Direction vector

Abstract: CURRENTS INSIDE a conducting body can be estimated by measuring the magnetic and/or the electric field at multiple locations outside and then constructing a solution to the inverse problem, i.e. determining a current configuration that could have produced the measured field. Unfortunately, there is no unique solution to this problem (HELMHOLTZ, 1853) unless restricting assumptions are made. The minimum-norm estimate (HAM/~.L,~INEN and ILMONIEMI, 1994) provides a solution with the smallest expected overall error when minimum a priori information about the source distribution is available. Other methods to estimate a continuous current distribution producing the measured signals have been studied (PASCUAL-MARQUI et al., 1994; WANG et aL, 1995; GORODNITSKY, et al., 1995). A different approach is to divide the brain activity into discrete components such as current dipoles (ScHERG, 1990; MOSHER et al., 1992). Here we widen this approach into arbitrary current configurations. In our signal-space projection (SSP) method, the signals measured by d sensors are considered to form a time-varying vector in a d-dimensional signal space. The component vectors,, i.e. the signals caused by the different neuronal sources, have different and fixed orientations in the signal space. In other words, each source has a distinct and stable field pattern. All the current eonfi~marations producing the same measured field pattern are indistinguishable on the basis of the field: they have the same vector direction in the signal space and thus belong to the same equivalence class of current configurations (TESCHE et al., 1995a). The angle in the signal space between vectors representing different equivalence classes, e.g. between component vectors, is a measure of similarity of the equivalence classes in signal space and a way to characterise the separability of sources. The cosine of this angle has previously been used as a numerical charaeterisation of the difference between topographical distributions (DESMEDT and CHALK[.IN, 1989). If the direction of at least one of the component vectors forming the measured multi-channel signal can be determined from the data, or is known otherwise, SSP can be used to simplify subsequent analysis. For example, if an early deflection in an evoked response is produced by one source, and the rest of the response is a mixture of signals from this and other sources, SSP can separate the data into two parts so that the early source contributes only to one part. In general, the signals are divided into two orthogonal parts: s~, including the time-varying contribution from sources with known signalspace directions; and s~_, including the rest of the signals. Both sl~ and s j_ can then be analysed separately in more detail. By analysing s t , we can detect activity originally masked by s~. On the other hand, the sources included in stl are seen with an enhanced signal-to-noise ratio. By forward modelling of sources in selected patches of cortex, it is possible to form a spatial filter that selectively passes only the signals that may have been generated by currents in the given patches. If the subspace defined by artefacts can be determined, the artefactflee S L can be analysed. In SSP, in contrast to PCA (HARRIS, 1975; MAIER et al., 1987) and other analysis methods (GRUMMICH et al., 1991; KOLES et aL, 1990; KOLES, 1991; SOONG and KOLES, 1995; BESA*), the source decomposition does not depend on the orthogonality of source components or the availability of source or conductivity models. No conductivity or source models are needed if the component vectors are estimated directly from the measured signals. This is useful when no source estimation is needed, e.g. when artefacts or somatomotor activity in a cogrritive study must be filtered out. The angles between the components provide an easy and illustrative way to characterise the linear dependence between the components and thus the separability of sources. The concept of signal space in MEG was introduced previously ([LMONIEMI, 1981; [LMONIEMI and WILLIAMSON,