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Physiology 472/572 - 2011 - Quantitative modeling of biological systems

Lecture 19: Harmonic oscillator, complex numbers, the inner ear

Harmonic oscillator (undamped, undriven)

• Newton’s Second Law: xmdt

xdmmaF 2

2

&&===

• Hooke’s Law: kxF −=

• 0kxxm =+&& - second order differential equation

• solution is oscillatory )tcos(Ax 0 φω += , natural frequency m/k0 =ω

Driven harmonic oscillator (undamped)

• sinusoidal driving force at frequency ω: )tcos(Fkxxm 0 ω=+&&

• assume oscillatory solution with frequency ω, ignoring initial transients: )tcos(Bx αω +=

• substituting gives α = 0 and 22

0

02

0 m/Fkm

FB

ωωω −=

+−=

• solution ))(tcos()(x ωαωωκ +=

where B)( =ωκ

• resonance when driven at natural

frequency

• phase shift of π at resonant frequency

Complex numbers (review)

• define 1i −=

• complex numbers are ibaz += where a, b are real

• complex conjugate ibaz −= , 22 bazz +=

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• represent in polar form: θθ sinAb,cosAa == where 22 baA += and )a/b(tan 1−=θ

• so )sini(cosAz θθ +=

• Euler's identity (1748): θθθ sinicosei +=

• θiAez = where zA = is the modulus and θ is the argument (phase) of z

Damped harmonic oscillator (undriven)

• 0kxxbxm =++ &&& where b is damping

• 0xxx 20 =++ ωγ&&& where γ = b/m

• this has no purely oscillatory solution, but try tiAex Ω= where A and Ω may be complex

• 0i 20

2 =+Ω+Ω− ωγ so ωγ ±=Ω 2/i where 4/220 γωω −=

• let αi0eAA = , then one solution is

)t(i2/t0

it)2/i(i0

ti eeAeeAAe αωγαωγ +−+Ω ==

• the real part is )tcos(eAx 2/t0 αωγ += − ; this is a

general solution

• solution is a sinusoidal oscillation with

exponential decay

Damped, driven harmonic oscillator

• eReF)tcos(Fkxxbxm ti00

ωω ==++ &&& where b is damping and Re denotes real part

• first solve ti0

20 e)m/F(xxx ωωγ =++ &&& where γ = b/m

• assume oscillatory solution, ignoring initial transients: tiAex ω= where A may be complex

• then m/F)i(A 022

0 =+− γωωω

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• note that 222220

220

220 )()i)(i( ωγωωγωωωγωωω +−=−−+−

• ti022222

0

2200 eA

)()i(

mF

A α

ωγωωγωωω

=+−

−−= where 2/122222

0

00 ])[(

m/FAωγωω +−

= is the amplitude

and 22

0

arctanωω

γωα−

−= is the phase

• solution is )tcos(Ax 0 αω +=

• resonance when ω = ω0

• quality factor, Q = ω/γ, shows how sharply tuned the system is

(i.e., its ability to resolve different frequencies)

The inner ear

• three fluid-filled compartments

• sensory complex (organ of Corti) sits atop a flexible membrane (basilar membrane, BM)

• sound reaching ear sets up a traveling wave along the BM

• BM is mechanically tuned, resonant frequency decreases with distance along the membrane

• membrane motion deflects hair cell bundles, which generate neural signals

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• see http://www.youtube.com/watch?v=dyenMluFaUw

Spontaneous otoacoustic emissions

• the ear emits sound, even in the absence of a stimulus

• this suggests that the hairs cells are not just passive receptors,

but act as amplifiers

• measurements on hair cells show force generation

• one model is a Van der Pol oscillator 0xx)1x(x 2 =+−+ &&& μ