No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
EEG Dipole Localization Bounds and MAP Algorithms for Head Models with Parameter Uncertainties
Abstract
The Cramer-Rao bound for unbiased dipole location estimation is derived under the assumption of a general head model parameterized by deterministic and stochastic parameters. The expression thus characterizes fundamental limits on EEG dipole localization performance due to the effects of both model uncertainty and statistical measurements noise. Expressions are derived for the cases of multivariate Gaussian and gamma distribution priors, and examples are given to illustrate the derived bounds when the radii and conductivities of a four-concentric sphere head model are allowed to be random. The joint MAP estimate of location/model parameters is then examined as a means of achieving robustness to deviations from an ideal head model. Random variations in both the multiple sphere radii and the layer conductivities are shown, via the stochastic Cramer-Rao bounds and Monte Carlo simulation of the MAP estimator, to have the most impact on localization performance in high SNR regions, where finite sample effects are not the limiting factors. This corresponds most often to spatial regions that are close to the scalp electrodes.
... We propose the use of the Cramér-Rao bound method (CRB) for quantifying the possible errors on the identified unknown parameters . CRB is widely-used in many engineering applications: heat transfer applications [11], biomedical engineering applications [12], and signal analysis applications [5]. The CRB offers the lower bound of the error within a rather small computational time compared to the well-know time-demanding techniques such as Monte Carlo simulations applied in, e.g., [13], stochastic finite element method [14], polynomial chaos decomposition [15]. ...
... So, the lower bound for the variances of the unknown parameter is [12]: ...
... Moreover, it is possible to use instead of Gaussian prior for also Gamma prior. needs to be changed then, see [12]. ...
Magnetic material properties of an electromagnetic device (EMD) can be estimated by solving an inverse problem where electromagnetic or mechanical measurements are adequately interpreted by a numerical forward model. Due to measurement noise and uncertainties in the forward model, errors are made in the reconstruction of the material properties. This paper describes the formulation and implementation of a time-efficient numerical error estimation procedure for predicting the optimal measurement modality that leads to minimal error resolution in magnetic material characterization. We extended the traditional Cramér-Rao bound technique for error estimation due to measurement noise only, with stochastic uncertain geometrical model parameters. Moreover, we applied the method onto the magnetic material characterization of a Switched Reluctance Motor starting from different measurement modalities: mechanical; local and global magnetic measurements. The numerical results show that the local magnetic measurement modality needs to be selected for this test case. Moreover, the proposed methodology is validated numerically by Monte Carlo simulations, and experimentally by solving multiple inverse problems starting from real measurements. The presented numerical procedure is able to determine a priori error estimation, without performing the very time consuming Monte Carlo simulations.
... Normally, the confidence volume is calculated for probabilities up to 95% or 99%. Several methods have been proposed for determining such a confidence interval and are available in the literature (Sarvas 1987;Hari et al. 1988;Radich 1995;Kuriki et al. 1989;Mosher et al. 1993). ...
... In both methods the variation of the source localization is performed in the three spatial dimensions. Other methods of parameter estimation based on Baysian statistics have also been suggested (Mosher et al. 1993;Radich 1995), but will not be discussed in this paper. ...
... As there are no conclusive strategies available to handle singular covariance matrices, Monte Carlo simulation is therefore the method of choice to estimate confidence volumes of source localization. The Cramer Rao lower bound method proposed by Radich (1995) and Mosher et al. 1993) considers only uncorrelated noise and is therefore not an alternative. The problem of correlated noise is more important in EEG than MEG, because the reference electrode in EEG introduces identical noise to all other channels, leading to a correlation between the electrodes (Huizenga and Molenaar 1995). ...
Noise in EEG and MEG measurements leads to inaccurate localizations of the sources. A confidence volume is used to describe the amount of localization error. Previous methods to estimate the confidence volume proved insufficient. Thus a new procedure was introduced and compared with previous ones. As one procedure, Monte Carlo simulations (MCS) were performed. The confidence volume was also estimated using two methods with different assumptions about a linear transfer function between source location and the distribution of the potential. One method used variable (LVM) and the other fixed dipole orientations (LFM). Finally, the confidence volume was estimated through a procedure in which there was no linearization of the transfer function. This procedure scans the confidence volume by varying the dipole location in multiple directions. Confidence volumes were calculated for simulated distributions of the electrical potential and for experimental data including somatosensory evoked responses to stimulation of lower lip, thumb, and little finger. Results from simulated data indicated that confidence volumes calculated with the MCS method were largest, and those calculated with the LFM method were smallest. For dipole locations close to the brain surface, the confidence volume was smaller than for a central deeper source. An increase in electrode density resulted in smaller confidence volumes. When the noise was correlated, only the method using the MCS produced acceptable results. Since the noise in experimental data is highly correlated, only the MCS method would appear to be useful in estimating the size of the confidence volume of the dipole locations. Thus, using real data with the MCS method, we easily distinguished separate and distinct representations of the thumb, little finger, and lower lip in the somatosensory cortex (SI). It was concluded that adequate estimation of confidence volumes is useful for localizing neural activity. On a practical level, this information can be used prior to an experiment for determining the conditions necessary to distinguish between different dipole sources, including the required signal to noise ratio and the minimum electrode density.
... Error bounds have been derived using an approximate model that represents the head as spherical layers of different known conductivities as in [31], assuming scalar magnetic sensors, and [18] for scalar or vector magnetic sensors. A spherically symmetric head with scalar magnetic sensors and random conductivities is assumed in [41] using a Bayesian approach. In [33], we introduced the basic ideas to derive the CRB for a realistic head model with possibly vector magnetic sensors. ...
... It is possible to handle anisotropic conductivities using the FEM expressions [50]. Unknown conductivities can be considered as random parameters, originating the approach of [41] to compute the estimation accuracy bound for the spherical head model using Bayesian methods. ...
... The matrix can be obtained from prior empirical studies [4], [13], [23], [29] or from modeling [8], [25]. A diagonal noise covariance matrix is assumed very often, representing noise and modeling errors that are uncorrelated between sensors [17], [31], [41]. Some experimental evidence that the covariance between the modality components of spontaneous brain noise cannot be assumed to be zero has been presented in [39]. ...
We derive Cramer-Rao bounds (CRBs) on the errors of estimating the
parameters (location and moment) of a static current dipole source using
data from electro-encephalography (EEG), magneto-encephalography (MEG),
or the combined EEG/MEG modality. We use a realistic head model based on
knowledge of surfaces separating tissues of different conductivities
obtained from magnetic resonance (MR) or computer tomography (CT)
imaging systems. The electric potentials and magnetic field components
at the respective sensors are functions of the source parameters through
integral equations. These potentials and field are formulated for
solving them by the boundary or the finite element method (BEM or FEM)
with a weighted residuals technique. We present a unified framework for
the measurements computed by these methods that enables the derivation
of the bounds. The resulting bounds may be used, for instance, to choose
the best configuration of the sensors for a given patient and region of
expected source location. Numerical results are used to demonstrate an
application for showing expected accuracies in estimating the source
parameters as a function of its position in the brain, based on real
EEG/MEG system and MR or CT images
... As we describe below, Bayesian approaches to the inverse electrocardiography problem have concentrated on the formulation of statistically-defined regularization methods. In the EEG/MEG literature, application of the performance analysis tools has been reported, but generally with relatively simple scenarios such as independent sources or the restriction of probabilistic parameters to the parameters of the medium [7], [8]. 0018-9294/$20.00 © 2005 IEEE In this paper, we extend the application of these statistical performance analysis tools to inverse ECG, a case in which the sources (in our formulation, the time-varying potentials at a mesh of nodes on the epicardium) are clearly correlated with each other across the surface. ...
... In addition to using a Bayesian estimation error metric, the authors also used the evidence to estimate these variances, and calculated the information content of the measurements using differential entropy and mutual information. Radich et al. [8] derived error bounds for unbiased dipole location estimation under a general head model parameterized by both deterministic and stochastic parameters. In their derivation, the only stochastic terms were the head model parameters, such as conductivities of various layers within a four-shell spherical head. ...
... Using this variance and the Bayesian MAP solution, one can compute confidence intervals for the estimate. For example, with 95% probability, the true solution at lead lies approximately within the range (8) If we map the error covariance or confidence intervals back onto the epicardium, we have a theoretical quantitative prediction of where on the epicardium we expect our method to give us more or less reliable results, and indeed an indication, in the appropriate physical unit, of this uncertainty. ...
In bioelectric inverse problems, one seeks to recover bioelectric sources from remote measurements using a mathematical model that relates the sources to the measurements. Due to attenuation and spatial smoothing in the medium between the sources and the measurements, bioelectric inverse problems are generally ill-posed. Bayesian methodology has received increasing attention recently to combat this ill-posedness, since it offers a general formulation of regularization constraints and additionally provides statistical performance analysis tools. These tools include the estimation error covariance and the marginal probability density of the measurements (known as the "evidence") that allow one to predictively quantify and compare experimental designs. These performance analysis tools have been previously applied in inverse electroencephalography and magnetoencephalography, but only in relatively simple scenarios. The main motivation here was to extend the utility of Bayesian estimation techniques and performance analysis tools in bioelectric inverse problems, with a particular focus on electrocardiography. In a simulation study we first investigated whether Bayesian error covariance, computed without knowledge of the true sources and based on instead statistical assumptions, accurately predicted the actual reconstruction error. Our study showed that error variance was a reasonably reliable qualitative and quantitative predictor of estimation performance even when there was error in the prior model. We also examined whether the evidence statistic accurately predicted relative estimation performance when distinct priors were used. In a simple scenario our results support the hypothesis that the prior model that maximizes the evidence is a good choice for inverse reconstructions.
... Therefore, the use of the Cramér-Rao bound method (CRB) is proposed for quantifying the possible uncertainties on the identified unknown parameter values u. CRB is widely-used in many engineering applications; heat transfer applications (Fadale et al., 1995a), biomedical engineering applications (Radich & Buckley, 1995), and signal analysis applications (Stoica & Nehorai, 1989). ...
... So, the lower bound for the p variances of the unknown parameter σ 2 u,M is (Radich & Buckley, 1995): ...
... From present and future measurements, a certain probability density function of the conductivities can be defined. Such a probability density function was also used in [5]. We propose to use so-called non-intrusive probabilistic algorithms to quantify the uncertainties on the location of the neural sources, which only assume the conductivity to be of finite variance [6], [7]. ...
... is the Moore-Penrose pseudo-inverse of the lead field matrix. In this way it is possible to redefine the inverse problem as (5) with ...
The electroencephalogram (EEG) is one of the techniques used for the non-invasive diagnosis of patients suffering from epilepsy. EEG source localization identifies the neural activity, starting from measured EEG. This numerical localization procedure has a resolution, which is difficult to determine due to uncertainties in the EEG forward models. More specifically, the conductivities of the brain and the skull in the head models are not precisely known. In this paper, we propose the use of a non-intrusive stochastic method based on a polynomial chaos decomposition for quantifying the possible errors introduced by the uncertain conductivities of the head tissues. The accuracy and computational advantages of this non-intrusive method for EEG source analysis is illustrated. Further, the method is validated by means of Monte Carlo simulations.
... If we consider that is distributed as , our stochastic CRB is given by (17) with as previously defined in (10). A derivation of (17) is presented in [23]. ...
... For the measurements, we used a standard 10-10 EEG configuration of . All these values were selected from previous studies in the area (see e.g., [23], [24]). The 10-10 arrangement allows consistent testing for EEG recordings as it complies with the international electrode placement system and is widely used in clinical applications [25]. ...
Techniques based on electroencephalography (EEG) measure the electric potentials on the scalp and process them to infer the location, distribution, and intensity of underlying neural activity. Accuracy in estimating these parameters is highly sensitive to uncertainty in the conductivities of the head tissues. Furthermore, dissimilarities among individuals are ignored when standarized values are used. In this paper, we apply the maximum-likelihood and maximum a posteriori (MAP) techniques to simultaneously estimate the layer conductivity ratios and source signal using EEG data. We use the classical 4-sphere model to approximate the head geometry, and assume a known dipole source position. The accuracy of our estimates is evaluated by comparing their standard deviations with the Cramér-Rao bound (CRB). The applicability of these techniques is illustrated with numerical examples on simulated EEG data. Our results show that the estimates have low bias and attain the CRB for sufficiently large number of experiments. We also present numerical examples evaluating the sensitivity to imprecise assumptions on the source position and skull thickness. Finally, we propose extensions to the case of unknown source position and present examples for real data.
... This sensitivity can be visualized at each possible source to obtain an impression for the expected error for different source locations. Radich et al. derived the Cramér-Rao bound for dipole location and gave an analytic expression for a four-shell spherical model [99]. Saturnino et al. used generalized polynomial chaos to implement efficient mapping from probability density functions for tissue parameters to probability densities in the resulting electric field during stimulation [100]. ...
Conventional transcranial electric stimulation(tES) using standard anatomical positions for the electrodes and standard stimulation currents is frequently not sufficiently selective in targeting and reaching specific brain locations, leading to suboptimal application of electric fields. Recent advancements in in vivo electric field characterization may enable clinical researchers to derive better relationships between the electric field strength and the clinical results. Subject-specific electric field simulations could lead to improved electrode placement and more efficient treatments. Through this narrative review, we present a processing workflow to personalize tES for focal epilepsy, for which there is a clear cortical target to stimulate. The workflow utilizes clinical imaging and electroencephalography data and enables us to relate the simulated fields to clinical outcomes. We review and analyze the relevant literature for the processing steps in the workflow, which are the following: tissue segmentation, source localization, and stimulation optimization. In addition, we identify shortcomings and ongoing trends with regard to, for example, segmentation quality and tissue conductivity measurements. The presented processing steps result in personalized tES based on metrics like focality and field strength, which allow for correlation with clinical outcomes.
... The above-mentioned studies considered only the experimental noise, while the uncertainties that might have existed in the known model parameters of heat transfer models were not taken into account, i.e., the predictions were assumed to be strictly accurate. Only a few research studies considered both the experimental noise and the uncertainties of model parameters, and the uncertainties of the retrieved properties were estimated using the Cramér-Rao lower bound (CRB)-based method [15][16][17][18][19][20][21][22]. These works, relative to inverse heat transfer problems, mainly focused on retrieving the thermal conductivity, thermal resistance, and heat transfer coefficient by solving inverse heat conduction problems [15][16][17]. ...
The conductive and radiative properties of participating medium can be estimated by solving an inverse problem that combines transient temperature measurements and a forward model to predict the coupled conductive and radiative heat transfer. The procedure, as well as the estimates of parameters, are not only affected by the measurement noise that intrinsically exists in the experiment, but are also influenced by the known model parameters that are used as necessary inputs to solve the forward problem. In the present study, a stochastic Cramér–Rao bound (sCRB)-based error analysis method was employed for estimation of the errors of the retrieved conductive and radiative properties in an inverse identification process. The method took into account both the uncertainties of the experimental noise and the uncertain model parameter errors. Moreover, we applied the method to design the optimal location of the temperature probe, and to predict the relative error contribution of different error sources for combined conductive and radiative inverse problems. The results show that the proposed methodology is able to determine, a priori, the errors of the retrieved parameters, and that the accuracy of the retrieved parameters can be improved by setting the temperature probe at an optimal sensor position.
... The radiation pattern is explicitly taken in account in some EEG localization methods. In particular, Cramér-Rao bounds for the localization of current dipoles in the skull are given in [6]. Results are however limited to pure dipoles. ...
... ∂ R ∂ x is the sensitivity of the model response R to the uncertain model parameter x. α needs to be calculated for a predefined value of the unknown parameter δ. Similarly, the lower bound of the variance of the unknown parameter σ 2 δ , in the adjusted CRLB approach, is given in [28] In general, the proposed methodology is a combined experimentalnumerical approach. The main part of the approach, in the case of the idealized model, is to calculate the sensitivity of the numerical model response that corresponds to the considered measurement modality to the unknown parameter, here the eccentricity level, i.e., ∂ R ∂ δ . ...
Detection of a static eccentricity fault in rotating electrical machines is possible through several measurement techniques, such as shaft voltages and flux probes. A predictive maintenance approach typically requires that the condition monitoring technique is online, accurate, and able to detect incipient faults. This paper presents a study of the optimal measurement modality that leads to a minimal identification error of the static eccentricity in synchronous machines. The Cramér-Rao lower bound technique is implemented by taking into account both measurement and model uncertainties. Numerical results are obtained using a two-dimensional finite element model and is experimentally validated on a synchronous two-pole generator. Results indicate that shaft voltage measurements are better suited to the detection of static eccentricity.
... Our proposed analysis can be easily extended to evaluate other factors, for example, the electrical properties of the surrounding tissues. It is well know that errors are introduced in the solution of the EEG inverse problem due to dissimilarities in the values of conductivities of the tissues among individuals [33][34][35], then it might be valuable to add those parameters in the optimization framework and evaluate the performance of metaheuristics using both the nonparametric statistical tests that are presented here and the stochastic Cramér-Rao bounds [36] in order to account also for noise effects. ...
We study the use of nonparametric multicompare statistical tests on the performance of simulated annealing (SA), genetic algorithm (GA), particle swarm optimization (PSO), and differential evolution (DE), when used for electroencephalographic (EEG) source localization. Such task can be posed as an optimization problem for which the referred metaheuristic methods are well suited. Hence, we evaluate the localization's performance in terms of metaheuristics' operational parameters and for a fixed number of evaluations of the objective function. In this way, we are able to link the efficiency of the metaheuristics with a common measure of computational cost. Our results did not show significant differences in the metaheuristics' performance for the case of single source localization. In case of localizing two correlated sources, we found that PSO (ring and tree topologies) and DE performed the worst, then they should not be considered in large-scale EEG source localization problems. Overall, the multicompare tests allowed to demonstrate the little effect that the selection of a particular metaheuristic and the variations in their operational parameters have in this optimization problem.
... As a community we have established electrical-property ranges for most head tissues in terms of conductivity σ and permittivity ε; however, we have to determine the actual electrical-conductivity distribution of an individual's head. As a result of these ranges, many historical studies assign average values to their tissues [15,[18][19][20][21][22][23][24][25][26]. Using an average value may not be appropriate for individualized models since those models may result in inaccurate solutions due to a function of position [27] or of age [28,29]. ...
We present the four key areas of research—preprocessing, the volume conductor, the forward problem, and the inverse problem—that affect the performance of EEG and MEG source
imaging. In each key area we identify prominent approaches and methodologies that have open
issues warranting further investigation within the community, challenges associated with certain
techniques, and algorithms necessitating clarification of their implications. More than providing
definitive answers we aim to identify important open issues in the quest of source localization.
... To fully understand the limits of dipole localization methods in EEG and MEG, further work is required in evaluating performance bounds and sensitivity to model assumptions for more complex spatiotemporal configurations. These bounds can also be used to compare the localizing abilities of the different sensor placements, gradiometer configurations, and parameter uncertainties (Radich and Buckley, 1995;Hochwald and Nehorai, 1997). ...
Equivalent current dipoles are a powerful tool for modeling focal sources. The dipole is often sufficient to adequately represent sources of measured scalp potentials, even when the area of activation exceeds 1 cm2 of cortex. Traditional least-squares fitting techniques involve minimization of an error function with respect to the location and orientation of the dipoles. The existence of multiple local minima in this error function can result in gross errors in the computed source locations. The problem is further compounded by the requirement that the model order, i.e. the number of dipoles, be determined before error minimization can be performed. An incorrect model order can produce additional errors in the estimated source parameters. Both of these problems can be avoided using alternative search strategies based on the MUSIC (multiple signal classification) algorithm. Here the authors review the MUSIC approach and demonstrate its application to the localization of multiple current dipoles from EEG data. The authors also show that the number of detectable sources can be determined in a recursive manner from the data. Also, in contrast to least-squares, the method can find dipolar sources in the presence of additional non-dipolar sources. Finally, extensions of the MUSIC approach to allow the modeling of distributed sources are discussed.
... Fast analytical methods can be applied in the calculation of geometrically simple volume conductor models, such as concentric and eccentric spheres (1)(2)(3)(4)(5)(6)(7)(8)(9). The shape and the tissue resistivities are major characteristics of the head as a volume conductor. ...
Two inverse algorithms were applied for solving the EEG inverse problem assuming a single dipole as a source model. For increasing the efficiency of the forward computations the lead field approach based on the reciprocity theorem was applied. This method provides a procedure to calculate the computationally heavy forward problem by a single solution for each EEG lead. A realistically shaped volume conductor model with five major tissue compartments was employed to obtain the lead fields of the standard 10-20 EEG electrode system and the scalp potentials generated by simulated dipole sources. A least-squares method and a probability-based method were compared in their performance to reproduce the dipole source based on the reciprocal forward solution. The dipole localization errors were 0 to 9 mm and 2 to 22 mm without and with added noise in the simulated data, respectively. The two different inverse algorithms operated mainly very similarly. The lead field method appeared applicable for the solution of the inverse problem and especially useful when a number of sources, e.g., multiple EEG time instances, must be solved.
... The sensitivity of EEG source localization to conductivity parameter uncertainties has been studied using the multilayer sphere model (Radich and Buckly, 1995; Stok, 1987; Zhang and Jewett, 1993). The sensitivity of the boundary potential to conductivity was recently investigated using a three-dimensional realistic model with the finite-element method (Haueisen et al. 1997). ...
EEG-based source localization techniques use scalp-potential data to estimate the location of underlying neural activity. EEG source location reconstruction requires the assumption of a source model and the assumption of a conductive head model. Brain lesions can present conductivity values that are dramatically different from those of surrounding normal tissues and have to be included in head models for accurate neural source reconstruction. It is therefore necessary to analyze subjects' anatomic images (using MRI or computed tomography) to identify lesion type and to assign the appropriate conductivity value. Source localization accuracy may be influenced by uncertainties in tissue conductivity assignment during head model construction. The authors present a sensitivity study quantifying the effect of uncertainty in brain lesion conductivity assignment on EEG dipole source localization. They adopted an eccentric-spheres head model in which an eccentric bubble approximated the effects of actual brain lesions. After simulating EEG signal measurement in 64 different pathologic situations, an inverse dipole fitting procedure was carried out, assuming an incorrect lesion conductivity assignment ranging from a half to twice the real value. Incorrect lesion conductivity assignment led to markedly wrong source reconstruction for highly conductive lesions like liquid-filled ones (localization errors as much as 1.7 cm). Conversely, low sensitivity to uncertainties in conductivity assignment was found for lesions with low conductivity like calcified tumors. The authors propose a method based on residual error analysis to improve the lesion conductivity estimate. This procedure can identify lesion tissue conductivity with only a few percent error and guarantees source localization errors less than 5 mm.
... Moreover, it may vary as a function of position (Ollikainen et al 1999). The effects of the difference in the tissue conductivity from the average values on the measurements have been investigated using different approaches (He et al 1987, Stok 1987, Hämäläinen and Sarvas 1989, Radich and Buckley 1995, Haueisen et al 1997, Van den Broek et al 1998, Huiskamp et al 1999, Vanrumste et al 2000). However, for a given source configuration, the sensitivity of a particular measurement to a conductivity perturbation (at a specific point) has not been investigated yet. ...
Monitoring the electrical activity inside the human brain using electrical and magnetic field measurements requires a mathematical head model. Using this model the potential distribution in the head and magnetic fields outside the head are computed for a given source distribution. This is called the forward problem of the electro-magnetic source imaging. Accurate representation of the source distribution requires a realistic geometry and an accurate conductivity model. Deviation from the actual head is one of the reasons for the localization errors. In this study, the mathematical basis for the sensitivity of voltage and magnetic field measurements to perturbations from the actual conductivity model is investigated. Two mathematical expressions are derived relating the changes in the potentials and magnetic fields to conductivity perturbations. These equations show that measurements change due to secondary sources at the perturbation points. A finite element method (FEM) based formulation is developed for computing the sensitivity of measurements to tissue conductivities efficiently. The sensitivity matrices are calculated for both a concentric spheres model of the head and a realistic head model. The rows of the sensitivity matrix show that the sensitivity of a voltage measurement is greater to conductivity perturbations on the brain tissue in the vicinity of the dipole, the skull and the scalp beneath the electrodes. The sensitivity values for perturbations in the skull and brain conductivity are comparable and they are, in general, greater than the sensitivity for the scalp conductivity. The effects of the perturbations on the skull are more pronounced for shallow dipoles, whereas, for deep dipoles, the measurements are more sensitive to the conductivity of the brain tissue near the dipole. The magnetic measurements are found to be more sensitive to perturbations near the dipole location. The sensitivity to perturbations in the brain tissue is much greater when the primary source is tangential and it decreases as the dipole depth increases. The resultant linear system of equations can be used to update the initially assumed conductivity distribution for the head. They may be further exploited to image the conductivity distribution of the head from EEG and/or MEG measurements. This may be a fast and promising new imaging modality.
... In a similar vein, it is also possible to incorporate parameters of the forward model into the full model (Fig. 1). For example, it has been shown that the standard head model parameters; i.e., conductivities and parameters, vary over subjects, e.g., Radich and Buckley (1995). This is important for EEG because in typical dipole fitting (including ours), these parameters are fixed to some standard values. ...
In magneto- and electroencephalography (M/EEG), spatial modelling of sensor data is necessary to make inferences about underlying brain activity. Most source reconstruction techniques belong to one of two approaches: point source models, which explain the data with a small number of equivalent current dipoles and distributed source or imaging models, which use thousands of dipoles. Much methodological research has been devoted to developing sophisticated Bayesian source imaging inversion schemes, while dipoles have received less such attention. Dipole models have their advantages; they are often appropriate summaries of evoked responses or helpful first approximations. Here, we propose a variational Bayesian algorithm that enables the fast Bayesian inversion of dipole models. The approach allows for specification of priors on all the model parameters. The posterior distributions can be used to form Bayesian confidence intervals for interesting parameters, like dipole locations. Furthermore, competing models (e.g., models with different numbers of dipoles) can be compared using their evidence or marginal likelihood. Using synthetic data, we found the scheme provides accurate dipole localizations. We illustrate the advantage of our Bayesian scheme, using a multi-subject EEG auditory study, where we compare competing models for the generation of the N100 component.
... However, real values vary from individual to individual [ll]. For the spherical model, reference [16] considers the conductivities as unknown random parameters, and computes the EEG bound using Bayesian methods. The same type of Bayesian CRB might be used for a realistic head model and combined modality. ...
Magnetoencephalography (MEG) and electroencephalography (EEG) are non-invasive methods for studying human brain activity, with a time resolution of milliseconds. Current dipoles are the most common model describing sources of brain activity. We show how to compute error bounds when estimating the dipole source parameters for a realistic head model, using measurements of electric potential (EEG), magnetic field (MEG) and their combination. The electric potentials and magnetic field components are obtained by discretizing the integral equations for the fields via the boundary element method (BEM) and a weighted residuals technique. This process requires an accurate representation of the surfaces separating the irregular head layers, as obtained from MR or CT imaging.
As known from physics, each current is accompanied by a magnetic field. So are the ionic currents within the brain and the nerves. Using highly sensitive sensors, so-called SQUIDs (“superconducting quantum interference device”) developed during the last three decades, magnetoencephalography (MEG) measures these extremely weak fields outside the head. MEG can pick up the fields associated with the concerted action of a few thousand neurons and thus non-invasively monitor brain activity. With good approximation these fields reflect only intracellular (mostly intradendritic) currents and are insensitive to the extracellular current distribution, in contrast to the scalp potentials measured with EEG (cf. Chapter 35).
Purpose
The purpose of this paper is to determine a priori the optimal needle placement so to achieve an as accurate as possible magnetic property identification of an electromagnetic device. Moreover, the effect of the uncertainties in the geometrical parameter values onto the optimal sensor position is studied.
Design/methodology/approach
The optimal needle placement is determined using the stochastic Cramér‐Rao lower bound method. The results obtained using the stochastic method are compared with a first order sensitivity analysis. The inverse problem is solved starting from real local magnetic induction measurements coupled with a 3D finite element model of an electromagnetic device (EI core inductor).
Findings
The optimal experimental design for the identification of the magnetic properties of an electromagnetic device is achieved. The uncertainties in the geometrical model parameters have a high effect on the inverse problem recovered solution.
Originality/value
The solution of the inverse problem is more accurate because the measurements are carried out at the optimal positions, in which the effects of the uncertainties in the geometrical model parameters are limited.
The measured voltage signals picked up by the needle probe method can be interpreted by a numerical method so as to identify the magnetic material properties of the magnetic circuit of an electromagnetic device. However, when solving this electromagnetic inverse problem, the uncertainties in the numerical method give rise to recovery errors since the calculated needle signals in the forward problem are sensitive to these uncertainties. This paper proposes a stochastic Cramér–Rao bound method for determining the optimal sensor placement in the experimental setup. The numerical method is computationally time efficient where the geometrical parameters need to be provided. We apply the method for the non-destructive magnetic material characterization of an EI inductor where we ascertain the optimal experiment design. This design corresponds to the highest possible resolution that can be obtained when solving the inverse problem. Moreover, the presented results are validated by comparison with the exact material characteristics. The results show that the proposed methodology is independent of the values of the material parameter so that it can be applied before solving the inverse problem, i.e. as a priori estimation stage.
We present here a quantitative characterization of the non-uniqueness for the inverse problem of electroencephalography (EEG). First, we identify the singular support of the electric potential generated by a dipolar current which is fired inside the spherical model of the brain. Next, we extend this result to a continuously distributed neuronal current and we derive the equivalent Green's integral representation. Then, using the Hansen representation of the current, we show that among the three scalar representation functions, only two are needed to represent the observed electric potential on the surface or outside the head. The scalar function that is missed by the EEG recordings is exactly the one that is recorded by magnetoencephalography (MEG). Finally, the solution of the inverse EEG problem is reduced to a specific moment problem, which is exactly solved under the minimum-current assumption.
Solution of the electro-magnetic source imaging (EMSI) problem requires an accurate representation of the head using a numerical model. Some of the errors in source estimation are due to the differences between this model and the actual head. This study investigates the effects of conductivity perturbations, that is, changing the conductivity of a small region by a small amount, on the EEG and MEG measurements. By computing the change in measurements for perturbations throughout the volume, it is possible to obtain a spatial distribution of sensitivity. Using this information, it is possible, for a given source configuration, to identify the regions to which a measurement is most sensitive. In this work, two mathematical expressions for efficient computation of the sensitivity distribution are presented. The formulation is implemented for a numerical head model using the finite element method (FEM). 3D sensitivity distributions are computed and analyzed for selected dipoles and sensors. It was observed that the voltage measurements are sensitive to the skull, the regions near the dipole and the electrodes. The magnetic field measurements are mostly sensitive to regions near the dipole. It could also be possible to use the computed sensitivity matrices to estimate or update the conductivity of the tissues from EEG and MEG measurements.
The uncertain conductivity value of skull and brain tissue influences the accuracy of the electroencephalogram (EEG) inverse problem solution. Indeed, when the assumed conductivity in the numerical procedure is different from the actual conductivity then a source localization error is introduced. When using traditional least-squares minimization methods, the number of electrodes in the EEG cap does not influence the spatial resolution. A recently developed reduced conductivity dependence (RCD) methodology, based on the selection of electrodes, is able to increase the spatial resolution of the EEG inverse problem. This paper presents the implications of the RCD method when using a large number of electrodes in the EEG cap on the spatial resolution of the EEG inverse solutions. We show by means of numerical experiments that in contrast to traditional methods, the RCD method enables to increase the spatial resolution. The computations show that the EEG hardware should be modified with as large as possible electrodes.
By interpreting electromagnetic or mechanical measurements with a numerical model of the considered electromagnetic device, magnetic properties of the magnetic circuit of that device can be estimated by solving an inverse numerical electromagnetic problem. Due to measurement noise and uncertainties in the forward model, errors are made in the reconstruction of the material properties. This paper describes the formulation and implementation of the error estimation and the prediction of which measurements that need to be carried out for accurate magnetic material characterization. Stochastic uncertainty analysis, based on Cramér-Rao bound (CRB), is introduced and applied to the magnetic material characterization of a Switched Reluctance Motor (SRM) starting from mechanical (torque) and local magnetic measurements. The traditional CRB method that estimates the error due to measurement noise is extended with the incorporation of stochastic uncertain geometrical model parameters.
We describe the use of the nonparametric bootstrap to investigate the accuracy of current dipole localization from magnetoencephalography (MEG) studies of event-related neural activity. The bootstrap is well suited to the analysis of event-related MEG data since the experiments are repeated tens or even hundreds of times and averaged to achieve acceptable signal-to-noise ratios (SNRs). The set of repetitions or epochs can be viewed as a set of independent realizations of the brain's response to the experiment. Bootstrap resamples can be generated by sampling with replacement from these epochs and averaging. In this study, we applied the bootstrap resampling technique to MEG data from somatotopic experimental and simulated data. Four fingers of the right and left hand of a healthy subject were electrically stimulated, and about 400 trials per stimulation were recorded and averaged in order to measure the somatotopic mapping of the fingers in the S1 area of the brain. Based on single-trial recordings for each finger we performed 5000 bootstrap resamples. We reconstructed dipoles from these resampled averages using the Recursively Applied and Projected (RAP)-MUSIC source localization algorithm. We also performed a simulation for two dipolar sources with overlapping time courses embedded in realistic background brain activity generated using the prestimulus segments of the somatotopic data. To find correspondences between multiple sources in each bootstrap, sample dipoles with similar time series and forward fields were assumed to represent the same source. These dipoles were then clustered by a Gaussian Mixture Model (GMM) clustering algorithm using their combined normalized time series and topographies as feature vectors. The mean and standard deviation of the dipole position and the dipole time series in each cluster were computed to provide estimates of the accuracy of the reconstructed source locations and time series.
The EEG is a neurological diagnostic tool with high temporal resolution. However, when solving the EEG inverse problem, its localization accuracy is limited because of noise in measurements and available uncertainties of the conductivity value in the forward model evaluations. This paper proposes the reduced conductivity dependence (RCD) method for decreasing the localization error in EEG source analysis by limiting the propagation of the uncertain conductivity values to the solutions of the inverse problem. We redefine the traditional EEG cost function, and in contrast to previous approaches, we introduce a selection procedure of the EEG potentials. The selected potentials are, as low as possible, affected by the uncertainties of the conductivity when solving the inverse problem. We validate the methodology on the widely used three-shell spherical head model with a single electrical dipole and multiple dipoles as source model. The proposed RCD method enhances the source localization accuracy with a factor ranging between 2 and 4, dependent on the dipole location and the noise in measurements.
The selected potentials are as low as possible affected
by the uncertainties of the conductivity when solving the inverse problem.
We validate the methodology on the widely-used three shell
spherical head model with a single electrical dipole and
multiple dipoles as source model. The proposed RCD method
enhances the source localization accuracy with a factor ranging
between 2 to 4, dependent on the dipole location and the noise
in measurements.
Thesis (Ph. D.)--Harvard-MIT Division of Health Sciences and Technology, 2000. Includes bibliographical references.
We study the effect of the head shape variations on the EEG/magnetoencephalography (MEG) forward and inverse problems. We build a random head model such that each sample represents the head shape of a different individual and solve the forward problem assuming this random head model, using a polynomial chaos expansion. The random solution of the forward problem is then used to quantify the effect of the geometry when the inverse problem is solved with a standard head model. The results derived with this approach are valid for a continuous family of head models, rather than just for a set of cases. The random model consists of three random surfaces that define layers of different electric conductivity, and we built an example based on a set of 30 deterministic models from adults. Our results show that for a dipolar source model, the effect of the head shape variations on the EEG/MEG inverse problem due to the random head model is slightly larger than the effect of the electronic noise present in the sensors. The variations in the EEG inverse problem solutions are due to the variations in the shape of the volume conductor, while the variations in the MEG inverse problem solutions, larger than the EEG ones, are caused mainly by the variations of the absolute position of the sources in a coordinate system based on anatomical landmarks, in which the magnetometers have a fixed position.
This paper presents a sensitivity study of electroencephalography-based source localization due to errors in the head-tissue conductivities and to errors in modeling the conductivity variation inside the brain and scalp. The study is conducted using a two-dimensional (2-D) finite element model obtained from a magnetic resonance imaging (MRI) scan of a head cross section. The effect of uncertainty in the following tissues is studied: white matter, gray matter, cerebrospinal fluid (CSF), skull, and fat. The distribution of source location errors, assuming a single-dipole source model, is examined in detail for different dipole locations over the entire brain region. We also present a detailed analysis of the effect of conductivity on source localization for a four-layer cylinder model and a four-layer sphere model. These two simple models provide insight into how the effect of conductivity on boundary potential translates into source location errors, and also how errors in a 2-D model compare to errors in a three-dimensional model. Results presented in this paper clearly point to the following conclusion: unless the conductivities of the head tissues and the distribution of these tissues throughout the head are modeled accurately, the goal of achieving localization accuracy to within a few millimeters is unattainable.
The consequences of artifact suppression by means of signal-space projection on dipole localization accuracy for magnetoencephalography measurements are studied. Approximate analytical formulas, equivalent to the Cramer-Rao bound, are presented and verified by Monte Carlo simulations which relate the increase of localization error for individual coordinates to the similarity of the artifact field and respective (contravariant) quadrupole fields obtained by differentiating the dipole field with respect to its origin. The expressions simplify significantly for dipoles placed below the center of the measuring system giving rise to highly symmetric field patterns. Formulas are presented both for single- and for multiple-artifact rejection. As illustrative examples artifact fields are constructed which a) lead to highly decreasing signal-to-noise ratio and goodness-of-fit (GOF), while the localization error is unaffected for all coordinates and b) lead to an increase of localization error while the SNR and the GOF stays constant. Finally, the rich structure of localization error increase is demonstrated for a class of artifact fields originating from artifact current dipoles.
The effects of tissue resistivities on EEG amplitudes were studied using an anatomically accurate computer model based on the finite difference method (FDM) and lead field analysis covering the whole brain area with 180,000 nodes. Five tissue types and three lead fields were considered for analysis. The changes in sensitivity distribution are directly comparable to changes in the potential distribution on the scalp. The results indicate that a 10% decrease in any tissue resistivity caused 3.0-4.1% differences in the sensitivity distributions of the selected EEG leads. The applied 10% decrease in the resistivity values covers only a fraction of the range of variation of 50% to 100% reported in the literature. The use of a 55% decreased skull resistivity value or a commonly applied three-compartment model increased the differences to 28% and 33%, respectively. In conclusion, both a realistic anatomy and accurate resistivity data are important in EEG head models.
We present a method that estimates three-dimensional statistical maps for electroencephalogram (EEG) source localization. The maps assess the likelihood that a point in the brain contains a dipolar source, under the hypothesis of one, two or three activated sources. This is achieved by examining all combinations of one to three dipoles on a coarse grid and attributing to each combination a score based on an F statistic. The probability density function of the statistic under the null hypothesis is estimated nonparametrically, using bootstrap resampling. A theoretical F distribution is then fitted to the empirical distribution in order to allow correction for multiple comparisons. The maps allow for the systematic exploration of the solution space for dipolar sources. They permit to test whether the data support a given solution. They do not rely on the assumption of uncorrelated source time courses. They can be compared to other statistical parametric maps such as those used in functional magnetic resonance imaging (fMRI). Results are presented for both simulated and real data. The maps were compared with LORETA and MUSIC results. For the real data consisting of an average of epileptic spikes, we observed good agreement between the EEG statistical maps, intracranial EEG recordings, and fMRI activations.
Compound statistical modelling of the uncompressed envelope of the backscattered signal has received much interest recently. In this note, a comprehensive collection of models is derived for the uncompressed envelope of the backscattered signal by compounding the Nakagami distribution with 13 flexible families. The corresponding estimation procedures are derived by the method of moments and the method of maximum likelihood. The sensitivity of the models to their various parameters is examined. It is expected that this work could serve as a useful reference and lead to improved modelling of the uncompressed envelope of the backscattered signal.
We derive Cramer-Rao bounds (CRBs) on the errors of estimating the
parameters (location and moment) of a current dipole source using data
from electro-encephalography (EEG), magneto-encephalography (MEG), or
the combined EEG/MEG modality. We use a realistic head model based on
knowledge of surfaces separating tissues of different conductivities,
obtained from magnetic resonance (MR) or computer tomography (CT)
imaging systems. The electric potentials and magnetic field components
at the respective sensors are functions of the source parameters through
integral equations. These potentials and field are computed using the
boundary or the finite element method (BEM or FEM), with a weighted
residuals technique. We present a unified framework for the measurements
computed by these methods that enables the derivation of the bounds. The
resulting bounds may be used, for instance, to choose the best
configuration of the sensors for a given patient and region of expected
source location. Numerical results are used to demonstrate an
application for shelving expected accuracies in estimating the source
parameters as a function of its position in the brain, based on real
EEG/MEG system and MR or CT images. The results include contours of
equal precision in the estimation and surfaces showing the size of the
90% confidence volume for a dipole on a sphere inside the brain
The authors present a sensitivity study of electroencephalography
based source localization due to errors in tissue conductivities and to
errors in modeling the conductivity variation inside the brain and
scalp. The study is conducted using a 2D finite element model obtained
from an MRI scan of a head cross-section. The effect of uncertainty in
the following tissues are studied: white matter, grey matter,
cerebrospinal fluid (CSF), skull, and fat. The distribution of source
location errors, assuming a single-dipole source model is examined in
detail for different dipole locations over the whole brain region.
Results presented in this paper clearly point to the following
conclusion. Unless the conductivities of the head tissues and the
distribution of these tissues throughout the head are modeled
accurately, the goal of achieving localization accuracy to within a few
millimeters is unattainable
Measures of performance for magnetoencephalography (MEG),
electroencephalography (EEG), and the combined MEG/EEG modality are
needed in order to compare, improve, and to optimize existing systems
and their configuration. We numerically compute bounds on the
mean-squared errors when estimating a current dipole's source parameters
for a three layered head model using measurements of EEG, MEG and their
combination. The electric potentials and magnetic field components are
related to the source through integral equations obtained via the
boundary element method (BEM). Discretization is achieved with a
weighted residuals technique, producing a center of gravity method. The
bounds obtained assume knowledge of the layer's conductivities and an
accurate representation of the interface surfaces, as obtained from MR
or CT imaging
The Cramer-Rao bound for unbiased location estimation of multiple
current dipoles is derived under the assumption of a general head model
parameterized by a combination of deterministic and stochastic
parameters. The expression (also applicable to MEG) thus characterizes
fundamental limits on EEG localization performance due to the effects of
both statistical measurement noise and model uncertainty
Consider the separable nonlinear least squares problem of finding a εRn and α εRk which, for given data (yi, ti), i=1,2,... m, and functions φ{symbol}j(α,t), j=1,2,..., n(m>n), minimize the functional {Mathematical expression} where θ(α)ij=φ{symbol}j(α, ti). Golub and Pereyra have shown that this problem can be reduced to a nonlinear least squares problem involving α only, and a linear least squares problem involving a only. In this paper we propose a new method for determining the optimal α which computationally has proved more efficient than the Golub-Pereyra scheme.
Localization error bounds are presented for both
electroencephalograms (EEGs) and magnetoencephalograms (MEGs) as
graphical error contours for a 37-sensor arrangement. Both one and two
dipole cases were examined for all possible dipole orientations and
locations within a head quadrant. The results show a strong dependence
on absolute dipole location and orientation. The results also show that
fusion of the EEG and MEG measurements into a combined model reduces the
lower bound. A Monte Carlo simulation was performed to check the
tightness of the bounds for a selected case. The simple head model, the
white and relatively low power noise, and the few relatively strong
dipoles were all selected in this study as optimistic conditions to
establish possibly fundamental resolution limits for any localization
effort
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.
It is investigated whether a mathematical dipole description is adequate for the localization of brain activity on the basis of visually evoked potentials (VEPs). Extended sources (dipole disks and dipole annuli) are stimulated and fitted with a mathematical dipole. It is found that the deviation between the positions of the disks and annuli and the equivalent dipole is very small. Also, the differences in the direction and amplitude may be neglected. The position of the extended source with respect to the electrode grid does not much influence these conclusions.
This text is the second edition of this book. It expands the widely acclaimed 1981 book, filling more gaps between EEG and the physical sciences. EEG opens a "window on the mind" by finding new connections between psychology and physiology. Topics include synaptic sources, electrode placement, choice of reference, volume conduction, power and coherence, projection of scalp potentials to dura surface, dynamic signatures of conscious experience, and neural networks immersed in global fields of synaptic action.
A new description of evoked potential activity in terms of ‘dipole source potentials’ is presented, based on the physical laws relating intracranial electrical activity and scalp potentials. Recorded at a sufficiently distant electrode, the electrical activity of a spatially restricted region can be approximated uniquely by a time varying dipole vector field with stationary equivalent dipole location. Each of its 3 projections on a 3-dimensional coordinate system presents an accordingly defined ‘dipole source component.’ Magnitudes of these components are functions of time, named ‘dipole source potentials.’
The spatial response of the magnetoencephalogram (MEG) to sources in the brain's cortex is compared with that of the electroencephalogram (EEG). This is done using computer modeling of the head which is approximated by 4 concentric spherical regions that represent the brain and surrounding bone and tissue. Lead fields are calculated at points on the cortex for unipolar, bipolar and quadrupolar MEG and EEG measurements. Since lead fields are patterns of the sensitivity of these measurements to a source at various locations and orientations, they provide a convenient means for comparison. It is found that a unipolar MEG has a very different lead field than a unipolar EEG. Hence, this type of MEG detects sources at different locations and orientations than this EEG. Although bipolar MEG and EEG lead fields are found to have similar patterns, the MEG lead field is narrower than that of the EEG and hence 'sees' a smaller area on the cortex than the EEG. This is because the potentials measured by the EEG are 'smeared' by the low-conductivity skull; the magnetic fields measured by the MEG are not smeared. Quadrupolar MEG and EEG lead fields are found to be about the same. The responses of bipolar MEGs and EEGs to distributed sources, which are composed of aligned and randomly oriented dipoles, are compared. It is found that for both types of sources, the MEG 'sees' an area on the cortex which is approximately 0.3 times that for the EEG. Hence, the MEG appears to be useful for detecting a more restricted group of sources than the EEG.
This article serves as an introduction to the other articles in this issue devoted to the problem of the localization of neural generators. Elements of the theory of electric volume conduction are briefly introduced, as far as these apply to the interpretation of observed scalp potentials. First, some basic methods for display of the different aspects of the spatiotemporal information are described. Next, the most prominent source and volume conductor models that have been postulated for the involved forward problem are summarized. The problems of source identification and source localization, known as the inverse problem, are then formulated in terms of a parameter estimation procedure. The importance of introducing a priori information in the inverse problem, aimed at stabilizing (regularizing) the obtained solution, is emphasized. Methods for imposing such constraints are briefly outlined.
In this paper, the reciprocity theorem is used to determine the sensitivity of EEG leads to the location and orientation of sources in the brain. Quantitative information used in determining the sensitivity is derived from constant potential plots of a three-concentric-sphere mathematical model of the head with current applied through surface leads (the reciprocal problem), and from an electrolytic tank employing a human skull. Advantages of the reciprocal or lead field approach are outlined and the following conclusions are drawn. 1) Leads placed at the end of a diameter through the center of the brain have a range of sensitivity due to source location of only 3 to 1. 2) For the same electrode placement, sensitivity is maximum to sources oriented parallel to the line of the electrodes regardless of source location. 3) Electrodes spaced 5 cm apart are about ten times more sensitive to proximal cortical sources (by virtue of position) than to sources near the center of the brain.
General formulas are presented for computing a lower bound on localization and moment error for electroencephalographic (EEG) or magnetoencephalographic (MEG) current source dipole models with arbitrary sensor array geometry. Specific EEG and MEG formulas are presented for multiple dipoles in a head model with 4 spherical shells. Localization error bounds are presented for both EEG and MEG for several different sensor configurations. Graphical error contours are presented for 127 sensors covering the upper hemisphere, for both 37 sensors and 127 sensors covering a smaller region, and for the standard 10-20 EEG sensor arrangement. Both 1- and 2-dipole cases were examined for all possible dipole orientations and locations within a head quadrant. The results show a strong dependence on absolute dipole location and orientation. The results also show that fusion of the EEG and MEG measurements into a combined model reduces the lower bound. A Monte Carlo simulation was performed to check the tightness of the bounds for a selected case. The simple head model, the low power noise and the few strong dipoles were all selected in this study as optimistic conditions to establish possibly fundamental resolution limits for any localization effort. Results, under these favorable assumptions, show comparable resolutions between the EEG and the MEG models, but accuracy for a single dipole, in either case, appears limited to several millimeters for a single time slice. The lower bounds increase markedly with just 2 dipoles. Observations are given to support the need for full spatiotemporal modeling to improve these lower bounds. All of the simulation results presented can easily be scaled to other instances of noise power and dipole intensity.
A mathematical procedure, which we call "Deblurring," was developed to reduce spatial blur distortion of scalp-recorded brain potentials due to transmission through the skull and other tissues. Deblurring estimates potentials at the superficial cerebral cortical surface from EEG's recorded at the scalp using a finite element model of each subject's scalp, skull and cortical surface constructed from their magnetic resonance images (MRI's). Simulations indicate that Deblurring is numerically stable, while a comparison of deblurred data with a direct cortical recording from a neurosurgery patient suggests that the procedure is valid. Application of Deblurring to somatosensory evoked potential data recorded at 124 scalp sites suggests that the method produces a dramatic improvement in spatial detail, and merits further development.
Many studies have been performed on the effects of various features of head geometry on electroencephalograms (EEG's) and magnetoencephalograms (MEG's) and on the accuracy with which electrical sources in the brain can be localized using these measurements. However, to date no studies have been performed of the effects of local variations in skull and scalp thickness. This paper presents a computer modeling study of the effects of such local variations. The results obtained in this study indicate that local variations in skull and scalp thickness have effects on EEG's and MEG's which range from a simple intuitive effect to complex effects which depend on such factors as source depth and orientation, the geometry of the variation in skull and scalp thickness, etc. These results also indicate that local variations in skull and scalp thickness cause EEG localization errors which are generally much less than 1 cm and MEG localization errors which are even smaller. These results also indicate that multichannel and single-channel MEG measurements will produce localization errors of approximately the same amplitude when there is a bump on the external surface of the head but that multichannel measurements will produce significantly smaller localization errors than single-channel measurements when a depression is present in that surface.
The electrical position of the heart with reference to the electrodes used in studying its field is unknown. For reasons presented, it is more likely eccentric; hence, the equation defining the field of an eccentric dipole in a spherical medium might be useful for projected experimental studies and for better understanding of the way in which a given electrical position determines the electrode potentials. A method introduced by Helmholtz was used for deriving the desired equation. It is discussed since its concepts are of considerable importance to other electrocardiographic problems, too. The more simple mathematical example dealing with the centric dipole in the sphere is discussed. The equation for the field of the eccentric dipole is given and data based upon it are presented in numerical and map form. The Helmholtz equation for the field in the spherical conductor produced by two small spherical electrodes arbitrarily located is also presented and briefly discussed.
The authors study the effects of sensor pattern and sensor
position perturbations on the angle estimation performance bound for
multiple narrow-band sources. The Cramer-Rao lower bound (CRLB) is used
with a probabilistic modeling of the perturbations. The CRLBs of a
linear uniform array under sensor position and pattern perturbations are
evaluated in detail for the case of two narrowband sources
Signal parameter estimators which are less sensitive to
perturbations in the array manifold are presented. A parametrized
stochastic model for the array uncertainties is introduced. The unknown
array parameters can include the individual gain and phase responses of
the sensors as well as their positions. Based on this model, a maximum a
posteriori (MAP) estimator is formulated. This results in a fairly
complex optimization problem which is computationally expensive. The MAP
estimator is simplified by exploiting properties of the weighted
subspace fitting method. An approximate method that further reduces the
complexity is also presented, assuming small array perturbations. A
compact expression for the MAP Cramer-Rao bound (CRB) on the signal and
array parameter estimates is derived. A simulation study indicates that
the proposed robust estimation procedures achieve the MAP-CRB even for
moderate sample sizes
The problem of signal parameter estimation of narrowband emitter
signals impinging on an array of sensors is addressed. A
multidimensional estimation procedure that applies to arbitrary array
structures and signal correlation is proposed. The method is based on
the recently introduced weighted subspace fitting (WSF) criterion and
includes schemes for both detecting the number of sources and estimating
the signal parameters. A Gauss-Newton-type method is presented for
solving the multidimensional WSF and maximum-likelihood optimization
problems. The global and local properties of the search procedure are
investigated through computer simulations. Most methods require
knowledge of the number of coherent/noncoherent signals present. A
scheme for consistently estimating this is proposed based on an
asymptotic analysis of the WSF cost function. The performance of the
detection scheme is also investigated through simulations
The performance of the MUSIC and ML methods is studied, and their
statistical efficiency is analyzed. The Cramer-Rao bound (CRB) for the
estimation problems is derived, and some useful properties of the CRB
covariance matrix are established. The relationship between the MUSIC
and ML estimators is investigated as well. A numerical study is reported
of the statistical efficiency of the MUSIC estimator for the problem of
finding the directions of two plane waves using a uniform linear array.
An exact description of the results is included
A quantitative evaluation has been made of methods designed to identify the intracranial sources of scalp recorded potentials. The scheme presented models the sources as equivalent dipoles and allows for the detection of multiple sources. The head is modeled by both homogeneous and inhomogeneous spheres. The results establish the technical feasibility of finding equivalent dipoles in the visual cortex for human visually evoked scalp potentials (VESP's).
This paper deals with source localization and strength estimation based on EEG and MEG data. It describes an estimation method (inverse procedure) which uses a four-spheres model of the head and a single current dipole. The dependency of the inverse solution on model parameters is investigated. It is found that sphere radii and conductivities influence especially the strength of the EEG equivalent dipole and not its location or direction. The influence on the equivalent dipole of the gradiometer is investigated. In general the MEG produces better location estimates than the EEG whereas the reverse is found for the component estimates. An inverse solution simultaneously based on EEG and MEG data appears slightly better than the average of separate EEG and MEG solutions. Variances of parameter estimators which can be calculated on the basis of a linear approximation of the model, were tested by Monte Carlo simulations.
The problem of locating the position of the source of evoked potentials from measurements on the surface of the scalp has been examined. It is shown that the position of the source in a head modeled by a sphere surrounded by two concentric shells of differing conductivities representing the skull and the scalp can be inferred from source localization calculations made on a homogeneous model. Sidman et al. proposed an approximate calculation to achieve the same goal, but it is shown that while their approximation is very good for sources located near the center of the head, such as in the midbrain or brainstem structures, it is less satisfactory for sources at an eccentricity of 0.6-0.9, which is the location of most cortical sources. In fact, over this range, their correction may introduce as many errors as it purports to remove. It is shown that midrange estimates of skull thickness and scalp thickness may introduce localization errors of ± 7 and ±3% of the outside radius of the scalp, respectively, but a poor estimate of skull conductivity introduces at most a 2% error.
A mathematical model consisting of a homogeneous spherical conductive medium was used to simulate the human head. A current dipole situated in the interior of the sphere produces, on the surface, a theoretical distribution of potential similar to certain EEG activities recorded on the human scalp. We propose a multi-stage method for computing the six parameters of a such virtual dipolar source by using only measurements on the surface of the scalp. Epileptic discharges observed on a young patient were thus analyzed. The computed dipole depends on the placement of pairs of electrodes used for the record. We compare the results of four ``assemblies'' using different pairings of electrodes. The center of the dipole varies within a volume of 2 cm 3 but its direction is stable. We discuss the importance and the influence of the approximations introduced by the model and the method. Although these approximations cannot be neglected, they do not change the signification of our results. The question is now raised: Has the virtual source a physical interpretation? Only intracerebral investigations can bring about a clear answer. However, whatever the answer may be, it appears that the method presented here can be helpful in the analysis of EEG surface situations.
On the forward and the inverse problem for EEG and MEG
- M Peters
- J De Munck
MUSIC, maximum likelihood, and Cramer-Rao bound
- P Stoica
- A Nchorai