Dipole fitting: The localization approach
Early quantitative source localization research in electro and magnetocardiography had promoted the equivalent current dipole as a generic model of massive electrophysiological activity. Before efficient estimation techniques and software were available, electrophysiologists would empirically solve the MEG/EEG forward and inverse problems to characterize the neural generators responsible for experimental effects detected on the scalp sensors.
This approach is exemplified in (Wood, Cohen, Cuffin, Yarita, & Allison, 1985), where terms such as ‘waveform morphology’ and ‘shape of scalp topography’ are used to discuss the respective sources of MEG and EEG signals. This empirical approach to localization has considerably benefited from the constant increase in the number of sensors of MEG and EEG systems.
Indeed, surface interpolation techniques of sensor data have gained considerable popularity in MEG and EEG research (Perrin, Pernier, Bertrand, Giard, & Echallier, 1987): investigators now can routinely access surface representations of their data on an approximation of the scalp surface – as a disc, a sphere – or on the very head surface extracted from the subject’s MRI. (Wood et al.., 1985) – like many others – used the distance between the minimum and maximum magnetic distribution of the dipolar-looking field topography to infer the putative depth of a dipolar source model of the data.
Computational approaches to source localization attempt to mimic the talent of electrophysiologists, with a more quantitative benefit though. We have seen that the current dipole model has been adopted as the canonical equivalent generator of the electrical activity of a brain region considered as a functional entity. Localizing a current dipole in the head implies that 6 unknown parameters be estimated from the data:
- 3 for location per se,
- 2 for orientation and
- 1 for amplitude.
Therefore, characterizing the source model by a restricted number of parameters was considered as a possible solution to the ill-posed inverse problem and has been attractive to many MEG/EEG scientists. Without additional prior information besides the experimental data, the number of unknowns in the source estimation problem needs to be smaller than that of the instantaneous observations for the inverse problem to be well-posed, in terms of uniqueness of a solution. Therefore, recent high-density systems with about 300 sensors would theoretically allow the unambiguous identification of 50 dipolar sources; a number that would probably satisfy the modeling of brain activity in many neuroscience questions.
It appears however, that most research studies using MEG/EEG source localization bear a more conservative profile, using much fewer dipole sources (typically <5). The reasons for this are both technical and proper to MEG/EEG brain signals as we shall now discuss.
Numerical approaches to the estimation of unknown source parameters are generally based on the widely-used least-squares (LS) technique which attempts to find the set of parameter values that minimize the (square of the) difference between observations and predictions from the model (Fig. 9). Biosignals such as MEG/EEG traces are naturally contaminated by nuisance components (e.g., environmental noise and physiological artifacts), which shall not be explained by the model of brain activity. These components however, contribute to some uncertainty on the estimation of the source model parameters. As a toy example, let us consider noise components that are independent and identically-distributed on all 300 sensors. One would theoretically need to adjust as many additional free parameters in the inverse model as the number of noise components to fully account for all possible experimental (noisy) observations. However, we would end up handling a problem with 300 additional unknowns, adding to the original 300 source parameters, with only 300 data measures available.
Hence, and to avoid confusion between contributions from nuisances and signals of true interest, the MEG/EEG scientist needs to determine the respective parts of interest (the signal) versus perturbation (noise) in the experimental data. The preprocessing steps we have reviewed in the earlier sections of this chapter are therefore essential to identify, attenuate or reject some of the nuisances in the data, prior to proceeding to inverse modeling.
Once the data has been preprocessed, the basic LS approach to source estimation aims at minimizing the deviation of the model predictions from the data: that is, the part in the observations that are left unexplained by the source model.
Let us suppose for the sake of further demonstration that the data is idealistically clean from any noisy disturbance, and that we are still willing to fit 50 dipoles to 300 data points. This is in theory an ideal case where there are as many unknowns as there are instantaneous data measures. However we shall discuss that unknowns in the models do not all share the same type of dependency to the data. In the case of a dipole model, doubling the amplitude of the dipole doubles the amplitude of the sensor data. Dipole source amplitudes are therefore said to be linear parameters of the model. Dipole locations however do not depend linearly on the data: the amplitude of the sensor data is altered non-linearly with changes in depth and position of the elementary dipole source. Source orientation is a somewhat hybrid type of parameter. It is considered that small, local displacements of brain activity can be efficiently modeled by a rotating dipole source at some fixed location. Though source orientation is a non-linear parameter in theory, replacing a free-rotating dipole by a triplet of 3 orthogonal dipoles with fixed orientations is a way to express any arbitrary source orientation by a set of 3 – hence linear – amplitude parameters. Non-linear parameters are more difficult to estimate in practice than linear unknowns. The optimal set of source parameters defined from the LS criterion exists and is theoretically unique when sources are constrained to be dipolar (see e.g. (Badia, 2004)). However in practice, non-linear optimization may be trapped by suboptimal values of the source parameters corresponding to a so-called local-minimum of the LS objective. Therefore the practice of multiple dipole fitting is very sensitive to initial conditions e.g., the values assigned to the unknown parameters to initiate the search, and to the number of sources in the model, which increases the possibility of the optimization procedure to be trapped in local, suboptimal LS minima.
In summary, even though localizing a number of elementary dipoles corresponding to the amount of instantaneous observations is theoretically well-posed, we are facing two issues that will drive us to reconsider the source-fitting problem in practice:
- The risk of overfitting the data: meaning that the inverse model may account for the noise components in the observations, and
- non-linear searches that tend to be trapped in local minima of the LS objective.
A general rule of thumb when the data is noisy and the optimization principle is ruled by non-linear dependency is to keep the complexity of the estimation as low as possible. Taming complexity starts with reducing the number of unknowns so that the estimation problem becomes overdetermined. In experimental sciences, overdeterminacy is not as critical as underdeterminacy. From a pragmatic standpoint, supplementary sensors provide additional information and allow the selection of subsets of channels, which may be less contaminated by noise and artifacts.
The early MEG/EEG literature is abundant in studies reporting on single dipole source models. The somatotopy of primary somatosensory brain regions (Okada, Tanenbaum, Williamson, & Kaufman, 1984, Meunier, Lehéricy, Garnero, & Vidailhet, 2003), primary, tonotopic auditory (Zimmerman, Reite, & Zimmerman, 1981) and visual (Lehmann, Darcey, & Skrandies, 1982) responses are examples of such studies where the single dipole model contributed to the better temporal characterization of primary brain responses.
Later components of evoked fields and potentials usually necessitate more elementary source to be fitted. However, this may be detrimental to the numerical stability and significance of the inverse model. The spatio-temporal dipole model was therefore developed to localize the sources of scalp waveforms that were assumed to be generated by multiple and overlapping brain activations (Scherg & Cramon, 1985). This spatio-temporal model and associated optimization expect that an elementary source is active for a certain duration – with amplitude modulations – while remaining at the same location with the same orientation. This is typical of the introduction of prior information in the MEG/EEG source estimation problem, and this will be further developed in the imaging techniques discussed below.
The number of dipoles to be adjusted is also a model parameter that needs to be estimated. However it leads to difficult, and usually impractical optimization (Waldorp, Huizenga, Nehorai, Grasman, & Molenaar, 2005). Therefore the number of elementary sources in the model is often qualitatively assessed by expert users, which may question the reproducibility of such user-dependent analyses. Hence, special care should be brought to the evaluation of the stability and robustness of the estimated source models. With all that in mind, source localization techniques have proven to be effective, even on complex experimental paradigms (see e.g., (Helenius, Parviainen, Paetau, & Salmelin, 2009)).
Signal classification and spatial filtering techniques are efficient alternative approaches in that respect. They have gained considerable momentum in the MEG/EEG community in the recent years. They are discussed in the following subsection.