Three Discontinuous Galerkin Methods for One- and Two-Dimensional Nonlinear Dirac Equations with a Scalar Self-Interaction

This paper develops three high-order accurate discontinuous Galerkin (DG) methods for the one-dimensional (1D) and two-dimensional (2D) nonlinear Dirac (NLD) equations with a general scalar self-interaction. They are the Runge-Kutta DG (RKDG) method and the DG methods with the one-stage fourth-order Lax-Wendroff type time discretization (LWDG) and the two-stage fourth-order accurate time discretization (TSDG). The RKDG method uses the spatial DG approximation to discretize the NLD equations and then utilize the explicit multistage high-order Runge-Kutta time discretization for the first-order time derivatives, while the LWDG and TSDG methods, on the contrary, first give the one-stage fourth-order Lax-Wendroff type and the two-stage fourth-order time discretizations of the NLD equations, respectively, and then discretize the first- and higher-order spatial derivatives by using the spatial DG approximation. The $L^2$ stability of the 2D semi-discrete DG approximation is proved in the RKDG methods for a general triangulation, and the computational complexities of three 1D DG methods are estimated. Numerical experiments are conducted to validate the accuracy and the conservation properties of the proposed methods. The interactions of the solitary waves, the standing and travelling waves are investigated numerically and the 2D breathing pattern is observed.     

胸主动脉数值模拟与数值模型比较

目的探究血管在不同情况下的性质。方法用有限元方法模拟建立胸主动脉直管段血管的线弹性小变形、线弹性大变形、非线性弹性和考虑动脉壁的残余应力因素的数值模型,分析了不同模型下动脉壁的应力分布和位移分布等力学性质,并对这几种模型进行讨论和比较。结果数值分析得出残余应力改变了周向应力的分布,没有改变周向应力水平,残余应力对血管的位移没有明显影响。结论本次研究得出考虑残余应力作用后,血管内壁的应力集中明显降低。     

Deterministic chaos and noise in three in vitro hippocampal models of epilepsy

Recent reports have suggested that chaos control techniques may be useful for electrically manipulating epileptiform bursting behavior in neuronal ensembles. Because the dynamics of spontaneous in vitro bursting had not been well determined previously, analysis of this behavior in the rat hippocampus was performed. Epileptiform bursting was induced in transverse rat hippocampal slices using three experimental methods. Slices were bathed in artificial cerebrospinal fluid containing: (1) elevated potassium ([K⁺]_o=10.5 mM), (2) zero magnesium, or (3) the GABA_A-receptor antagonists bicuculline (20 M) and picrotoxin (250 M). The existence of chaos and determinism was assessed using two different analytical techniques: unstable periodic orbit (UPO) analysis and a new technique for estimating Lyapunov exponents. Significance of these results was assessed by comparing the calculations for each experiment with corresponding randomized surrogate data. UPOs of multiple periods were highly prevalent in experiments from all three epilepsy models: 73% of all experiments contained at least one statistically significant period-1 or period-2 orbit. However, the expansion rate analysis did not provide any evidence of determinism in the data. This suggests that the system may be globally stochastic but contains local pockets of determinism. Thus, manipulation of bursting behavior using chaos control algorithms may yet hold promise for reverting or preventing epileptic seizures. © 2001 Biomedical Engineering Society. PAC01: 8719Nn, 8719Xx, 0545Gg, 8717Nn, 8719La     

血流的非线性因素与血流波的畸变

本文用血流波形畸变的实测资料和非线性动力学理论分析得出了血流波形畸变是较普遍的现象,其畸变的原因主要是非线性因素。     

Prediction of the concentrations in rat tissues of the intravenous anaesthetic methohexital from a nonlinear pharmacokinetic model

A nonlinear pharmacokinetic model that describes the tissue distribution of intravenous anaesthetics was evaluated against experimental values for methohexital in the rat. There was excellent agreement between experimental and theoretical values for brain tissue, and good agreement for blood and adipose tissue. Agreement for lean tissue was good if it was assumed that some adipose tissue was present in skeletal muscle. Agreement was poor for all other visceral tissues. The experimental results justify further development of this mathematical model for use in accounting for differences in tissue distribution of anaesthetics, especially under various physiological conditions.     

Improving nonlinear modeling capabilities of functional link adaptive filters

The functional link adaptive filter (FLAF) represents an effective solution for online nonlinear modeling problems. In this paper, we take into account a FLAF-based architecture, which separates the adaptation of linear and nonlinear elements, and we focus on the nonlinear branch to improve the modeling performance. In particular, we propose a new model that involves an adaptive combination of filters downstream of the nonlinear expansion. Such combination leads to a cooperative behavior of the whole architecture, thus yielding a performance improvement, particularly in the presence of strong nonlinearities. An advanced architecture is also proposed involving the adaptive combination of multiple filters on the nonlinear branch. The proposed models are assessed in different nonlinear modeling problems, in which their effectiveness and capabilities are shown.     

ACL deficiency affects stride-to-stride variability as measured using nonlinear methodology

Previous studies suggested that the small fluctuations present in movement patterns from one stride to the next during walking can be useful in the investigation of various pathological conditions. Previous studies using nonlinear measures have resulted in the development of the “loss of complexity hypothesis” which states that disease can affect the variability and decrease the complexity of a system, rendering it less able to adjust to the ever changing environmental demands. The nonlinear measure of the Lyapunov Exponent (LyE) has already been used for the assessment of stride-to-stride variability in the anterior cruciate ligament (ACL) deficient knee in comparison to the contralateral intact knee. However, there is biomechanical evidence that after ACL rupture, adaptations are also present in the contralateral intact knee. Thus, our goal was to investigate stride-to-stride variability in the ACL deficient knee as compared to a healthy control knee. Seven subjects with unilateral ACL deficiency and seven healthy controls walked at their self-selected speed on a treadmill, while three-dimensional knee kinematics was collected for 80 consecutive strides. A nonlinear measure, the largest LyE was calculated from the resulted knee joint flexion-extension data of both groups. Larger LyE values signify increased variability and increased sensitivity to initial conditions. Our results showed that the ACL deficient group exhibited significantly less variable walking patterns than the healthy control. These changes are not desirable because they reflect decreases in system’s complexity, which indicates narrowed functional responsiveness, according to the “loss of complexity hypothesis.” This may be related with the increased future pathology found in ACL deficient patients. The methods used in the present paper showed great promise to assess the gait handicap in knee injured patients.     

The NIG‐S&ARCH model: a fat‐tailed,stochastic, and autoregressive conditional heteroskedastic volatility model

This paper examines the capabilities of the Normal Inverse Gaussian distribution as a model for stock returns. We extend the model of Barndorff‐Nielsen (1997) to allow for a richer volatility structure and compare with the existing GARCH‐type models. We conclude that the proposed model outperforms some of the most praised GARCH‐M models. In particular, we make a big gain in modelling the skewness of equity returns.     

Effect of gap-filling technique and gap location on linear and nonlinear calculations of motion during locomotor activities

BackgroundMarker occlusion during camera-based movement analysis is common. Different interpolation techniques are available for estimating location of missing marker trajectories.Research questionWhat is the effect of gap location and interpolation technique on linear and nonlinear measures for a given kinematic time series?MethodsKinematic data were recorded during motor-assisted elliptical training and treadmill walking. Gap-filling techniques (i.e., Cubic, Makima, Autoregressive, Nearest Neighbor, and No Interpolation) and gap locations experimentally applied to each cycle across initially complete time series (Gap 1: local minimum and maximum peaks; Gap 2: maximum peaks; Gap 3: maximum peaks at negative slope; Gap 4: random locations) were examined during linear (Maxima and Minima joint angles) and nonlinear [maximum Lyapunov exponent (LyE)] measures.ResultsGap-filling technique and gap location influenced values calculated for linear and nonlinear measures of joint motions. When referenced to the gold standard (original data series without gaps), across all joints studied the average % error of Maxima and Minima joint angles and LyE % error were lower when applying Cubic, Makima, Autoregressive, and Nearest Neighbor techniques compared to No Interpolation (p < 0.0001). The % error of Maxima joint angles was lower for Gaps 1, 3, and 4 compared to Gap 2 (p = 0.0003), while % error of Minima joint angles was lower for Gaps 2 and 3, compared to Gaps 1 and 4 (p < 0.0001). An interaction between gap-filling technique and gap location was identified for LyE % error, in which Gap 4 % error was significantly greater during No Interpolation compared to other gap-filling techniques (p < 0.0001).SignificanceFindings can guide selection of appropriate techniques to manage missing kinematic data points in camera-based motion analysis time series. Gap-filling techniques significantly reduced error in calculating select linear and nonlinear measures of variability, with Cubic most consistently resulting in the greatest reduction in error.     

Integration of Pharmacometric and Statistical Analyses Using Clinical Trial Simulations to Enhance Quantitative Decision Making in Clinical Drug Development

This article outlines a general framework in which clinical trial simulations (CTS) are employed integrating both pharmacometric and statistical analyses to support trial design and quantitative decision making in drug development. Specifically, predictive pharmacometric models are used as data-generation models to simulate data, while data-analytic models as specified in the statistical analysis plan are used to analyze the simulated data and to apply a quantitative data-analytic decision rule. Various probability metrics including probability of achieving the target value, probability of success, and probability of a correct decision are proposed to support study design recommendations and quantitative decision-making. A case study is presented to illustrate the CTS methods and procedures described in this article.