Application of the Richardson extrapolation in simulation studies of EEGs

Meijs, J. W. H.; Boom, H.B.K.; Peters, M.J.; Oosterom, A. van

doi:10.1007/bf02442855

Cited by 12 publications

(9 citation statements)

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“…However, modeling the head by a poor conductor (the skull) between two much better conductors (the brain tissue and the scalp) requires modifications in the standard numerical approach in order to produce accurate results. Meijs and Peters (1987) found that numerical errors in computing V may be excessive because the ratio in the conductivity of the skull to that of the brain and the scalp is less than 0.1. The first suggested solution for this numerical instability was the use of the Richardson extrapolation (Richardson and Guant, 1927;Meijs, Peters, et al , 1987).…”

Section: Realistically Shaped Conductor Modelsmentioning

confidence: 99%

“…Meijs and Peters (1987) found that numerical errors in computing V may be excessive because the ratio in the conductivity of the skull to that of the brain and the scalp is less than 0.1. The first suggested solution for this numerical instability was the use of the Richardson extrapolation (Richardson and Guant, 1927;Meijs, Peters, et al , 1987). This approach relies on the assumption that the correct potential distribution can be obtained by estimating an asymptote on the basis of two solutions with different grid densities.…”

Section: Realistically Shaped Conductor Modelsmentioning

confidence: 99%

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Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain

et al. 1993

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Magnetoencephalography(MEG) is a noninvasive technique for investigating neuronal activity in the living human brairi. The time resolution of the method is better than 1 ms and the spatial discrimination is, under favorable circumstances, 2 -3 mm for sources in the cerebral cortex. In MEG studies, the weak 10 fT -1 pT magnetic fields produced by electric currents fiowing in neurons are measured with multichannel SQUID {superconducting quantum interference device) gradiometers. The sites in the cerebral cortex that are activated by a stimulus can be found from the detected magnetic-field distribution, provided that appropriate assumptions about the source render the solution of the inverse problem unique. Many interesting properties of the working human brain can be studied, including spontaneous activity and signal processing following external stimuli. For clinical purposes, determination of the locations of epileptic foci is of interest. The authors begiri with a general introduction and a short discussion of the neural basis of MEG. The mathematical theory of the method is then explained in detail, followed by a thorough description of MEG instrumentation, data analysis, and practical construction of multi-SQUID devices. Finally, several MEG experiments performed in the authors laboratory are described, covering studies of evoked responses and of spontaneous activity in both healthy and diseased brains. Many MEG studies by other groups are discussed briefiy as well.

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Section: Realistically Shaped Conductor Modelsmentioning

confidence: 99%

Section: Realistically Shaped Conductor Modelsmentioning

confidence: 99%

Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain

et al. 1993

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“…However, this procedure leads to a practical problem since the number of discrete points needed to obtain an accuracy of 5% in the final field distribution would result in a CPU time in the order of 200 h for the DEC-20 computer used for the computations. The amount of computational work can be substantially decreased by making use of the Richardson extrapolation technique, as described elsewhere (Meijs et al 1987). Nevertheless, the present study demonstrates the usefulness of the new model of the volume conductor.…”

Section: Modelmentioning

confidence: 91%

On the magnetic field distribution generated by a dipolar current source situated in a realistically shaped compartment model of the head

Meijs

Bosch

Peters

et al. 1987

Electroencephalography and Clinical Neurophysiology

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SummaryThe magnetic field distribution around the head is simulated using a realistically shaped compartment model of the head. The model is based on magnetic resonance images. The 3 compartments describe the brain, the skull and the scalp. The source is represented by a current dipole situated in the visual cortex. The magnetic field distribution due to the source and that due to the volume currents are calculated separately. The simulations are carried out in order to ascertain which matrix of grid points is suitable as a measuring grid. The possibilities studied are grid points situated in a plane, in a surface which follows the contours of the head and in a sphere. This sphere is taken concentric to the sphere which is the best possible fit for the head. Taking into account the relative contribution of the volume currents and the possible accuracy in the positioning of the magnetic field detector, it can be concluded that the best choice is to measure the normal component of the magnetic field at points which are situated in the spherical surface. The results of this study also show that the magnetic field distribution based on a realistically shaped compartment model differs from that based on a compartment model consisting of concentric spheres. In the spherical model of the head no contribution of the volume currents to the component of the field normal to the sphere can be expected. The difference between the results obtained with these two volume conductor models increases with source depth.Key words: magnetoencephalography; visual cortex; simulation; volume conductor shape; recording surfacesThe accuracy of source localization within the brain based on magnetoencephalographic (MEG) measurements around the head depends on the adequacy of the models used to represent both the source and the volume conductor (i.e., the head). The most commonly used model for a source is a single-current dipole or a current dipole layer. Up to now mathematical models of the head have been confined to spherical models (Cohen and Cuffin 1983), although the influence of the shape of an isolated human skull on the magnetic field was measured by Weinberg et al. (1984) and Barth et al. (1986).In some cases the adequacy of a model used for the localization of a source may be confirmed, for instance, by X-ray tomographs in studies of focal epilepsy (Barth et al. 1984). However, in most other cases such a confirmation cannot be obtained. Although not conclusive, a model inspires confidence when source localization, using this model, leads to the same results when the localization is based on the measured EEG distribution as in the case where the localization is based on the measured MEG distribution.For sources in the visual cortex the estimated source location based on the visual evoked potentials did not coincide with the location based on the visual evoked magnetic fields (Stok et al. 1984). The model used for the computations consisted of 4 concentric spheres, where the spheres were fitted in the section of the head near the vi...

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“…If one of the components inside the volume conductor has a relatively low conductivity compared with the conductivities of the other components, the numerical errors have been shown to be blown up [ 5 ] , [ 8 ] . Let us assume that only one compartment inside the volume conductor has a relatively low conductivity, not being the outer shell, i.e., the ( m -1 )th compartment of the volume conductor consisting of N ( N > 2 ) , nested, compartments ( N I m , Fig.…”

Section: The Isolated Problem Approachmentioning

confidence: 99%

“…However, during simulation studies using a head model consisting of four realistically shaped compartments, the discrete boundary element (BE) method used to compute the EEG's was found to generate numerical errors, which were blown up due to the small conductivity of the skull [5]. To study this problem, simulations of EEG's based on the four concentric spheres model were carried out by means of the BE method.…”

Section: Introduction or A Quantitative Interpretation Of Electroementioning

confidence: 99%

On the numerical accuracy of the boundary element method (EEG application)

Meijs

Weier

Peters

et al. 1989

IEEE Trans. Biomed. Eng.

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282

169

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The numerical accuracy of the boundary element (BE) method used to solve the volume conduction problem of nested compartments, each having a homogeneous conductivity, is studied. The following techniques for improving this accuracy are discussed: the handling of the auto solid angle element omega ii, the overall refinement of the level of discreteness, the use of a locally refined discrete grid, the isolated problem approach, and an adaptive refined computation of the discrete surface integrals involved in the BE method. The effects of these techniques on the numerical accuracy of the computed electrical potentials are illustrated by taking a volume conductor consisting of four concentric spheres representing the head since for this model an analytical (exact) solution is available. The techniques are of importance for numerically computed electroencephalograms (EEG's) since the numerically computed surface EEG's are severely affected by the relatively low conductivity of the compartment representing the skull.

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Application of the Richardson extrapolation in simulation studies of EEGs

Cited by 12 publications

References 13 publications

Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain

Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain

On the magnetic field distribution generated by a dipolar current source situated in a realistically shaped compartment model of the head

On the numerical accuracy of the boundary element method (EEG application)

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