A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Regulation of early steps of GPVI signal transduction by phosphatases: a systems biology approach

We present a data-driven mathematical model of a key initiating step in platelet activation, a central process in the prevention of bleeding following Injury. In vascular disease, this process is activated inappropriately and causes thrombosis, heart attacks and stroke. The collagen receptor GPVI is the primary trigger for platelet activation at sites of injury. Understanding the complex molecular mechanisms initiated by this receptor is important for development of more effective antithrombotic medicines. In this work we developed a series of nonlinear ordinary differential equation models that are direct representations of biological hypotheses surrounding the initial steps in GPVI-stimulated signal transduction. At each stage model simulations were compared to our own quantitative, high-temporal experimental data that guides further experimental design, data collection and model refinement. Much is known about the linear forward reactions within platelet signalling pathways but knowledge of the roles of putative reverse reactions are poorly understood. An initial model, that includes a simple constitutively active phosphatase, was unable to explain experimental data. Model revisions, incorporating a complex pathway of interactions (and specifically the phosphatase TULA-2), provided a good description of the experimental data both based on observations of phosphorylation in samples from one donor and in those of a wider population. Our model was used to investigate the levels of proteins involved in regulating the pathway and the effect of low GPVI levels that have been associated with disease. Results indicate a clear separation in healthy and GPVI deficient states in respect of the signalling cascade dynamics associated with Syk tyrosine phosphorylation and activation. Our approach reveals the central importance of this negative feedback pathway that results in the temporal regulation of a specific class of protein tyrosine phosphatases in controlling the rate, and therefore extent, of GPVI-stimulated platelet activation.

Optimal control of nonlinear differential algebraic equation systems

A novel iterative procedure is described for solving nonlinear optimal control problems subject to differential algebraic equations. The procedure iterates on an integrated modified linear quadratic model based problem with parameter updating in such a manner that the correct solution of the original non-linear problem is achieved. The resulting algorithm has a particular advantage in that the solution is achieved without the need to solve the differential algebraic equations . Convergence aspects are discussed and a simulation example is described which illustrates the performance of the technique. 1. Introduction When modelling industrial processes often the resulting equations consist of coupled differential and algebraic equations (DAEs). In many situations these equations are nonlinear and cannot readily be directly reduced to ordinary differential equations.

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

Searching efficiency and the functional response of a pause-travel forager

1. The feeding rates of many predators and parasitoids exhibit type II functional responses, with a decelerating rate of increase to reach an asymptotic value as the density of their prey or hosts increases. Holling's disc equation describes such relationships and predicts that the asymptotic feeding rate at high prey densities is set by handling time, while the rate at which feeding rate increases with increased prey density is determined by searching efficiency. Searching efficiency and handling time are also parameters in other models which describe the functional response. Models which incorporate functional responses in order to make predictions of the effects of food shortage thus rely upon a clear understanding and accurate quantification of searching efficiency and handling time. 2. Blackbird Turdus merula exhibit a type II functional response and use pause-travel foraging, a foraging technique in which animals search for prey while stationary and then move to capture prey. Pause-travel foraging allows accurate direct measurement of feeding rate and both searching efficiency and handling time. We use Blackbirds as a model species to: (i) compare observed measures of both searching efficiency and handling time with those estimated by statistically fitting the disc equation to the observed functional response; and (ii) investigate alternative measures of searching efficiency derived by the established method where search area is assumed to be circular and a new method that we propose where it is not. 3. We find that the disc equation can adequately explain the functional response of blackbirds feeding on artificial prey. However, this depends critically upon how searching efficiency is measured. Two variations on the previous method of measuring search area (a component of searching efficiency) overestimated searching efficiency, and hence predicted feeding rates higher than those observed. Two variations of our alternative approach produced lower estimates of searching efficiency, closer to that estimated by fitting the disc equation, and hence more accurately predicted feeding rate. Our study shows the limitations of the previous method of measuring searching efficiency, and describes a new method for measuring searching efficiency more accurately.

Measuring the functional responses of farmland birds: an example for a declining seed-feeding bunting

1. Many farmland bird species have undergone significant declines. It is important to predict the effect of agricultural change on these birds and their response to conservation measures. This requirement could be met by mechanistic models that predict population size from the optimal foraging behaviour and fates of individuals within populations. A key component of these models is the functional response, the relationship between food and competitor density and feeding rate. 2. This paper describes a method for measuring functional responses of farmland birds, and applies this method to a declining farmland bird, the corn bunting Miliaria calandra L. We derive five alternative models to predict the functional responses of farmland birds and parameterize these for corn bunting. We also assess the minimum sample sizes required to predict accurately the functional response. 3. We show that the functional response of corn bunting can be predicted accurately from a few behavioural parameters (searching rate, handling time, vigilance time) that are straightforward to measure in the field. These parameters can be measured more quickly than the alternative of measuring the functional response directly. 4. While corn bunting violated some of the assumptions of Holling's disk equation (model 1 in our study), it still provided the most accurate fit to the observed feeding rates while remaining the most statistically simple model tested. Our other models may be more applicable to other species, or corn bunting feeding in other locations. 5. Although further tests are required, our study shows how functional responses can be predicted, simplifying the development of mechanistic models of farmland bird populations.

Bone morphogenetic protein (BMP) ligands and receptors in bovine ovarian follicle cells: actions of BMP-4,-6 and-7 on granulosa cells and differential modulation of Smad-1 phosphorylation by follistatin

Given the paucity of information on the potential roles of bone morphogenetic proteins (BMPs) in the ruminant ovary we conducted immunolocalization and functional studies on cells isolated from bovine antral follicles. Immunocytochemistry revealed expression of BMP-4 and -7 in isolated theca cells whereas granulosa cells and oocytes selectively expressed RMP-6. All three cell types expressed a range of BMP-responsive type-I (BMPRIB, ActRI) and type-II (BMPRII, ActRII, ActRIIB) receptors supporting autocrine/paracrine roles within the follicle. This was reinforced by functional experiments on granulosa cells which showed that BMP-4, -6 and -7 promoted cellular accumulation of phosphorylated Smad-1 but not Smad-2 and enhanced 'basal' and IGF-stimulated secretion of oestradiol (E2), inhibin-A, activin-A and follistatin (FS). Concomitantly, each BMP suppressed 'basal' and IGF-stimulated progesterone secretion, consistent with an action to prevent or delay atresia and/or luteinization. BMPs also increased viable cell number under 'basal' (BMP-4 and -7) and IGF-stimulated (BMP-4, -6 and -7) conditions. Since FS, a product of bovine granulosa cells, has been shown to bind several BMPs, we used the Biacore technique to compare its binding affinities for activin-A (prototype FS ligand) and BMP-4, -6 and -7. Compared with activin-A (K-d 0.28 +/- 0.02 nM; 100%), the relative affinities of FS for BMP-4, -6 and -7 were 10, 5 and 1% respectively. Moreover, studies on granulosa cells showed that preincubation of ligand with excess FS abolished activin-A-induced phosphorylation of Smad-2 and BMP-4-induced phosphorylation of Smad-1. However, FS only partially reversed BMP-6-induced Smad-1 phosphorylation and had no inhibitory effect on BMP-7-induced Smad-1 phosphorylation. These findings support functional roles for BMP-4, -6 and -7 as paracrine/autocrine modulators of granulosa cell steroidogenesis, peptide secretion and proliferation in bovine antral follicles. The finding that FS can differentially modulate BMP-induced receptor activation and that this correlates with the relative binding affinity of FS for each BMP type implicates FS as a potential modulator of BMP action in the ovary.

Evidence that GABA rho subunits contribute to functional ionotropic GABA receptors in mouse cerebellar Purkinje cells

Ionotropic gamma-amino butyric acid (GABA) receptors composed of heterogeneous molecular subunits are major mediators of inhibitory responses in the adult CNS. Here, we describe a novel ionotropic GABA receptor in mouse cerebellar Purkinje cells (PCs) using agents reported to have increased affinity for rho subunit-containing GABA(C) over other GABA receptors. Exogenous application of the GABA(C)-preferring agonist cis-4-aminocrotonic acid (CACA) evoked whole-cell currents in PCs, whilst equimolar concentrations of GABA evoked larger currents. CACA-evoked currents had a greater sensitivity to the selective GABA(C) antagonist (1,2,5,6-tetrahydropyridin-4-yl)methylphosphinic acid (TPMPA) than GABA-evoked currents. Focal application of agonists produced a differential response profile; CACA-evoked currents displayed a much more pronounced attenuation with increasing distance from the PC soma, displayed a slower time-to-peak and exhibited less desensitization than GABA-evoked currents. However, CACA-evoked currents were also completely blocked by bicuculline, a selective agent for GABA(A) receptors. Thus, we describe a population of ionotropic GABA receptors with a mixed GABA(A)/GABA(C) pharmacology. TPMPA reduced inhibitory synaptic transmission at interneurone-Purkinje cell (IN-PC) synapses, causing clear reductions in miniature inhibitory postsynaptic current (mIPSC) amplitude and frequency. Combined application of NO-711 (a selective GABA transporter subtype 1 (GAT-1) antagonist) and SNAP-5114 (a GAT-(2)/3/4 antagonist) induced a tonic GABA conductance in PCs; however, TPMPA had no effect on this current. Immunohistochemical studies suggest that rho subunits are expressed predominantly in PC soma and proximal dendritic compartments with a lower level of expression in more distal dendrites; this selective immunoreactivity contrasted with a more uniform distribution of GABA(A) alpha 1 subunits in PCs. Finally, co-immunoprecipitation studies suggest that rho subunits can form complexes with GABA(A) receptor alpha 1 subunits in the cerebellar cortex. Overall, these data suggest that rho subunits contribute to functional ionotropic receptors that mediate a component of phasic inhibitory GABAergic transmission at IN-PC synapses in the cerebellum.

On the spectra of certain integro-differential-delay problems with applications in neurodynamics

We investigate the spectrum of certain integro-differential-delay equations (IDDEs) which arise naturally within spatially distributed, nonlocal, pattern formation problems. Our approach is based on the reformulation of the relevant dispersion relations with the use of the Lambert function. As a particular application of this approach, we consider the case of the Amari delay neural field equation which describes the local activity of a population of neurons taking into consideration the finite propagation speed of the electric signal. We show that if the kernel appearing in this equation is symmetric around some point a= 0 or consists of a sum of such terms, then the relevant dispersion relation yields spectra with an infinite number of branches, as opposed to finite sets of eigenvalues considered in previous works. Also, in earlier works the focus has been on the most rightward part of the spectrum and the possibility of an instability driven pattern formation. Here, we numerically survey the structure of the entire spectra and argue that a detailed knowledge of this structure is important within neurodynamical applications. Indeed, the Amari IDDE acts as a filter with the ability to recognise and respond whenever it is excited in such a way so as to resonate with one of its rightward modes, thereby amplifying such inputs and dampening others. Finally, we discuss how these results can be generalised to the case of systems of IDDEs.

Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations

This paper describes a method for the state estimation of nonlinear systems described by a class of differential-algebraic equation models using the extended Kalman filter. The method involves the use of a time-varying linearisation of a semi-explicit index one differential-algebraic equation. The estimation technique consists of a simplified extended Kalman filter that is integrated with the differential-algebraic equation model. The paper describes a simulation study using a model of a batch chemical reactor. It also reports a study based on experimental data obtained from a mixing process, where the model of the system is solved using the sequential modular method and the estimation involves a bank of extended Kalman filters.

Velocity-based moving mesh methods for nonlinear partial differential equations

This article describes a number of velocity-based moving mesh numerical methods formultidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

Elevated CO2 enrichment induces a differential biomass response in a mixed species temperate forest plantation

• In a free-air CO2 enrichment study (BangorFACE) Alnus glutinosa, Betula pendula and Fagus sylvatica were planted in areas of one, two and three species mixtures (n=4). The trees were exposed to ambient or elevated CO2 (580 µmol mol-1) for four years, and aboveground growth characteristics measured. • In monoculture, the mean effect of CO2 enrichment on aboveground woody biomass was +29, +22 and +16% for A. glutinosa, F. sylvatica, and B. pendula respectively. When the same species were grown in polyculture, the response to CO2 switched to +10, +7 and 0%, for A. glutinosa, B. pendula, and F. sylvatica respectively. • In ambient atmosphere our species grown in polyculture increased aboveground woody biomass from 12.9 ± 1.4 kg m-2 to 18.9 ± 1.0 kg m-2, whereas in an elevated CO2 atmosphere aboveground woody biomass increased from 15.2 ± 0.6 kg m-2 to 20.2 ± 0.6 kg m-2. The overyielding effect of polyculture was smaller (+7%) in elevated CO2 than in an ambient atmosphere (+18%). • Our results show that the aboveground response to elevated CO2 is significantly affected by intra- and inter-specific competition, and that elevated CO2 response may be reduced in forest communities comprised of tree species with contrasting functional traits.

Mapping the platelet profile for functional genomic studies and demonstration of the effect size of the GP6 locus

BACKGROUND: Evidence suggests the wide variation in platelet response within the population is genetically controlled. Unraveling the complex relationship between sequence variation and platelet phenotype requires accurate and reproducible measurement of platelet response. OBJECTIVE: To develop a methodology suitable for measuring signaling pathway-specific platelet phenotype, to use this to measure platelet response in a large cohort, and to demonstrate the effect size of sequence variation in a relevant model gene. METHODS: Three established platelet assays were evaluated: mobilization of [Ca(2+)](i), aggregometry and flow cytometry, each in response to adenosine 5'-diphosphate (ADP) or the glycoprotein (GP) VI-specific crosslinked collagen-related peptide (CRP). Flow cytometric measurement of fibrinogen binding and P-selectin expression in response to a single, intermediate dose of each agonist gave the best combination of reproducibility and inter-individual variability and was used to measure the platelet response in 506 healthy volunteers. Pathway specificity was ensured by blocking the main subsidiary signaling pathways. RESULTS: Individuals were identified who were hypo- or hyper-responders for both pathways, or who had differential responses to the two agonists, or between outcomes. 89 individuals, retested three months later using the same methodology, showed high concordance between the two visits in all four assays (r(2) = 0.872, 0.868, 0.766 and 0.549); all subjects retaining their phenotype at recall. The effect of sequence variation at the GP6 locus accounted for approximately 35% of the variation in the CRP-XL response. CONCLUSION: Genotyping-phenotype association studies in a well-characterized, large cohort provides a powerful strategy to measure the effect of sequence variation in genes regulating the platelet response.