Random matrix theory tools for the predictive analysis of functional magnetic resonance imaging examinations.

Purpose: Random matrix theory (RMT) is an increasingly useful tool for understanding large, complex systems. Prior studies have examined functional magnetic resonance imaging (fMRI) scans using tools from RMT, with some success. However, RMT computations are highly sensitive to a number of analytic choices, and the robustness of findings involving RMT remains in question. We systematically investigate the usefulness of RMT on a wide variety of fMRI datasets using a rigorous predictive framework. Approach: We develop open-source software to efficiently compute RMT features from fMRI images and examine the cross-validated predictive potential of eigenvalue and RMT-based features ("eigenfeatures") with classic machine-learning classifiers. We systematically vary pre-processing extent, normalization procedures, RMT unfolding procedures, and feature selection and compare the impact of these analytic choices on the distributions of cross-validated prediction performance for each combination of dataset binary classification task, classifier, and feature. To deal with class imbalance, we use the area under the receiver operating characteristic curve (AUROC) as the main performance metric. Results: Across all classification tasks and analytic choices, we find RMT- and eigenvalue-based "eigenfeatures" to have predictive utility more often than not (82.4% of median AUROCs>0.5; median AUROC range across classification tasks 0.47 to 0.64). Simple baseline reductions on source timeseries, by contrast, were less useful (58.8% of median AUROCs>0.5, median AUROC range across classification tasks 0.42 to 0.62). Additionally, eigenfeature AUROC distributions were overall more right-tailed than baseline features, suggesting greater predictive potential. However, performance distributions were wide and often significantly affected by analytic choices. Conclusions: Eigenfeatures clearly have potential for understanding fMRI functional connectivity in a wide variety of scenarios. The utility of these features is strongly dependent on analytic decisions, suggesting caution when interpreting past and future studies applying RMT to fMRI. However, our study demonstrates that the inclusion of RMT statistics in fMRI investigations could improve prediction performances across a wide variety of phenomena.

Spatial correlation of the dynamic propensity of a glass-forming liquid.

We present computer simulation results on the dynamic propensity (as defined by Widmer-Cooper et al 2004 Phys. Rev. Lett. 93 135701) in a Kob-Andersen binary Lennard-Jones liquid system consisting of 8788 particles. We compute the spatial correlation function for the dynamic propensity as a function of both the reduced temperature T, and the time scale on which the particle displacements are measured. For T ≤ 0.6, we find that non-zero correlations occur at the largest length scale accessible in our system. We also show that a cluster-size analysis of particles with extremal values of the dynamic propensity, as well as 3D visualizations, reveal spatially correlated regions that approach the size of our system as T decreases, consistently with the behavior of the spatial correlation function. Next, we define and examine the 'coordination propensity', the isoconfigurational average of the coordination number of the minority B particles around the majority A particles. We show that a significant correlation exists between the spatial fluctuations of the dynamic and coordination propensities. In addition, we find non-zero correlations of the coordination propensity occurring at the largest length scale accessible in our system for all T in the range 0.466 < T < 1.0. We discuss the implications of these results for understanding the length scales of dynamical heterogeneity in glass-forming liquids.

Spectral statistics of the quenched normal modes of a network-forming molecular liquid.

We evaluate the density of states of the quenched normal modes of ST2 water, and their statistical fluctuations, for a range of densities spanning three regimes of behavior of a hydrogen bonded liquid: a lower-density regime of random tetrahedral network formation, in the vicinity of a liquid-liquid critical point, and in a higher-density regime of fragile glass-forming behavior. For all cases we find that the fluctuations around the mean spectral densities obey the predictions of the Gaussian orthogonal ensemble of random matrix theory. We also measure the participation ratio of the normal modes across the entire frequency range, and find behavior consistent with the majority of modes being of an extended nature, rather than localized.

Universality in the vibrational spectra of weakly-disordered two-dimensional clusters.

We numerically investigate the vibrational spectra of single-component clusters in two dimensions. Stable configurations of clusters at local energy minima are obtained, and for each the Hessian matrix is evaluated and diagonalized to obtain eigenvalues as well as eigenvectors. We study the density of states so obtained as a function of the width of the potential well describing the two-body interaction. As the width is reduced, as in three dimensions, we find that the density of states approaches a common form but the two-peak behavior survives. Further, calculations of the participation ratio show that most states are extended, although a smaller fraction of the degrees of freedom are involved in these modes compared to three dimensions. We show that the fluctuation properties of these modes converge to those of the Gaussian orthogonal ensemble of random matrices, in common with previous results on three-dimensional amorphous clusters and molecular liquids.

Structural and dynamical heterogeneity in a glass-forming liquid.

We use the "isoconfigurational ensemble" [Phys. Rev. Lett. 93, 135701 (2004)] to analyze both dynamical and structural properties in simulations of a glass-forming molecular liquid. We show that spatially correlated clusters of low-potential-energy molecules are observable on the time scale of structural relaxation, despite the absence of spatial correlations of potential energy in the instantaneous structure of the system. We find that these structural heterogeneities correlate with dynamical heterogeneities in the form of clusters of low molecular mobility.

Universality in the vibrational spectra of single-component amorphous clusters.

We have numerically investigated the vibrational spectra of amorphous single-component clusters for several types of interactions among the particles. For all the potentials we have studied, we find that the density of states can be described, except at the two ends of the spectrum, by the same functional form to a very good approximation, and that the fluctuation properties of the spectra in this central region converge to those of the Gaussian orthogonal ensemble of random matrices with increasing system size. We conjecture that this scenario is true for a broad class of potentials.