Modeling of Biological Systems Discrete A Mathematical model

Modeling of Biological Systems

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

Logistic map : Two fixed points :

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

Fixed points in one dimensional space : (MIT Open. Course. Ware » Mathematics » Mathematical Exposition, Spring 2005) Fixed points in two dimensional space :

Stable Neutrally Stable Unstable

a=3. 5 a=3. 8

Bifurcation Diagram : Universality :

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

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 -1 S S(1) K f R R(1)

f -1 S S(1) K f f -1 R S R(1) S(1) K f R P R(1) P(n 0) 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(n 0); 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);

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

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.

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

Simple Signatures of Chaos By Power Spectrum :

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

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

Simple Signatures of Chaos Lyapunov Exponent :

Simple Signatures of Chaos

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

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