Hexagonal patterns in finite domains

In many mathematical models for pattern formation, a regular hexagonal pattern is stable in an infinite region. However, laboratory and numerical experiments are carried out in finite domains, and this imposes certain constraints on the possible patterns. In finite rectangular domains, it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational. In practice, it is found experimentally that in a rectangular region, patterns of irregular hexagons are often observed. This work analyses the geometry and dynamics of irregular hexagonal patterns. These patterns occur in two different symmetry types, either with a reflection symmetry, involving two wavenumbers, or without symmetry, involving three different wavenumbers. The relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case. It is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls. Numerical simulations of a model partial differential equation are also presented to illustrate the behaviour.

Waves, bumps, and patterns in neural field theories

Neural field models of firing rate activity have had a major impact in helping to develop an understanding of the dynamics seen in brain slice preparations. These models typically take the form of integro-differential equations. Their non-local nature has led to the development of a set of analytical and numerical tools for the study of waves, bumps and patterns, based around natural extensions of those used for local differential equation models. In this paper we present a review of such techniques and show how recent advances have opened the way for future studies of neural fields in both one and two dimensions that can incorporate realistic forms of axo-dendritic interactions and the slow intrinsic currents that underlie bursting behaviour in single neurons.

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. Article published and (c) American Physical Society 2007

Instabilities in threshold-diffusion equations with delay

The introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio-temporal chaos. In this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria, periodic orbits and travelling waves. Importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function. Using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback. Direct numerical simulations are used to confirm our bifurcation analysis, and to probe some of the rich behaviour possible for mixed feedback.

Cerebral misery perfusion diagnosed using hypercapnic blood-oxygenation-level-dependent contrast functional magnetic resonance imaging: a case report

Introduction Cerebral misery perfusion represents a failure of cerebral autoregulation. It is animportant differential diagnosis in post-stroke patients presenting with collapses in the presence of haemodynamically significant cerebrovascular stenosis. This is particularly the case when cortical or internal watershed infarcts are present. When this condition occurs, further investigation should be done immediately. Case presentation A 50-year-old Caucasian man presented with a stroke secondary to complete occlusion of his left internal carotid artery. He went on to suffer recurrent seizures. Neuroimaging demonstrated numerous new watershed-territory cerebral infarcts. No source of arterial thromboembolism was demonstrable. Hypercapnic blood-oxygenation-level-dependent-contrast functional magnetic resonance imaging was used to measure his cerebrovascular reserve capacity. The findings were suggestive of cerebral misery perfusion. Conclusions Blood-oxygenation-level-dependent-contrast functional magnetic resonance imaging allows the inference of cerebral misery perfusion. This procedure is cheaper and more readily available than positron emission tomography imaging, which is the current gold standard diagnostic test. The most evaluated treatment for cerebral misery perfusion is extracranial-intracranial bypass. Although previous trials of this have been unfavourable, the results of new studies involving extracranial-intracranial bypass in high-risk patients identified during cerebral perfusion imaging are awaited. Cerebral misery perfusion is an important and under-recognized condition in which emerging imaging and treatment modalities present the possibility of practical and evidence-based management in the near future. Physicians should thus be aware of this disorder and of recent developments in diagnostic tests that allow its detection.