinear optimization procedures such as simplex or gradient descent allow the estimation of a dipole source; current orientation and strength are linear parameters that can be optimized separately. If a single focal current source is active in a simple medium, such procedures can localize it very precisely. However, at any given instant in time, many sources may contribute to a given response topography. Because of superposition, a combination of several sources may give rise to a distribution that is adequately modeled by a single source at an entirely different location. Knowledge of the number and nature of active sources can reduce the ambiguity of the source localization problem. Spatiotemporal source localization techniques attempt to find a minimal set of sources, each with an associated time course, that explains the observed distribution across a defined interval within an event-related response. If the sources are activated asynchrously, this strategy can be very effective for decomposing and localizing the contributions of individual sources.

Nonlinear optimization methods can be used to find locations for a collection of simple current sources that might explain changing field topographies observed across time. Because the time course is an estimate of the source current amplitude as a function of time, linear methods can be used to optimize the estimates between iterative nonlinear optimization steps. This general strategy has been the most widely used approach for NEM source localization for over a decade, but modifications, extensions, or alternatives to the method can provide enhanced performance. A common problem with nonlinear methods is that they require a starting estimate of model parameters. This may be provided by an informed analyst or by a stochastic process. If the starting estimate is close enough, an optimization procedure that follows the error gradient can find the best-fitting model. If the starting estimates are far off and the error surface is complex (often the case with multiple-source models), the procedure may fall into a local minimum that is not globally optimal. A conscientious analysis can combat this problem by running the optimization procedure with multiple sets of starting parameters. Multistart procedures automate this process, employing a numerical algorithm to generate random starting parameter estimates. Such methods often find a consistent set of best-fitting source models that are globally optimal. Other methods such as simulated annealing or genetic algorithms employ alternative strategies to address the problem of local minima within the model parameter error space. Examples of these approaches are illustrated in Fig. 6.

The multiple signal classification (MUSIC) algorithm operates within the same framework (i.e., a multiple-dipole spatiotemporal model) but employs a more systematic strategy for finding sources. The array of measured potential or field values at any given instant in time is treated as a multidimensional vector in the space of possible measurements. Similarly, the topography associated with any given dipole is another vector in the same space. The algorithm operates by systematically stepping through a set of possible sources (e.g., a grid of locations within the brain) and evaluates the match between the source field vector and the collection of signal vectors across time. The sources that most closely match the observed signals across time are considered the most likely. The method is very effective and can be exhaustive, avoiding problems of local minima seen with nonlinear optimization techniques. Multiple sources with highly correlated time courses create problems for the algorithm, but enhanced methods such as recursively applied (RAP) MUSIC address this and other concerns.

A second general strategy is to use linear inverse techniques to solve a large and general source model. The reconstruction space is defined by a regular grid, a collection of voxels, or vertices from a computational mesh. One to three current elements are associated with each possible source location. The reconstruction procedure employs the Moore-Penrose pseudo-inverse to assign a current value to each model element. This procedure, based on singular value decomposition, estimates a source distribution with minimum power over the collection of driving currents. A number of variants on this minimum norm procedure have been described, mostly based on different strategies for weighting the lead field basis matrix in order to select a solution with desired properties. Anatomical constraints based on cortical geometry can improve accuracy and efficiency. However, even with substantial reductions in the source space based on anatomy, the inverse is a highly underdetermined problem. There are many more source model parameters to estimate than the number of independent measures available from MEG or EEG.

The major problem with minimum norm procedures is that there is no guarantee that the solution of minimum Euclidean norm (i.e., the sum of squared currents) will be representative of the true solution. Because of the strong dependence of measured magnetic field on distance from the source, the basic minimum norm procedure tends to produce diffuse, superficial current reconstructions, even when the reconstruction space is constrained to the cortical surface. Currents closer to the sensor array can account for more power in the field map with less current and therefore are favored by the method. However, more current elements are required to account for the shape of a field distribution that may actually arise from a more focal but deeper source.

Figure 6 Source localization by multiple dipole spatiotemporal estimation procedures. (a) Dipole source locations and time courses estimated from visual evoked response data. This analysis used a multistart algorithm; the best 10 solutions from 10,000 generated are tightly clustered, suggesting that the algorithm has found a global best fit. (b) Dipole locations and time courses associated with an epileptic response from a photosensitive child. Dipole locations were estimated with a genetic algorithm. Note that the magnetic response of the eye contains evidence of the strobe flashes that triggered the response.

Figure 6 Source localization by multiple dipole spatiotemporal estimation procedures. (a) Dipole source locations and time courses estimated from visual evoked response data. This analysis used a multistart algorithm; the best 10 solutions from 10,000 generated are tightly clustered, suggesting that the algorithm has found a global best fit. (b) Dipole locations and time courses associated with an epileptic response from a photosensitive child. Dipole locations were estimated with a genetic algorithm. Note that the magnetic response of the eye contains evidence of the strobe flashes that triggered the response.

In order to combat this tendency, it is possible to scale the field distributions (or, alternatively, the strength of the unit currents) in order to normalize the field power associated with each elemental source. Pseudo-inverse procedures based on a normalized basis matrix offer some improvement in the fidelity of reconstructions. Explicit or implicit basis matrix weighting procedures have proven to be a useful general strategy for modifying the properties of reconstruction algorithms. The FOCUSS algorithm employs an iterative reweighting procedure to derive sparse reconstructions based on focal activated patches. The LORETA algorithm uses an alternative weighting scheme to find current reconstructions that are maximally smooth.

Given the fundamental ambiguity of the inverse problem and the complex error surface associated with the parameter space, there is no guarantee that the proper form of source model (e.g., the number of active sources) can be determined or that a single global minimum will be found. The estimated parameter values critically depend on model assumptions and may vary widely as a function of small amounts of noise in the data.

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