Continuous current source inversion of evoked potential fields in a spherical model head

Evaluation of L1 and L2 minimum norm performances on EEG localizations

OBJECTIVE

In this work we study the performance of minimum norm methods to estimate the localization of brain electrical activity. These methods are based on the simplest forms of L(1) and L(2) norm estimates and are applied to simulated EEG data. The influence of several factors like the number of electrodes, grid density, head model, the number and depth of the sources and noise levels was taken into account. The main objective of the study is to give information about the dependence, on these factors, of the localization sources, to allow for proper interpretation of the data obtained in real EEG records.

METHODS

For the tests we used simulated dipoles and compared the localizations predicted by the L(1) and L(2) norms with the location of these point-like sources. We varied each parameter separately and evaluated the results.

RESULTS

From this work we conclude that, the grid should be constructed with approximately 650 points, so that the information about the orientation of the sources is preserved, especially for L(2) norm estimates; in favorable noise conditions, both L(1) and L(2) norm approaches are able to distinguish between more than one point-like sources.

CONCLUSIONS

The critical dependence of the results on the noise level and source depth indicates that regularized and weighted solutions should be used. Finally, all these results are valid both for spherical and for realistic head models.

Numerical aspects of spatio-temporal current density reconstruction from EEG-/MEG-data

The determination of the sources of electric activity inside the brain from electric and magnetic measurements on the surface of the head is known to be an ill-posed problem. In this paper, a new algorithm which takes temporal a priori information modeled by the smooth activation model into account is described and compared with existing algorithms such as Tikhonov-Phillips.

Modeling extended sources of event-related potentials using anatomical and physiological constraints

For the study of functional organization and reorganization of the human cortex by means of electromagnetic source imaging, a measure of the location and spatial extent of neural sources is of interest. This study evaluates the cortical patch method (CPM), an iterative procedure introduced by Lütkenhöner et al. [1995] that models EEG/MEG activity by means of extended cortical patches. Anatomical information is used to constrain estimates of location and extent of neural sources that generate the measured evoked potential. Whereas minimum norm approaches use mathematical constraints to solve the ambiguity of the inverse problem, the CPM introduces constraints based on anatomical and physiological knowledge about neural mass activity. In order to test the proposed method, the simulated activity in an artificial sulcus was subjected to the CPM. The results show that even activity on opposing walls of a sulcus can be well reconstructed. The simulations demonstrate the usefulness and limits of the CPM in estimating the spatial extent of neural sources in the cerebral cortex. As an example, an application of the method on experimental somatosensory evoked potentials is presented in the Appendix.

A Bayesian approach to introducing anatomo-functional priors in the EEG/MEG inverse problem

In this paper, we present a new approach to the recovering of dipole magnitudes in a distributed source model for magnetoencephalographic (MEG) and electroencephalographic (EEG) imaging. This method consists in introducing spatial and temporal a priori information as a cure to this ill-posed inverse problem. A nonlinear spatial regularization scheme allows the preservation of dipole moment discontinuities between some a priori noncorrelated sources, for instance, when considering dipoles located on both sides of a sulcus. Moreover, we introduce temporal smoothness constraints on dipole magnitude evolution, at time scales smaller than those of cognitive processes. These priors are easily integrated into a Bayesian formalism, yielding a maximum a posteriori (MAP) estimator of brain electrical activity. Results from EEG simulations of our method are presented and compared with those of classical quadratic regularization and a now popular generalized minimum-norm technique called low-resolution electromagnetic tomography (LORETA).

Estimating cortical activity from VEPS with the shrinking ellipsoid inverse

An iterative inverse method using Tikhonov regularization (the shrinking ellipsoid method) previously tested in a model system is used to invert the sequence of bioelectric scalp fields evoked by the onset of a checkerboard pattern in either the right or left lower hemifield. The shrinking ellipsoid method is modified from its original description to accommodate simultaneously inverting a sequence of thirteen VEP scalp fields measured from 65 to 125 ms after stimulus onset. This allows the evoked cortical activity to be tracked in 5-ms intervals without distortion due to occasional VEP scalp fields in the sequence that have too low a signal-to-noise ratio to be reliably inverted in isolation. A new method is described to identify the surface of the cortex from MRI data. This is required to implement the shrinking ellipsoid inverse. Results from two subjects studied in detail are presented. The earliest cortical activity occurs either in area MT (the middle temporal area) or simultaneously in MT and striate cortex (V1). However when it does occur in both areas, the activity in V1 is relatively weak and quickly subsides. Seventy-five ms after stimulus onset activity is seen mainly near MT corresponding to a region identified from PET studies as one that subserves motion processing. Activity moves to V1 by 90-100 ms after stimulus onset. Near 120 ms after stimulus onset, cortical activity returns to the region near MT. Virtually all activity identified in this time epoch occurs in the cortical hemisphere contralateral to the location of the stimulus in the visual field.

High resolution EEG: a new model-dependent spatial deblurring method using a realistically-shaped MR-constructed subject's head model

This paper presents a new model-dependent method for the spatial deblurring of scalp-recorded EEG potentials based on boundary-element and cortical imaging techniques. This model-dependent spatial deblurring (MDSD) method used MR images for the reconstruction of the subject's head model, and a layer of 364 radially-oriented equivalent current dipoles as a source model. The validation of the MDSD method was performed on simulated potential distributions generated from equivalent dipoles oriented radially, obliquely, and tangentially to the head surface. Furthermore, this method was used to localize neocortical sources of human movement-related and somatosensory-evoked potentials. It was shown that the new MDSD method improved markedly the spatial resolution of the simulated surface potentials and scalp-recorded event-related potentials. The spatial information content of the scalp-recorded EEG potentials increased progressively by increasing the spatial sampling from 28 to 128 channels. These results indicate that the new method could be satisfactorily used for high resolution EEG studies.

Postcentral origin of P22: evidence from source reconstruction in a realistically shaped head model and from a patient with a postcentral lesion

The source of the radial field of P22 was previously attributed either to the precentral (area 4) or postcentral (area 1) gyrus, on the basis of interpretation of potential maps recorded on the skin or cortex, respectively. The present study used dipole localization within realistically shaped head models and constrained the inverse solution by using the individual cortex and the normals on it, as derived from MR-tomography. In all normal subjects in which a sufficient solution was obtained (6 of 10, goodness of fit above 90%, and relative power of above 94.4% in principal-component analysis) the P22 source resided at the crown of the postcentral gyrus. Further evidence came from a patient with a postcentral lesion (area 1) and loss of P22, while N20-P20 was preserved.

Iterative refinement of the minimum norm solution of the bioelectric inverse problem

Functional brain imaging based on the bioelectric scalp field depends on the solution of the inverse problem. The object is to estimate the current distribution in the cortex from a measured noisy scalp potential field. A minimum norm least squares solution is often used for this purpose. Although this approach leads to a unique solution, it represents only one of a set of feasible solutions. In particular, it leads to a solution in which current is found to be generated widely in the cortex. There are other feasible minimum norm solutions for which cortical current is generated only in a restricted region, and it is likely that in many cases, these solutions are of physiological importance. This study presents a method to uncover these solutions. It is based on the idea that a minimum norm solution can be used to define a region of interest, an ellipsoid, within which it is worthwhile to search for another feasible solution. The minimum norm approach is used iteratively and the ellipsoid shrinks. As it shrinks it provides evidence pointing to the location of the active cortical region. The method explicitly recognizes the role of measurement noise in defining feasible solutions. It is tested in a model of the human head that incorporates a realistic cortex in a three-shell sphere. The method improves the localization of cortical activity mimicking physiological processing.

An iterative approach to the solution of the inverse problem

The bioelectric inverse problem is framed as a search through a feasible set of solutions for one that is physiologically plausible. The definition of the feasible set of solutions takes into account the important effects of measurement noise. This leads to an iterative approach. At each step, a regularized inverse is used to limit the amount of brain that need be searched in the next step. The approach is tested in a model system that incorporates a digitized cadaver cortex into a 3-shell spherical replica of the head.

The Duffing oscillator: a model for the dynamics of the neuronal groups comprising the transient evoked potential

Thirty years ago Zeeman conjectured that the dynamics of the EEG might be modeled by the "equation of motion" of a Duffing oscillator, a pendulum with a nonlinear, cubic, restoring force. In this study the idea is extended to the evoked potential (EP). A transient sensory EP reflects activity from several neuronal groups, pools of neurons that fire in synchrony, with voltage-time curves that overlap appreciably. When the dynamics of each neuronal group is modeled by a Duffing oscillator, multi-electrode transient VEPs are well predicted. Predictions based on Duffing oscillator dynamics are substantially better than those based on the assumption that each neuronal group follows a simpler exponentially damped sinusoid or a function that simulates a post-synaptic potential. The component voltage-time curves are reasonably consistent over 7 subjects, suggesting sequential activation of neuronal groups with delays of several tens of milliseconds between them. The scalp topographies of the components suggest their origins in the occipital cortex.