Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities

In this paper we consider instabilities of localised solutions in planar neural field firing rate models of Wilson-Cowan or Amari type. Importantly we show that angular perturbations can destabilise spatially localised solutions. For a scalar model with Heaviside firing rate function we calculate symmetric one-bump and ring solutions explicitly and use an Evans function approach to predict the point of instability and the shapes of the dominant growing modes. Our predictions are shown to be in excellent agreement with direct numerical simulations. Moreover, beyond the instability our simulations demonstrate the emergence of multi-bump and labyrinthine patterns. With the addition of spike-frequency adaptation, numerical simulations of the resulting vector model show that it is possible for structures without rotational symmetry, and in particular multi-bumps, to undergo an instability to a rotating wave. We use a general argument, valid for smooth firing rate functions, to establish the conditions necessary to generate such a rotational instability. Numerical continuation of the rotating wave is used to quantify the emergent angular velocity as a bifurcation parameter is varied. Wave stability is found via the numerical evaluation of an associated eigenvalue problem.

Telmisartan to prevent recurrent stroke and cardiovascular events

Background Prolonged lowering of blood pressure after a stroke reduces the risk of recurrent stroke. In addition, inhibition of the renin–angiotensin system in high-risk patients reduces the rate of subsequent cardiovascular events, including stroke. However, the effect of lowering of blood pressure with a renin–angiotensin system inhibitor soon after a stroke has not been clearly established. We evaluated the effects of therapy with an angiotensin-receptor blocker, telmisartan, initiated early after a stroke. Methods In a multicenter trial involving 20,332 patients who recently had an ischemic stroke, we randomly assigned 10,146 to receive telmisartan (80 mg daily) and 10,186 to receive placebo. The primary outcome was recurrent stroke. Secondary outcomes were major cardiovascular events (death from cardiovascular causes, recurrent stroke, myocardial infarction, or new or worsening heart failure) and new-onset diabetes. Results The median interval from stroke to randomization was 15 days. During a mean followup of 2.5 years, the mean blood pressure was 3.8/2.0 mm Hg lower in the telmisartan group than in the placebo group. A total of 880 patients (8.7%) in the telmisartan group and 934 patients (9.2%) in the placebo group had a subsequent stroke (hazard ratio in the telmisartan group, 0.95; 95% confidence interval [CI], 0.86 to 1.04; P = 0.23). Major cardiovascular events occurred in 1367 patients (13.5%) in the telmisartan group and 1463 patients (14.4%) in the placebo group (hazard ratio, 0.94; 95% CI, 0.87 to 1.01; P = 0.11). New-onset diabetes occurred in 1.7% of the telmisartan group and 2.1% of the placebo group (hazard ratio, 0.82; 95% CI, 0.65 to 1.04; P = 0.10). Conclusions Therapy with telmisartan initiated soon after an ischemic stroke and continued for 2.5 years did not significantly lower the rate of recurrent stroke, major cardiovascular events, or diabetes. (ClinicalTrials.gov number, NCT00153062.)