Quantitative modeling of biological systems (Physiology 472)University of ArizonaFall 200910/27/09Instructors: Dr. Tim Secomb ([email protected]) Dr. Chris Bergevin ([email protected])Schedule: Tuesdays & Thursdays 9:30-10:45 (Optical Sciences 432)Website: http://www.physiology.arizona.edu/people/secomb/472572info09Lecture 19Mathematical topics we will cover:- Fourier analysis- Second order differential equations- Complex numbers and applications to solutions of ODEs/PDEshttp://www6.miami.edu/UMH/CDA/UMH_Main/SOAE - Spontaneous otoacoustic emission, recorded in the absence of any external stimulationtime waveform recorded from ear canal... zoomed inFourier transformTime DomainSpectral DomainMotivation: One of the ear’s primaryfunctions is to act as a Fourier‘transformer’Tone-like sounds spontaneously emitted by the earBlackbird (Turdus merula)SpectrogramTime Waveformtimefrequencyamplitudehttp://www.birdsongs.it/Square Wave is Comprised of Sinusoidal (Odd) Harmonicshttp://en.wikipedia.org/wiki/Square_wavehttp://mathworld.wolfram.com/SquareWave.htmlInner ear OAEs generated here Middle ear Outer earOAEs measured hereStarting Point: Damped, Driven Harmonic OscillatorCase 1: Undamped, Undriven Newton’s Second LawHooke’s LawSecond order differentialequation Solution is oscillatory!System has a natural frequencyCase 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 solutionCase 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 frequencyCase 3: Damped, UndrivenPurely sinusoidal solution no longer works!Change variablesAssumption: Form of solution is acomplex exponentialTrigonometry review  Sinusoids (e.g. tones)Sinusoid has 3 basic properties:i. Amplitude - heightii. 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 FormPolar FormCase 3: Damped, Undriven(slightly lower frequency of oscillation due to damping)[A and  are constants of integration, depending upon initial conditions] Damping causesenergy loss from systemCase 4: Damped, DrivenSinusoidal 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 tunedthe system is (i.e., ability toresolve 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 storageNotational/Mathematical Asides1. (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