Quantitative modeling of biological systems (Physiology 472)

University of Arizona

Fall 200910/27/09

Instructors: Dr. Tim Secomb (secomb@email.arizona.edu)

Dr. Chris Bergevin (cbergevin@math.arizona.edu)

Schedule: Tuesdays & Thursdays 9:30-10:45 (Optical Sciences 432)

Website: http://www.physiology.arizona.edu/people/secomb/472572info09

Lecture 19

Mathematical topics we will cover:

- Fourier analysis

- Second order differential equations

- Complex numbers and applications to solutions of ODEs/PDEs

http://www6.miami.edu/UMH/CDA/UMH_Main/

SOAE - Spontaneous otoacoustic emission, recorded in the absence of any external stimulation

time waveform recorded from ear canal

... zoomed in

Fourier transform

Time DomainSpectral Domain

Motivation: One of the ear’s primary

functions is to act as a Fourier

‘transformer’

Tone-like sounds

spontaneously emitted

by the ear

Blackbird (Turdus merula)

Spectrogram

Time Waveform

time

frequency

amplitude

http://www.birdsongs.it/

Square Wave is Comprised of Sinusoidal (Odd) Harmonics

http://en.wikipedia.org/wiki/Square_wave

http://mathworld.wolfram.com/SquareWave.html

Inner ear OAEs generated here

Middle ear Outer earOAEs measured here

Starting Point: Damped, Driven Harmonic Oscillator

Case 1: Undamped, Undriven

Newton’s Second Law

Hooke’s Law

Second order differential

equation

Solution is oscillatory!

System has a

natural frequency

Case 1: Undamped, Undriven (cont.)

Consider the system’s energy:

- Two means to store energy: mass and spring

- Oscillation results as energy transfers back

and forth between these two modes

(i.e., system is considered second-order)

phase plane portrait for H.O.

Case 2: Undamped, Driven

Sinusoidal driving force atfrequency

Assumption: Ignore onset behavior and that system oscillates at frequency

Assumed form of solution

Case 2: Undamped, Driven (cont.)

Two Important Concepts Demonstrated Here:

- Resonance when system is driven at natural frequency

- Phase shift of 1/2 cycle about resonant frequency

Case 3: Damped, Undriven

Purely sinusoidal solution

no longer works!

Change variables

Assumption: Form of solution is a

complex exponential

Trigonometry review Sinusoids (e.g. tones)

Sinusoid has 3 basic properties:

i. Amplitude - height

ii. Frequency = 1/T [Hz]

iii. Phase - tells you where thepeak is (needs a reference)

Phase reveals timing information

(x2)

Case 3: Damped, Undriven (cont.)

Motivation for complex solution:

Complex solution contain both magnitude and phase information

Cartesian Form Polar Form

Case 3: Damped, Undriven

(slightly lower frequency of

oscillation due to damping)[A and are constants of integration, depending upon initial conditions]

Damping causes

energy loss from system

Case 4: Damped, Driven

Sinusoidal driving force atfrequency

Assumption: Ignore onset behavior and that system oscillates at frequency

Assumed form of solution

(magnitude)

(phase)

Case 4: Damped, Driven (cont.)

Second-order oscillator behaves as

as band-pass filter

(i.e., it is a mechanical Fourier transformer tuned to a specific frequency)

Case 4: Damped, Driven (cont.)

- Can find general solution (e.g., transient behavior at onset) by considering a particular

solution and the solution to the homogeneous equation

- Quality Factor (Q):Reveals how sharply tuned

the system is (i.e., ability to

resolve different frequencies)

- Impedance (Z):

Sharply tuned oscillators have long build-up times

Real part of Z (resistance) describes energy loss while imaginary

part (reactance) describes energy storage

Notational/Mathematical Asides

1. (notational difference, used primarily by E.E.

folks to avoid confusion with electric current)

2. consider i as an exponential via Euler’s formula

(or more simply, as a point in the complex plane)

three of my favorite #s!