Fourier space solution to the magnetostatic imaging problem

Biomagnetism: The First Sixty Years

Biomagnetism is the measurement of the weak magnetic fields produced by nerves and muscle. The magnetic field of the heart-the magnetocardiogram (MCG)-is the largest biomagnetic signal generated by the body and was the first measured. Magnetic fields have been detected from isolated tissue, such as a peripheral nerve or cardiac muscle, and these studies have provided insights into the fundamental properties of biomagnetism. The magnetic field of the brain-the magnetoencephalogram (MEG)-has generated much interest and has potential clinical applications to epilepsy, migraine, and psychiatric disorders. The biomagnetic inverse problem, calculating the electrical sources inside the brain from magnetic field recordings made outside the head, is difficult, but several techniques have been introduced to solve it. Traditionally, biomagnetic fields are recorded using superconducting quantum interference device (SQUID) magnetometers, but recently, new sensors have been developed that allow magnetic measurements without the cryogenic technology required for SQUIDs.

A comparison of two-dimensional techniques for converting magnetocardiogram maps into effective current source distributions

The aim of this study was to develop a method for converting the pseudo two-dimensional current given by a current-arrow map (CAM) into the physical current. The physical current distribution is obtained by the optimal solution in a least mean square sense with Tikhonov regularization (LMSTR). In the current dipole simulation, the current pattern differences (ΔJ) between the results of the CAM and the LMSTR with several regularization parameters (α = 10(-1)-10(-15)) are calculated. In magnetocardiographic (MCG) analysis, the depth (z(d)) of a reconstruction plane is chosen by using the coordinates of the sinus node, which is estimated from MCG signals at the early p-wave. The ΔJs at p-wave peaks, QRS-complex peaks, and T-wave peaks of MCG signals for healthy subjects are calculated. Furthermore, correlation coefficients and regression lines are also calculated from the current values of the CAM and the LMSTR during p-waves, QRS-complex, and T-waves of MCG signals. In the simulation, the ΔJs (α ≈ 10(-10)) had a minimal value. The ΔJs (α = 10(-10)) at p-wave peaks, QRS-complex peaks, and T-wave peaks of MCG signals for healthy subjects also had minimal value. The correlation coefficients of the current values given by the CAM and the LMSTR (α = 10(-10)) were greater than 0.9. Furthermore, slopes (y) of the regression lines are correlated with the depth (z(d)) (r = -0.93). Consequently, the CAM value can be transformed into the LMSTR current value by multiplying it by the slope (y) obtained from the depth (z(d)). In conclusion, the result given by the CAM can be converted into an effective physical current distribution by using the depth (z(d)).

Surface visualization of electromagnetic brain activity

Advances in hardware and software have made possible the reconstruction of brain activity from non-invasive electrophysiological measurements over a large part of the brain. The appreciation of the information content in the data is enhanced when relevant anatomical detail is also available for visualization. Different neuroscientific questions give rise to different requirements for optimal superposition of structure and function. Most available software deal with scalar measures of activity, especially hemodynamic changes. In contrast, the electrophysiological observables are generated by electrical activity, which depends on the synchrony of neuronal assemblies and the geometry of the local cortical surface. We describe methods for segmentation and visualization of spatio-temporal brain activity, which allow the interplay of geometry and scalar as well as vector properties of the current density directly in the representations. The utility of these methods is demonstrated through displays of tomographic reconstructions of early sensory processing in the somatosensory and visual modality extracted from magnetoencephalography (MEG) data. The activation course characteristic to a specific area could be observed as current density or statistical maps independently and/or contrasted to the activity in other areas or the whole brain. MEG and functional magnetic resonance imaging (fMRI) activations were simultaneously visualized. Integrating and visualizing complementary functional data into a single environment helps evaluating analysis and understanding structure/function relationships in normal and diseased brain.

Linear and nonlinear current density reconstructions

Minimum norm algorithms for EEG source reconstruction are studied in view of their spatial resolution, regularization, and lead-field normalization properties, and their computational efforts. Two classes of minimum norm solutions are examined: linear least squares methods and nonlinear L1-norm approaches. Two special cases of linear algorithms, the well known Minimum Norm Least Squares and an implementation with Laplacian smoothness constraints, are compared to two nonlinear algorithms comprising sparse and standard L1-norm methods. In a signal-to-noise-ratio framework, two of the methods allow automatic determination of the optimum regularization parameter. Compensation methods for the different depth dependencies of all approaches by lead-field normalization are discussed. Simulations with tangentially and radially oriented test dipoles at two different noise levels are performed to reveal and compare the properties of all approaches. Finally, cortically constrained versions of the algorithms are applied to two epileptic spike data sets and compared to results of single equivalent dipole fits and spatiotemporal source models.

Mathematical analysis of lead field expansions

The solution to the bioelectromagnetic inverse problem is discussed in terms of a generalized lead field expansion, extended to weights depending polynomially on the current strength. The expansion coefficients are obtained from the resulting system of equations which relate the lead field expansion to the data. The framework supports a family of algorithms which include the class of minimum norm solutions and those of weighted minimum norm, including FOCUSS (suitably modified to conform to requirements of rotational invariance). The weighted-minimum-norm family is discussed in some detail, making explicit the dependence (or independence) of the weighting scheme on the modulus of the unknown current density vector. For all but the linear case, and with a single power in the weight, a highly nonlinear system of equations results. These are analyzed and their solution reduced to tractable problems for a finite number of degrees of freedom. In the simplest magnetic field tomography (MFT) case, this is shown to possess expected properties for localized distributed sources. A sensitivity analysis supports this conclusion.

Selective minimum-norm solution of the biomagnetic inverse problem

A new multidipole estimation method which gives a sparse solution of the biomagnetic inverse problem is proposed. This solution is extracted from the basic feasible solutions of linearly independent data equations. These feasible solutions are obtained by selecting exactly as many dipole-moments as the number of magnetic sensors. By changing the selection, we search for the minimum-norm vector of selected moments. As a result, a practically sparse solution is obtained; computer-simulated solutions for Lp-norm (p = 2, 1, 0.5, 0.2) have a small number of significant moments around the real source-dipoles. In particular, the solution for L1-norm is equivalent to the minimum-L1-norm solution of the original inverse problem. This solution can be uniquely computed by using Linear Programming.

Multiple current dipole estimation using simulated annealing

A method for estimating electrical current distribution in the human brain using a multiple current dipole model is presented. A cost function for estimating multiple dipoles is proposed and a simulated annealing algorithm is used to obtain an acceptable solution. Computer simulation is used to evaluate the effectiveness of this method.

MNLS inverse discriminates between neuronal activity on opposite walls of a simulated sulcus of the brain

The minimum-norm least-squares inverse for magnetic field measurements is applied to a representation of a sulcus of the human brain, where one or both walls have regions of neuronal activity. Simulations indicate that the magnetic source image (MSI) is largely confined to the appropriate wall of the sulcus, even for a depth of 4 cm where the distance between walls is only 3 mm. Two nearly oppositely oriented dipoles located 3 mm apart are found to be distinguished. Influences on the quality of the MSI by measurement noise and inaccuracy in determining the image surface are discussed in detail.

Interpreting magnetic fields of the brain: minimum norm estimates

The authors have applied estimation theory to the problem of determining primary current distributions from measured neuromagnetic fields. In this procedure, essentially nothing is assumed about the source currents, except that they are spatially restricted to a certain region. Simulation experiments show that the results can describe the structure of the current flow fairly well. By increasing the number of measurements, the estimate can be made more localised. The current distributions may be also used as an interpolation and an extrapolation for the measured field patterns.

Uniqueness of the generators of brain evoked potential maps

This study considers the uniqueness of neuronal generators of human brain evoked potentials measured on the scalp using the physical and mathematical properties of the volume conductor model. The results are applicable to a realistic, nonhomogeneous head shape where the potential map is known on a continuous set of points on the scalp. It is shown that sources which occupy "zero volume" in space such as point dipoles or sources distributed on an open surface or a line are uniquely defined by the potential maps. Finite volume nonoverlapping sources are also uniquely defined by their potential map. However, there are infinitely many different but overlapping sources which can create the same map. Several examples of such sources are provided. It is shown that there is a unique, minimum volume source which can be defined in this case. Results suggest that if a reconstruction of the sources starts from a continuous scalp map (obtained by interpolation of the data between electrode sites), one can obtain unique results concerning the source parameters that are not available in a search for a source whose potential map fits only at a discrete set of points.

Minimum-norm least-squares estimation: magnetic source images for a spherical model head

This paper extends the minimum-norm least-squares inverse approach to a local spherical model for the conductivity geometry of the human head. In simulations of cortical activity of the human brain, the magnetic field pattern across the scalp is interpreted with prior knowledge of anatomy, and the properties of intraneuronal current flow to yield a unique magnetic source image across a portion of cerebral cortex. Influences on the quality of magnetic source images from the noise in measurements, the position error in determining the image surface, and the number of sensors are evaluated in detail.

Imaging regional changes in the spontaneous activity of the brain: an extension of the minimum-norm least-squares estimate

This paper describes methods for inferring mathematically unique local distributions of primary cortical current that underly changes in the average pattern of power of the ongoing ("spontaneous") extracranial magnetic field of the brain. In previous work we demonstrated that mathematically unique solutions to the inverse problem are possible for current sources of the brain's field, without assuming a small set of current dipoles as a source model. In principle, it is possible to locate and delineate patterns of current of any configuration. In practice this approach applies to synchronized neuronal activity, e.g., activity which is known to underly average evoked or event-related brain responses. This paper extends that approach to local changes in incoherent activity, e.g., activity yielding fields or potentials that tend to be self-cancelling when averaged over time. This includes the spontaneous brain activity normally treated as background noise when it accompanies event-related responses. We demonstrate that local changes in this ongoing incoherent activity may also be uniquely delineated in space and time. The solution is a covariance matrix characterizing activity across an image surface. Its diagonal elements represent the spatial pattern of mean current power. Evidence is reviewed indicating that the distribution of the brain's magnetic field, due to both its synchronized and incoherent neural activity, is affected by early sensory-perceptual processes and by higher cognitive processes. Hence, in principle, the ability to delineate both kinds of sources in space and time makes it possible to form more comprehensive dynamic functional images of the human brain.

Estimation of the spatio-temporal correlations of biological electrical sources from their magnetic fields

Quasi-static electromagnetic systems, such as those found in biological systems, produce electric and magnetic fields whose temporal and spatial correlations reflect the source correlations in a straightforward manner. These fields can be noninvasively measured, providing information about the coherence properties of the source, which may directly represent ordered physiological processes of the organism. The description "biocoherence" will be adopted here to refer to the manifestation of the coherence in the magnetic measurements of these sources due solely to physiological processes. In this paper a general formulation linking the spatial and temporal coherence of measurable magnetic fields with the corresponding spatial and temporal coherence of the inaccessible current sources is derived in the quasi-static model. A method for reconstructing the spatial and temporal coherence of the source distribution is then presented. Such coherence maps would be useful descriptors of physiological processes occurring over time and space, and would represent more information than an image of the current sources frozen in time, or even a temporal sequence of such images.

Magnetic source images determined by a lead-field analysis: the unique minimum-norm least-squares estimation

The minimum norm least-squares approach based on lead field theory provides a unique inverse solution for a magnetic source image that is the best estimate in the least-squares sense. This has been applied to determine the source current distribution when the primary current is confined to a surface or set of surfaces. In model simulations of cortical activity of the human brain, the magnetic field pattern across the scalp is interpreted with prior knowledge of anatomy to yield a unique magnetic source image across a portion of cerebral cortex, without resort to an explicit source model.

Multiple dipole modeling and localization from spatio-temporal MEG data

An array of biomagnetometers may be used to measure the spatio-temporal neuromagnetic field or magnetoencephalogram (MEG) produced by neural activity in the brain. A popular model for the neural activity produced in response to a given sensory stimulus is a set of current dipoles, where each dipole represents the primary current associated with the combined activation of a large number of neurons located in a small volume of the brain. An important problem in the interpretation of MEG data from evoked response experiments is the localization of these neural current dipoles. We present here a linear algebraic framework for three common spatio-temporal dipole models: i) unconstrained dipoles, ii) dipoles with a fixed location, and iii) dipoles with a fixed orientation and location. In all cases, we assume that the location, orientation, and magnitude of the dipoles are unknown. With a common model, we show how the parameter estimation problem may be decomposed into the estimation of the time invariant parameters using nonlinear least-squares minimization, followed by linear estimation of the associated time varying parameters. A subspace formulation is presented and used to derive a suboptimal least-squares subspace scanning method. The resulting algorithm is a special case of the well-known MUltiple SIgnal Classification (MUSIC) method, in which the solution (multiple dipole locations) is found by scanning potential locations using a simple one dipole model. Principal components analysis (PCA) dipole fitting has also been used to individually fit single dipoles in a multiple dipole problem. Analysis is presented here to show why PCA dipole fitting will fail in general, whereas the subspace method presented here will generally succeed. Numerically efficient means of calculating the cost functions are presented, and problems of model order selection and missing moments are discussed. Results from a simulation and a somatosensory experiment are presented.

Neuromagnetic localization using magnetic resonance images

Ionic flow associated with neural activation of the brain produces a magnetic field, called the neuromagnetic field, that can be measured outside the head using a highly sensitive superconducting quantum interference device (SQUID)-based neuromagnetometer. Under certain conditions, the sources producing the neuromagnetic field can be localized from a sampling of the neuromagnetic field. Neuromagnetic measurements alone, however, do not contain sufficient information to visualize brain structure. Thus, it is necessary to combine neuromagnetic localization with an anatomical imaging technique such as magnetic resonance imaging (MRI) to visualize both function and anatomy in vivo. Using experimentally measured human neuromagnetic fields and magnetic resonance images, the authors have developed a technique to register accurately these two modalities and have applied the registration procedure to portray the spatiotemporal distribution of neural activity evoked by auditory stimulation.

Measurement of nonuniform current density by magnetic resonance

A noninvasive tissue current measurement technique and its use in measuring a nonuniform current density are described. This current density image is created by measuring the magnetic field arising from these currents and taking its curl. These magnetic fields are proportional to the phase component of a complex magnetic resonance image. Measurements of all three components of a quasistatic nonuniform current density in a phantom are described. Expected current density calculations from a numerical solution for the magnetic field which was created by the phantom are presented for comparison. The results of a numerical simulation of the experiment, which used this field solution and which included the effects of slice selection and sampling, are also presented. The experimental and simulated results are quantitatively compared. It is concluded that the principle source of systematic error was the finite slice thickness, which causes blurring of boundaries.

Linear estimation theory applied to the reconstruction of a 3-D vector current distribution

Linear estimation theory incorporating statistical a priori knowledge is applied to the inverse problem of reconstructing a static 3-D vector source field from another 3-D vector measurement field. The motivation for this development is to reconstruct 3-D electric current distributions from a set of magnetic measurements. Such a capability would be useful for the clinical determination of neural currents, for example. A simulation is presented to demonstrate the reconstruction of a class of simple nonbiological source objects, and to show the dependence of these reconstructions on the data taking configuration and the statistical a priori knowledge that is incorporated into the reconstruction process.

Biomagnetic Fourier imaging [current density reconstruction]

A technique for reconstructing a current density distribution from measurements of its magnetic field is described. The technique assumes that the current distribution is confined to a single plane. The data it requires are measurements of the magnetic flux on a plane. These can be provided by an integrated planar array of superconducting quantum interference device magnetometers. The approach is based on the magnetic lead field which is derived in a simple way based on energy concepts. Using the lead field and conservation of charge conditions provides two linear, spatially invariant imaging equations relating the current density and flux measurements. These equations are solved using Fourier techniques. The validity of the resulting reconstruction technique is shown both analytically and with a computer model. The effects of not satisfying the planar assumption are described for the case where the currents are parallel but not in the same plane.