Direct reconstruction algorithm of current dipoles for vector magnetoencephalography and electroencephalography

Abstract

This paper presents a novel algorithm to reconstruct parameters of a sufficient number of current dipoles that describe data (equivalent current dipoles, ECDs, hereafter) from radial/vector magnetoencephalography (MEG) with and without electroencephalography (EEG). We assume a three-compartment head model and arbitrary surfaces on which the MEG sensors and EEG electrodes are placed. Via the multipole expansion of the magnetic field, we obtain algebraic equations relating the dipole parameters to the vector MEG/EEG data. By solving them directly, without providing initial parameter guesses and computing forward solutions iteratively, the dipole positions and moments projected onto the xy-plane (equatorial plane) are reconstructed from a single time shot of the data. In addition, when the head layers and the sensor surfaces are spherically symmetric, we show that the required data reduce to radial MEG only. This clarifies the advantage of vector MEG/EEG measurements and algorithms for a generally-shaped head and sensor surfaces. In the numerical simulations, the centroids of the patch sources are well localized using vector/radial MEG measured on the upper hemisphere. By assuming the model order to be larger than the actual dipole number, the resultant spurious dipole is shown to have a much smaller strength magnetic moment (about 0.05 times smaller when the SNR = 16 dB), so that the number of ECDs is reasonably estimated. We consider that our direct method with greatly reduced computational cost can also be used to provide a good initial guess for conventional dipolar/multipolar fitting algorithms.

Electro–magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries

“…the case of a collection of dipoles, is analysed in Dassios & Fokas (preprint a,b) for spherical and ellipsoidal geometries, respectively. For other related important works, see El Badia & Ha-Duong (2000), Jerbi et al (2002), Nara & Ando (2003), Nolte & Dassios (2005), Albanese & Monk (2006), Peng et al (2006), Nara et al (2007) and Leblond et al (preprint). This paper is organized as follows: the equations needed for EEG and MEG in a three-shell model are derived in §2; this is done for the sake of completeness so that this paper is self-contained.…”

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

“…(10) from c m (m = 0, 1, ···) appears in many inverse problems such as computed tomography (Golub et al (1997)), EEG inversion (El-Badia et al (2000); Nara et al (2003)), MEG inversion (Nara et al (2007)), and locating the zeros of analytic functions (Kravanja et al (1994) …”

“…Ω 3 represents the head. We assume that the radial component of the magnetic field is measured on the sphere Γ = ∂Ω 4 with the radius of R. Although we use this simple head modelaswellasthesphericalsensorsurface,themethodcanbeextendedtoamorerealistic case when the head is modeled by a piecewise homogeneous layered domain and the sensors are set on an arbitrarily shaped surface (Nara et al (2007)). Let us assume that the neural current J p is supported on several domains D k (⊂ Ω 1 ) for k = 1, 2, ··· , N. The 'center' position of D k is denoted by r k =( x k , y k , z k ) T ;t h isist h ema in parameter to be reconstructed.…”

“…In Nara (2008b), we proposed an explicit method for 3D case when the dipoles were distributed in a plane parallel to the xy-plane. The aim of this chapter is to describe explicit methods for the equivalent current dipole model and the equivalent current dipole-quadrupole model in Nara et al (2007), Nara (2008a), and Nara (2008b) and compare them using numerical simulations. Here, the term 'explicit' means that we have an analytic and explicit form of the solution to the inverse problem.…”

Computation of multipole moments from incomplete boundary data for Magnetoencphalography inverse problem

“…Although the usual algorithm for this source model is the non-linear least-squares method that minimizes the squared error of the data and the forward solution, it has a problem that an initial parameter estimate close to the true one is required without which the algorithm often converges to a local minimum. To address this issue, several researchers have proposed a direct method [1][2][3][4] which reconstructs the source parameters directly and algebraically from the data. From the efficiency of the algorithm, it is expected to be used for real-time monitoring of the brain activity.…”