Interface dynamics in planar neural field models

Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localized solutions in the form of traveling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyze neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and traveling waves.

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

The Empirical Forcing Function as a tool for the diagnosis of large-scale atmospheric anomalies

The time-mean quasi-geostrophic potential vorticity equation of the atmospheric flow on isobaric surfaces can explicitly include an atmospheric (internal) forcing term of the stationary-eddy flow. In fact, neglecting some non-linear terms in this equation, this forcing can be mathematically expressed as a single function, called Empirical Forcing Function (EFF), which is equal to the material derivative of the time-mean potential vorticity. Furthermore, the EFF can be decomposed as a sum of seven components, each one representing a forcing mechanism of different nature. These mechanisms include diabatic components associated with the radiative forcing, latent heat release and frictional dissipation, and components related to transient eddy transports of heat and momentum. All these factors quantify the role of the transient eddies in forcing the atmospheric circulation. In order to assess the relevance of the EFF in diagnosing large-scale anomalies in the atmospheric circulation, the relationship between the EFF and the occurrence of strong North Atlantic ridges over the Eastern North Atlantic is analyzed, which are often precursors of severe droughts over Western Iberia. For such events, the EFF pattern depicts a clear dipolar structure over the North Atlantic; cyclonic (anticyclonic) forcing of potential vorticity is found upstream (downstream) of the anomalously strong ridges. Results also show that the most significant components are related to the diabatic processes. Lastly, these results highlight the relevance of the EFF in diagnosing large-scale anomalies, also providing some insight into their interaction with different physical mechanisms.

Constructing the frequency and wave normal distribution of whistler-mode wave power

We introduce a new methodology that allows the construction of wave frequency distributions due to growing incoherent whistler-mode waves in the magnetosphere. The technique combines the equations of geometric optics (i.e. raytracing) with the equation of transfer of radiation in an anisotropic lossy medium to obtain spectral energy density as a function of frequency and wavenormal angle. We describe the method in detail, and then demonstrate how it could be used in an idealised magnetosphere during quiet geomagnetic conditions. For a specific set of plasma conditions, we predict that the wave power peaks off the equator at ~15 degrees magnetic latitude. The new calculations predict that wave power as a function of frequency can be adequately described using a Gaussian function, but as a function of wavenormal angle, it more closely resembles a skew normal distribution. The technique described in this paper is the first known estimate of the parallel and oblique incoherent wave spectrum as a result of growing whistler-mode waves, and provides a means to incorporate self-consistent wave-particle interactions in a kinetic model of the magnetosphere over a large volume.

Nonlinear symmetric stability of planetary atmospheres

The energy–Casimir method is applied to the problem of symmetric stability in the context of a compressible, hydrostatic planetary atmosphere with a general equation of state. Formal stability criteria for symmetric disturbances to a zonally symmetric baroclinic flow are obtained. In the special case of a perfect gas the results of Stevens (1983) are recovered. Finite-amplitude stability conditions are also obtained that provide an upper bound on a certain positive-definite measure of disturbance amplitude.

Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics. In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms. The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent. This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii). In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

A general method for finding extremal states of Hamiltonian dynamical systems, with applications to perfect fluids

In addition to the Hamiltonian functional itself, non-canonical Hamiltonian dynamical systems generally possess integral invariants known as ‘Casimir functionals’. In the case of the Euler equations for a perfect fluid, the Casimir functionals correspond to the vortex topology, whose invariance derives from the particle-relabelling symmetry of the underlying Lagrangian equations of motion. In a recent paper, Vallis, Carnevale & Young (1989) have presented algorithms for finding steady states of the Euler equations that represent extrema of energy subject to given vortex topology, and are therefore stable. The purpose of this note is to point out a very general method for modifying any Hamiltonian dynamical system into an algorithm that is analogous to those of Vallis etal. in that it will systematically increase or decrease the energy of the system while preserving all of the Casimir invariants. By incorporating momentum into the extremization procedure, the algorithm is able to find steadily-translating as well as steady stable states. The method is applied to a variety of perfect-fluid systems, including Euler flow as well as compressible and incompressible stratified flow.

Remarks concerning the double cascade as a necessary mechanism for the instability of steady equivalent-barotropic flows

n a recent paper, Petroniet al. claim that a necessary condition for the instability of two-dimensional steady flows is a «double cascade» of energy and enstrophy respectively to larger and to smaller scales of motion. It is shown here that the analytical reasoning employed by Petroniet al. is flawed and that their conclusions are incorrect. What is true is that in any scale interaction (whether an instability or not), neither energy nor enstrophy can be transferred in one spectral direction only, but this result is extremely well known.

A spectral view of nonlinear fluxes and stationary-transient interaction in the atmosphere

Nonlinear spectral transfers of kinetic energy and enstrophy, and stationary-transient interaction, are studied using global FGGE data for January 1979. It is found that the spectral transfers arise primarily from a combination, in roughly equal measure, of pure transient and mixed stationary-transient interactions. The pure transient interactions are associated with a transient eddy field which is approximately locally homogeneous and isotropic, and they appear to be consistently understood within the context of two-dimensional homogeneous turbulence. Theory based on spatial wale separation concepts suggests that the mixed interactions may be understood physically, to a first approximation, as a process of shear-induced spectral transfer of transient enstrophy along lines of constant zonal wavenumber. This essentially conservative enstrophy transfer generally involves highly nonlocal stationary-transient energy conversions. The observational analysis demonstrates that the shear-induced transient enstrophy transfer is mainly associated with intermediate-scale (zonal wavenumber m > 3) transients and is primarily to smaller (meridional) scales, so that the transient flow acts as a source of stationary energy. In quantitative terms, this transient-eddy rectification corresponds to a forcing timescale in the stationary energy budget which is of the same order of magnitude as most estimates of the damping timescale in simple stationary-wave models (5 to 15 days). Moreover, the nonlinear interactions involved are highly nonlocal and cover a wide range of transient scales of motion.

Eye candy for the blind: re-introducing Lyotard's Acinema into discourses on excess, motion and spectacle in contemporary Hollywood

Jean-François Lyotard's 1973 essay ‘Acinema’ is explicitly concerned with the cinematic medium, but has received scant critical attention. Lyotard's acinema conceives of an experimental, excessive form of film-making that uses stillness and movement to shift away from the orderly process of meaning-making within mainstream cinema. What motivates this present paper is a striking link between Lyotard's writing and contemporary Hollywood production; both are concerned with a sense of excess, especially within moments of motion. Using Charlie's Angels (McG, 2000) as a case study – a film that has been critically dismissed as ‘eye candy for the blind’ – my methodology brings together two different discourses, high culture theory and mainstream film-making, to test out and propose the value of Lyotard's ideas for the study of contemporary film. Combining close textual analysis and engagement with key scholarship on film spectacle, I reflexively engage with the process of film analysis and re-direct attention to a neglected essay by a major theorist, in order to stimulate further engagement with his work.

Multiscale geostatistical analysis of AVHRR, SPOT-VGT, and MODIS global NDVI products

Global NDVI data are routinely derived from the AVHRR, SPOT-VGT, and MODIS/Terra earth observation records for a range of applications from terrestrial vegetation monitoring to climate change modeling. This has led to a substantial interest in the harmonization of multisensor records. Most evaluations of the internal consistency and continuity of global multisensor NDVI products have focused on time-series harmonization in the spectral domain, often neglecting the spatial domain. We fill this void by applying variogram modeling (a) to evaluate the differences in spatial variability between 8-km AVHRR, 1-km SPOT-VGT, and 1-km, 500-m, and 250-m MODIS NDVI products over eight EOS (Earth Observing System) validation sites, and (b) to characterize the decay of spatial variability as a function of pixel size (i.e. data regularization) for spatially aggregated Landsat ETM+ NDVI products and a real multisensor dataset. First, we demonstrate that the conjunctive analysis of two variogram properties – the sill and the mean length scale metric – provides a robust assessment of the differences in spatial variability between multiscale NDVI products that are due to spatial (nominal pixel size, point spread function, and view angle) and non-spatial (sensor calibration, cloud clearing, atmospheric corrections, and length of multi-day compositing period) factors. Next, we show that as the nominal pixel size increases, the decay of spatial information content follows a logarithmic relationship with stronger fit value for the spatially aggregated NDVI products (R2 = 0.9321) than for the native-resolution AVHRR, SPOT-VGT, and MODIS NDVI products (R2 = 0.5064). This relationship serves as a reference for evaluation of the differences in spatial variability and length scales in multiscale datasets at native or aggregated spatial resolutions. The outcomes of this study suggest that multisensor NDVI records cannot be integrated into a long-term data record without proper consideration of all factors affecting their spatial consistency. Hence, we propose an approach for selecting the spatial resolution, at which differences in spatial variability between NDVI products from multiple sensors are minimized. This approach provides practical guidance for the harmonization of long-term multisensor datasets.

Thermobaric control of gravitational potential energy generation by diapycnal mixing in the deep ocean

Sources and sinks of gravitational potential energy (GPE) play a rate-limiting role in the large scale ocean circulation. A key source is turbulent diapycnal mixing, whereby irre- versible mixing across isoneutral surfaces is enhanced by turbulent straining of these surfaces. This has motivated international observational efforts to map diapycnal mixing in the global ocean. However, in order to accurately relate the GPE supplied to the large scale circulation by diapycnal mixing to the mixing energy source, it is first necessary to determine the ratio, ξ , of the GPE generation rate to the available potential energy dissipation rate associated with turbulent mixing. Here, the link between GPE and hydro- static pressure is used to derive the GPE budget for a com- pressible ocean with a nonlinear equation of state. The role of diapycnal mixing is isolated and from this a global cli- matological distribution of ξ is calculated. It is shown that, for a given source of mixing energy, typically three times as much GPE is generated if the mixing takes place in bottom waters rather than in the pycnocline. This is due to GPE destruction by cabbelling in the pycnocline, as opposed to thermobaric enhancement of GPE generation by diapycnal mixing in the deep ocean.

On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti- mately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non- linear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.

Evaluation of biospheric components in Earth system models using modern and palaeo-observations: the state-of-the-art

Earth system models (ESMs) are increasing in complexity by incorporating more processes than their predecessors, making them potentially important tools for studying the evolution of climate and associated biogeochemical cycles. However, their coupled behaviour has only recently been examined in any detail, and has yielded a very wide range of outcomes. For example, coupled climate–carbon cycle models that represent land-use change simulate total land carbon stores at 2100 that vary by as much as 600 Pg C, given the same emissions scenario. This large uncertainty is associated with differences in how key processes are simulated in different models, and illustrates the necessity of determining which models are most realistic using rigorous methods of model evaluation. Here we assess the state-of-the-art in evaluation of ESMs, with a particular emphasis on the simulation of the carbon cycle and associated biospheric processes. We examine some of the new advances and remaining uncertainties relating to (i) modern and palaeodata and (ii) metrics for evaluation. We note that the practice of averaging results from many models is unreliable and no substitute for proper evaluation of individual models. We discuss a range of strategies, such as the inclusion of pre-calibration, combined process- and system-level evaluation, and the use of emergent constraints, that can contribute to the development of more robust evaluation schemes. An increasingly data-rich environment offers more opportunities for model evaluation, but also presents a challenge. Improved knowledge of data uncertainties is still necessary to move the field of ESM evaluation away from a "beauty contest" towards the development of useful constraints on model outcomes.

Dissociation between the impact of evidence on eye movement target choice and confidence judgements

It has been suggested that the evidence used to support a decision to move our eyes and the confidence we have in that decision are derived from a common source. Alternatively, confidence may be based on further post-decisional processes. In three experiments we examined this. In Experiment 1, participants chose between two targets on the basis of varying levels of evidence (i.e., the direction of motion coherence in a Random-Dot-Kinematogram). They indicated this choice by making a saccade to one of two targets and then indicated their confidence. Saccade trajectory deviation was taken as a measure of the inhibition of the non-selected target. We found that as evidence increased so did confidence and deviations of saccade trajectory away from the non-selected target. However, a correlational analysis suggested they were not related. In Experiment 2 an option to opt-out of the choice was offered on some trials if choice proved too difficult. In this way we isolated trials on which confidence in target selection was high (i.e., when the option to opt-out was available but not taken). Again saccade trajectory deviations were found not to differ in relation to confidence. In Experiment 3 we directly manipulated confidence, such that participants had high or low task confidence. They showed no differences in saccade trajectory deviations. These results support post-decisional accounts of confidence: evidence supporting the decision to move the eyes is reflected in saccade control, but the confidence that we have in that choice is subject to further post-decisional processes.