Evans functions for integral neural field equations with Heaviside firing rate function

In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model and a limiting case is shown to recover recent results of L. Zhang [On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16, (2003), pp.513-536.]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speed. Such fronts may be connected and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses.

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

Dynamic instabilities in scalar neural field equations with space-dependent delays

In this paper we consider a class of scalar integral equations with a form of space-dependent delay. These non-local models arise naturally when modelling neural tissue with active axons and passive dendrites. Such systems are known to support a dynamic (oscillatory) Turing instability of the homogeneous steady state. In this paper we develop a weakly nonlinear analysis of the travelling and standing waves that form beyond the point of instability. The appropriate amplitude equations are found to be the coupled mean-field Ginzburg-Landau equations describing a Turing-Hopf bifurcation with modulation group velocity of O(1). Importantly we are able to obtain the coefficients of terms in the amplitude equations in terms of integral transforms of the spatio-temporal kernels defining the neural field equation of interest. Indeed our results cover not only models with axonal or dendritic delays but those which are described by a more general distribution of delayed spatio-temporal interactions. We illustrate the predictive power of this form of analysis with comparison against direct numerical simulations, paying particular attention to the competition between standing and travelling waves and the onset of Benjamin-Feir instabilities.

Gender and Citizenship under New Labour

Abstract To what extent has citizenship been transformed under the New Labour government to include women as equal citizens? This chapter will examine New Labour’s record in terms of alternative conceptions of citizenship: a model based on equal obligations to paid work, a model based on recognising care and gender difference, and a model of universal citizenship, underpinning equal expectations of care work and paid work with rights to the resources needed for individuals to combine both. It will argue that, while New Labour has signed up to the EU resolution on work-life balance, which includes commitment to a ‘new social contract on gender’, and has significantly increased resources for care, obligations to work are at the heart of New Labour ideas of citizenship, with work conceived as paid employment: policies in practice have done more to bring women into employment than men into care. Women’s citizenship is still undermined – though less than under earlier governments - by these unequal obligations and their consequences in social rights.

Relationship between therapeutic changes in blood pressure and outcomes in acute stroke: a metaregression

Both low and high blood pressure (BP) during the acute phase of stroke are associated independently with a poor outcome. Several small clinical trials have involved the alteration of BP and this study assessed the relationship between change in BP and functional outcome. Randomised controlled trials of interventions that would be expected, on pharmacological grounds, to alter BP in patients within one week of the onset of acute ischaemic or haemorrhagic stroke were sought using electronic searches. Data were collected on BP and clinical outcome. The relationship between the difference in on-treatment BP and odds ratios (OR) for outcomes was assessed using meta-regression. Thirty-seven trials involving 9,008 patients were included. A ‘U’ or ‘J’ shaped relationship were found between on-treatment BP difference and early death, death at the end of 90 day follow up, and combined death or dependency at the end of follow up. Although outcomes were not significantly reduced at any level of change in BP, the lowest odds occurred at: early death (OR 0.87, 95% confidence interval, CI 0.54 to 1.23) - 8.1 mmHg; death at end of follow up (OR 0.96, 95% CI 0.31 to 1.65) - 14.4 mmHg; and combined death or dependency at end of follow up (OR 0.95, 95% CI 0.11 to 1.72) - 14.6 mmHg. Although large falls or increases in BP are associated with a worse outcome, modest reductions may reduce death, and combined death or dependency, although the confidence intervals are wide and compatible with overall benefit or hazard.