Seminar, October 20th & 21th 2014
Monday 20th ReaDi-EHESS, 190-198 av. de France, 75013 Paris, Room 466
16 h : Jack Cowan (University of Chicago)
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Geometric Visual Hallucinations: Past, Present, and Future
In 1979 Ermentrout and Cowan showed how to analyze spatial pattern formation in the form of perceived geometric hallucinations in a simplified model of visual cortex, using equivariant bifurcation theory applied to the mean-field Wilson-Cowan equations. In doing so they introduced the idea that the visual cortex possessed the Euclidean symmetry of the plane E(2), and had the Cartesian coordinates {x,y} at every point of the plane ℜ2. They then provided the first account of how geometric visual hallucinations might be generated in the brain. In 2001 Bressloff, Cowan, Golubitsky, Thomas, and Wiener introduced a more elaborate model of visual cortex, in which the coordinates of any cortical location were now {x,y,ϕ} where ϕ is the orientation preference at the point {x,y}. The space is now ℜ2xS1. In order to preserve Euclidean symmetry on mapping ℜ2xS1 onto a discrete lattice, the rotation operator θ maps {x,y,ϕ} into {Rθ{x,y},ϕ+θ}. In computer vision this is called a shift-twist operator. They then used the equivariant branching lemma of Golubitsky, Stewart, and Schaeffer (1988), to extend the classes of hallucinatory images generated by such models. In 2002 Bressloff and Cowan showed that this model could in turn be extended to include spatial frequency preference by representing the cortical space as ℜ2xS2 in which orientation and spatial frequency preferences are represented by the angular coordinates of the sphere S2. In this lecture I hope to show that the remaining features of early vision: color, depth and motion can also be represented in similar fashion, and that visual hallucinations involving such features can be incorporated into the geometric patterns seen as visual hallucinations. |
Tuesday 21th ReaDi-EHESS, 190-198 av. de France, 75013 Paris, Room 466
11 h : Hirokazu Ninomiya (Meiji University)
Traveling pulses and traveling spots of a free boundary problem arising from excitable systems
In this talk we consider the singular limit problems arising from FitzHugh-Nagumo type model, which is a free boundary problem.
The singular limit problem of FHN system has been studied by many authors. However, the dynamics of the free boundary problem are not well known yet. To understand the dynamics, we first consider this problem in one dimensional space. In this case we can show that all solutions can be represented by the sum of traveling fronts and the traveling pulses as t tends to infinity. In the two dimensional space, the problem becomes difficult.
Thus we consider the dynamics in some special setting. I explain the existence of the two-dimensional traveling spots including the front and the back. Using this traveling spots, I will explain some mathematical understanding of the formation of spirals which is induced by obstacles. The formation of the spiralsdepends on the shapes of the obstacle. This spiral formation is deeply related to the ventricular fibrillation.
In this talk we consider the singular limit problems arising from FitzHugh-Nagumo type model, which is a free boundary problem.
The singular limit problem of FHN system has been studied by many authors. However, the dynamics of the free boundary problem are not well known yet. To understand the dynamics, we first consider this problem in one dimensional space. In this case we can show that all solutions can be represented by the sum of traveling fronts and the traveling pulses as t tends to infinity. In the two dimensional space, the problem becomes difficult.
Thus we consider the dynamics in some special setting. I explain the existence of the two-dimensional traveling spots including the front and the back. Using this traveling spots, I will explain some mathematical understanding of the formation of spirals which is induced by obstacles. The formation of the spiralsdepends on the shapes of the obstacle. This spiral formation is deeply related to the ventricular fibrillation.