Applications of Cellular Automata in Cardiac
and Ecological Systems
國立東華大學物理系
蕭又新
4/28/2006
Outline: Cardiac Systems
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Heart rate variability
Action potential and Cardiac cells
Arrhythmias and spiral waves
Spiral waves described by partial differential
equations
Cellular automata approach
Cardiac activity and ECG
正常人的心率及R-R分佈圖
食用搖頭丸的女性患者
Action Potential in a Ventricular Cell
動
作
電
位
週
期
舒
張
區
間
APD versus DI
Restitution Curve by Experiment
Restitution Curve in canine endocardial muscle
Koller, Marcus L. et al. Dynamic restitution of action
potential duration during electrical alternans and
ventricular fibrillation. Am. J. Physiol. 275(Heart Circ.
Physiol. 44): H1635-H1642, 1998.
APD、DI and T(CL)
An 1  R( Dn )
An  Dn  T
Restitution Curve
Conduction Block
Spatially distributed action potential dynamics in a
canine cardiac Purkinje fiber
Jeffrey J. Fox et al. Spatiotemporal Transition to
Conduction Block in Canine Ventricle. Circ Res.
2002;90:289-296
Normal Rhythm and Arrhythmias
Normal sinus rhythm
60~100 beats per minute
 Ectopic rhythms
For examples：
Ventricular tachycardia(心室頻脈)
Ventricular fibrillation(心室顫動)
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Ventricular Tachycardia (VT)
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Ventricular tachycardia (VT) is a
tachydysrhythmia originating from a
ventricular ectopic focus, characterized by a
rate typically greater than 120 beats per
minute and wide QRS complexes.
VT may be monomorphic or polymorphic.
Nonsustained VT is defined as a run of
tachycardia of less than 30 seconds duration;
a longer duration is considered sustained VT.
Reference
http://www.emedicine.com/
Ventricular Fibrillation (VF)
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What is ventricular fibrillation? The heart
beats when electrical signals move through it.
Ventricular fibrillation is a condition in which
the heart's electrical activity becomes
disordered. When this happens, the heart's
lower chambers contract in a rapid,
unsynchronized way. The heart pumps little
or no blood.
VT and VF in Electrocardiogram
Reference: Chaos, Solitons and Fractals Vol.13 (2002) 1755.
Normal Rhythm
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ventricle cells
2.5 days in culture
8 day old embryo
0.8 ml plating density
recorded temp: 36 deg.
C
each frame is
approximately 1 cm
square
Reference：
Optical Mapping Image Database
http://www.cnd.mcgill.ca/bios/bub/imagebase.html
Spiral Waves
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ventricle cells
2 days in culture
8 day old embryo
recorded temp: 36 deg
C.
each frame is
approximately 1 cm
square
Reference：
Optical Mapping Image Database
http://www.cnd.mcgill.ca/bios/bub/imagebase.html
Spiral Waves Breakup
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ventricle cells
2 days in culture
8 day old embryo
0.4 ml plating density
alphaGA acid 50ul
recorded temp: 36 deg C.
each frame is approximately
1 cm square
Reference：
Optical Mapping Image Database
http://www.cnd.mcgill.ca/bios/bub/imagebase.html
Experimental Results for Multi-armed
Spirals in Cardiac Tissue
Reference: PNAS, vol. 101, p15530 (2004).
Aliev-Panfilov Model
2
2

e
 e  e
 d  2  2   ke(e  a )( e  1)  er
t
y 
 x
r 
1 r 
  
  r  ke(e  b  1)
t 
( 2  e ) 
Cable Theory
 2V
 raxial I m
2
x
I m  I c  I ion ; I c  Cm
dV
dt
1  2V
V
 Cm
 I ion
2
raxial x
t
V
 2V
 I ion / Cm  D 2
t
x
D  1 /(Cm raxial )
Normal Rhythm and Conduction Block
Simulation results of normal rhythm and conduction block
Spiral Waves Formation and Breakup
Action Potential in Cardiac Muscle
Cellular Automata in Cardiac Tissue
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Activation state (6 time
units)
Refractory state (3 time
units)
Rest state
Nearest-neighbor coupling
Target Waves
Spiral Waves Formation (I)
Spiral Waves Formation (II)
Wave Breaks by Considering SpatialModulation of the Refractory Period
Wave Breaks Occurring by Heterogeneity :
Alain Karma, PNAS 97, 5687 (2000)
Simulated 3D Spirals Based on MRI
Images
256X256 grids for each frame
Enjoy Music Coming from Your
Heart
Outline: Ecological Systems
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Complexity in laboratory insect populations
Extinction in spatially structured
populations
Cellular automata approach in a modeling
ecology: grass, rabbit, and wolf
Time-domain analysis: Hurst exponent
Future works: computational epidemiology
Laboratory Insect Populations
Proc. R. Lond. B 264, 481 (1997)
Food Chain
Predator-Prey Mechanism
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Species: grass, rabbit,
and wolf
Season effect
Nearest-neighbor and
next nearest-neighbor
coupling: 8 cells
50x50 cells
Rules of Cellular Automata
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The frame of CA (50 X 50).
The components of the ecosystem.
0 ~ Carnivores ~ 1
0 ~ Herbivores ~ 3
0~
Plants
~9
Update the Population
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The next population in a cell.
Time step = 1.
now
next
Value(next) = Value(now) + Changes
The Rules of Plants
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Plants dominated by season and herbivores.
Roughly separating the season into two parts.
Pla.(next) = Pla.(now) + Changes
{
Changes
Summer
Winter
+1
–Her.
–Her.
The Neighbors of a Fixed Cell
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The affection coming from neighboring cells.
Define the local sum (L) of the population
densities.
L(i) = Value(i) + Value(j)
j = Neibors
Eight Neighbors
The Rules of Herbivores
Her.(next) = Her.(now) + Changes
If Pla. GE. Her.
Car. = 0 ; H0~L(H)~H1
+1
Otherwise
-1
{
Changes
The Rules of Herbivores
Her.(next) = Her.(now) + Changes
If Pla. LT. Her.
Changes
{
-(Her. – Pla.) – Car.
The Rules of Carnivores
Her.(next) = Her.(now) + Changes
{
Changes
Her. > 0 ; C0~L(C)~C1 +1
Otherwise
-1
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No wolf
Summer period
Complicated fluctuations
Anti-correlation in
between grass and rabbit
No extinction
Spatiotemporal Plot for Grass Evolution
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Considering wolf
Summer period
Complicated fluctuations
Positive correlation in
between rabbit and wolf
No extinction
Spatiotemporal Plots for Grass Evolution
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No wolf
Considering winter effect
(W=1, S+W=10)
Complicated fluctuations
No extinction
Anti-correlation in
between grass and rabbit
Spatiotemporal Plots for Grass Evolution
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Considering wolf
Considering longer winter
(W=3, S+W=10)
Complicated fluctuations
Wolf extinction
Anti-correlation in between
grass and rabbit
Complicated correlation in
between wolf and rabbit
Spatiotemporal Evolution of Grass
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No wolf
Considering spatial effect:
uniformly distributed rabbit
(R=1)
Summer period
Complicated fluctuations
In early stage rabbits
increase fast
Rabbit extinction
Anti-correlation in between
grass and rabbit
Spatiotemporal Plots for Grass Evolution
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No wolf
Considering spatial effect:
uniformly distributed rabbit
(R=3)
Considering winter effect
(W=1, S+W=10)
Complicated fluctuations
Surprise! slow down rabbit
extinction
Anti-correlation in between
grass and rabbit
Spatiotemporal Plots for Grass Evolution
It might be a good way to design tiles as well as carpets!
Spiral Waves in Ecology: SURPRISE!
Random Noise and Brownian Diffusion
60
50
40
4
30
3
20
10
2
0
1
-10
0
-20
0
-1
500
1000
1500
2000
2500
3000
Brownian trajectory
-2
-3
-4
0
500
1000
1500
2000
2500
Gaussian random noise
3000
 x  x 2 
1
p( x, t ) 
exp 
2 
2 t
 2 t 
 
Hurst Exponent (I)
Hurst Exponent (II)
H=0.8
H=0.6
Persistent noise: H>0.5
Random noise: H=0.5
Anti-persistent noise: H<0.5
S(f) ~ f-b, b = 2H – 1
H=0.4
H=0.2
Extinction Characterized by H: OK
Extinction Characterized by H: NOT OK
Computational Epidemiology
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S: susceptible state
(latent period)
I: Infectious state
(infectious period)
R: recovery period
Measles and Vaccination