Traveling spots in an excitable medium and ventricular fibrillation

Transcription

1 Meiji University School of Interdisciplinary Mathematical Sciences 1 Traveling spots in an excitable medium and ventricular fibrillation Hirokazu Ninomiya Meiji University School of Interdisciplinary Mathematical Sciences Interna'onal Conference on Mathema'cal Modeling and Applica'ons, Meiji University, Nov. 26,

2 Motivations Traveling spots Singular limit problem Existence of traveling spot Contents Application to ventricular fibrillation Influence of obstacles Tip formation Joint work with Y.- Y. Chen and Y. Kohsaka Joint work with M. Kaihara, T. Shimizu, N. J. Suematsu Collaborators Y.- Y. Chen (Meiji University) Y. Kohsaka (Muroran Ins:tude of Technology) M. Kaihara (Meiji University) T. Shimizu (Meiji University) N. J. Suematsu (Meiji University) 2

3 Patterns in an excitable medium To capture a solution of PDE (especially RD) in multidimensional space means to characterize its pattern There are many results for 1D Bifurcation of traveling spots Ohta, Mimura, Kobayashi (1989): radially symmetric SS Krischer, Mikhailov (1994): Bifurca'on of TS from SS Or- Guil, Bode, Schenk, Purwins (1998): 3 components system Pismen (2001): TS with sharp interface and bifurca'on Hagberg, Meron (2003), Nishiura, Teremoto, Ueda (2005) Photosensitive BZ reaction Mihaliuk, Sakurai, Chirila, Showalter (2001) Sakurai- Mihaliuk- Chirila- Showalter (2002, SCIENCE) Zykov, Showalter (2005): wave front interac'on model in an excitable media Guo, Ninomiya, Tsai (2010) As the first step, we need to get more informa'on of non- radial TS. 3

4 FitzHugh-Nagumo equation and its singular limit problem FitzHugh-Nagumo equation (ODE) where u t = u + 1 (f (u) 2 v) v t = g(u, v) f " (u) =u(1 g(u, v) =g 1 u u)(u g 2 v g 0 + g 3 v u t = 1 (f (u) 2 v) v t = g(u, v) " ) := g 2 g 1 g 3 1 2, g i > 0 Excitable system (mono- stable) 2 u t = 2 u + (f (u) v) v t = g(u, v) f 0 (u) = 0 u = 0, 1 2, 1 Interface appears u res'ng region (t) exci'ng region 4

5 FitzHugh-Nagumo equation and its singular limit problem FitzHugh-Nagumo equation u t = u + 1 (f (u) 2 v) v t = g(u, v) f 0 (u) = 0 u = 0, 1 2, 1 v Interface appears u (t) V exci'ng region (t) res'ng region V = C(v) v t = g( (t), v) where C(v) = a bv a = 2, b = 6 2 Planar waves with speed a V: normal velocity h: a root of v=f(u) κ:curvature χ: characteris'c func'on 5

6 Theorem (Chen-Kohsaka-Ninomiya 2013) (,v) V = a bv v t = g( (t), v) For any 0<c<a, there are b and a traveling spot with speed c such that lim v(x, y, 0) = 0, v(x, y, 0) = 0 if y Y M. x 2 +y 2 Moreover, (i) 0 c c (ii) c < c < a There is a convex traveling spot Non-convex traveling spot circle c c c planar There is a traveling spot with any large size. 6

7 V = a bv v t = g( (t), v) Outine of proof Traveling wave solu'on (t) = {(x + ct, y) (x, y) (0)} v(x, y, t) = v(x ct, y) Tip V c c cos = a bv cv x = g( (0), v) Back X (y) (t) X + (y) Front We denote the interface where V>0 (resp. V<0, V=0) by the front (resp. back x = X (y), 'p ) Due to the effect of v, V is nega've. { d dx ± ds = sin ± dy ± ds = cos ± ± ds = apple ± = a bv ± c cos ± x = X + (y) 7

8 Numerical solutions of FitzHugh- Nagumo equation u t = u v t = g(u, v) (f (u) v) Profile of u Profile of v 8

9 Application to ventricular fibrillation (VF) 心室細動 ut vt = u + u u3 = u av + b Wavebreaks may take place by an obstacle Jalife et al (1998) Tanaka (2008) v

10 Formation of wavebreaks Shimizu, Kaihara, Suematsu and N. (2013) Heart as an excitable medium Obstacle Wave moves downward Fibrilla'on occurs by obstacle Using numerical calcula'on, we can characterize the rela'onship between the shape of domain and the 'p forma'on. 10