ERC project ReaDi
  • Home
    • Directions & Maps
  • The project
  • The team
  • Jobs
  • News and Links
  • Untitled
    • Reaction-diffusion, propagation and modeling
    • ReaDi workshop 2018 >
      • 1st session, September 24th & 25th 2013
      • 2nd session, November 12th & 13th 2013
      • 3rd session, December 9th & 10th 2013
      • 4th session, February 24th & 25th 2014
      • 5th session, March 25th 2014
      • 6th session, October 6th & 7th 2014
      • 7th session, October 27th & 28th 2014
      • 8th session, December 16th & 17th 2014
      • 9th session, March 16th & 17th 2015
      • 10th session, April 13th 2015
      • 11th session, May 18th & 19th 2015
      • 12th session, June 23rd 2015
    • Between discrete and continuous
    • ReaDi punctual sessions >
      • June 10th 2014
      • October 20th & 21th 2014
      • November 6th 2014
      • May 12th 2015
      • June 16th 2015
    • PDE seminar >
      • January 26th 2016
      • February 10th 2016
      • February 18th 2016
      • February 25th 2016
      • March 9th 2016
      • March 16th 2016
      • March 17th 2016
      • March 23th 2016
      • March 31st 2016
      • April 7th 2016
      • April 14th 2016
      • April 28th 2016
      • May 12th 2016
      • May 18th 2016
      • May 26th 2016
      • Previous years >
        • February 11th 2015
        • February 18th 2015
        • February 25th 2015
        • March 11th 2015
        • March 25th 2015
        • April 1st 2015
        • April 8th 2015
        • April 22th 2015
        • May 13th 2015
    • Workshop on Mathematics and Social Sciences
  • Chairs and Visitors
  • Papers
  • ReaDi workshop 2018

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)
Photo
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.
Powered by
✕