Identifying causal networks of neuronal sources from EEG/MEG data with the phase slope index: a simulation study

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.The investigation of functional neuronal synchronization has recently become a growing field of research. With high temporal resolution, electroencephalography and magnetoencephalography are well-suited measurement techniques to identify networks of interacting sources underlying the recorded data. The analysis of the data in terms of effective connectivity, nevertheless, contains intrinsic issues such as the problem of volume conduction and the non-uniqueness of the inverse solution. Here, we briefly introduce a series of existing methods assessing these problems. To determine the locations of interacting brain sources robust to volume conduction, all computations are solely based on the imaginary part of the cross-spectrum as a trustworthy source of information. Furthermore, we demonstrate the feasibility of estimating causal relationships of systems of neuronal sources with the phase slope index in realistically simulated data. Finally, advantages and drawbacks of the applied methodology are highlighted and discussed

Neue Methoden für Quellenlokalisierung und die Analyse von Hirnkonnektivität : mit Invarianz gegenüber Artefakten der Volumenleitung

Due to the artifacts of volume conduction, localization of interacting brain sources is an intricate issue in inverse calculations of EEG and MEG data. Since non-interacting brain sources do not contribute systematically, i.e. apart from random fluctuations around zero, to the imaginary part of cross-spectrum calculated from EEG/MEG data, these measures are powerful tools to study functional brain connectivity from noninvasive electrophysiological data. MUltiple SIgnal Classification (MUSIC), is a standard localization method.In one of the MUSIC variants called Recursively Applied and Projected MUSIC (RAP-MUSIC), multiple iterations are proposed in order to decrease the location estimation uncertainties introduced by subspace estimation errors. Since we are interested in the interacting sources, I propose to apply the existing subspace method “RAP-MUSIC” to the subspace found from the dominant singular vectors of the imaginary part of cross-spectrum. Secondly, to estimate the specific sources interacting with each other, I use a modified LCMV-beamformer approach in which the source direction for each voxel is determined by maximizing the imaginary part of coherency with respect to a given reference. Subspace based algorithms, such as MUSIC and RAP-MUSIC are very sensitive to the choice of subspace. In case the subspace is not accurately estimated, the sources which best explain the data are not localized optimally. RAP-MUSIC is therefore applicable in this form, i.e. on the subspace spanned by the eigenvectors of the imaginary part of cross-spectrum rather than the eigenvectors of covariance matrix, only if the number of interacting sources is even. The reason is that the imaginary part of cross-spectrum is antisymmetric and all eigenvalues occur in pairs. To solve this issue, a new method called Self-Consistent MUSIC (SC-MUSIC) is suggested which is based on the idea that the presence of several sources has a bias on the localization of each source through the bias on the estimation of the subspace. This bias can be reduced by projecting out all other sources mutually rather than iteratively. Further in this thesis, I introduce a new method which tests the sensitivity of connectivity measures for artifacts of volume conduction. The idea is to construct surrogate data which are statistically as close as possible to the original data but are superpositions of independent sources. For any connectivity which is noticed in the surrogate data, we can claim that there is not enough evidence to show that it is generated by real interactions rather than by artifacts of volume conduction. The Bispectrum, another measure of interaction which is applied in the estimation of non-linear interactions, is discussed in this thesis. A new normalization factor called univariate normalization is introduced which is unique in the sense that satisfies two fundamental requirements: 1. The absolute value of the normalized bispectrum is bounded by zero and one. 2. The normalization value by itself is only a measure of the signal strength rather than the interactions between signals.Aufgrund von Artefakten bei der Volumenleitung ist die Lokalisation von interagierenden Hirnquellen ein schwieriges Problem bei Inversrechnungen in EEG und MEG Daten. Da nicht-interagierende Hirnquellen nicht systematisch, d.h. außer von zufälligen Fluktuationen um den Nullpunkt, zum Imaginärteil des Kreuzspektrums von EEG/MEG beitragen, ist dieses Maß ein mächtiges Instrument, um funktionelle Konnektivität von nicht-invasiven eletrophysiologischen Daten zu untersuchen. MUltiple SIgnal Classification (MUSIC) ist eine Standardlokalisierungsmethode. In einer der MUSIC Varianten names Recursively Applied and Projected MUSIC (RAP-MUSIC), werden multiple Iterationen vorgeschlagen, um die Unsicherheit bei der Lokalisierung durch Schätzungsfehler beim Unterraum zu verringern. Da wir an interagierenden Quellen interessiert sind, schlage ich vor, das existierende Unterraum-Verfahren auf den dominanten Singulärvektor des Imaginärteils des Kreuzspektrums anzuwenden. Zweitens, um die spezifisch miteinander interagierenden Quellen zu finden, nutze ich einen modifizierten LCMV beamformer, in dem die Richtung der Quelle in jedem Voxel so bestimmt wird, dass sie den Imaginärteil der Kohärenz in Bezug auf eine gegebene Referenz maximiert. Unterraum-basierte Verfahren, wie MUSIC und RAP-MUSIC, sind sehr sensibel in Bezug auf die Wahl des Unterraums. Falls der Unterraum nicht akkurat geschätzt wird, so werden die Quellen, die die Daten am besten erklären, nicht optimal lokalisiert. RAP-MUSIC ist daher nur dann in dieser Form anwendbar, d.h. auf den Unterraum, der von den Eigenvektoren des Imaginärteils des Kreuzspektrums aufgespannt anstatt der Eigenvektoren der Kovarianzmatrix, wenn die Anzahl der interagierenden Quellen gerade ist. Der Grund ist der, dass der Imaginärteil des Kreuzspektrums anti-symmetrisch ist, und damit alle Eigenwerte in Paaren auftauchen. Um dieses Problem zu lösen, führen wir eine neue Methode namens Self-Consistent Music (SC-MUSIC) ein, die auf der Idee basiert ist, dass die Anwesenheit von mehreren Quellen die Lokalisierung jeder einzelnen Quelle beinflusst, in dem die Schätzung des Unterraums verzerrt wird. Dieser Effekt kann reduziert werden, indem man alle Quellen gleichzeitig anstatt von iterativ herausprojiziert. Weiterhin führe ich eine neue Methode ein, die die Sensiblität von Konnektivitätsmaßen gegenüber Artefakten der Volumenleitung testet. Die Idee besteht darin, Surrogatdaten zu bestimmen, die statistisch so nah wie möglich an den Originaldaten liegen, die aber Summen von unabhängigen Quellen sind. Für jedes Maß an Konnektivität, das in den Surrogatdaten entdeckt wird, gibt es somit nicht genug Beweise, um nachzuweisen, dass es sich um reale Interaktionen handelt anstatt von Artefakten der Volumenleitung. Das Bispektrum, ein weiteres Maß von Interaktion, dass bei der Schätzung von nicht-linearen Interaktionen angewandt wird, wird in dieser Arbeit ebenfalls behandelt. Ein neuer Normalisierungsfaktor namens "univariate Normalisation" wird eingeführt, der eindeutig ist in dem Sinne, dass er zwei fundamentelle Forderungen erfüllt: 1. Der Absolutwert des normalisierten Bispektrums liegt zwischen null und eins. 2. Der Normalisierungsfaktor ist nur ein Maß der Signalstärke, nicht der Stärke der Interaktion zwischen Signalen

Localizing True Brain Interactions from EEG and MEG Data with Subspace Methods and Modified Beamformers

To address the problem of mixing in EEG or MEG connectivity analysis we exploit that noninteracting brain sources do not contribute systematically to the imaginary part of the cross-spectrum. Firstly, we propose to apply the existing subspace method “RAP-MUSIC” to the subspace found from the dominant singular vectors of the imaginary part of the cross-spectrum rather than to the conventionally used covariance matrix. Secondly, to estimate the specific sources interacting with each other, we use a modified LCMV-beamformer approach in which the source direction for each voxel was determined by maximizing the imaginary coherence with respect to a given reference. These two methods are applicable in this form only if the number of interacting sources is even, because odd-dimensional subspaces collapse to even-dimensional ones. Simulations show that (a) RAP-MUSIC based on the imaginary part of the cross-spectrum accurately finds the correct source locations, that (b) conventional RAP-MUSIC fails to do so since it is highly influenced by noninteracting sources, and that (c) the second method correctly identifies those sources which are interacting with the reference. The methods are also applied to real data for a motor paradigm, resulting in the localization of four interacting sources presumably in sensory-motor areas

The LDA beamformer: Optimal estimation of ERP source time series using linear discriminant analysis

We introduce a novel beamforming approach for estimating event-related potential (ERP) source time series based on regularized linear discriminant analysis (LDA). The optimization problems in LDA and linearly-constrained minimum-variance (LCMV) beamformers are formally equivalent. The approaches differ in that, in LCMV beamformers, the spatial patterns are derived from a source model, whereas in an LDA beamformer the spatial patterns are derived directly from the data (i.e., the ERP peak). Using a formal proof and MEG simulations, we show that the LDA beamformer is robust to correlated sources and offers a higher signal-to-noise ratio than the LCMV beamformer and PCA. As an application, we use EEG data from an oddball experiment to show how the LDA beamformer can be harnessed to detect single-trial ERP latencies and estimate connectivity between ERP sources. Concluding, the LDA beamformer optimally reconstructs ERP sources by maximizing the ERP signal-to-noise ratio. Hence, it is a highly suited tool for analyzing ERP source time series, particularly in EEG/MEG studies wherein a source model is not available

Localizing bicoherence from EEG and MEG

We propose a new method for the localization of nonlinear cross-frequency coupling in EEG and MEG data analysis, based on the estimation of bicoherences at the source level. While for the analysis of rhythmic brain activity, source directions are commonly chosen to maximize power, we suggest to maximize bicoherence instead. The resulting nonlinear cost function can be minimized effectively using a gradient approach. We argue, that bicoherence is also a generally useful tool to analyze phase-amplitude coupling (PAC), by deriving formal relations between PAC and bispectra. This is illustrated in simulated and empirical LFP data. The localization method is applied to EEG resting state data, where the most prominent bicoherence signatures originate from the occipital alpha rhythm and the mu rhythm. While the latter is hardly visible using power analysis, we observe clear bicoherence peaks in the high alpha range of sensorymotor areas. We additionally apply our method to resting-state data of subjects with schizophrenia and healthy controls and observe significant bicoherence differences in motor areas which could not be found from analyzing power differences