复杂系统研究的科学基础

1. 复杂系统研究的科学基础 • 自组织理论与动力学 狄增如 北京师范大学管理学院系统科学系 北京师范大学复杂性研究中心 2010.7

2. I. Prigogine • H. Haken

3. 自组织理论 I. Progogine: Far-from-equilibrium studies led me to the conviction that irreversibility has a constructive role. It makes form. It makes human beings. 自组织系统的主要特征： 1、开放系统，与环境有物质和能量交流； 2、组元众多，且存在非线性相互作用； 3、远离平衡态； 4、涨落是有序结构形成的触发器。

4. 分支图 参数

5. 生物进化树 生物、社会、经济等领域中的应用

6. 系统演化的动力学描述 • 关心系统演化的极限行为及其随环境条件的变化所导致的分支行为 • 线性稳定性分析，分支理论，突变论

7. 关于混沌 Chaos 古希腊与中国：混沌初开 20世纪80年代，混沌理论成为一个 新的、激动人心的科研领域 它将深刻地改变我们对自然及人类的认识

8. 种群增长与倍周期分叉 混沌区

10. 混沌的基本性质 对初值的敏感依赖性——蝴蝶效应 差之毫厘，失之千里 确定性系统中的随机行为 在不引入任何随机因素的情况下，一个 简单的系统可以产生非常不规则的行为 一种没有周期性的有序

11. 埃农（Henon）映射

12. 分形结构 Fractal 分形：Logistic map, Lorenz attractor, Mandelbrot set, Julia set

13. 生物体中的分形 人体血管总体积<5% 肺的总面积>网球场 • 股票分时走势图

14. 为什么混沌如此引人注目？  混沌揭示了简单性和复杂性、有序和无序 之间的精妙关联，从而沟通了科学与生活；  一个遵循基本物理规律、确定论性的世界， 可以是无序的，具有复杂性和不可预测性；  复杂现象的背后可能具有简单的规律；  在任何层次上，我们对未来的理解和预测都是 有限的；  混沌是非常漂亮的。

15. 二十世纪物理学的重要进展 * 量子力学——微观世界 * 狭义与广义相对论——宏观世界 * 自组织与混沌——生命现象

17. The Cell Cycle Nurse P. The incredible life and times of biological cells. Science 289:1711, 2000

18. Protein Micro-arrays Detecting Protein Interaction/Biochemistry Service RF. Protein arrays step out of the shadows. Science 289:1673, 2000

19. The city dynamics

20. The division of labor

21. Stock Market： Levy分布——正态分布

22. Minority Game Agent-based Computational Economics

23. 案例研究：心脏中的动力学 UCLA Cardiology Division

24. CARDIAC FIBRILLATION • Ventricular fibrillation • 220,000 sudden deaths annually in U.S. • Atrial fibrillation • 6% of population over age 65 • 1/3 of all strokes over age 65 • doubled mortality rate

25. VT to VF transition VF maintenance VT initiation ? ? PVC Hypothesis CAST SWORD SUDDEN CARDIAC DEATH

26. heart

28. The Basic Unit 10 mm

29. Ca2+ Na+ K+ -80 mV T-tubules T tubule myofilaments

30. Ca2+ (10-20%) Extracellular space Ca channel T-tubule membrane Ca release channel (Ryanodine receptor) Ca2+ (80-90%) SR Ca ATPase Sarcoplasmic reticulum Ca2+

31. 3D Confocal Image of T-tubule System Courtesy of Joy Frank, PhD & Alan Garfinkel, PhD UClA Cardiovascular Research Laboratory

32. Courtesy of Joy Frank, PhD UCLA Cardiovascular Research Laboratory

33. RyRs DHPR Ca DHPRs Ca RyRs Ca SR Ca stores

34. Na+-Ca2+ Exchanger SL Ca2+- ATPase Ca2+ (10-20%) 3Na+ Calsequestrin Ca channel T-tubular membrane Ca release channel (Ryanodine receptor) Ca2+ Ca2+ (80-90%) Sarcoplasmic reticulum SR Ca ATPase

36. Hodgkin AL, Huxley AF. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 1952;117:500-544.

37. mV time(ms)

38. Cardiac Action Potential Model dVm/dt = -S (Iionic + Iext)/ Cm Zeng J, Laurita KR, Rosenbaum DS, Rudy Y. Circ. Res.77:140-152, (1995)

39. 50000 steps in 4.43 seconds 20 0 -20 V (mV) -40 -60 -80 -100 0 100 200 300 400 500 600 700 800 900 1000 TIME (msec) 2 ms 15 mA = Runge-Kutta 4th order, DT = .02 ms

40. FitzHugh-NagumoModel: Barkley Dynamics: du/dt= f(u,v)=u(1-u)[u-(v+b)/a]/, dv/dt=g(u,v)=u-v u 1 g(u,v)=0 0.9 v 0.8 0.7 0.6 v= au-b x, y 0.5 0.4 b 0.3 v 0.2 0.1 u 0 0 2 4 6 8 10 12 14 16 18 20 TIME

41. vanCapelle FJL, Durrer D. Computer simulation of arrhythmias in a network of coupled excitable elements. Circ. Res. 1980;47:454-466.

42. ¶ V = - + I / C D V Ñ . Ñ ion m t ¶ å å = = I I f ( V ) ion k k Neumann boundary condition : r . = n V 0 Ñ

43. P Plane Wave S Spiral Wave SK Spiral Wave Breakup

44. 1 2 What Causes The Waves To Break? Traditional Answer: Pre-existing Tissue Heterogeneities (anatomic or electrophysiological) Slope < 1

45. APD S2 S1 Diastolic Interval

46. Electrical Restitution (S1S2 Method) APD Restitution CV Restitution THE SLOPE! >1 : + gain amplifier <1 : - gain attentuator Wavelength Is Also Controlled Dynamically by Electrical Restitution (in the absence of pre-existing heterogeneities)

47. 2 1 3 Dynamic Wavebreak: The Role of APD Restitution Steepness Slope < 1 Slope > 1

48. X Steep Slope Shallow Slope Y  X Y Y Y < X X

49. A B a a b 150 b 100 100 c APD (ms) APD (ms) 50 c 50 0 0 0 50 100 0 50 100 150 DI (ms) DI (ms) b b c c d d 0 0 -40 -40 V (mV) V (mV) -80 -80 0 200 400 600 800 1000 1200 1400 0 200 400 600 800 1000 1200 1400 t (ms) t (ms)