Structural Susceptibility and Separation of Time Scales in the van der Pol Oscillator

We use an extension of the van der Pol oscillator as an example of a system with multiple time scales to study the susceptibility of its trajectory to polynomial perturbations in the dynamics. A striking feature of many nonlinear, multi-parameter models is an apparently inherent insensitivity to large magnitude variations in certain linear combinations of parameters. This phenomenon of "sloppiness" is quantified by calculating the eigenvalues of the Hessian matrix of the least-squares cost function which typically span many orders of magnitude. The van der Pol system is no exception: Perturbations in its dynamics show that most directions in parameter space weakly affect the limit cycle, whereas only a few directions are stiff. With this study we show that separating the time scales in the van der Pol system leads to a further separation of eigenvalues. Parameter combinations which perturb the slow manifold are stiffer and those which solely affect the transients in the dynamics are sloppier.Comment: 7 pages, 4 figure

ALL RIGHTS RESERVEDSCIENCE IN HIGH DIMENSIONS: MULTIPARAMETER MODELS AND BIG DATA

Complex multiparameter models such as in climate science, economics, systems biology, materials science, neural networks and machine learning have a large-dimensional space of undetermined parameters as well as a large-dimensional space of predicted data. These high-dimensional spaces of inputs and outputs pose many challenges. Recent work with a diversity of nonlinear predictive models, microscopic models in physics, and analysis of large datasets, has led to important insights. In particular, it was shown that nonlinear fits to data in a variety of multiparameter models largely rely on only a few stiff directions in parameter space. Chapter 2 explores a qualitative basis for this compression of parameter space using a model nonlinear system with two time scales. A systematic separation of scales is shown to correspond to an increasing insensitivity of parameter space directions that only affect the fast dynamics. Chapter 3 shows with the help of microscopic physics models that emergent theories in physics also rely on a sloppy compression of the parameter space where macroscopically relevant variables form the stiff directions. Lastly, in chapter 4, we will learn that the data space of historical daily stock returns of US public companies has an emergent simplex structure that makes it amenable to a low-dimensional representation. This leads t

Uncertainty quantification for classical effective potentials

Effective potentials are an essential ingredient of classical molecular dynamics simulations. Little is understood of the errors incurred in representing the complex energy landscape of an atomic configuration by an effective potential containing considerably fewer parameters. This thesis details the introduction of an uncertainty quantification framework into the potential fitting process within the potfit force matching code. The probabilistic sloppy model method has been implemented within potfit as a means to quantify the uncertainties in analytic potential parameters, and in subsequent quantities measured using the fitted potential. Uncertainties in the effective potential are propagated through molecular dynamics simulations to obtain uncertainties in quantities of interest, which are a measure of the confidence in the model predictions. The implementation has been designed to fit flexibly within the existing potfit workflow, and is generalised to work with any potential model or material. The uncertainty quantification software contains a variety of controllable parameters, which provide the user with diagnostic capabilities to understand the nature of the fitting landscape defined by their potential model and reference data. The implementation is available for use by the materials modelling community as part of the open source potfit software. The uncertainty quantification technique is demonstrated using three potentials for nickel: two simple pair potentials, Lennard-Jones and Morse, and a local density dependent EAM potential. A sloppy model fit to ab initio reference data is constructed for each potential to calculate the uncertainties in lattice constants, elastic constants and thermal expansion. These can be used to show the unsuitability of pair potentials for nickel. In contrast, with EAM we observe a decreased uncertainty in the model predictions. This shows that our method can capture the effects of the error incurred in the potential generation process without resorting to comparison with experiment or ab inito calculations, which is an essential part to assess the predictive power of molecular dynamics simulations

Bard Observer, Vol. 14, No. 6 (May 18, 1971)

https://digitalcommons.bard.edu/observer/1154/thumbnail.jp

Current, September 02, 1982

https://irl.umsl.edu/current1980s/1068/thumbnail.jp

The role of extrinsic noise in biomolecular information processing systems: an in silico analysis

The intrinsic stochasticity of biomolecular systems is a well studied phe- nomenon. Less attention has been paied to other sources of variability, so called extrinsic noise. While the precise definition of extrinsic noise de- pends on the system in question, it affects all cells and its significance has been demonstrated experimentally. Information theory provides a rigorous mathematical framework for quan- tifying both the amount of information available to a signalling system and its ability to transmit this information. Intracellular signal transduction re- mains a relatively unexplored frontier for the application of information theory. In this thesis, we rely on a metric called mutual information to quantify in- formation flow in models of biochemical signalling systems. After briefly discussing the theoretical background and some of the practical difficulties of estimating mutual information in Chapter 2, we apply it in the context of simplified models of intracellular signalling, referred to as motifs. Using a comprehensive set of two-node motifs we explore the effects of extrin- sic noise, model parameters and various combinations of interaction, on the system’s ability to transmit information about an input signal, repre- sented by a telegraph process. Our results illustrate the importance of the system’s response time and demonstrate a trade-off in transmitting infor- mation about the current state of the input or its average intensity over a period of time. In Chapter 4, we address the problem of determining the magnitude of ex- trinsic noise in the presence of intrinsic stochasticity. Using the Approxi- mate Bayesian Computation - sequential Monte Carlo algorithm, together with published experimental data, we infer parameters describing extrinsic noise in a model of E. coli gene expression. Lastly, in Chapter 5, we construct and analyse models of bacterial two- component signalling, bringing together insights gleaned from earlier work. The results show how the abundances of different molecular species in the system may transmit information about the input signal despite its stochas-tic nature and considerable variation in the numbers of protein molecules present.Open Acces

Sweetwater Blues

A novel, Sweetwater Blues, by the author