A Posteriori Error Analysis of hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic Problems

We develop the a-posteriori error analysis of hp-version interior-penalty discontinuous Galerkin finite element methods for a class of second-order quasilinear elliptic partial differential equations. Computable upper and lower bounds on the error are derived in terms of a natural (mesh-dependent) energy norm. The bounds are explicit in the local mesh size and the local degree of the approximating polynomial. The performance of the proposed estimators within an automatic hp-adaptive refinement procedure is studied through numerical experiments.

Toward blueprints for network architecture, biophysical dynamics, and signal transduction

We review mathematical aspects of biophysical dynamics, signal transduction and network architecture that have been used to uncover functionally significant relations between the dynamics of single neurons and the networks they compose. We focus on examples that combine insights from these three areas to expand our understanding of systems neuroscience. These range from single neuron coding to models of decision making and electrosensory discrimination by networks and populations, as well as coincidence detection in pairs of dendrites and the dynamics of large networks of excitable dendritic spines. We conclude by describing some of the challenges that lie ahead as the applied mathematics community seeks to provide the tools that will ultimately underpin systems neuroscience.

Chiral polymerisation and the RNA world

The purpose of this paper is to review two mathematical models: one for the formation of homochiral polymers from an originally chirally symmetric system; and the other, to show how, in an RNA-world scenario, RNA can simultaneously act both as information storage and a catalyst for its own production. We note the similarities and differences in chemical mechanisms present in the systems. We review these two systems, analysing steady-states, interesting kinetics and the stability of symmetric solutions. In both systems we show that there are ranges of parameter values where some chains increase their own concentrations faster than others.

The Role of Cannabinoids in the Neurobiology of Sensory Gating: A firing rate model study

Gating of sensory (e.g. auditory) information has been demonstrated as a reduction in the auditory-evoked potential responses recorded in the brain of both normal animals and human subjects. Auditory gating is perturbed in schizophrenic patients and pharmacologically by drugs such as amphetamine, phencyclidine or ketamine, which precipitate schizophrenic-like symptoms in normal subjects. The neurobiological basis underlying this sensory gating can be investigated using local field potential recordings from single electrodes. In this paper we use such technology to investigate the role of cannabinoids in sensory gating. Cannabinoids represent a fundamentally new class of retrograde messengers which are released postsynaptically and bind to presynaptic receptors. In this way they allow fine-tuning of neuronal response, and in particular can lead to so-called depolarization-induced suppression of inhibition (DSI). Our experimental results show that application of the exogenous cannabinoid WIN55, 212-2 can abolish sensory gating as measured by the amplitude of local field responses in rat hippocampal region CA3. Importantly we develop a simple firing rate population model of CA3 and show that gating is heavily dependent upon the presence of a slow inhibitory (GABAB) pathway. Moreover, a simple phenomenological model of cannabinoid dynamics underlying DSI is shown to abolish gating in a manner consistent with our experimental findings.

Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes

In this paper we consider the a posteriori and a priori error analysis of discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. Based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement. The theoretical results are illustrated by a series of numerical experiments.

Vitamins Reverse Endothelial Dysfunction Through Regulation of eNOS and NAD(P)H Oxidase Activities

Antioxidant vitamins C and E have protective properties in genetic hypertension associated with enhanced oxidative stress. This study investigated whether vitamins C and/or E modulate vascular function by regulating enzymatic activities of endothelial nitric oxide synthase (eNOS) and NAD(P)H oxidase using thoracic aortas of 20- to 22-week-old male spontaneously hypertensive rats (SHR) and their matched normotensive counterparts, Wistar-Kyoto rats (WKY). SHR aortas had impaired relaxant responses to acetylcholine but not to sodium nitroprusside, despite an 2-fold increase in eNOS activity and NO release. The levels of superoxide anion (O2 ), a potent NO scavenger, and NAD(P)H oxidase activity were also 2-fold higher in SHR aortas. Mechanical but not pharmacological inactivation of endothelium (by rubbing and 100 mol/L L-NAME, respectively) significantly abrogated O2 in both strains. Treatments of SHR aortas with NAD(P)H oxidase inhibitors, namely diphenyleneiodinium and apocynin, significantly diminished O2 production. The incubation of SHR aortas with different concentrations of vitamin C (10 to 100 mol/L) and specifically with high concentrations of vitamin E (100 mol/L) improved endothelial function, reduced superoxide production as well as NAD(P)H oxidase activity, and increased eNOS activity and NO generation in SHR aortas to the levels observed in vitamin C- and E-treated WKY aortas. Our results reveal endothelial NAD(P)H oxidase as the major source of vascular O2 in SHR and also show that vitamins C and E are critical in normalizing genetic endothelial dysfunction through regulation of eNOS and NAD(P)H oxidase activities.

Molecular Characterization and Localization of the NAD(P)H Oxidase Components gp91-phox and p22-phox in Endothelial Cells

The production of reactive oxygen species (ROS) within endothelial cells may have several effects, including alterations in the activity of paracrine factors, gene expression, apoptosis, and cellular injury. Recent studies indicate that a phagocyte-type NAD(P)H oxidase is a major source of endothelial ROS. In contrast to the high-output phagocytic oxidase, the endothelial enzyme has much lower biochemical activity and a different substrate specificity (NADH.NADPH). In the present study, we (1) cloned and characterized the cDNA and predicted amino acid structures of the 2 major subunits of rat coronary microvascular endothelial cell NAD(P)H oxidase, gp91-phox and p22-phox; (2) undertook a detailed comparison with phagocytic NADPH oxidase sequences; and (3) studied the subcellular location of these subunits in endothelial cells. Although these studies revealed an overall high degree of homology (.90%) between the endothelial and phagocytic oxidase subunits, the endothelial gp91-phox sequence has potentially important differences in a putative NADPH-binding domain and in putative glycosylation sites. In addition, the subcellular location of the endothelial gp91-phox and p22-phox subunits is significantly different from that reported for the neutrophil oxidase, in that they are predominantly intracellular and collocated in the vicinity of the endoplasmic reticulum. This first detailed characterization of gp91-phox and p22-phox structure and location in endothelial cells provides new data that may account, in part, for the differences in function between the phagocytic and endothelial NAD(P)H oxidases.

Discontinuous Galerkin Methods on hp-Anisotropic Meshes I: A Priori Error Analysis

We consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local finite element spaces employed. In particular, we prove a priori hp-error bounds for linear target functionals of the solution, on (possibly) anisotropic computational meshes with anisotropic tensor-product polynomial basis functions. The theoretical results are illustrated by a numerical experiment.