The generalized shrinkage estimator for the analysis of functional connectivity of brain signals

We develop a new statistical method for estimating functional connectivity between neurophysiological signals represented by a multivariate time series. We use partial coherence as the measure of functional connectivity. Partial coherence identifies the frequency bands that drive the direct linear association between any pair of channels. To estimate partial coherence, one would first need an estimate of the spectral density matrix of the multivariate time series. Parametric estimators of the spectral density matrix provide good frequency resolution but could be sensitive when the parametric model is misspecified. Smoothing-based nonparametric estimators are robust to model misspecification and are consistent but may have poor frequency resolution. In this work, we develop the generalized shrinkage estimator, which is a weighted average of a parametric estimator and a nonparametric estimator. The optimal weights are frequency-specific and derived under the quadratic risk criterion so that the estimator, either the parametric estimator or the nonparametric estimator, that performs better at a particular frequency receives heavier weight. We validate the proposed estimator in a simulation study and apply it on electroencephalogram recordings from a visual-motor experiment.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS396 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonlinear spectral analysis: A local Gaussian approach

The spectral distribution $f(\omega)$ of a stationary time series $\{Y_t\}_{t\in\mathbb{Z}}$ can be used to investigate whether or not periodic structures are present in $\{Y_t\}_{t\in\mathbb{Z}}$, but $f(\omega)$ has some limitations due to its dependence on the autocovariances $\gamma(h)$. For example, $f(\omega)$ can not distinguish white i.i.d. noise from GARCH-type models (whose terms are dependent, but uncorrelated), which implies that $f(\omega)$ can be an inadequate tool when $\{Y_t\}_{t\in\mathbb{Z}}$ contains asymmetries and nonlinear dependencies. Asymmetries between the upper and lower tails of a time series can be investigated by means of the local Gaussian autocorrelations introduced in Tj{\o}stheim and Hufthammer (2013), and these local measures of dependence can be used to construct the local Gaussian spectral density presented in this paper. A key feature of the new local spectral density is that it coincides with $f(\omega)$ for Gaussian time series, which implies that it can be used to detect non-Gaussian traits in the time series under investigation. In particular, if $f(\omega)$ is flat, then peaks and troughs of the new local spectral density can indicate nonlinear traits, which potentially might discover local periodic phenomena that remain undetected in an ordinary spectral analysis.Comment: Version 4: Major revision from version 3, with new theory/figures. 135 pages (main part 32 + appendices 103), 11 + 16 figure

Nonlinear cross-spectrum analysis via the local Gaussian correlation

Spectrum analysis can detect frequency related structures in a time series $\{Y_t\}_{t\in\mathbb{Z}}$, but may in general be an inadequate tool if asymmetries or other nonlinear phenomena are present. This limitation is a consequence of the way the spectrum is based on the second order moments (auto and cross-covariances), and alternative approaches to spectrum analysis have thus been investigated based on other measures of dependence. One such approach was developed for univariate time series in Jordanger and Tj{\o}stheim (2017), where it was seen that a local Gaussian auto-spectrum $f_{v}(\omega)$, based on the local Gaussian autocorrelations $\rho_v(\omega)$ from Tj{\o}stheim and Hufthammer (2013), could detect local structures in time series that looked like white noise when investigated by the ordinary auto-spectrum $f(\omega)$. The local Gaussian approach in this paper is extended to a local Gaussian cross-spectrum $f_{kl:v}(\omega)$ for multivariate time series. The local cross-spectrum $f_{kl:v}(\omega)$ has the desirable property that it coincides with the ordinary cross-spectrum $f_{kl}(\omega)$ for Gaussian time series, which implies that $f_{kl:v}(\omega)$ can be used to detect non-Gaussian traits in the time series under investigation. In particular: If the ordinary spectrum is flat, then peaks and troughs of the local Gaussian spectrum can indicate nonlinear traits, which potentially might discover local periodic phenomena that goes undetected in an ordinary spectral analysis.Comment: 41 pages, 12 figure

The evolution of vibrational excitations in glassy systems

The equations of the mode-coupling theory (MCT) for ideal liquid-glass transitions are used for a discussion of the evolution of the density-fluctuation spectra of glass-forming systems for frequencies within the dynamical window between the band of high-frequency motion and the band of low-frequency-structural-relaxation processes. It is shown that the strong interaction between density fluctuations with microscopic wave length and the arrested glass structure causes an anomalous-oscillation peak, which exhibits the properties of the so-called boson peak. It produces an elastic modulus which governs the hybridization of density fluctuations of mesoscopic wave length with the boson-peak oscillations. This leads to the existence of high-frequency sound with properties as found by X-ray-scattering spectroscopy of glasses and glassy liquids. The results of the theory are demonstrated for a model of the hard-sphere system. It is also derived that certain schematic MCT models, whose spectra for the stiff-glass states can be expressed by elementary formulas, provide reasonable approximations for the solutions of the general MCT equations.Comment: 50 pages, 17 postscript files including 18 figures, to be published in Phys. Rev.

Speech rhythms and multiplexed oscillatory sensory coding in the human brain

Cortical oscillations are likely candidates for segmentation and coding of continuous speech. Here, we monitored continuous speech processing with magnetoencephalography (MEG) to unravel the principles of speech segmentation and coding. We demonstrate that speech entrains the phase of low-frequency (delta, theta) and the amplitude of high-frequency (gamma) oscillations in the auditory cortex. Phase entrainment is stronger in the right and amplitude entrainment is stronger in the left auditory cortex. Furthermore, edges in the speech envelope phase reset auditory cortex oscillations thereby enhancing their entrainment to speech. This mechanism adapts to the changing physical features of the speech envelope and enables efficient, stimulus-specific speech sampling. Finally, we show that within the auditory cortex, coupling between delta, theta, and gamma oscillations increases following speech edges. Importantly, all couplings (i.e., brain-speech and also within the cortex) attenuate for backward-presented speech, suggesting top-down control. We conclude that segmentation and coding of speech relies on a nested hierarchy of entrained cortical oscillations