No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
An Explicit Method for Inverse Reconstruction of Equivalent Current Dipoles and Quadrupoles
... The aim of the current paper is to extend our algebraic method from the 2D case to the 3D case so that the dipole and quadrupoles, which equivalently represent the neural current, can be reconstructed from the magnetic field data without an initial parameter estimate or iterative computing of the forward solution. In [19,20], we proposed an algebraic method when the dipoles were distributed in a plane parallel to the -plane, which is a very special case and is severely restricted in its practical usage. This paper derives a method for the general case. ...
... . See Appendix C for the derivation of (21). Substituting (20) and (21) into (18) gives We now put ...
... Comparing with the special case where the dipoles were distributed in a plane parallel to the -plane in [20], one finds that has an extra term: ( , − , )( + ). Equation (24) has the same form as that of (6) in [17], and consequently the dipole-quadrupole position projected on the -plane, ( = 1, 2, . . . ...
This paper presents an algebraic method for an inverse source problem for the Poisson equation where the source consists of dipoles and quadrupoles. This source model is significant in the magnetoencephalography inverse problem. The proposed method identifies the source parameters directly and algebraically using data without requiring an initial parameter estimate or iterative computation of the forward solution. The obtained parameters could be used for the initial solution in an optimization-based algorithm for further refinement.
Measurement of external magnetic fields provides information on electric current distribution inside an object. For example, in magnetoencephalography modern measurement devices sample the magnetic field produced by the brain in several hundred distinct locations around the head. The signal space separation (SSS) method creates a fundamental linear basis for all measurable multichannel signal vectors of magnetic origin. The SSS basis is based on the fact that the magnetic field can be expressed as a combination of two separate and rapidly converging expansions of harmonic functions with one expansion for signals arising from inside of the measurement volume of the sensor array and another for signals arising from outside of this volume. The separation is based on the different convergence volumes of the two expansions and on the fact that the sensors are located in a source current-free volume between the interesting and interfering sources. Individual terms of the expansions are shown to contain uncorrelated information of the underlying source distribution. SSS provides a stable decomposition of the measurement into a fundamental device-independent form when used with an accurately calibrated multichannel device. The external interference signals are elegantly suppressed by leaving the interference components out from the reconstruction based on the decomposition. Representation of multichannel data with the SSS basis is shown to provide a large variety of applications for improved analysis of multichannel data.
Magnetoencephalography (MEG) is a non-invasive functional imaging modality based on the measurement of the external magnetic field produced by neural current sources within the brain. The reconstruction of the underlying sources is a severely ill-posed inverse problem typically tackled using either low-dimensional parametric source models, such as an equivalent current dipole (ECD), or high-dimensional minimum-norm imaging techniques. The inability of the ECD to properly represent non-focal sources and the over-smoothed solutions obtained by minimum-norm methods underline the need for an alternative approach. Multipole expansion methods have the advantages of the parametric approach while at the same time adequately describing sources with significant spatial extent and arbitrary activation patterns. In this paper we first present a comparative review of spherical harmonic and Cartesian multipole expansion methods that can be used in MEG. The equations are given for the general case of arbitrary conductors and realistic sensor configurations and also for the special cases of spherically symmetric conductors and radially oriented sensors. We then report the results of computer simulations used to investigate the ability of a first-order multipole model (dipole and quadrupole) to represent spatially extended sources, which are simulated by 2D and 3D clusters of elemental dipoles. The overall field of a cluster is analysed using singular value decomposition and compared to the unit fields of a multipole, centred in the middle of the cluster, using subspace correlation metrics. Our results demonstrate the superior utility of the multipolar source model over ECD models in providing source representations of extended regions of activity.
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.
We have presented techniques [1] - [6] based on linear prediction (LP) and singular value decomposition (SVD) for accurate estimation of closely spaced frequencies of sinusoidal signals in noise. In this note we extend these techniques to estimate the parameters of exponentially damped sinusoidal signals in noise. The estimation procedure presented here makes use of "backward prediction" in addition to SVD. First, the method is applied to data consisting of one and two exponentially damped sinusoids. The choice of one and two signal components facilitates the comparison of estimation error in pole damping factors and pole frequencies to the appropriate Cramer-Rao (CR) bounds and to traditional methods of linear prediction. Second, our method is applied to an example due to Steiglitz [8] in which the data consists of noisy values of the impulse response samples (composed of many exponentially damped sinusoids) of a linear system having both poles and zeros. The poles of the system are accurately determined by our method and the zeros are obtained subsequently, using Shanks' method.
In this paper, we clearly demonstrate that the electric potential and the magnetic field can contain different information about current sources in three-dimensional conducting media. Expressions for the magnetic fields of electric dipole and quadrupole current sources immersed in an infinite conducting medium are derived, and it is shown that two different point dipole distributions that are electrically equivalent have different magnetic fields. Although measurements of the electric potential are not sufficient to determine uniquely the characteristics of a quadrupolar source, the radial component of the magnetic field can supply the additional information needed to resolve these ambiguities and to determine uniquely the configuration of dipoles required to specify the electric quadrupoles. We demonstrate how the process can be extended to even higher-order terms in an electrically silent series of magnetic multipoles. In the context of a spherical brain source model, it has been mathematically demonstrated that the part of the neuronal current generating the electric potential lives in the orthogonal complement of the part of the current generating the magnetic potential. This implies a mathematical relationship of complementarity between electroencephalography and magnetoencephalography, although the theoretical result in question does not apply to the nonspherical case [G. Dassios, Math. Med. Biol. 25, 133 (2008)]. Our results have important practical applications in cases where electrically silent sources that generate measurable magnetic fields are of interest. Moreover, electrically silent, magnetically active moments of higher order can be useful when cancellation due to superposition of fields can occur, since this situation leads to a substantial reduction in the measurable amplitude of the signal. In this context, information derived from magnetic recordings of electrically silent, magnetically active multipoles can supplement electrical recordings for the purpose of studying the physiology of the brain. Magnetic fields of the electric multipole sources in a conducting medium surrounded by an insulating spherical shell are also presented and the relevance of this calculation to cardiographic and encephalographic experimentation is discussed.
Spatially restricted biological current distributions, like the primary neuronal response in the human somatosensory cortex evoked by electric nerve stimulation, can be described adequately by a current multipole expansion. Here analytic formulas are derived for computing magnetic fields induced by current multipoles in terms of an nth-order derivative of the dipole field. The required differential operators are given in closed form for arbitrary order. The concept is realized in different forms for an expansion of the scalar as well as the dyadic Green's function, the latter allowing for separation of those multipolar source components that are electrically silent but magnetically detectable. The resulting formulas are generally applicable for current sources embedded in arbitrarily shaped volume conductors. By using neurophysiologically relevant source parameters, examples are provided for a spherical volume conductor with an analytically given dipole field. An analysis of the signal-to-noise ratio for multipole coefficients up to the octapolar term indicates that the lateral extent of cortical current sources can be detected by magnetoencephalographic recordings.
A technique for calculating the magnetostatic multipole moments
(and therefore complete biomagnetic information) from measurements of
the normal component of the magnetic field on a plane is described. The
technique involves multiplying the measured values by a set of unique
filtering functions and integrating. A new formulation and method for
calculating the filtering functions is introduced. The method uses
expansions in basis sets and matrix inversion. Proofs are given of the
nonsingularity of the matrices. Computer simulations of the application
of the technique to a circular current loop and a current dipole in a
spherical conductor are described
This paper considers the problem of data continuation for an explicit identification method for electric dipoles situated in a conductor, and shows that the magnetic field measured over a part of a closed surface (which encloses the conductor) can be analytically continued to the rest of the surface. Such a continuation problem has a unique solution for an arbitrarily shaped conductor and observation surface. The paper proposes two straightforward methods for the continuation - the spherical harmonic expansion and the integral equation approach - and presents continuation examples for the case of a single dipole in a spherical conductor with the data measured on a part of a spherical observation surface. The spherical harmonic expansion method works well when the field is continued relatively far from the conductor and when the measured data are clean; however, the spherical harmonic expansion fails when the data are noisy. On the other hand, the integral equation approach, although it suffers from numerical integration errors, shows a much better continuation with quite noisy data. (A method to reduce the numerical integration errors is proposed.) The latter approach is also much more accurate than the spherical harmonic expansion for continuing the field in the proximity of the conductor. Finally, the paper discusses the applicability of both methods to an explicit identification method.
High-resolution magnetoencephalography (MEG) allows for a detailed
description of focal neuronal current sources going far beyond the
dipole approximation which merely indicates the center and magnitude of
neuronal activity. Higher order multipole coefficients can be related to
other bulk properties, like spatial extent or curvature. The possibility
and limitations of measuring spatial extent by interpreting
reconstructed multipole coefficients was tested under realistic noise
conditions and for model misspecifications; for this analysis the
primary cortical response (“N20”) to electric median nerve
stimulation was modeled by a one dimensional source distribution. The
forward calculation was done analytically up to octapolar order for a
spherical volume conductor. The multipole expansion is shown to estimate
the lateral source extent with negligible bias; this estimate is to
first-order stable against additional source features, like gyral
curvature or spatial extent in a second direction (gyral depth, neuronal
length). For a dipole moment of 20 nAm a lateral extent of 2 cm can be
detected for a realistic noise level with large but experimentally still
reasonable effort. Approximating a realistic head model by a sphere
results in errors larger than the extent to be estimated; accordingly,
studies on human cortical evoked responses will require multipole
fitting in realistic head models