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The purpose of the neuromagnetic inverse problem is to reconstruct primary neural current from measured MEG data. It is known that this inverse problem is ill-posed: uniqueness of the solution to the inverse problem is not guaranteed without a priori assumptions on the current source (Fokas et al. (2004)), and, even when using the source model that can be uniquely reconstructed, the obtained solution changes very sensitively depending on the noise contained in MEG data. Thus, employment of a stable algorithm is highly required in order that MEG becomes a non-invasive brain monitoring tool with not only high temporal resolution but also high spatial resolution. Basically, conventional methods are categorized into two groups: parametric approaches and imaging approaches. See the detailed list of references in Baillet et al. (2001). The former methods assume that the current source can be represented by a relatively small number of equivalent current dipoles (ECDs). This source model is shown to be uniquely reconstructed from radial MEG data, except the radial component of the dipole moment, when the head is assumed to consist of concentric spheres. The usual algorithm for this source model is the non-linear least-squares method that minimizes the squared error of data and the forward solution. An advantage of this algorithm is that the parametric description allows us the accurate estimation of the center position of the activated region. However, the problems of this algorithm are: 1) it requires an initial solution, 2) it requires an iterative computing of forward solution, 3) it is often trapped by the local minimum of the cost function, which gives a solution far from the true one, 4) estimation of the number of ECDs is difficult, and 5) spatial extent of the source is not considered. The secondmethods, imaging approaches, assume that there exist current dipoles at the nodes of artificial meshes on the cerebral surface, and solve a system of linear equations for the dipole moments at the fixed positions. An advantage of this method is that it can describe the spatial distribution of the neural current. However, the problems of this algorithm are: 1) the solution is not unique, 2) adding a regularization term often gives a unique but over-smoothed solution, 3) choice of the regularization parameters strongly affects the solution. Recently, a method with the multipolar representation of the source has been developed that incorporates some of the advantages of the above twomethods, and has attracted considerable attention (Irimia et al. (2009); Jerbi et al. (2002; 2004); Nara (2008a); Nolte et al. (1997; 2000)). In this model, instead of expressing the current source by an equivalent current dipole, 6

An Explicit Method for Inverse Reconstruction of Equivalent Current Dipoles and Quadrupoles