Modeling the effects of anesthesia on the electroencephalogram

Changes to the electroencephalogram (EEG) observed during general anesthesia are modeled with a physiological mean field theory of electrocortical activity. To this end a parametrization of the postsynaptic impulse response is introduced which takes into account pharmacological effects of anesthetic agents on neuronal ligand-gated ionic channels. Parameter sets for this improved theory are then identified which respect known anatomical constraints and predict mean firing rates and power spectra typically encountered in human subjects. Through parallelized simulations of the eight nonlinear, two-dimensional partial differential equations on a grid representing an entire human cortex, it is demonstrated that linear approximations are sufficient for the prediction of a range of quantitative EEG variables. More than 70 000 plausible parameter sets are finally selected and subjected to a simulated induction with the stereotypical inhaled general anesthetic isoflurane. Thereby 86 parameter sets are identified that exhibit a strong “biphasic” rise in total power, a feature often observed in experiments. A sensitivity study suggests that this “biphasic” behavior is distinguishable even at low agent concentrations. Finally, our results are briefly compared with previous work by other groups and an outlook on future fits to experimental data is provided.

Multi-mode approximations to wave scattering by an uneven bed

Approximations to the scattering of linear surface gravity waves on water of varying quiescent depth are Investigated by means of a variational approach. Previous authors have used wave modes associated with the constant depth case to approximate the velocity potential, leading to a system of coupled differential equations. Here it is shown that a transformation of the dependent variables results in a much simplified differential equation system which in turn leads to a new multi-mode 'mild-slope' approximation. Further, the effect of adding a bed mode is examined and clarified. A systematic analytic method is presented for evaluating inner products that arise and numerical experiments for two-dimensional scattering are used to examine the performance of the new approximations.

Approximations to the scattering of water waves by steep topography

A new method is developed for approximating the scattering of linear surface gravity waves on water of varying quiescent depth in two dimensions. A conformal mapping of the fluid domain onto a uniform rectangular strip transforms steep and discontinuous bed profiles into relatively slowly varying, smooth functions in the transformed free-surface condition. By analogy with the mild-slope approach used extensively in unmapped domains, an approximate solution of the transformed problem is sought in the form of a modulated propagating wave which is determined by solving a second-order ordinary differential equation. This can be achieved numerically, but an analytic solution in the form of a rapidly convergent infinite series is also derived and provides simple explicit formulae for the scattered wave amplitudes. Small-amplitude and slow variations in the bedform that are excluded from the mapping procedure are incorporated in the approximation by a straightforward extension of the theory. The error incurred in using the method is established by means of a rigorous numerical investigation and it is found that remarkably accurate estimates of the scattered wave amplitudes are given for a wide range of bedforms and frequencies.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular, we are able to treat “patchy” connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a “lattice-directed” traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs.

Representation errors and retrievals in linear and nonlinear data assimilation

This article shows how one can formulate the representation problem starting from Bayes’ theorem. The purpose of this article is to raise awareness of the formal solutions,so that approximations can be placed in a proper context. The representation errors appear in the likelihood, and the different possibilities for the representation of reality in model and observations are discussed, including nonlinear representation probability density functions. Specifically, the assumptions needed in the usual procedure to add a representation error covariance to the error covariance of the observations are discussed,and it is shown that, when several sub-grid observations are present, their mean still has a representation error ; socalled ‘superobbing’ does not resolve the issue. Connection is made to the off-line or on-line retrieval problem, providing a new simple proof of the equivalence of assimilating linear retrievals and original observations. Furthermore, it is shown how nonlinear retrievals can be assimilated without loss of information. Finally we discuss how errors in the observation operator model can be treated consistently in the Bayesian framework, connecting to previous work in this area.

Model-based vehicle detection and classification using orthographic approximations

This paper reports the current state of work to simplify our previous model-based methods for visual tracking of vehicles for use in a real-time system intended to provide continuous monitoring and classification of traffic from a fixed camera on a busy multi-lane motorway. The main constraints of the system design were: (i) all low level processing to be carried out by low-cost auxiliary hardware, (ii) all 3-D reasoning to be carried out automatically off-line, at set-up time. The system developed uses three main stages: (i) pose and model hypothesis using 1-D templates, (ii) hypothesis tracking, and (iii) hypothesis verification, using 2-D templates. Stages (i) & (iii) have radically different computing performance and computational costs, and need to be carefully balanced for efficiency. Together, they provide an effective way to locate, track and classify vehicles.

Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

Linear viscoelasticity from molecular dynamics simulation of entangled polymers

The linear viscoelastic (LVE) spectrum is one of the primary fingerprints of polymer solutions and melts, carrying information about most relaxation processes in the system. Many single chain theories and models start with predicting the LVE spectrum to validate their assumptions. However, until now, no reliable linear stress relaxation data were available from simulations of multichain systems. In this work, we propose a new efficient way to calculate a wide variety of correlation functions and mean-square displacements during simulations without significant additional CPU cost. Using this method, we calculate stress−stress autocorrelation functions for a simple bead−spring model of polymer melt for a wide range of chain lengths, densities, temperatures, and chain stiffnesses. The obtained stress−stress autocorrelation functions were compared with the single chain slip−spring model in order to obtain entanglement related parameters, such as the plateau modulus or the molecular weight between entanglements. Then, the dependence of the plateau modulus on the packing length is discussed. We have also identified three different contributions to the stress relaxation:  bond length relaxation, colloidal and polymeric. Their dependence on the density and the temperature is demonstrated for short unentangled systems without inertia.

Linear amplification of marginally neutral baroclinic waves

Baroclinic wave development is investigated for unstable parallel shear flows in the limit of vanishing normal-mode growth rate. This development is described in terms of the propagation and interaction mechanisms of two coherent structures, called counter-propagating Rossby waves (CRWs). It is shown that, in this limit of vanishing normal-mode growth rate, arbitrary initial conditions produce sustained linear amplification of the marginally neutral normal mode (mNM). This linear excitation of the mNM is subsequently interpreted in terms of a resonance phenomenon. Moreover, while the mathematical character of the normal-mode problem changes abruptly as the bifurcation point in the dispersion diagram is encountered and crossed, it is shown that from an initial-value viewpoint, this transition is smooth. Consequently, the resonance interpretation remains relevant (albeit for a finite time) for wavenumbers slightly different from the ones defining cut-off points. The results are further applied to a two-layer version of the classic Eady model in which the upper rigid lid has been replaced by a simple stratosphere.

Boundary value problems for third-order linear PDEs in time-dependent domains

We present the extension of a methodology to solve moving boundary value problems from the second-order case to the case of the third-order linear evolution PDE qt + qxxx = 0. This extension is the crucial step needed to generalize this methodology to PDEs of arbitrary order. The methodology is based on the derivation of inversion formulae for a class of integral transforms that generalize the Fourier transform and on the analysis of the global relation associated with the PDE. The study of this relation and its inversion using the appropriate generalized transform are the main elements of the proof of our results.

Decadal predictability of the Atlantic Ocean in a coupled GCM: forecast skill and optimal perturbations using Linear Inverse Modelling

The decadal predictability of three-dimensional Atlantic Ocean anomalies is examined in a coupled global climate model (HadCM3) using a Linear Inverse Modelling (LIM) approach. It is found that the evolution of temperature and salinity in the Atlantic, and the strength of the meridional overturning circulation (MOC), can be effectively described by a linear dynamical system forced by white noise. The forecasts produced using this linear model are more skillful than other reference forecasts for several decades. Furthermore, significant non-normal amplification is found under several different norms. The regions from which this growth occurs are found to be fairly shallow and located in the far North Atlantic. Initially, anomalies in the Nordic Seas impact the MOC, and the anomalies then grow to fill the entire Atlantic basin, especially at depth, over one to three decades. It is found that the structure of the optimal initial condition for amplification is sensitive to the norm employed, but the initial growth seems to be dominated by MOC-related basin scale changes, irrespective of the choice of norm. The consistent identification of the far North Atlantic as the most sensitive region for small perturbations suggests that additional observations in this region would be optimal for constraining decadal climate predictions.