Building Models of Functional Interactions Among Brain Domains that Encode Varying Information Complexity: A Schizophrenia Case Study

ABSTRACTRevealing associations among various structural and functional patterns of the brain can yield highly informative results about the healthy and disordered brain. Studies using neuroimaging data have more recently begun to utilize the information within as well as across various functional and anatomical domains (i.e., groups of brain networks). However, most whole-brain approaches assume similar complexity of interactions throughout the brain. Here we investigate the hypothesis that interactions between brain networks capture varying amounts of complexity, and that we can better capture this information by varying the complexity of the model subspace structure based on available training data. To do this, we employ a Bayesian optimization-based framework known as the Tree Parzen Estimator (TPE) to identify, exploit and analyze patterns of variation in the information encoded by temporal information extracted from functional magnetic resonance imaging (fMRI) subdomains of the brain. Using a repeated cross-validation procedure on a schizophrenia classification task, we demonstrate evidence that interactions between specific functional subdomains are better characterized by more sophisticated model architectures compared to less complicated ones required by the others for optimally contributing towards classification and understanding the brain’s functional interactions. We show that functional subdomains known to be involved in schizophrenia require more complex architectures to optimally unravel discriminatory information about the disorder. Our study points to the need for adaptive, hierarchical learning frameworks that cater differently to the features from different subdomains, not only for a better prediction but also for enabling the identification of features predicting the outcome of interest.

A periodically varying code for improving deblending of simultaneous sources in marine acquisition

We have designed a periodically varying code that can avoid the problem of the local coherency and make the interference distribute uniformly in a given range; hence, it was better at suppressing incoherent interference (blending noise) and preserving coherent useful signals compared with a random dithering code. We have also devised a new form of the iterative method to remove interference generated from the simultaneous source acquisition. In each iteration, we have estimated the interference using the blending operator following the proposed formula and then subtracted the interference from the pseudodeblended data. To further eliminate the incoherent interference and constrain the inversion, the data were then transformed to an auxiliary sparse domain for applying a thresholding operator. During the iterations, the threshold was decreased from the largest value to zero following an exponential function. The exponentially decreasing threshold aimed to gradually pass the deblended data to a more acceptable model subspace. Two numerically blended synthetic data sets and one numerically blended practical field data set from an ocean bottom cable were used to demonstrate the usefulness of our proposed method and the better performance of the periodically varying code over the traditional random dithering code.

Composition of Inner Functions

We study the image of the model subspace Kθ under the composition operator C, where and θ are inner functions, and find the smallest model subspace which contains the linear manifold CKθ. Then we characterize the case when C maps Kθ into itself. This case leads to the study of the inner functions and ψ such that the composition ψ ∘ is a divisor of ψ in the family of inner functions.

INTEGRAL MEANS OF THE DERIVATIVES OF BLASCHKE PRODUCTS

We study the rate of growth of some integral means of the derivatives of a Blaschke product and we generalize several classical results. Moreover, we obtain the rate of growth of integral means of the derivative of functions in the model subspaceKBgenerated by the Blaschke productB.

Admissible Majorants for Model Subspaces of H2, Part I: Slow Winding of the Generating Inner Function

A model subspace Kϴ of the Hardy space H2 = H2(ℂ+) for the upper half plane ℂ+ is H2(ℂ+) ϴ ϴH2(ℂ+) where ϴ is an inner function in ℂ+. A function ω: ⟼ [0,∞) is called an admissible majorant for Kϴ if there exists an f ∈ Kϴ, f ≢ 0, |f(x)| ≤ ω(x) almost everywhere on ℝ. For some (mainly meromorphic) ϴ's some parts of Adm ϴ (the set of all admissible majorants for Kϴ) are explicitly described. These descriptions depend on the rate of growth of argϴ along ℝ. This paper is about slowly growing arguments (slower than x). Our results exhibit the dependence of Adm B on the geometry of the zeros of the Blaschke product B. A complete description of Adm B is obtained for B's with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.