1. Modeling of Biological Systems

2. A Mathematical model of Biological Population Growth :
Discrete
A Mathematical model of Biological Population Growth :
A>1
explosion
extinction
A<1
&

3. Logistic map :
Two fixed points :

4. General Concepts :
Fixed Points :
Attractors :
One-D.
Two-D.

5. Fixed points in one dimensional space :
(MIT OpenCourseWare » Mathematics » Mathematical Exposition, Spring 2005)
Fixed points in two dimensional space :

6. Stable
Neutrally Stable
Unstable

8. a=3.5
a=3.8

9. Bifurcation Diagram :
Universality :

10. Chaos in Mathematical Models
Discrete
Continuous
Logistic map & Population growth & Sleep Oscillator
Lorenz & Rossler & Duffing Oscillator &…

11. A Simple Model of Waking & Sleeping :
Discrete
A Simple Model of Waking & Sleeping :
S(k+1)=.8*R(k)-tansig((1/.01)*R(k))+S(1);
R(k+1)=.8*S(k)-tansig((1/.01)*S(k))+R(1); f f -1 S R K
S(1)
R(1)

16. S(k+1)=.8*R(k)-tansig((1/.01)*R(k))+S(1); f f f f -1 -1 S R S R P K K
S(1)
R(1)
S(1)
R(1)
P(n0)
S(k+1)=.8*R(k)-tansig((1/.01)*R(k))+S(1);
R(k+1)=.8*S(k)-tansig((1/.01)*S(k))+R(1);
P(k+1)=.8*R(k)-tansig((1/eps)*S(k))+P(n0);
S(k+1)=.8*P(k)-tansig((1/eps)*R(k))+S(1);
R(k+1)=.8*S(k)-tansig((1/eps)*P(k))+R(1);

19. A Model of Convecting Fluids & The Lorenz Equations :
Continuous
A Model of Convecting Fluids & The Lorenz Equations :
1. the fluid remain stationary
fixed points :
2. a circulation pattern that is stable in time
3.the circulating currents & the resulting temperature differences within the fluid start to vary in time

20. relaxes to the steady non-convective state
for r<1 , all of the xyz space is the basin of attraction for the attractor at the origin .

22. stability analysis :
stable node ; r<1
saddle point ; r>1
are stable ;
are unstable ;

23. Simple Signatures of Chaos
By Power Spectrum :

24. Simple Signatures of Chaos
Poincare Map & Structure in Phase Space:

25. Simple Signatures of Chaos
Sensitivity to initial conditions & Lyapunov Exponent :

26. Simple Signatures of Chaos
Sensitivity to initial conditions & Lyapunov Exponent :

27. Simple Signatures of Chaos
Lyapunov Exponent :

28. Simple Signatures of Chaos

29. Simple Signatures of Chaos

30. The other Signatures of Chaos :
Kolmogorov-Sinai Entropy
Fractal Dimension
Correlation Dimension
Capacity
Similarity Dimension
Hausdorff Dimension
Box counting Dimension

31. Chaos in the Other Biological Systems
dynamics of human illness
human cardiac signal
human EEG
Protein Ion Channels
Action Potential
ERP signal
Pacemaker