Uniqueness of the generators of brain evoked potential maps

Stable Numerical Identification of Sources in Non-Homogeneous Media

In this work, we present a numerical algorithm to solve the inverse problem of volumetric sources from measurements on the boundary of a non-homogeneous conductive medium, which is made of conductive layers with constant conductivity in each layer. This inverse problem is ill-posed since there is more than one source that can generate the same measurement. Furthermore, the ill-posedness is due to the fact that small variations (or errors) in the measurement (input data) can produce substantial variations in the identified source location. We propose two steps to solve this inverse problem in some classes of sources: we first recover the harmonic part of the volumetric source, and, in a second step, we compute the non-harmonic part of the source. For the reconstruction of the harmonic part of the source, we follow a variational approach based on the reformulation of the inverse problem as a distributed control problem, for which the cost function incorporates a penalized term with the input data on the boundary. This cost function is minimized by a conjugate gradient algorithm in combination with a finite element discretization. We recover the non-harmonic component of the source using a priori information and an iterative algorithm for some particular classes of sources. To validate the numerical methodology, we develop synthetic examples both in circular (simple) and irregular (complex) regions. The numerical results show that the proposed methodology allows to recover the complete source and produce stable and accurate numerical solutions.

Partial independence of bioelectric and biomagnetic fields and its implications for encephalography and cardiography

In this paper, we clearly demonstrate that the electric potential and the magnetic field can contain different information about current sources in three-dimensional conducting media. Expressions for the magnetic fields of electric dipole and quadrupole current sources immersed in an infinite conducting medium are derived, and it is shown that two different point dipole distributions that are electrically equivalent have different magnetic fields. Although measurements of the electric potential are not sufficient to determine uniquely the characteristics of a quadrupolar source, the radial component of the magnetic field can supply the additional information needed to resolve these ambiguities and to determine uniquely the configuration of dipoles required to specify the electric quadrupoles. We demonstrate how the process can be extended to even higher-order terms in an electrically silent series of magnetic multipoles. In the context of a spherical brain source model, it has been mathematically demonstrated that the part of the neuronal current generating the electric potential lives in the orthogonal complement of the part of the current generating the magnetic potential. This implies a mathematical relationship of complementarity between electroencephalography and magnetoencephalography, although the theoretical result in question does not apply to the nonspherical case [G. Dassios, Math. Med. Biol. 25, 133 (2008)]. Our results have important practical applications in cases where electrically silent sources that generate measurable magnetic fields are of interest. Moreover, electrically silent, magnetically active moments of higher order can be useful when cancellation due to superposition of fields can occur, since this situation leads to a substantial reduction in the measurable amplitude of the signal. In this context, information derived from magnetic recordings of electrically silent, magnetically active multipoles can supplement electrical recordings for the purpose of studying the physiology of the brain. Magnetic fields of the electric multipole sources in a conducting medium surrounded by an insulating spherical shell are also presented and the relevance of this calculation to cardiographic and encephalographic experimentation is discussed.

An automated method for micro-state segmentation of evoked potentials

We present a method for segmenting evoked potentials into functional micro-states. The method is based on measuring the similarity between all the topographic maps in the evoked potential and grouping them into functional micro-states based on minimizing an error function. The similarity is measured as the normalized cross-correlation coefficient. The method was validated on simulated data and tested on its ability to segment a visual evoked potential. On simulated data the method missed from 1% to 8.5% of the micro-state boundaries for evoked potentials with a signal-to-noise ratio of 20-1dB, respectively. The proposed segmentation method was compared with segmentation based on K-mean clustering. It was found that the proposed method was better at detecting the correct number of micro-states and was computationally more efficient. The automatic segmentation of the visual evoked potential was compared to the manual segmentation performed by eleven EEG specialists. No significant difference in the deviation of micro-state boundaries was observed between two random EEG specialists and between a random EEG specialist and the automatic method. Thus it was found that the method could reliably segment evoked potentials into their functional micro-states.

Variational, geometric, and statistical methods for modeling brain anatomy and function

We survey the recent activities of the Odyssée Laboratory in the area of the application of mathematics to the design of models for studying brain anatomy and function. We start with the problem of reconstructing sources in MEG and EEG, and discuss the variational approach we have developed for solving these inverse problems. This motivates the need for geometric models of the head. We present a method for automatically and accurately extracting surface meshes of several tissues of the head from anatomical magnetic resonance (MR) images. Anatomical connectivity can be extracted from diffusion tensor magnetic resonance images but, in the current state of the technology, it must be preceded by a robust estimation and regularization stage. We discuss our work based on variational principles and show how the results can be used to track fibers in the white matter (WM) as geodesics in some Riemannian space. We then go to the statistical modeling of functional magnetic resonance imaging (fMRI) signals from the viewpoint of their decomposition in a pseudo-deterministic and stochastic part that we then use to perform clustering of voxels in a way that is inspired by the theory of support vector machines and in a way that is grounded in information theory. Multimodal image matching is discussed next in the framework of image statistics and partial differential equations (PDEs) with an eye on registering fMRI to the anatomy. The paper ends with a discussion of a new theory of random shapes that may prove useful in building anatomical and functional atlases.

High-resolution EEG mapping: an equivalent charge-layer approach

Brain electrical signal is one of the windows to understanding neural activities. Various high-resolution imaging techniques have been developed to reveal the electrical activities underneath the cortical surface from scalp electroencephalographic recordings, such as scalp Laplacian, cortical surface potential, equivalent charge layer (ECL) and equivalent dipole layer (EDL). In this work, we develop forward density formulae for the ECL and the EDL of neural electric sources in a 4-concentric-sphere head model, and compare ECL with EDL in theory, simulation and real evoked data tests. The results confirm that the ECL map may be of higher spatial resolution than the EDL map.

An electrophysiological study of scene effects on object identification

The meaning of a visual scene influences the identification of visual objects embedded in it. We investigated the nature and time course of scene effects on object identification by recording event-related brain potentials (ERPs) and response times (RTs). In three experiments, participants identified objects within a scene that were either semantically congruous (e.g., a pot in a kitchen) or incongruous (e.g., a desk in a river). As expected, RTs were faster for congruous than incongruous objects. The earliest sign of reliable scene congruity effects in the ERPs (greater positivity for congruous pictures between 300 and 500 ms) was around 300 ms. Both the morphology and time course of the N390 scene congruity effect are reminiscent of the N400 sentence congruity effect typically observed in sentence context paradigms, suggesting a functional similarity of the neural processes involved. Overall, these results support theories postulating that visual scenes do not appreciably affect object identification processes before associated semantic information is activated. We speculate that the N390 scene congruity effect reflects the action of visual scene schemata stored in the anterior temporal lobe.

On the uniqueness of the surface sources of evoked potentials

The uniqueness of a surface density of sources localized inside a spatial region R and producing a given electric potential distribution in its boundary B0 is revisited. The situation in which R is filled with various subregions, each one having a definite constant value for the electric conductivity is considered. It is argued that the knowledge of the potential in all B0 fully determines the surface-located sources for a general class of surfaces supporting them and also a wide type of those sources. The class of surfaces can be defined as a union of an arbitrary but finite number of open or closed surfaces. The only restriction upon them is that no one of the closed surfaces contains inside it another (nesting) of the closed or open ones. The types of sources are surface charge densities and double layer (dipolar) densities for the open surfaces and more restrictively, only surface charge densities for the closed ones. A two-dimensional analytically solvable example illustrating the drastic appearance of uniqueness after arbitrarily small holes are opened in nested surfaces is discussed.

A study of equivalent source techniques for high-resolution EEG imaging

High-resolution EEG imaging has been an important topic in recent EEG research, and much work has been done on the two equivalent source imaging techniques: the equivalent distributed dipole-layer source imaging technique (EST) and the equivalent multipole source imaging technique (SAT). In this paper we first develop a forward density formula for a spherical equivalent distributed dipole layer of an arbitrary dipole in a three-concentric-sphere head model. It is clarified using the derived forward formula that the equivalent dipole-layer source and equivalent multipole source are interrelated in theory. Finally, simulation comparisons are conducted, the results of which suggest that EST has a higher spatial resolution than SAT when both of them are implemented by a truncated singular value decomposition algorithm. This is due to the different singularities of the inversion equations involved in the two techniques. An empirical VEP data study also shows that EST is better than SAT in providing higher spatial resolution EEG imaging.

Localization of brain electrical activity via linearly constrained minimum variance spatial filtering

A spatial filtering method for localizing sources of brain electrical activity from surface recordings is described and analyzed. The spatial filters are implemented as a weighted sum of the data recorded at different sites. The weights are chosen to minimize the filter output power subject to a linear constraint. The linear constraint forces the filter to pass brain electrical activity from a specified location, while the power minimization attenuates activity originating at other locations. The estimated output power as a function of location is normalized by the estimated noise power as a function of location to obtain a neural activity index map. Locations of source activity correspond to maxima in the neural activity index map. The method does not require any prior assumptions about the number of active sources of their geometry because it exploits the spatial covariance of the source electrical activity. This paper presents a development and analysis of the method and explores its sensitivity to deviations between actual and assumed data models. The effect on the algorithm of covariance matrix estimation, correlation between sources, and choice of reference is discussed. Simulated and measured data is used to illustrate the efficacy of the approach.

A sampling theorem for EEG electrode configuration

An analytical tool to help in selecting the number of electrodes required for recording electroencephalogram (EEG) signals is presented. The main assumption made is that the scalp can be modeled as a hemispherical surface. The number of sensors required to sample a surface is derived by using a mean square error (MSE) measure to approximate the continuous potential functions on the hemispherical surface. An algorithm for selecting the number of electrodes for arbitrary head geometries is also proposed. A sampling theorem is then derived with conditions on the sampling points for electrode placement.

Event-related potential scalp fields during parallel and serial visual searches

Event-related potential (ERP) scalp fields generated during parallel and serial searches were compared using a bootstrap resampling technique. Two different parallel search tasks required the detection of a single feature either color or orientation. A serial search task required the detection of the feature conjunction target: color and orientation. Identical stimuli were used for both parallel searches and a similar stimulus for the serial search. No evidence for scalp field differences earlier than 150 ms were discovered, suggesting that "low-level' visual processing is the same in both types of searches. ERP scalp fields that distinguished parallel from serial searches were identified between 150 and 250 ms. It is proposed that these different scalp fields represent timing and/or magnitude differences in the regions of cortex activated during parallel and serial searches.

The equivalent source technique and cortical imaging

As an equivalent source localisation technique, the radial dipole source has been chosen as the equivalent source of the actual neural source in the recently developed cortical imaging technique (CIT). In this short paper, the point current source is numerically tested as a new kind of equivalent source for implementing CIT. The results confirm the efficiency of the new equivalent source.

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.

Insidious errors in dipole parameters due to shell model misspecification using multiple time-points

Insidious errors in dipole modeling due to shell model misspecification in a spherical model were examined analyzing multiple time-points using the constraints of a commonly-used DSL (Dipole Source Localization) method. The computer simulation examined the differences in the fit dipole parameters for the same generator under two circumstances: 1) when computed as a single dipole active alone, and 2) when computed as a member of a simultaneously-active dipole pair. The computations were done using a simplification by which the dipole parameters computed from multiple time-points can be correctly assessed by computing dipole parameters at only two virtual time-points. Using multiple time-points in the DSL generally resulted in less error than if only a single time-point was used. However, how much improvement cna be achieved by using multiple time-points, as compared with a single time-point, is a function of many factors, such as the location and orientation of the dipoles, and the relative magnitudes and overlap of the waveforms (i.e., time-varying magnitudes) of the dipoles, as well as the model used in the fitting. Further, it was shown that it is incorrect to assume that a multiple-time-point DSL will compute a zero magnitude for generators during quiescent intervals. Additionally, it was shown that a "correction" to reduce error for one pair of waveforms will not be applicable to other waveforms. Also, even if location errors are eliminated, magnitude and orientation errors can still be shown to be present. Finally, iterative reduction of the least-square error between the observed and predicted surface maps leads to increasing errors in dipole parameters. We conclude that a DSL with model misspecification can contain insidious (undetectable) errors.