1. Applications of Cellular Automata in Cardiac and Ecological Systems 國立東華大學物理系 蕭又新 4/28/2006

2. Outline: Cardiac Systems Heart rate variability Action potential and Cardiac cells Arrhythmias and spiral waves Spiral waves described by partial differential equations Cellular automata approach

3. Cardiac activity and ECG

5. 正常人的心率及 R-R 分佈圖 食用搖頭丸的女性患者

6. Action Potential in a Ventricular Cell

7. APD versus DI

8. 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.

9. APD 、 DI and T(CL)

10. Restitution Curve A+D=T 1:1 2:2

11. 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

12. 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 )

13. Ventricular Tachycardia (VT) 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/

14. Ventricular Fibrillation (VF) 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.

15. VT and VF in Electrocardiogram Reference: Chaos, Solitons and Fractals Vol.13 (2002) 1755. Normal Beats (NB) NB to VT VT to VF

16. Normal Rhythm 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

17. Spiral Waves 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

18. Spiral Waves Breakup 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

19. Experimental Results for Multi-armed Spirals in Cardiac Tissue Reference: PNAS, vol. 101, p15530 (2004).

20. Aliev-Panfilov Model

21. Cable Theory

22. Normal Rhythm and Conduction Block Simulation results of normal rhythm and conduction block

23. Spiral Waves Formation and Breakup

24. Action Potential in Cardiac Muscle

25. Cellular Automata in Cardiac Tissue Activation state (6 time units) Refractory state (3 time units) Rest state Nearest-neighbor coupling

26. Target Waves

27. Spiral Waves Formation (I)

28. Spiral Waves Formation (II)

29. Wave Breaks by Considering Spatial- Modulation of the Refractory Period

30. Wave Breaks Occurring by Heterogeneity : Alain Karma, PNAS 97, 5687 (2000)

31. Simulated 3D Spirals Based on MRI Images 256X256 grids for each frame

33. Enjoy Music Coming from Your Heart

34. Outline: Ecological Systems 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

36. Laboratory Insect Populations

37. Proc. R. Lond. B 264, 481 (1997)

39. Food Chain

40. Predator-Prey Mechanism Species: grass, rabbit, and wolf Season effect Nearest-neighbor and next nearest-neighbor coupling: 8 cells 50x50 cells

41. The frame of CA (50 X 50). The components of the ecosystem. 0 ~ Carnivores ~ 1 0 ~ Herbivores ~ 3 0 ~ Plants ~ 9 Rules of Cellular Automata

42. The next population in a cell. Time step = 1. = Value(now) + Changes Value(next) = Value(now) + Changes now next Update the Population

43. Plants dominated by season and herbivores. Roughly separating the season into two parts. Pla.(next) = Pla.(now) + Changes Changes Summer +1 –Her. Winter –Her. The Rules of Plants

44. The affection coming from neighboring cells. Define the local sum (L) of the population densities. Eight Neighbors L(i) = Value(i) + Value(j) j = Neibors j = Neibors The Neighbors of a Fixed Cell

45. Her.(next) = Her.(now) + Changes If Pla. GE. Her. Changes Car. = 0 ; H 0 ~L(H)~H 1 +1 Otherwise -1 The Rules of Herbivores

46. Her.(next) = Her.(now) + Changes If Pla. LT. Her. Changes -(Her. – Pla.) – Car. -(Her. – Pla.) – Car. The Rules of Herbivores

47. Her.(next) = Her.(now) + Changes Changes Her. > 0 ; C 0 ~L(C)~C 1 +1 Otherwise -1 The Rules of Carnivores

48. No wolf Summer period Complicated fluctuations Anti-correlation in between grass and rabbit No extinction

49. Spatiotemporal Plot for Grass Evolution

50. Considering wolf Summer period Complicated fluctuations Positive correlation in between rabbit and wolf No extinction

51. Spatiotemporal Plots for Grass Evolution

52. No wolf Considering winter effect (W=1, S+W=10) Complicated fluctuations No extinction Anti-correlation in between grass and rabbit

53. Spatiotemporal Plots for Grass Evolution

54. 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

55. Spatiotemporal Evolution of Grass

56. 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

57. Spatiotemporal Plots for Grass Evolution

58. 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

59. Spatiotemporal Plots for Grass Evolution It might be a good way to design tiles as well as carpets!

60. Spiral Waves in Ecology: SURPRISE!

61. Random Noise and Brownian Diffusion Gaussian random noise Brownian trajectory

62. Hurst Exponent (I)

63. Hurst Exponent (II) H=0.8 H=0.6 H=0.4 H=0.2 Persistent noise: H>0.5 Random noise: H=0.5 Anti-persistent noise: H<0.5 S(f) ~ f-b, b = 2H – 1

64. Extinction Characterized by H: OK

65. Extinction Characterized by H: NOT OK

66. Computational Epidemiology S: susceptible state (latent period) I: Infectious state (infectious period) R: recovery period

67. Measles and Vaccination