Attention through self-synchronisation in the spiking neuron stochastic diffusion network

The paper discusses ensemble behaviour in the Spiking Neuron Stochastic Diffusion Network, SNSDN, a novel network exploring biologically plausible information processing based on higher order temporal coding. SNSDN was proposed as an alternative solution to the binding problem [1]. SNSDN operation resembles Stochastic Diffusin on Search, SDS, a non-deterministic search algorithm able to rapidly locate the best instantiation of a target pattern within a noisy search space ([3], [5]). In SNSDN, relevant information is encoded in the length of interspike intervals. Although every neuron operates in its own time, ‘attention’ to a pattern in the search space results in self-synchronised activity of a large population of neurons. When multiple patterns are present in the search space, ‘switching of at- tention’ results in a change of the synchronous activity. The qualitative effect of attention on the synchronicity of spiking behaviour in both time and frequency domain will be discussed.

Time complexity analysis of the stochastic diffusion search

The Stochastic Diffusion Search algorithm -an integral part of Stochastic Search Networks is investigated. Stochastic Diffusion Search is an alternative solution for invariant pattern recognition and focus of attention. It has been shown that the algorithm can be modelled as an ergodic, finite state Markov Chain under some non-restrictive assumptions. Sub-linear time complexity for some settings of parameters has been formulated and proved. Some properties of the algorithm are then characterised and numerical examples illustrating some features of the algorithm are presented.

Plane wave approximation in linear elasticity

We consider the approximation of solutions of the time-harmonic linear elastic wave equation by linear combinations of plane waves. We prove algebraic orders of convergence both with respect to the dimension of the approximating space and to the diameter of the domain. The error is measured in Sobolev norms and the constants in the estimates explicitly depend on the problem wavenumber. The obtained estimates can be used in the h- and p-convergence analysis of wave-based finite element schemes.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Superfast non-linear diffusion: capillary transport in particulate porous media

The migration of liquids in porous media, such as sand, has been commonly considered at high saturation levels with liquid pathways at pore dimensions. In this letter we reveal a low saturation regime observed in our experiments with droplets of extremely low volatility liquids deposited on sand. In this regime the liquid is mostly found within the grain surface roughness and in the capillary bridges formed at the contacts between the grains. The bridges act as variable-volume reservoirs and the flow is driven by the capillary pressure arising at the wetting front according to the roughness length scales. We propose that this migration (spreading) is the result of interplay between the bridge volume adjustment to this pressure distribution and viscous losses of a creeping flow within the roughness. The net macroscopic result is a special case of non-linear diffusion described by a superfast diffusion equation (SFDE) for saturation with distinctive mathematical character. We obtain solutions to a moving boundary problem defined by SFDE that robustly convey a time power law of spreading as seen in our experiments.

Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. For large-scale atmospheric dynamics, the Eliassen–Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions, their application to a horizontally homogeneous background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Earlier three-dimensional results based on linear WKB theory considered only Doppler-shifted gravity waves, not waves in a stratified shear flow. Consideration of a background flow depending only on altitude is motivated by the parameterization of subgrid-scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances to a horizontally homogeneous background flow. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional problem can be treated as a limiting case of the three-dimensional problem. The results also generalize earlier three-dimensional results in that there is no slowly varying WKB-type requirement on the background flow, and the results are extendable to finite amplitude. The relationship A E =cA P between pseudoenergy A E and pseudomomentum A P, where c is the horizontal phase speed in the direction of symmetry associated with A P, has important applications to gravity-wave parameterization and provides a generalized statement of the first Eliassen–Palm theorem.

Estimation of optimal gravity wave parameters for climate models using data assimilation

There is a current need to constrain the parameters of gravity wave drag (GWD) schemes in climate models using observational information instead of tuning them subjectively. In this work, an inverse technique is developed using data assimilation principles to estimate gravity wave parameters. Because mostGWDschemes assume instantaneous vertical propagation of gravity waves within a column, observations in a single column can be used to formulate a one-dimensional assimilation problem to estimate the unknown parameters. We define a cost function that measures the differences between the unresolved drag inferred from observations (referred to here as the ‘observed’ GWD) and the GWD calculated with a parametrisation scheme. The geometry of the cost function presents some difficulties, including multiple minima and ill-conditioning because of the non-independence of the gravity wave parameters. To overcome these difficulties we propose a genetic algorithm to minimize the cost function, which provides a robust parameter estimation over a broad range of prescribed ‘true’ parameters. When real experiments using an independent estimate of the ‘observed’ GWD are performed, physically unrealistic values of the parameters can result due to the non-independence of the parameters. However, by constraining one of the parameters to lie within a physically realistic range, this degeneracy is broken and the other parameters are also found to lie within physically realistic ranges. This argues for the essential physical self-consistency of the gravity wave scheme. A much better fit to the observed GWD at high latitudes is obtained when the parameters are allowed to vary with latitude. However, a close fit can be obtained either in the upper or the lower part of the profiles, but not in both at the same time. This result is a consequence of assuming an isotropic launch spectrum. The changes of sign in theGWDfound in the tropical lower stratosphere, which are associated with part of the quasi-biennial oscillation forcing, cannot be captured by the parametrisation with optimal parameters.

Constraints on wave drag parameterization schemes for simulating the quasi-biennial oscillation. Part I: gravity wave forcing

Parameterization schemes for the drag due to atmospheric gravity waves are discussed and compared in the context of a simple one-dimensional model of the quasi-biennial oscillation (QBO). A number of fundamental issues are examined in detail, with the goal of providing a better understanding of the mechanism by which gravity wave drag can produce an equatorial zonal wind oscillation. The gravity wave–driven QBOs are compared with those obtained from a parameterization of equatorial planetary waves. In all gravity wave cases, it is seen that the inclusion of vertical diffusion is crucial for the descent of the shear zones and the development of the QBO. An important difference between the schemes for the two types of waves is that in the case of equatorial planetary waves, vertical diffusion is needed only at the lowest levels, while for the gravity wave drag schemes it must be included at all levels. The question of whether there is downward propagation of influence in the simulated QBOs is addressed. In the gravity wave drag schemes, the evolution of the wind at a given level depends on the wind above, as well as on the wind below. This is in contrast to the parameterization for the equatorial planetary waves in which there is downward propagation of phase only. The stability of a zero-wind initial state is examined, and it is determined that a small perturbation to such a state will amplify with time to the extent that a zonal wind oscillation is permitted.

Whistler mode wave growth and propagation in the prenoon magnetosphere

Pitch-angle scattering of electrons can limit the stably trapped particle flux in the magnetosphere and precipitate energetic electrons into the ionosphere. Whistler-mode waves generated by a temperature anisotropy can mediate this pitch-angle scattering over a wide range of radial distances and latitudes, but in order to correctly predict the phase-space diffusion, it is important to characterise the whistler-mode wave distributions that result from the instability. We use previously-published observations of number density, pitch-angle anisotropy and phase space density to model the plasma in the quiet pre-noon magnetosphere (defined as periods when AE<100nT). We investigate the global propagation and growth of whistler-mode waves by studying millions of growing ray paths and demonstrate that the wave distribution at any one location is a superposition of many waves at different points along their trajectories and with different histories. We show that for observed electron plasma properties, very few raypaths undergo magnetospheric reflection, most rays grow and decay within 30 degrees of the magnetic equator. The frequency range of the wave distribution at large L can be adequately described by the solutions of the local dispersion relation, but the range of wavenormal angle is different. The wave distribution is asymmetric with respect to the wavenormal angle. The numerical results suggest that it is important to determine the variation of magnetospheric parameters as a function of latitude, as well as local time and L-shell.

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.