PET imaging of pathological tau correlates more closely with Alzheimer’s disease–related cognitive impairment than does imaging of β-amyloid.

Measurement of correlations between brain regions (functional connectivity) using blood oxygen level dependent (BOLD) fMRI has proven to be a powerful tool for studying the functional organization of the brain. Recently, dynamic functional connectivity has emerged as a major topic in the resting-state BOLD fMRI literature. Here, using simulations and multiple sets of empirical observations, we confirm that imposed task states can alter the correlation structure of BOLD activity. However, we find that observations of “dynamic” BOLD correlations during the resting state are largely explained by sampling variability. Beyond sampling variability, the largest part of observed “dynamics” during rest is attributable to head motion. An additional component of dynamic variability during rest is attributable to fluctuating sleep state. Thus, aside from the preceding explanatory factors, a single correlation structure—as opposed to a sequence of distinct correlation structures—may adequately describe the resting state as measured by BOLD fMRI. These results suggest that resting-state BOLD correlations do not primarily reflect moment-to-moment changes in cognitive content. Rather, resting-state BOLD correlations may predominantly reflect processes concerned with the maintenance of the long-term stability of the brain's functional organization.

Project Euclid - mathematics and statistics online

We define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.

We give necessary and sufficient conditions to solve an interpolation problem for free holomorphic functions bounded in norm on a free polynomial polyhedron. As an application, we prove that every bounded holomorphic function on a polynomial polyhedron extends to a bounded free function.