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7/29/2019 Modelling Biological Oscillations
1/19
Modelling biological oscillations
Modelling biological oscillations
Shan He
School for Computational Science
University of Birmingham
Module 06-23836: Computational Modelling with MATLAB
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Modelling biological oscillations
Outline
Outline of Topics
Limit circle oscillation: Van der Pol equationVan der Pol equation
Van der Pol equation and Hopf bifurcationClassification of oscillationsThe forced Van der Pol oscillator and chaos
Oscillations in biology
Circadian clocks and Van der Pol equation
Modelling neural oscillation
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Modelling biological oscillations
Limit circle oscillation: Van der Pol equation
Van der Pol equation
Van der Pol equation
Discovered by Dutch electrical engineer Balthasar Van der Polin 1927.
The first experimental report about chaos.
The Van der Pol equation has been widely used in thephysical and biological sciences.
The equation:
d2x
dt2 (1
x2)
dx
dt + x = 0
where x is the position coordinate, which is a dynamicalvariable, and is a scalar parameter which controls thenonlinearity and the strength of the damping.
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Modelling biological oscillations
Limit circle oscillation: Van der Pol equation
Van der Pol equation
Van der Pol equation: two dimensional form
We can transform the above second order ODE as a firstorder system:
dxdt
= ydydt
= (1 x2)y x.
By Lienard transform, the equation can be also written as:dxdt
= (x 13x3 y)
dydt
= 1x.
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Modelling biological oscillations
Limit circle oscillation: Van der Pol equation
Van der Pol equation
MATLAB implementation
18 function dy=vandePolfun(t,y)
19 m u = 1 ;
20 dy = [mu*(y(1) - 1/3*y(1)3 - y(2));
21 1/mu*y(1)]22 end
23
24 function dy=vandePolfun2(t,y)
25 m u = 1 ;
26 dy = [y(2); mu*(1-y(1)2)*y(2)-y(1)]
27 end
28
29 end
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Modelling biological oscillations
Limit circle oscillation: Van der Pol equation
Van der Pol equation
MATLAB implementation
1 function VanderPol
2 clear all; % clear all variables
3 tspan = [0, 20];
4 y0 = [20; 0];
5
6[t,Y] = ode45(@vandePolfun,tspan,y0);
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Modelling biological oscillations
Limit circle oscillation: Van der Pol equation
Van der Pol equation and Hopf bifurcation
Bifurcation in the Van der Pol oscillator
By change the scalar parameter , we can observe bifurcation.
The bifurcation is Hopf bifurcation: a spiral point of adynamical system switch from stable to unstable, or viceverse, and a periodic solution, e.g., limit circle appears.
Without any external force, the Van der Pol oscillator will runinto limit cycle.
The limit cycle frequency depends on constant
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Modelling biological oscillations
Limit circle oscillation: Van der Pol equation
Classification of oscillations
Limit cycle oscillation vs harmonic oscillation
Harmonic oscillator: a system that executes a periodicbehaviour, the amplitude of the oscillations depends on theinitial conditions.
The examples of harmonic oscillators: swinging pendulum andthe Lotka-Volterra model
This contrasts with limit cycle oscillators, where the amplitudeis not determined by the initial conditions.
Instead, limit cycle oscillators will automatically come back toa limit cycle after perturbation. In other words, they have acharacteristic amplitude
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Modelling biological oscillations
Limit circle oscillation: Van der Pol equation
Classification of oscillations
Limit cycle oscillators
Limit cycle has an isolated closed trajectory, e.g., itsneighbouring trajectories are not closed but spiral eithertowards or away from the limit cycle.
Stable limit cycle: it attracts all neighbouring trajectories. A system with a stable limit cycle exhibits self-sustained
oscillation.
self-sustained oscillations can be seen in many biological
systems, e.g., electroencephalogram (EEG), mammaliancircadian clocks, stem cell differentiation, etc.
Essentially involves a dissipative mechanism to damposcillators that grow too large and a source of energy to pumpup those that become too small.
M d ll b l l ll
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Modelling biological oscillations
Limit circle oscillation: Van der Pol equation
The forced Van der Pol oscillator and chaos
The forced Van der Pol oscillator
By adding a driving function A sin(t), the new equation is:
d2x
dt2 (1 x2)
dx
dt+ x = A sin(t)
where A is the amplitude (displacement) of the wave functionand is its angular frequency.
There are two frequencies in the oscillator: The frequency of self-oscillation determined by ; The frequency of the periodic forcing .
We can transform the above second order ODE as a firstorder system:
dxdt
= ydy
dt = (1
x2
)y
x
A sin(t).
M d lli bi l i l ill i
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Modelling biological oscillations
Limit circle oscillation: Van der Pol equation
The forced Van der Pol oscillator and chaos
Chaotic behaviour of the forced Van der Pol oscillator
By setting the parameters to = 8.53, amplitude A = 1.2 andangular frequency = 2
10we can observe chaotic behaviour
Source code for the forced Van der Pol oscillator
M delli bi l i l s ill ti s
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Modelling biological oscillations
Oscillations in biology
Oscillations in biology
We have seen oscillation in predator-prey interaction system,
e.g., Lotka-Volterra equations. Oscillations are prevalent, e.g., swinging pendulum, Business
cycle, AC power, etc.
Oscillations play an important role in many cellular dynamic
process such as cell cycle
Modelling biological oscillations
http://www.physorg.com/news190878712.htmlhttp://www.physorg.com/news190878712.html7/29/2019 Modelling Biological Oscillations
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Modelling biological oscillations
Oscillations in biology
Oscillations in cell biology
Novak, Tyson (2008) Design principles of biochemical oscillators.
Nature review 9:981-991.
Modelling biological oscillations
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Modelling biological oscillations
Oscillations in biology
Circadian clocks and Van der Pol equation
Circadian clock Circadian clock: or circadian rhythm, is an endogenously
driven, roughly 24-hour cycle in biochemical, physiological, orbehavioural processes.
Source: wiki edia
Modelling biological oscillations
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Modelling biological oscillations
Oscillations in biology
Circadian clocks and Van der Pol equation
Circadian clock
Circadian clock had been widely observed in bacteria, fungi,plants and animals.
Biological mechanism partially understood, used mathematicalmodels to piece together diverse experimental data and guidefuture research.
Applications: jet lag countermeasures, schedules of shift
workers, treat circadian disorders. Circadian clock acts as a limit cycle oscillator.
Modelling biological oscillations
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Modelling biological oscillations
Oscillations in biology
Circadian clocks and Van der Pol equation
Circadian clock and Van der Pol oscillator
Only has one nonlinear term, Van der Pol oscillator is thesimplest possible two-dimensional limit cycle oscillator.
Many studies of circadian clock used Van der Pol oscillator forsimulation
Modelling biological oscillations
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g g
Oscillations in biology
Circadian clocks and Van der Pol equation
Circadian clock of Drosophila
Van der Pol oscillator cannot accurately describe the internalmechanism of the circadian clock
Goldbeter model of circadian clock of Drosophila:
Goldbeter (1995), A model for circadian oscillations in theDrosophila period protein (PER), Proc. Roy. Soc. London B, 261
Modelling biological oscillations
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Modelling neural oscillation
FitzHughNagumo oscillators Proposed by Richard FitzHugh in 1961 and implemented by
Nagumo using electronic circuit in 1962. Used to describes how action potentials in neurons are
initiated and propagated.
FitzHughNagumo model is a simplified version of the famousHodgkinHuxley model, which won a Nobel Prize. FitzHughNagumo model equation:
dv
dt
= vv3
3
w+ Iext
dw
dt= (v a bw)
where v is the membrane potential, w is a recovery variable,Iext is the magnitude of stimulus current, and a, b and areconstants.
Modelling biological oscillations
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Modelling neural oscillation
Assignment
Search literature for the parameters a, b and .
Implement the model in MATLAB.
Try different value of Iext and observe output v of the neuron.
If you have more time, try to find the chaotic behaviour andbifurcation of the model.