On Local Bifurcations in Neural Field Models with Transmission Delays

Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results

Towards a computational model for stimulation of the Pedunculopontine nucleus

The pedunculopontine nucleus (PPN) has recently been suggested as a new therapeutic target for deep brain stimulation (DBS) in patients suffering from Parkinson's disease, particularly those with severe gait and postural impairment [1]. Stimulation at this site is typically delivered at low frequencies in contrast to the high frequency stimulation required for therapeutic benefit in the subthalamic nucleus (STN) [1]. Despite real therapeutic successes, the fundamental physiological mechanisms underlying the effect of DBS are still not understood. A hypothesis is that DBS masks the pathological synchronized firing patterns of the basal ganglia that characterize the Parkinsonian state with a regularized firing pattern. It remains unclear why stimulation of PPN should be applied with low frequency in contrast to the high frequency stimulation of STN. To get a better understanding of PPN stimulation we construct a computational model for the PPN Type I neurons in a network

Ashkin-Teller universality in a quantum double model of Ising anyons

We study a quantum double model whose degrees of freedom are Ising anyons. The terms of the Hamiltonian of this system give rise to a competition between single and double topologies. By studying the energy spectra of the Hamiltonian at different values of the coupling constants, we find extended gapless regions which include a large number of critical points described by conformal field theories with central charge c=1. These theories are part of the Z_2 orbifold of the bosonic theory compactified on a circle. We observe that the Hilbert space of our anyonic model can be associated with extended Dynkin diagrams of affine Lie algebras which yields exact solutions at some critical points. In certain special regimes, our model corresponds to the Hamiltonian limit of the Ashkin-Teller model, and hence integrability over a wide range of coupling parameters is established.Comment: 11 pages, minor revision

A Framework for Directional and Higher-Order Reconstruction in Photoacoustic Tomography

Photoacoustic tomography is a hybrid imaging technique that combines high optical tissue contrast with high ultrasound resolution. Direct reconstruction methods such as filtered backprojection, time reversal and least squares suffer from curved line artefacts and blurring, especially in case of limited angles or strong noise. In recent years, there has been great interest in regularised iterative methods. These methods employ prior knowledge on the image to provide higher quality reconstructions. However, easy comparisons between regularisers and their properties are limited, since many tomography implementations heavily rely on the specific regulariser chosen. To overcome this bottleneck, we present a modular reconstruction framework for photoacoustic tomography. It enables easy comparisons between regularisers with different properties, e.g. nonlinear, higher-order or directional. We solve the underlying minimisation problem with an efficient first-order primal-dual algorithm. Convergence rates are optimised by choosing an operator dependent preconditioning strategy. Our reconstruction methods are tested on challenging 2D synthetic and experimental data sets. They outperform direct reconstruction approaches for strong noise levels and limited angle measurements, offering immediate benefits in terms of acquisition time and quality. This work provides a basic platform for the investigation of future advanced regularisation methods in photoacoustic tomography.Comment: submitted to "Physics in Medicine and Biology". Changes from v1 to v2: regularisation with directional wavelet has been added; new experimental tests have been include

On the reduction of the degree of linear differential operators

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let k^a be the algebraic closure of k. For a solution y, Ly=0, we determine the linear differential operator of minimal degree M and coefficients in k^a, such that My=0. This result is then applied to some Picard-Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka-Volterra type

Oscillatory eigenmodes and stability of one and two arbitrary fractional vortices in long Josephson 0-kappa-junctions

We investigate theoretically the eigenmodes and the stability of one and two arbitrary fractional vortices pinned at one and two $\kappa$-phase discontinuities in a long Josephson junction. In the particular case of a single $\kappa$-discontinuity, a vortex is spontaneously created and pinned at the boundary between the 0 and $\kappa$-regions. In this work we show that only two of four possible vortices are stable. A single vortex has an oscillatory eigenmode with a frequency within the plasma gap. We calculate this eigenfrequency as a function of the fractional flux carried by a vortex. For the case of two vortices, pinned at two $\kappa$-discontinuities situated at some distance $a$ from each other, splitting of the eigenfrequencies occur. We calculate this splitting numerically as a function of $a$ for different possible ground states. We also discuss the presence of a critical distance below which two antiferromagnetically ordered vortices form a strongly coupled ``vortex molecule'' that behaves as a single object and has only one eigenmode.Comment: submitted to Phys. Rev. B (

Controllable plasma energy bands in a 1D crystal of fractional Josephson vortices

We consider a 1D chain of fractional vortices in a long Josephson junction with alternating $\pm\kappa$ phase discontinuities. Since each vortex has its own eigenfrequency, the inter-vortex coupling results in eigenmode splitting and in the formation of an oscillatory energy band for plasma waves. The band structure can be controlled at the design time by choosing the distance between vortices or \emph{during experiment} by varying the topological charge of vortices or the bias current. Thus one can construct an artificial vortex crystal with controllable energy bands for plasmons.Comment: 4 pages, 2 Fig