A Spectral Method for Generating Surrogate Graph Signals

The increasing availability of network data is leading to a growing interest in processing of signals on graphs. One notable tool for extending conventional signal-processing operations to networks is the graph Fourier transform that can be obtained as the eigendecomposition of the graph Laplacian. In this letter, we used the graph Fourier transform to define a new method for generating surrogate graph signals. The approach is based on sign-randomization of the graph Fourier coefficients and, therefore, the correlation structure of the surrogate graph signals (i.e., smoothness on the graph topology) is imposed by the measured data. The proposed method of surrogate data generation can be widely applied for nonparametric statistical hypothesis testing. Here, we showed a proof-of-concept with a high-density electroencephalography dataset

Decoupling of brain function from structure reveals regional behavioral specialization in humans

The brain is an assembly of neuronal populations interconnected by structural pathways. Brain activity is expressed on and constrained by this substrate. Therefore, statistical dependencies between functional signals in directly connected areas can be expected higher. However, the degree to which brain function is bound by the underlying wiring diagram remains a complex question that has been only partially answered. Here, we introduce the structural-decoupling index to quantify the coupling strength between structure and function, and we reveal a macroscale gradient from brain regions more strongly coupled, to regions more strongly decoupled, than expected by realistic surrogate data. This gradient spans behavioral domains from lower-level sensory function to high-level cognitive ones and shows for the first time that the strength of structure-function coupling is spatially varying in line with evidence derived from other modalities, such as functional connectivity, gene expression, microstructural properties and temporal hierarchy

A New Surrogating Algorithm by the Complex Graph Fourier Transform (CGFT)

[EN] The essential step of surrogating algorithms is phase randomizing the Fourier transform while preserving the original spectrum amplitude before computing the inverse Fourier transform. In this paper, we propose a new method which considers the graph Fourier transform. In this manner, much more flexibility is gained to define properties of the original graph signal which are to be preserved in the surrogates. The complex case is considered to allow unconstrained phase randomization in the transformed domain, hence we define a Hermitian Laplacian matrix that models the graph topology, whose eigenvectors form the basis of a complex graph Fourier transform. We have shown that the Hermitian Laplacian matrix may have negative eigenvalues. We also show in the paper that preserving the graph spectrum amplitude implies several invariances that can be controlled by the selected Hermitian Laplacian matrix. The interest of surrogating graph signals has been illustrated in the context of scarcity of instances in classifier training.This research was funded by the Spanish Administration and the European Union under grant TEC2017-84743-P.Belda, J.; Vergara Domínguez, L.; Safont Armero, G.; Salazar Afanador, A.; Parcheta, Z. (2019). A New Surrogating Algorithm by the Complex Graph Fourier Transform (CGFT). Entropy. 21(8):1-18. https://doi.org/10.3390/e21080759S118218Schreiber, T., & Schmitz, A. (2000). Surrogate time series. Physica D: Nonlinear Phenomena, 142(3-4), 346-382. doi:10.1016/s0167-2789(00)00043-9Miralles, R., Vergara, L., Salazar, A., & Igual, J. (2008). Blind detection of nonlinearities in multiple-echo ultrasonic signals. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 55(3), 637-647. doi:10.1109/tuffc.2008.688Mandic, D. ., Chen, M., Gautama, T., Van Hulle, M. ., & Constantinides, A. (2008). 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On the Nature of X-ray Variability in Ark 564

We use data from a recent long ASCA observation of the Narrow Line Seyfert 1 Ark 564 to investigate in detail its timing properties. We show that a thorough analysis of the time series, employing techniques not generally applied to AGN light curves, can provide useful information to characterize the engines of these powerful sources.We searched for signs of non-stationarity in the data, but did not find strong evidences for it. We find that the process causing the variability is very likely nonlinear, suggesting that variability models based on many active regions, as the shot noise model, may not be applicable to Ark 564. The complex light curve can be viewed, for a limited range of time scales, as a fractal object with non-trivial fractal dimension and statistical self-similarity. Finally, using a nonlinear statistic based on the scaling index as a tool to discriminate time series, we demonstrate that the high and low count rate states, which are indistinguishable on the basis of their autocorrelation, structure and probability density functions, are intrinsically different, with the high state characterized by higher complexity.Comment: 13 pages, 13 figures, accepted for publication in A&

A simple method for detecting chaos in nature

Chaos, or exponential sensitivity to small perturbations, appears everywhere in nature. Moreover, chaos is predicted to play diverse functional roles in living systems. A method for detecting chaos from empirical measurements should therefore be a key component of the biologist's toolkit. But, classic chaos-detection tools are highly sensitive to measurement noise and break down for common edge cases, making it difficult to detect chaos in domains, like biology, where measurements are noisy. However, newer tools promise to overcome these limitations. Here, we combine several such tools into an automated processing pipeline, and show that our pipeline can detect the presence (or absence) of chaos in noisy recordings, even for difficult edge cases. As a first-pass application of our pipeline, we show that heart rate variability is not chaotic as some have proposed, and instead reflects a stochastic process in both health and disease. Our tool is easy-to-use and freely available

EEG-Based Quantification of Cortical Current Density and Dynamic Causal Connectivity Generalized across Subjects Performing BCI-Monitored Cognitive Tasks.

Quantification of dynamic causal interactions among brain regions constitutes an important component of conducting research and developing applications in experimental and translational neuroscience. Furthermore, cortical networks with dynamic causal connectivity in brain-computer interface (BCI) applications offer a more comprehensive view of brain states implicated in behavior than do individual brain regions. However, models of cortical network dynamics are difficult to generalize across subjects because current electroencephalography (EEG) signal analysis techniques are limited in their ability to reliably localize sources across subjects. We propose an algorithmic and computational framework for identifying cortical networks across subjects in which dynamic causal connectivity is modeled among user-selected cortical regions of interest (ROIs). We demonstrate the strength of the proposed framework using a "reach/saccade to spatial target" cognitive task performed by 10 right-handed individuals. Modeling of causal cortical interactions was accomplished through measurement of cortical activity using (EEG), application of independent component clustering to identify cortical ROIs as network nodes, estimation of cortical current density using cortically constrained low resolution electromagnetic brain tomography (cLORETA), multivariate autoregressive (MVAR) modeling of representative cortical activity signals from each ROI, and quantification of the dynamic causal interaction among the identified ROIs using the Short-time direct Directed Transfer function (SdDTF). The resulting cortical network and the computed causal dynamics among its nodes exhibited physiologically plausible behavior, consistent with past results reported in the literature. This physiological plausibility of the results strengthens the framework's applicability in reliably capturing complex brain functionality, which is required by applications, such as diagnostics and BCI

A Spectral Graph Uncertainty Principle

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral "spreads" are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within $\varepsilon$ by a fast approximation algorithm requiring $O(\varepsilon^{-1/2})$ typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.Comment: 40 pages, 8 figure