the starting line - best of me

done! (i might proofread it sometime before class...)


this one's 4 pages double spaced. you have absolutely no obligation to read it...some fucked up math in it that even i barely understand....ugh.


Modeling Cochlear Mechanics

The inner ear is one of the many organs in the human body that to this day, in all our technological advancement, is still not completely understood. Through mechanical modeling, however, the many mysteries of the cochlea are being solved, one by one.
As Paul J. Kolston points out in his paper, ¡§The Importance of Phase Data and Model Dimensionality to Cochlear Mechanics,¡¨ the cochlea is an extremely fragile organ. Because of this, real life (or ¡§in vivo¡¨ as Kolston puts it) observation of its mechanics is almost impossible. Lacking sufficient data from observation, scientists needed to find ways to model the inner ear to further their studies and especially to improve technology in treating hearing disorders that inevitably occur with such an fragile organ.
Study of real cochlea has provided accurate responses due to various types of stimulation. The mysteries arise regarding what happens in between stimulation and response. Analogously, one could put an input into a ¡§black box¡¨ and only be allowed to see the output in order to determine what was going on inside. Unfortunately, the output of the cochlea suggests an extremely complicated ¡§black box¡¨.
Peterson and Bogert¡¦s give a good diagram along with an extensive mental picture of the properties of the cochlea in their paper, ¡§A Dynamical Theory of the Cochlea.¡¨ Essentially, the cochlea is a pair of liquid filled tubes (the scalas) in parallel separated by a much smaller tube called the cochlear duct. The cochlear duct consists of very flexible membrane and, of more importance, the basilar membrane(often abbreviated simply BM). The tubes are connected at one end (the ¡§far¡¨ end) where both the BM and cochlear duct end. The ¡§front¡¨ ends of one of these tubes represent the division between the middle and inner ear. One duct is connected to the stapes at the ¡§oval window¡¨, which receives the actual audio signal and transmits it into the inner ear. A flexible membrane at the ¡§round window¡¨ covers the other duct. Simplistically, the two tubes represent one long balloon bent in half. When pressure is exerted at one end (at the stapes from the middle ear), the pressure is transmitted through the fluid in the balloon around the bend where the two tubes are connected, and down to the other end (the round window), where the excess pressure bulges the thin membrane and exerts pressure back into the middle ear.
Branching momentarily from the mechanics of the inner ear, an important aspect of the cochlea is the organ of Corti, an extremely complicated organ stretching the length of the cochlea on the BM. The organ of Corti is where the mechanical signal from the rest of the ear is translated into an electrical signal the brain can process. The entire cochlear structure is coiled up into something that looks like a seashell and nerves run up the middle to get information from different portions of the organ of Corti.
My ¡§simple¡¨ balloon case gets infinitely more complicated, unfortunately. For one thing, and possibly most importantly, the BM is flexible. Pressure will not just be transmitted all the way around my balloon. Instead, at certain points, the energy will tend to build up and transmit the pressure through the BM; therefore giving the organ of Corti (sitting on the BM) the information it needs to process the sound. Different frequencies excite different parts of the BM, so the result is a biomechanical spectrum analyzer for the sound coming in through the stapes.
One of the most interesting aspects of the cochlear response to stimulation is remarkable frequency sensitivity. A pure tone fed into a cochlea results in a sharp, asymmetrical displacement peak in the BM (a good picture of which is seen in Kolston Fig. 7A) which rises exponentially to a peak and the drops off almost immediately (varying with increasing x). This is unlike a simple travelling wave on, say, a string, which would remain periodic on the string (either standing or travelling waves). The reason for this is that in a string, the ¡§wavenumber¡¨ k is fixed. The waves peed c on the string becomes c = w/k where w is the angular velocity and so on a string, the wave speed is constant. In the cochlea, however, because of nonuniformities in the BM width, BM stiffness, cross-sectional area (in scalas), and fluid compressibility, the value of k changes depending on where on the cochlea you are and the frequency being dealt with. The value k(w, x) is expressed in Lighthill¡¦s ¡§Advantages from Describing Cochlear Mechanics in Terms of Energy Flow¡¨ as a measure of the ¡§¡¦crinkliness¡¦ or ¡¥waviness¡¦¡¨ of a wave at a certain part of the cochlea.
It is readily observed in cochlear mechanics that there is a progressive reduction in the spatial phase you get farther from the base. This means that with a pure tone, the vibration is still sinusoidal with respect to time, but not with respect to place. Travelling waves vary according to the frequency, w, and the wavenumber, k. Lighthill uses equation (5):
H = a(x) cos [wt + ć(x)], where dć/dx = -k
where ¡¥a¡¦ represents the amplitude. This equation agrees with the observation that the spatial phase is progressively reduced only if k increases with increasing x indicating that k has the dimension mm-1 and indicates the rate at which the phase is changing (let¡¦s say in radians). Attempting to help visualize this, see the variable k as the reciprocal of the wavelength, given in mm/radian. This means that as k increases to infinity, the wavelength decreases to zero. It would be like taking each period of the wave and smashing it into a tighter and tighter length on x.
Now imagine that as a wave is excited in the cochlea it travels down the BM and k is constantly increasing. This means that as the wave travels, it starts getting compressed. Because of the compression, the wave starts slowing down longitudinally as it progresses along the BM. Lighthill states that the energy propagates at a velocity U = dw/dk (keeping x constant) in equation 4 along with E = (1/2)sa2 (energy) from equation 5 to say that energy flows at UE per second towards the apex. This gives the the equations in (6):
d(UE)/dx = -DE (where D is dissipation of vibrational energy per length) givng
E(x) = [U(0)E(0)/U(x)] exp[-(integral (Ddx/U) from x to 0).
This equation assumes light damping (D) and therefore shows that because the waves is slowing down (U(x) is decreasing), E(x) rises until it hits a ¡§critical layer¡¨ where the magic happens. In this narrow segment of the BM, while U is slow, but not zero, the integral in equation 6 increases without bound (because U is so small) and D, no matter how small, comes into play and dissipates all the energy. Following this graphically, E(x) is increasing up until the wave hits the critical layer and the integral kicks in dissipating all the energy and giving us our asymmetrical peak, dropping off almost immediately.
With such a well-defined peak response on our BM, the organ of Corti is given a very precise frequency spectrum which it can then pass on to the brain and gives us our extraordinary frequency sensitivity. The discussion above shows in one dimension how our ¡§black box¡¨ works to provide the peaking response that has been observed. For simplicity we only considered the longitudinal motion of the wave, though. Rest assured that more complex models exist that add more dimensions to the calculations, starting with hydrodynamic properties in three dimensions which vastly improve the models scientists use today.


cheers!