In this paper, we establish a new global phenomenological model of ischemic stroke. It takes into account local ischemia, energy reduction, propagation of spreading depressions (SD), damages to the cells and cellular death by apoptosis or necrosis. The spatial diffusion of the ions in the extracellular space which triggers the propagation of SD is a central point here. First we expose the various biological hypotheses that we have made in this model, and then we explain how to determine the parameters and solve the system of equations that we obtain. Next we present some results of this model: we simulate a KCl injection and then a local ischemia. Finally we discuss results and propose some improvements for this model.

In this paper, we give examples of the influence of the domain of propagation on progressive waves. More precisely, we numerically investigate the propagation of reaction diffusion waves in cylinders with variable radius. We show that, when the radius rapidly expands from a very small radius to a larger one, depending on the viscosity and the nonlinearity, the travelling wave may be blocked. The aim of this paper is to give numerical illustrations and quantifications of this effect, and to propose some conjectures which could be interesting subjects for further mathematical investigations.This work is linked to the study of spreading depression (SD), a propagative mechanism in brain and various tissues which has been observed in vivo and in vitro in many species since their discovery in 1944 by Leao. As a matter of fact, their direct observation in Man is still controversial. The complex structure of gray and white matter in humans may block the propagation of SD over large distances in brain and thus explain the difficulty of observing it. Medical consequences of the current numerical studies are detailed in [M.A. Dronne, et al., Influence of brain geometry on spreading depressions: a computationnal study, Progress in Biophysics and Molecular Biology 97 (1) (2008) 54-59] and a first mathematical approach given in [M.A. Dronne, E. Grenier, H. Gilquin, Modelization of spreading depressions following Nedergaard, preprint, 2003]. (C) 2008 Elsevier Ltd. All rights reserved.

Migraine with aura is a complex phenomena, which remains still not completely understood. A striking fact is that its clinical manifestations may change from one patient to another. Migraine with aura may only consist in visual hallucinations, but may as well go on to temporary aphasy. However, for all the patients it always stops before it goes from area 3 to area 4, thus just before crossing Rolando sulcus. In this paper, we give arguments showing that the detailed geometry of Rolando sulcus in human cortex may by itself explain that migraine attack never crosses Rolando sulcus.

In this paper, we study the influence of the intensity and the duration of blood flow reduction on necrosis and apoptosis during an acute ischemic stroke. We use in silico experiments to show that it influences a lot the size of the infarcted area. The stronger the intensity of the the blood flow reduction is, the bigger is the infarcted area. Both apoptotic and necrostic area are increased. Th duration of the bloodf flow reduction has a more complicate role on the size of the infarcted area. If the return to a normal blood flow is badly synchronised with the propagation of spreading depressions, the size of the apoptotic area can increase a lot. These in silico results are then discussed.