Numerical Stability Analysis of PTW Solu7ons in Excitable...

Numerical Stability Analysis of PTW Solu7ons in Excitable RD Systems

M. Osman Gani Meiji University

Joint work Professor Toshiyuki Ogawa (Meiji Uni.)

反応拡散現象にみられる境界層とその周辺の数理 November 28, 2014

Outline •  Background and Mo:va:on (Some basics about heart and heart cells contrac:on) •  Exis:ng mathema:cal models for cardiac electrical ac:vi:es •  Existence and stability of PTWs for FHN model •  Existence and stability of PTWs for a proposed model •  Eckhaus and Hopf type instability •  Spiral wave instability in 2D •  Summary

Excitable media

•  The system which has one stable rest state (monostable) with small perturbations is called excited system.

•  Examples: 1. The axon of a nerve cell or neuron 2. The Belousov-‐Zhabo:nsky (BZ) reac:on 3. The predator-‐prey models of popula:on dynamics 4. Cardiac cell etc.

Common PaTerns in Excitable media

Rota:ng Spiral Single wave front

Target PaTern More than one wave front

Periodic Traveling Wave

Scroll Wave

Winfree(1974)

1D:

Traveling Pulse Traveling Front

2D: 3D:

Spiral PaTerns in Nature 1. Spiral PaTerns in nature

2. Spiral paTerns in BZ reac:on

E. Mihailov et al., Faraday Discuss (120),2001

3. Wave in Cardiac :ssue

www.rit.edu/cos/sms/acm/research/cardiac.php

hTp://thevirtualheart.org

hTp://en.wikipedia.org /wiki/Logarithmic_spiral

hTp://www.messersmith.name/ wordpress/2010/10/24/ of-‐turbans-‐and-‐alien-‐wri:ng/

Winfree (1974)

What is the heart?

Flow of blood:= inferior vena cava and superior vena cava (from the body) è right atrium èright ventricleè lungs è le_ atrium è le_ ventricle è aorta (to the body).

Electrical Ac:vi:es in Cardiac Cells

(Silverthorn (2012), Figure 14: 13)

Cardiac Conduc:on System

Gordon (2013)

Cardiac Ac:on Poten:al

Gordon (2013) Klabunde(2011)

Electrocardiogram (ECG/EKG)

Gordon (2013)

•  P wave: depolariza:on of the atria during atrial systole •  QRS complex: depolariza:on of the ventricles during ventricular systole •  T wave : repolariza:on of the ventricles during the relaxa:on phase

u Normal Heart Cycle=70/min.

Electrical signal on the surface of body as a consequence of the ac:on poten:als inside the heart

ECG abnormali:es

Silverthorn (2012), Fig. 14.15h

1D and 2D wave PaTerns

www.thevirtualheart.org

Fast-‐slow system

Here ε is very small :me constant and t is a slow :me

Now if we scale :me by t = ετ ,, then the above equa:ons will take the form

(2)

dudτ

= f (u,v)

dvdτ

= ε g(u,v)

!

"##

$##

(1)εdudt= f (u,v); u∈ Rn

dvdt=g(u,v); v ∈ Rm

"

#$$

%$$

0 ≤ ε <<1, , is a fast :me τ

Many models in mathema:cal physiology have a system of differen:al equa:ons:

Fast-‐slow system

(2)0

dudτ

= f (u,v)

dvdτ

= 0

!

"##

$##

(1)00 = f (u,v)dvdt=g(u,v)

!

"#

$#

Now , if we take the formal limit ε = 0 ,then (1) and (2) implies: ,which is slow dynamics and solu:ons lie on the slow manifold (cubic curve) or cri:cal manifold

dudτ

= f (u,v),which is fast dynamics and we say for fixed , the equa:on is a fast dynamics or fast subsystem

v

ε u=u(1−u)(u− a)− vv = u−γv

As for example: •  FitzHugh-‐Nagumo Equa:on:

v

u0 1a Fast

Fast

Slo

w

Slo

w

tv =0

u =0t

Hodgkin-‐Huxley Model (1952)

Cmdvdt= −gNam

3h(v− vNa )− gKn4 (v− vK )− gL (v− vL )+ Iapp

dmdt

=αm (v)(1−m)−βm (v)m

dhdt=αh (v)(1− h)−βh (v)h

dndt=αn (v)(1− n)−βn (v)n

where “v” is the membrane poten:al (fast variable), “m” is the sodium ac:va:on variable (fast variable), “h” is the sodium inac:va:on variable (slow variable), “n” is the potassium ac:va:on variable (fast variable).

m, n, and h are called ga:ng variables, is the applied current, is the membrane capacitance, are constant equilibrium poten:als and the other parameters are constant.

VNa,VK ,VLCm

Iapp

Keener and Sneyd (2010)

In order to describe the ac:on poten:al propaga:on on squids (a large nerve cell)

Noble Model (1962) (For the Cardiac Cell Dybnamics)


Based on the Hodgkin-‐Huxley model Noble in 1962 proposed the first physiological model for the dynamics of Purkinje fiber cardiac cells.

The main mo:va:on with the Noble model was to have a prolong plateau phase in the ac:on poten:al.

Several Ionic Models: McAllister et al. (1975); Beeler and Reuter (1977); Sharp and Joyner (1980); Ebihara and Johnson (1980); Luo and Rudy (1991); Winslow et al. (1993); Luo and Rudy (1994a,b), etc.

Two-‐variable Reduc:on of HH Model [FitzHugh’s Fast-‐Slow approach(1960,1961)]

Set and . h+ n = 0.8 m =m∞(v)This reduces to a two-‐variable fast-‐slow system as follows:

Cmdvdt= −gNam∞(v)(0.8− n)(v− vNa )− gKn

4 (v− vK )− gL (v− vL )

dndt=αn (v)(1− n)−βn (v)n


Two-‐variable FitzHugh-‐Nagumo Model (Simplified HH Model)

dudt= u(1−u)(u− a)− v

dvdt= ε(u−γv)

0 < a <1/ 20 < ε <<1

u: fast ac:vator (excitable) variable, v: slow inhibitor (recovery) variable

u works like membrane poten:al and v plays the role of ga:ng variables Iapp = 0

v

u0 1a Fast

Fast

Slo

w

Slo

w

tv =0

u =0t

a = 0.07,ε = 0.001,γ = 2.0

0 100 200 300 400 500 600 700 800 900 1000−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

v(t)

u(t)

Other two-‐variable models: Pertsov et al. (1984); Karma (1993); Panfilov and Hogeweg (1993); Aliev and Panfilov (1996); Panfilov (1998); etc.

Goal

The purpose of this work is to study the existence and stability of Periodic Traveling Wave solu:ons (PTWs) in the FHN model and in a proposed variant of the FHN model.

Standard FitzHugh-‐Nagumo model

g(u,v) = 0f (u,v) = 0

Fig: Nullclines of the FHN model

•

ut = duΔu+u(1−u)(u− a)− vvt = dvΔv+ε(u−γv)

0 < a <1/ 20 < ε <<1du >> dv

“u”: fast ac:vator (excitable) variable, “v”: slow inhibitor (recovery) variable

“u” works like membrane poten:al and “v” plays the role of ga:ng variables

0 20 40 60 80 100 120 140 160 180 200−0.2

0

0.2

0.4

0.6

0.8

1

1.2

v(x)

u(x)

Pulse profile

a = 0.2,ε = 0.001, γ = 2.5,du = 0.05,dv = 0.005

What is PTW solu:on?

u(x, t) =U(z) v(x, t) =V (z),and Subs:tu:ng where z = x − ctis a travelling wave coordinate, in the model equa:ons we have a system of ODE of the form :

du !!U + c !U + f (U,V ) = 0dv !!V + c !V + g(U,V ) = 0

where ' denotes w.r.to z

A_er reducing the first order the above system yields:

!U = P!P = (−cP − f (U,V )) / du!V =Q!Q = (cQ− g(U,V )) / dv

where ‘c’ is the wave speed

A PTW solu:on is a limit cycle solu:on of this ODE system.

Stability of PTWs

In order to understand the stability of PTWs our interest is to calculate the essen:al spectra of the PTWs.

u(x, t) =U(z) v(x, t) =V (z)and Again Subs:tu:ng where z = x − ctand neglec:ng the non-‐linear terms yields a linearized PDE :

∂∂tulin = D

∂2

∂x2ulin +

ulin. ΔF

ulin (x, t) =u(x, t)−

U(z).where

We obtain the eigenvalue problem: Again subs:tu:ng

ulin (x, t) = eλtU(z)

λU = c d

dzU +D d 2

dz2U +U . Δ

F

with boundary condi:on U(L) =

U(0)exp(iγ ), for some γ ∈ R,

Where L is the period of the PTW, is the phase shi_ across one period of the wave, is the Jacobian matrix of along the PTW, D is the diffusion matrix , is the eigenfunc:on and is the eigenvalue.

γ ΔF

( f (u,v), g(u,v)) U

λ

Eckhaus and Hopf Instabili:es of the PTWs

J. A. SherraT, Adv Comput Math (2013)

Method of Con:nua:on (FHN Model) !U = P!P = (−cP −U(1−U)(U − a)+V ) / du!V =Q!Q = (cQ−ε(U −γV )) / dv

1. 4-‐dim ODE system: u(x, t) =U(z)v(x, t) =V (z)z = x − ct

Where,

2. Linearized PDE:

∂ulin∂t

= du∂2ulin∂x2

+ulin (−3U2 + 2(1+ a)U − a)+ vlin (−1)

∂vlin∂t

= dv∂2vlin∂x2

+ulin (ε)+ vlin (−εγ )

ulin (x, t) = u(x, t)−U(z)vlin (x, t) = v(x, t)−V (z)

where

3. First order eigenvalue problem: ddz

Ulin

Mlin

VlinNlin

!

"

#####

$

%

&&&&&

= (A(z)+λB)

Ulin

Mlin

VlinNlin

!

"

#####

$

%

&&&&&

A =

0 1 0 0

−1du(−3U 2 + 2(1+ a)U − a) −c

du1du

0

0 0 0 1−εdv

0 εγdv

−cdv

"

#

$$$$$$$

%

&

'''''''

where

and B =

0 0 0 01du

0 0 0

0 0 0 0

0 0 1dv

0

!

"

#######

$

%

&&&&&&&

Existence of PTWs as a func:on of the parameter “ ”

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

c

gamma

8.9e+1

8.9e+1

9.0e+1

9.2e+1

1.0e+2

1.0e+2

1.0e+2

1.1e+2

1.2e+2

1.3e+2

6.5e+1

6.5e+1

6.6e+1

6.8e+1

7.0e+1

7.4e+1

7.8e+1

8.3e+1

9.0e+1

1.0e+2

4.7e+1

4.8e+1

4.8e+1

5.0e+1

5.2e+1

5.4e+1

5.7e+1

6.1e+1

6.6e+1

7.4e+1

3.3e+1

3.4e+1

3.4e+1

3.5e+1

3.6e+1

3.8e+1

4.0e+1

4.3e+1

4.7e+1

5.3e+1

2.2e+1

2.2e+1

2.3e+1

2.3e+1

2.4e+1

2.5e+1

2.7e+1

2.9e+1

3.2e+1

3.5e+1

1.4e+1

1.4e+1

1.4e+1

1.4e+1

1.5e+1

1.6e+1

1.7e+1

1.8e+1

2.0e+1

2.2e+1

7.5e00

7.6e00

7.8e00

8.0e00

8.3e00

8.7e00

9.3e00

9.9e00

1.1e+1

1.2e+1

4.0e00

4.0e00

γ

=No PTWs

-0.4-0.2

0 0.2 0.4 0.6 0.8

1

U

-1-0.8-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

1

P

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.1 0.11 0.12

V-0.012

-0.01-0.008-0.006-0.004-0.002

0 0.002 0.004 0.006 0.008

0.01

0 0.2 0.4 0.6 0.8 1Q

z/Period

Stability of PTWs and Essen:al Spectra

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

c

gamma

Since the curvature of the spectrum changes sign at the origin, This indicate a stability change of Eckhaus type.

=Stable PTW =Unstable PTW

-0.005 -0.004 -0.003 -0.002 -0.001 0

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Im(e

igen

valu

e)

0 2

Re(eigenvalue)

Re=0.0E-1

-0.0005 -0.0004 -0.0003 -0.0002 -0.0001 0-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Im(e

igen

valu

e)

0 2

Re(eigenvalue)

Re=6.720E-6

Stability Boundary of Eckhaus type

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

c

gamma

Stable

Unstable

0

10

20

30

40

50

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

pe

rio

d

c

Fold: c=0.1906E-01

Eckhaus:c=0.3438626E-01

Min. stable period=9.24

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

-0.02 -0.015 -0.01 -0.005 0

Im(e

igenvalu

e)

Re(eigenvalue)

Re=0.0E-1

-0.04

-0.02

0

0.02

0.04

-0.01 -0.008 -0.006 -0.004 -0.002 0 0.002

Im(e

igenvalu

e)

Re(eigenvalue)

Re=9.763E-4

iii γ = 2.5

Our proposed model ut = duΔu+u(1−u)(u− a)− vvt = dvΔv+ε(du(b−u)(u+ c)− v)

0 < a <1/ 20 < ε <<1b >1

u: fast ac:vator (excitable) variable, v: slow inhibitor (recovery) variable

a,b,c,d,ε,du,dv are parameters; du >> dv

0 5 10 15 20 25 30 35 40 45 50−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

2

u(x)

v(x)

3

4

1 1

Pulse Profile Nullclines of the model

g(u,v) = 0

f (u,v) = 0 more slow

Neuron and Cardiac Ac:on Poten:als membrane

•  An ac7on poten7al is a short-‐las:ng event in which the electrical membrane poten:al of a cell rapidly rises and falls

•  Cardiac AP has a prolonged plateau phase las:ng around 300 millisecond (ms) compared with 1 ms in nerves.

Neuron AP Cardiac AP

Plateau Phase

(FHN) m-‐FHN

Ac:on Poten:als as a func:on of parameter “ ” ba b c d du dv ε

0.07 free 3.0 0.21 0.05 0.005 0.011

0 50 100 150 200 250 300 350 400 450

−0.2

0

0.2

0.4

0.6

0.8

1

t

u

b=1.05b=1.1b=1.2

Ac:on Poten:als Nullclines

−0.2 0 0.2 0.4 0.6 0.8 1

0

0.05

0.1

0.15

0.2

0.25b=1.1

b=1.2

b=1.05

u

v

g=0

f=0

PDE Simula:on (Bifurca:on Diagram)

System size = Spa:al period, i.e. L= l=25

The bifurca:on point is found at b=1.0295. Let us say this cri:cal value b1

1.026 1.027 1.028 1.029 1.03 1.031 1.032 1.03317.5

18

18.5

19

19.5

20

20.5

21

21.5

22

22.5

b

max

and

min

wid

th o

f wav

e pu

lses

maximum widthminimum width

0

5

10

15

20

8500

9000

9500

10000

tx

0

5

10

15

20

80008200

84008600

88009000

92009400

96009800

10000

tx

0 5 10 15 20 25

−0.2

0

0.2

0.4

0.6

0.8

1

x

u

Bifurca:on Diagram (L=50, l=25)

1.029 1.03 1.031 1.032 1.033 1.034 1.035 1.036 1.037 1.038

12

14

16

18

20

22

24

26

28

30

b

max

. and

min

. wid

th o

f wav

e pu

lses

maximum widthminimum width

0 10 20 30 40 50−0.4

−0.2

0

0.2

0.4

0.6

0.8

1b=1.035

x

u

b2The bifurca:on point is detected here at b = 1.0343. Let us say this cri:cal value

b1 < b2 <So one can expect:

b=1.038

b=1.025

0 10 20 30 40 50−0.4

−0.2

0

0.2

0.4

0.6

0.8

1b=1.03

x

u

Stable and Oscillatory PaTern of Solu:ons

b=1.038 b = 1.03

Stable paTern Oscillatory PaTern

For Con:nua:on Technique in the Proposed Model

!U = P!P = (−cP −U(1−U)(U − a)+V ) / du!V =Q!Q = (−cQ−ε(dU(b−U)(U + c)−V )) / dv

1. 4-‐dim ODE system: u(x, t) =U(z)v(x, t) =V (z)z = x − ct

Where,

2. Linearized PDE:

∂ulin∂t

= du∂2ulin∂x2

+ulin (−3U2 + 2(1+ a)U − a)+ vlin (−1)

∂vlin∂t

= dv∂2vlin∂x2

+ulin (ε(−3dU2 + 2(db− dc)U + dbc))+ vlin (−ε)

ulin (x, t) = u(x, t)−U(z)vlin (x, t) = v(x, t)−V (z)

where

3. First order eigenvalue problem: ddz

Ulin

Mlin

VlinNlin

!

"

#####

$

%

&&&&&

= (A(z)+λB)

Ulin

Mlin

VlinNlin

!

"

#####

$

%

&&&&&

A =

0 1 0 0

−1du(−3U 2 + 2(1+ a)U − a) −c

du1du

0

0 0 0 1−εdv[−3dU 2 + 2(db− dc)U + dbc] 0 ε

dv−cdv

"

#

$$$$$$$

%

&

'''''''

where

and A =

0 0 0 01du

0 0 0

0 0 0 0

0 0 1dv

0

!

"

#######

$

%

&&&&&&&

Existence of PTWs in the Proposed Model

= No PTWs

0.04

0.06

0.08

0.1

0.12

0.14

1 1.1 1.2 1.3 1.4 1.5

c

b

1.4e+1

1.5e+1

1.5e+1

1.6e+1

1.7e+1

1.9e+1

2.1e+1

2.6e+1

3.8e+1

1.2e+1

1.2e+1

1.2e+1

1.3e+1

1.4e+1

1.5e+1

1.7e+1

2.1e+1

3.2e+1

9.5e00

9.8e00

1.0e+1

1.1e+1

1.1e+1

1.2e+1

1.4e+1

1.7e+1

2.5e+1

7.7e00

7.9e00

8.2e00

8.6e00

9.0e00

9.7e00

1.1e+1

1.3e+1

1.9e+1

6.3e00

6.5e00

6.8e00

7.1e00

7.6e00

8.3e00

9.8e00

1.4e+1

5.7e00

6.1e00

7.0e00

9.5e00

5.7e00

-0.4-0.2

0 0.2 0.4 0.6 0.8

1

U

-0.8-0.6-0.4-0.2

0 0.2 0.4 0.6 0.8

1

P

0 0.02 0.04 0.06 0.08

0.1 0.12 0.14

V

-0.04-0.03-0.02-0.01

0 0.01 0.02 0.03 0.04

0 0.2 0.4 0.6 0.8 1

Q

z/Period

Stability of the PTWs Through Essen:al Spectra

0.04

0.06

0.08

0.1

0.12

0.14

1 1.1 1.2 1.3 1.4 1.5

c

b

= Stable PTWs = Unstable PTWs = No PTWs = Stability boundary (Eckhaus Type)

-0.025 -0.02 -0.015 -0.01 -0.005 0

-0.1

-0.05

0

0.05

0.1

Im(e

igen

valu

e)

0 2

Re(eigenvalue)

Re=0.0E-1

-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Im(e

igen

valu

e)

0 2

Re(eigenvalue)

Re=7.73E-3

Fast Family of PTWs

0.04

0.06

0.08

0.1

0.12

0.14

1 1.1 1.2 1.3 1.4 1.5

c

b

25

15

10 Unstable(U)

U Stable

0.04

0.06

0.08

0.1

0.12

0.14

1 1.01 1.02 1.03 1.04 1.05 1.06

c

b

25

15

10

StableUnstable

b2b1

Magnifica:on

•  Fast Family crosses the stability boundary in our model •  The bifurca:on points b_1 and b_2 lie in the unstable area

= Fast family with constant period

Bifurca:on Diagrams or Dispersion Curves

0

10

20

30

40

50

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13

perio

d

c

Fold: c=0.4350E-01

Fold: c=0.6494E-01

Fold: c=0.1269E+00

E

Min. stable period=6.51 0

10

20

30

40

50

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13

perio

d

c

Fold: c=0.3794E-01

Fold: c=0.6293E-01E

Min. stable period=6.88 0

10

20

30

40

50

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13

perio

dc

Fold: c=0.3731E-01

Fold: c=0.6274E-01


E

b=1.1 b=1.04 b = b2 =1.0343

0

10

20

30

40

50

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12

perio

d

c

Fold: c=0.5076E-01

Fold: c=0.6815E-01

Fold: c=0.1255E+00

0

10

20

30

40

50

0.06 0.07 0.08 0.09 0.1 0.11 0.12

perio

d

c

Fold: c=0.5394E-01

Fold: c=0.6971E-01

Fold: c=0.1247E+00

0

10

20

30

40

50

0.055 0.06 0.065 0.07 0.075

perio

d

cb=2.2 b=1.25 b=1.2

Essen:al Spectra ( Eckhaus and Hopf instability)

0.016

0.018

0.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

1 1.02 1.04 1.06 1.08 1.1 1.12 1.14

c

b

f

e

cd

a

b

-0.005 -0.004 -0.003 -0.002 -0.001 0 0.001 0.002

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Im(e

igen

valu

e)

0 2

Re(eigenvalue)

Re=9.821E-4

-0.003 -0.0025 -0.002 -0.0015 -0.001 -0.0005 0-0.03

-0.02

-0.01

0

0.01

0.02

0.03

Im(e

igen

valu

e)

0 2

Re(eigenvalue)

Re=0.0E-1

-0.0002 -0.00015 -0.0001 -5e-05 0 5e-05 0.0001 0.00015

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Im(e

igen

valu

e)

0 2

Re(eigenvalue)

Re=8.787E-5

-0.0001 -8e-05 -6e-05 -4e-05 -2e-05 0 2e-05-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Im(e

igen

valu

e)0 2

Re(eigenvalue)

Re=-1.127E-5

-0.0001 -5e-05 0 5e-05 0.0001 0.00015 0.0002

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Im(e

igen

valu

e)

0 2

Re(eigenvalue)

Re=1.688E-4

Essen:al Spectra in the different parts of a dispersion curve

4

6

8

10

12

14

16

18

20

0.025 0.03 0.035 0.04

peri

od

c

Fold: c=0.2364E-01

E

H

Min. stable period=10.85(H)

Min. stable period=8.10(E)

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

-0.005 -0.004 -0.003 -0.002 -0.001 0

Im(e

igenvalu

e)

Re(eigenvalue)

Re=0.0E-1

-0.02

-0.01

0

0.01

0.02

-0.0005 -0.0004 -0.0003 -0.0002 -0.0001 0

Im(e

igenval

ue)

Re(eigenvalue)

Re=-3.433E-7

-0.02

-0.01

0

0.01

0.02

-0.0002-0.0001 0 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006

Im(e

igenv

alue

)

Re(eigenvalue)

Re=3.213E-4

-0.02

-0.01

0

0.01

0.02

0 0.0005 0.001 0.0015 0.002

Im(e

igenvalu

e)

Re(eigenvalue)

Re=1.713E-3

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

-0.0005 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

Im(e

igen

valu

e)

Re(eigenvalue)

Re=2.883E-3

iii

iv

ii

i

v

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Im(e

igenv

alue)

Re(eigenvalue)

Re=-4.309E-3

vi

Dispersion curve for b=1.05

PDE Simula:on as a consequence of the Hopf instability

0 20 40 60 80 8440

8460

8480

8500

x

t

L=86.96, l=10.87

0 20

40 60

80

4860

4880

4900

x

t

L=86.4, l=10.8

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16 1.18 1.2

c

b

period=10.85U S

U

FHN vs m-‐FHN

0

10

20

30

40

50

0.055 0.06 0.065 0.07 0.075

perio

d

c

b=2.2

0

10

20

30

40

50

0.048 0.05 0.052 0.054 0.056 0.058 0.06 0.062 0.064

perio

d

c

Eckhaus:c=0.561E-1


a=0.24

0.04

0.06

0.08

0.1

0.12

0.14

1 1.1 1.2 1.3 1.4 1.5

c

b

25

15

10 Unstable(U)

U Stable

0

0.05

0.1

0.15

0 0.05 0.1 0.15 0.2 0.25

c

a

25

25

U

S

•  Thus in our model by controlling the parameter “b” we control the stability of solu:ons •  The fast family with sufficiently large period always stable in the FHN model, whereas in our model they crosses the stability boundary and this is responsible for the oscillatory paTern of solu:on in the full PDE simula:on

Remarks:

Spiral wave forma:on

b =1.05

Spiral pulses widths increasing as b decreased

Ini:al data b=1.2

b=1.05 b=1.04

b=1.3

b=1.03

b=1.1

b=1.035

Calcula:on of Spiral Pulse widths and onset of oscilla:on

Figure: Max and Min width of spiral pulses as a funcMon of b.

Pulse width = x1 − x2

0 10 20 30 40 50 60 70 80 90 100−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

u

x1 x2

BA

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

b

Wid

ths

of s

pira

l pul

ses

Maximum widthMinimum width

b=1.038

Speed of parallel wave in 2D as a func:on of b (asterisk line)

0.04

0.06

0.08

0.1

0.12

0.14

1 1.1 1.2 1.3 1.4 1.5

c

b

15

U S

U

Stability of PTWs when “ “

0.04

0.06

0.08

0.1

0.12

0.14

1 1.1 1.2 1.3 1.4 1.5

c

b

25

15

UStable

Unstable(U)

dv = 0.005

0.04

0.06

0.08

0.1

0.12

0.14

1 1.1 1.2 1.3 1.4 1.5

c

b

25

15

U

S

U

dv = 0

0

10

20

30

40

50

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13

perio

d

c

Fold: c=0.3952E-01

Fold: c=0.6553E-01

E(0.05143,8.86)

b=1.05dv = 0

0

10

20

30

40

50

0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13

perio

d

c

Fold: c=0.3899E-01

Fold: c=0.6327E-01

b=1.05

EE(0.0438,6.52)

dv = 0.005

dv = 0

PDE Simula:on, when dv = 0

0 30 60 90 120 6000

6030

6060

6090

x

t

b=1.03

0 30 60 90 120 6000

6030

6060

6090

x

t

b=1.055 b=1.045 Onset of instability

However, for d_v=0.005, the onset of instability b = 1.0355

Spiral Dynamics, when d_v=0

Ini:al data

b=1.05

b=1.2 b=1.1

b=1.04

Thus the stable spiral paTern bifurcate to an unstable paTern between b=1.1 and b=1.05

Summary •  We have studied the stability of PTWs numerically in FHN-‐type RD systems for excitable media

•  We found two types of instability in the proposed model: Eckhaus type and Hopf type •  The fast family is stable in the FHN model, whereas it crosses the stability boundary in the proposed model.

•  The alternans, i.e. the oscilla:on of pulse width, is a consequence of the instability of PTWs.

•  The onset of spiral wave instability was observed as a consequence of Eckhaus instability of the PTWs.

Thank you for your aQen7on!!

Numerical Stability Analysis of PTW Solu7ons in Excitable...

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