Meiji University
School of Interdisciplinary Mathematical Sciences

1Traveling spots in an excitablemedium 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,	  2013	 1	

Contents• Motivations• Traveling spots– Singular limit problem– Existence of traveling spot
Joint	  work	  with	  Y.-­‐Y.	  Chen	  and	  Y.	  Kohsaka	• Application to ventricular fibrillation– Influence of obstacles– Tip formation

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	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.	  

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FitzHugh-Nagumo equation and its
singular limit problem• FitzHugh-Nagumo equation
ut

=u+vt

= g(u, v)f" (u) = u(1where	g(u, v) = g1 u(ODE)	

2utvt

112

(f (u)

1u)(u+ "↵)g v22

g0 + g3 v

ut=vt= g(u, v)==u + (f (u)g(u, v)

f0 (u) = 022v)(f (u)

g2
:=
g1 g3

1, gi > 02v)
Excitable	  system	  (mono-­‐stable)	v)

1
u = 0, , 1
2Interface	  appears	
u	res'ng	  region	
⌦(t)

exci'ng	  region	

4	

FitzHugh-Nagumo equation and its
singular limit problem• FitzHugh-Nagumo equation
ut

=u+vt

= g(u, v)

f0 (u) = 0
Interface	  appears	
u	v	⌦(t)V

exci'ng	  region	⌦(t)2(f (u)v)

1u = 0, , 1
2

V = C(v)vt = g( (t) , v)
where	res'ng	  region	

1C(v) = aa=	  V:	  normal	  velocity	  	  h:	  a	  root	  of	  v=f(u)	  κ:Curvature	  	χ:	  characteris'c	  func'on	  bv2 ,b = 6 2

Planar	  waves	  with	  speed	  a	5	

Theorem (Chen-Kohsaka-Ninomiya 2013)

V = a bv
vt = g( (t) , v)

For any 0<c<a, there are b and a traveling spot (⌦, v) with speedc such thatlim

x2 +y 2v(x, y, 0) = 0,Moreover,(i) 0cv(x, y, 0) = 0 if |y|(ii) c < c < ac

There is a convex traveling spotcircle	c

YM .

cc

There	  is	  a	  traveling	  spot	  with	  any	  large	  size.	

Non-convex traveling spot

planar	6	Outine of proof
V = a bv
vt = g( (t) , v)
TipV(t) = {(x + ct, y) | (x, y)

cBackFront

X (y)Traveling	  wave	  solu'on	

X+ (y)

v(x, y, t) = v(xct, y)c cos = acvx = g(bv(0)}(0) , v)

We	  denote	  the	  interface	  where	  V>0	  (resp.	  V<0,	  V=0)	  by	  the	  front	  	  	  	  	   x = X+ (y)(resp.	  back	  	  	  x	  	  	  =	  	  	  	  X	  	  	  	  	  (y)	  	  	  	  	  ,	  'p	  )	Due	  to	  the	  eﬀect	  of	  v,	  V	  is	  nega've.	

⌦(t){dx±= sin ✓±ds
dy±= cos ✓±dsd✓±= ± = ads

bv±

c cos ±

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Numerical solutions of FitzHughNagumo equation

Proﬁle	  of	  u	ut

=u+vt

= g(u, v)

12(f (u)v)

Proﬁle	  of	  v	

8	Application to ventricularfibrillation (VF)
心室細動	utvt=u + u u3
= u av + b

Wavebreaks	  may	  take	  place	  by	  an	  obstacle	  	• Jalife et al (1998)• Tanaka (2008)v

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.	
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