412 research outputs found
Dynamic Decomposition of Spatiotemporal Neural Signals
Neural signals are characterized by rich temporal and spatiotemporal dynamics
that reflect the organization of cortical networks. Theoretical research has
shown how neural networks can operate at different dynamic ranges that
correspond to specific types of information processing. Here we present a data
analysis framework that uses a linearized model of these dynamic states in
order to decompose the measured neural signal into a series of components that
capture both rhythmic and non-rhythmic neural activity. The method is based on
stochastic differential equations and Gaussian process regression. Through
computer simulations and analysis of magnetoencephalographic data, we
demonstrate the efficacy of the method in identifying meaningful modulations of
oscillatory signals corrupted by structured temporal and spatiotemporal noise.
These results suggest that the method is particularly suitable for the analysis
and interpretation of complex temporal and spatiotemporal neural signals
On dynamical low-rank integrators for matrix differential equations
This thesis is concerned with dynamical low-rank integrators for matrix differential equations, typically stemming from space discretizations of partial differential equations. We first construct and analyze a dynamical low-rank integrator for second-order matrix differential equations, which is based on a Strang splitting and the projector-splitting integrator, a dynamical low-rank integrator for first-order matrix
differential equations proposed by Lubich and Osedelets in 2014. For the analysis, we derive coupled recursive inequalities, where we express the global error of the scheme in terms of a time-discretization error and a low-rank error contribution. The first can be treated with Taylor series expansion of the exact solution. For the latter, we make use of an induction argument and the convergence result derived by Kieri, Lubich, and Walach in 2016 for the projector-splitting integrator.
From the original method, several variants are derived which are tailored to, e.g., stiff or highly oscillatory second-order problems. After discussing details on the implementation of dynamical low-rank schemes, we turn towards rank-adaptivity. For the projector-splitting integrator we derive both a technique to realize changes in the approximation ranks efficiently and a heuristic to choose the rank appropriately over time. The core idea is to determine the rank such that the error of the low-rank
approximation does not spoil the time-discretization error. Based on the rank-adaptive pendant of the projector-splitting integrator, rank-adaptive dynamical low-rank integrators for (stiff and non-stiff) first-order and second-order matrix differential equations are derived. The thesis is concluded with numerical experiments to confirm our theoretical findings
WavePacket: A Matlab package for numerical quantum dynamics. II: Open quantum systems, optimal control, and model reduction
WavePacket is an open-source program package for numeric simulations in
quantum dynamics. It can solve time-independent or time-dependent linear
Schr\"odinger and Liouville-von Neumann-equations in one or more dimensions.
Also coupled equations can be treated, which allows, e.g., to simulate
molecular quantum dynamics beyond the Born-Oppenheimer approximation.
Optionally accounting for the interaction with external electric fields within
the semi-classical dipole approximation, WavePacket can be used to simulate
experiments involving tailored light pulses in photo-induced physics or
chemistry. Being highly versatile and offering visualization of quantum
dynamics 'on the fly', WavePacket is well suited for teaching or research
projects in atomic, molecular and optical physics as well as in physical or
theoretical chemistry. Building on the previous Part I which dealt with closed
quantum systems and discrete variable representations, the present Part II
focuses on the dynamics of open quantum systems, with Lindblad operators
modeling dissipation and dephasing. This part also describes the WavePacket
function for optimal control of quantum dynamics, building on rapid
monotonically convergent iteration methods. Furthermore, two different
approaches to dimension reduction implemented in WavePacket are documented
here. In the first one, a balancing transformation based on the concepts of
controllability and observability Gramians is used to identify states that are
neither well controllable nor well observable. Those states are either
truncated or averaged out. In the other approach, the H2-error for a given
reduced dimensionality is minimized by H2 optimal model reduction techniques,
utilizing a bilinear iterative rational Krylov algorithm
MATLAB
A well-known statement says that the PID controller is the "bread and butter" of the control engineer. This is indeed true, from a scientific standpoint. However, nowadays, in the era of computer science, when the paper and pencil have been replaced by the keyboard and the display of computers, one may equally say that MATLAB is the "bread" in the above statement. MATLAB has became a de facto tool for the modern system engineer. This book is written for both engineering students, as well as for practicing engineers. The wide range of applications in which MATLAB is the working framework, shows that it is a powerful, comprehensive and easy-to-use environment for performing technical computations. The book includes various excellent applications in which MATLAB is employed: from pure algebraic computations to data acquisition in real-life experiments, from control strategies to image processing algorithms, from graphical user interface design for educational purposes to Simulink embedded systems
The Sixth Copper Mountain Conference on Multigrid Methods, part 1
The Sixth Copper Mountain Conference on Multigrid Methods was held on 4-9 Apr. 1993, at Copper Mountain, CO. This book is a collection of many of the papers presented at the conference and as such represents the conference proceedings. NASA LaRC graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth
Harmonized-Multinational qEEG Norms (HarMNqEEG)
This paper extends the frequency domain quantitative electroencephalography (qEEG) methods pursuing higher sensitivity to detect Brain Developmental Disorders. Prior qEEG work lacked integration of cross-spectral information omitting important functional connectivity descriptors. Lack of geographical diversity precluded accounting for site-specific variance, increasing qEEG nuisance variance. We ameliorate these weaknesses. i) Create lifespan Riemannian multinational qEEG norms for cross-spectral tensors. These norms result from the HarMNqEEG project fostered by the Global Brain Consortium. We calculate the norms with data from 9 countries, 12 devices, and 14 studies, including 1564 subjects. Instead of raw data, only anonymized metadata and EEG cross-spectral tensors were shared. After visual and automatic quality control, developmental equations for the mean and standard deviation of qEEG traditional and Riemannian DPs were calculated using additive mixed-effects models. We demonstrate qEEG "batch effects" and provide methods to calculate harmonized z-scores. ii) We also show that the multinational harmonized Riemannian norms produce z-scores with increased diagnostic accuracy to predict brain dysfunction at school-age produced by malnutrition only in the first year of life. iii) We offer open code and data to calculate different individual z-scores from the HarMNqEEG dataset. These results contribute to developing bias-free, low-cost neuroimaging technologies applicable in various health settings
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