modelling biological oscillationsszh/teaching/matlabmodeling/lecture10_body.pdf · modelling...

Modelling biological oscillations


Shan He

School for Computational ScienceUniversity of Birmingham

Module 06-23836: Computational Modelling with MATLAB


Outline

Outline of Topics

Limit circle oscillation: Van der Pol equationVan der Pol equationVan der Pol equation and Hopf bifurcationClassification of oscillationsThe forced Van der Pol oscillator and chaos

Oscillations in biologyCircadian clocks and Van der Pol equation

Modelling neural oscillation


Limit circle oscillation: Van der Pol equation

Van der Pol equation


I Discovered by Dutch electrical engineer Balthasar Van der Polin 1927.

I The first experimental report about chaos.

I The Van der Pol equation has been widely used in thephysical and biological sciences.

I The equation:

d2x

dt2− µ(1 − x2)

dx

dt+ x = 0

where x is the position coordinate, which is a dynamicalvariable, and µ is a scalar parameter which controls thenonlinearity and the strength of the damping.




Van der Pol equation: two dimensional form

I We can transform the above second order ODE as a firstorder system: {

dxdt = ydydt = µ(1 − x2)y − x .

I By Lienard transform, the equation can be also written as:{dxdt = µ(x − 1

3x3 − y)

dydt = 1

µx .




MATLAB implementation

18 function dy=vandePolfun(t,y)19 mu = 1;20 dy = [mu*(y(1) - 1/3*y(1)ˆ3 - y(2));21 1/mu*y(1)]22 end23

24 function dy=vandePolfun2(t,y)25 mu = 1;26 dy = [y(2); mu*(1-y(1)ˆ2)*y(2)-y(1)]27 end28

29 end




MATLAB implementation

1 function VanderPol2 clear all; % clear all variables3 tspan = [0, 20];4 y0 = [20; 0];5

6 [t,Y] = ode45(@vandePolfun,tspan,y0);



Van der Pol equation and Hopf bifurcation

Bifurcation in the Van der Pol oscillator

I By change the scalar parameter µ, we can observe bifurcation.

I 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.

I Without any external force, the Van der Pol oscillator will runinto limit cycle.

I The limit cycle frequency depends on constant µ



Classification of oscillations

Limit cycle oscillation vs harmonic oscillation

I Harmonic oscillator: a system that executes a periodicbehaviour, the amplitude of the oscillations depends on theinitial conditions.

I The examples of harmonic oscillators: swinging pendulum andthe Lotka-Volterra model

I This contrasts with limit cycle oscillators, where the amplitudeis not determined by the initial conditions.

I Instead, limit cycle oscillators will automatically come back toa limit cycle after perturbation. In other words, they have acharacteristic amplitude



Classification of oscillations

Limit cycle oscillators

I Limit cycle has an isolated closed trajectory, e.g., itsneighbouring trajectories are not closed but spiral eithertowards or away from the limit cycle.

I Stable limit cycle: it attracts all neighbouring trajectories.

I A system with a stable limit cycle exhibits self-sustainedoscillation.

I self-sustained oscillations can be seen in many biologicalsystems, e.g., electroencephalogram (EEG), mammaliancircadian clocks, stem cell differentiation, etc.

I Essentially involves a dissipative mechanism to damposcillators that grow too large and a source of energy to pumpup those that become too small.



The forced Van der Pol oscillator and chaos

The forced Van der Pol oscillator

I By adding a driving function A sin(ωt), the new equation is:

d2x

dt2− µ(1 − x2)

dx

dt+ x = A sin(ωt)

where A is the amplitude (displacement) of the wave functionand ω is its angular frequency.

I There are two frequencies in the oscillator:I The frequency of self-oscillation determined by µ;I The frequency of the periodic forcing ω.

I We can transform the above second order ODE as a firstorder system:{

dxdt = ydydt = µ(1 − x2)y − x − A sin(ωt).



The forced Van der Pol oscillator and chaos

Chaotic behaviour of the forced Van der Pol oscillator

I By setting the parameters to µ = 8.53, amplitude A = 1.2 andangular frequency ω = 2π

10 we can observe chaotic behaviour

I Source code for the forced Van der Pol oscillator


Oscillations in biology


I We have seen oscillation in predator-prey interaction system,e.g., Lotka-Volterra equations.

I Oscillations are prevalent, e.g., swinging pendulum, Businesscycle, AC power, etc.

I Oscillations play an important role in many cellular dynamicprocess such as cell cycle

http://www.physorg.com/news190878712.html



Oscillations in cell biology

Novak, Tyson (2008) Design principles of biochemical oscillators.Nature review 9:981-991.



Circadian clocks and Van der Pol equation

Circadian clockI Circadian clock: or circadian rhythm, is an endogenously

driven, roughly 24-hour cycle in biochemical, physiological, orbehavioural processes.

Source: wikipedia




Circadian clock

I Circadian clock had been widely observed in bacteria, fungi,plants and animals.

I Biological mechanism partially understood, used mathematicalmodels to piece together diverse experimental data and guidefuture research.

I Applications: jet lag countermeasures, schedules of shiftworkers, treat circadian disorders.

I Circadian clock acts as a limit cycle oscillator.




Circadian clock and Van der Pol oscillator

I Only has one nonlinear term, Van der Pol oscillator is thesimplest possible two-dimensional limit cycle oscillator.

I Many studies of circadian clock used Van der Pol oscillator forsimulation




Circadian clock of Drosophila

I Van der Pol oscillator cannot accurately describe the internalmechanism of the circadian clock

I 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



FitzHugh–Nagumo oscillatorsI Proposed by Richard FitzHugh in 1961 and implemented by

Nagumo using electronic circuit in 1962.I Used to describes how action potentials in neurons are

initiated and propagated.I FitzHugh–Nagumo model is a simplified version of the famous

Hodgkin–Huxley model, which won a Nobel Prize.I FitzHugh–Nagumo model equation:

dv

dt= v − v3

3− w + Iext

dw

dt= ε(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 ε areconstants.



Assignment

I Search literature for the parameters a, b and ε.

I Implement the model in MATLAB.

I Try different value of Iext and observe output v of the neuron.

I If you have more time, try to find the chaotic behaviour andbifurcation of the model.

modelling biological oscillationsszh/teaching/matlabmodeling/lecture10_body.pdf · modelling...

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