Oscillation in Biological
Systems
School of Mathematics, University of Tehran.
School of Mathematics, IPM.
[email protected]
[email protected]
1. Mechanism of Oscillation in One
Dimensional Systems
2. Example of Oscillations in One
dimensional Systems
Mechanisms of Oscillation in One
Dimensional Systems
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•
•
•
time-dependent systems
delayed systems
Hysteresis
Bifurcations
Mechanisms of Oscillation in One
Dimensional Systems
time-dependent systems
System under real time control
System under force
System under slowly time varying nature
System under time dependent perturbation
x = f ( x) + u (t )
x = f ( x) + g ( x, t )
x = f ( x) + h( x, a (t ))
x = f ( x) + εF ( x, t , ε )
Mechanisms of Oscillation in One
Dimensional Systems
delayed systems
Discrete delay
Continuous delay
Dirac function, strong kernels, weak kernels
x = rx(1 − w ∗ x)
Mechanisms of Oscillation in One
Dimensional Systems
Hysteresis
Memory
Residual Energy
Mechanisms of Oscillation in One
Dimensional Systems
Bifurcations
Hopf Bifurcation caused by delay
Cusp Bifurcation
Multistability
x
parameter
Mechanisms of Oscillation in two
Dimensional Systems
• Three dimensional centre manifold of Hopf
bifurcation
• Predator-Pray
Kuznetsov Y..A. Elements of applied bifurcation theory, Springer, 1998.
Mechanisms of Oscillation in two
Dimensional Systems
• Homoclinic bifurcation
Kuznetsov Y.A. Elements of applied bifurcation theory, Springer, 1998.
Mechanisms of Oscillation in
two Dimensional Systems
• Slow fast Dynamics
Murrar, J. D., Mathematical Biology I, Springer, 2003.
Mechanisms of Oscillation in
two Species Systems
• Slow fast Dynamics
Murrar, J. D., Mathematical Biology I, Springer, 2003.
Mechanisms of Oscillation in
two Species Systems
• Slow fast Dynamics (eg. BZ)
f c1 < f < f c2
Murrar, J. D., Mathematical Biology I, Springer, 2003.
Other methods
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•
•
•
•
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Poincare – Bendixon theorem
La’Sall Invariant Principle
Shil’nikov
Poincare method
Averaging
KAM
Other Kind of Oscillation
• Oscillation in phase shifting equation(BZ)
χ = ψ 1 −ψ 2
Murrar, J. D., Mathematical Biology I, Springer, 2003.
A practical example
Dynamics of Thyroid
Dynamics of Thyroid
• Definition of terms
• Invariant region of the whole system
• Fixed Point Problem
• Subsystems
Rokni Lamooki, et al. (2011)-(2012)
• Input Iodide
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
100
200
300
400
Rokni Lamooki, et al. (2011)-(2012)
500
600
700
800
900
1000
• Relaxation oscillation of blood iodine
20
18
16
14
12
10
8
6
4
2
0
0
500
1000
1500
2000
Rokni Lamooki, et al. (2011)-(2012)
2500
3000
3500
4000
4500
5000
• Oscillation of T3
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
4
x 10
Rokni Lamooki et al. (2011)-(2012)
• Oscillation of T4
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
4
x 10
Rokni Lamooki et al. (2011)-(2012)
• Oscillation of TSH
3
2.5
2
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
4
x 10
Rokni Lamooki et al. (2011)-(2012)
• Projection on (T3,T4)-space
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
Rokni Lamooki et al. (2011)-(2012)
0.4
0.5
0.6
0.7
0.8
0.9
1