1. If the upward deection of the basilar membrane does lead, in fact, to a downward force, the negative-feedback model of Mountain, Hubbard, and McMullen would be vindicated, and Ashmore's micromechanical argument would have to be revised.
2. If Ashmore's positive-feedback description is correct, the negative damping still may be achieved through the application of a three-quarter-cycle phase shift. In this case, the negative-feedback model would have to be rejected.
3. It is possible that a completely dierent mechanism is dominant. Kolston, for example, has argued on the basis of energy eciency that the outer hair cells should act so as to aect only the stiness or mass components of the basilar-membrane impedance [55]. Zweig has suggested that an outer-hair-cell force with a slow-acting and a fast- acting component may be more appropriate [133]; this suggestion nds support in the measurements of Ashmore [4].
Still other ingenious arguments may surface before the issue is resolved.
Nearly all of the models use pure delays in the implementation of the outer-hair-cell mechanical input-output relationships. While pure delays are very easy to implement in a computer simulation, they are very dicult to implement in a physical medium, and it is most unlikely that the outer hair cells can be modeled accurately in such a way. A more reasonable assumption would be that the outer hair cells respond with a rst- or second-order ltered version of their input stimulus.
feeding back the ltered and saturated signal as a force acting on the basilar membrane. A exible model would allow for both positive and negative feedback.
5.2.1 Mathematical Description
Recall that the basilar membrane boundary condition for the passive cochlear model was given in Equation 3.8:
M(x)@2
@t2 = 2@@t S(x) (x)@
@t at y=h: (5:1)
In the active case, we require an additional term FH to represent the force generated by the outer hair cells:
M(x)@2
@t2 = 2@@t S(x) (x)@
@t +FH(;!;H(x);QH;s;G(x)) at y =h: (5:2) The forceFH should be proportional to a delayed and saturated version of the displacement signal. We now consider each of these features in turn.
The conceptual model calls for a frequency- and place-specic delay, which can be mod- eled most reasonably by a low-pass or band-pass lter. The parameters of the low-pass or band-pass lter areH(x) andQH, where we have implicitly assumed a second-order lter.
Since the outer hair cells and their stereocilia increase in length from base to apex, it is reasonable to assume that the time-constantH increases with position x. For simplicity, we will assume thatQH is constant everywhere.
The conceptual model calls for a saturating nonlinearity, since there must be an upper limit on the magnitude of the force that can be produced by an outer hair cell. A simple saturating nonlinearity is the hyperbolic tangent function (also called a Boltzmann function [130], and closely related to the Fermi-Dirac distribution function [117, p. 71]), which will require a parameters to control the magnitude of displacements at which the saturating eect becomes noticeable|that is, the \width of the tanh." Finally, the scaling factorG(x) controls the magnitude of the delayed and saturated force.
Combining these terms leads to the following form of the outer hair cell force:
FH =GtanhF(;!;H;QH) s
; (5:3)
It is tempting to assume a harmonic time dependence with frequency !, and to write the lter operator explicitly:
FH =Gtanh
"
s 1
1 +i!H=QH !22H
!#
: (5:4)
However, this step is not justied because of the nonlinearity. For this reason, we cannot speak of the \impedance of the organ of Corti," since the notion of an impedance is a linear-systems concept.
Diependaal and Viergever point out that the driving function at the stapes must be handled with considerably more care in the nonlinear model, since the harmonics generated within the cochlea may have an eect on the stapes motion [24]. A truly realistic model must include a model of the middle ear.
5.2.2 Analysis and Simulation
The new boundary-value problem contains a nonlinear boundary condition. Whitham [126]
describes solution methods for other nonlinear wave problems; however, many of the meth- ods, such as the method for nding soliton solutions of the Korteweg{de Vries equation, capitalize on the special form of the nonlinearities. The application of analytical techniques to the nonlinear propagation of waves in cochlear mechanics has hardly been touched, and represents a challenging and potentially rewarding research opportunity.
Yates has drawn an illuminating analogy between the behavior of the cochlea and the behavior of a simple feedback system containing a saturating nonlinear element (tanh func- tion) in the feedback loop [130], as shown in Figure 5.1. We may use the behavior of the simple feedback system to reason about the real cochlea, which is a distributed system with many nonlinear elements that contribute to the traveling wave. For very soft sounds, many elements will be able to contribute their high gain to amplify the wave considerably as it travels. For moderate sounds, early elements will contribute high gain, until the sound is amplied so much that later elements can have little further eect. The overall eect is to broaden the compressively nonlinear range, so that virtually all of the large input dynamic
Vin Vout
jVoutj(dB)
jVinj(dB)
(a) (b)
Figure 5.1 Yates' nonlinear feedback system. (a) A simple feedback system with a saturating nonlinear element in the feedback loop has many important analogies with cochlear behavior. (b) At low input signal levels, the feedback element is nearly linear, so the large closed-loop gain is applied to the signal, and the input-output relation is nearly linear. At high input signal levels, the feedback element is saturated at a relatively small value, so the feedback path is ineective, resulting in a low open-loop gain, and a nearly linear input- output relation. At moderate input signal levels, the input-output relation is compressive. The dashed lines indicate linear behavior. Adapted from Yates [130].
a large-signal linear range will not be of much value|nonlinearities are signicant over the entire range of interest.
In the meantime, we need an experimental medium in which to investigate the be- havior of the model. Virtually all researchers are using numerical simulation on a digital computer for this task, with the notable exception of Zwicker and colleagues, who use an analog electrical circuit [136]. However, numerical solution of the problem is computation- ally demanding. Diependaal and Viergever reported in 1989 that time-domain solution of the two-dimensional problem with 256 points on the basilar membrane and 2560 time steps (corresponding to 40 ms of real time) requires 8.5 hours of CPU time on an HP9050 computer, using a very ecient integral-equation method [24].
The use of an analog circuit to model the nonlinear and active cochlea has important advantages over digital simulations. The analog circuit can be made to operate in real time, and since the circuit operates in continuous time, there are no stability problems associated with discrete time steps. In the hopes of exploiting these advantages, we proceed with the development of the circuit model of the outer hair cell.
5.2.3 The Circuit Model
The circuit model of the outer hair cell is shown in Figure 5.2; for context, the original basilar-membrane circuit is also shown with a single resistor from the resistive network.
The outer-hair-cell circuit breaks down into three functional blocks.
The rst block converts the current Ivel, which is analogous to membrane velocity, into a voltage Vd, which is analogous to membrane displacement. The relationship between the circuit variables is
CdVdtd +g1(Vd Vref) = Ivel: (5:5) For a steady-state inputIvel, we may neglect theCdVd=dtterm, and thus the resting value of Vd is Vref g1Ivel. For a quickly varying input Ivel, we may neglect the g1(Vd Vref) term, and thus
CdVdtd = Ivel; (5:6)
OriginalMembraneCircuit toDisplacementConversionfromVelocity Second-OrderFilterSaturatingNonlinearity ConversiontoCurrent Feedback
Vref
Vref
IvelVd
Vd2 g1
g2 g3 g4
CCC Ivel Vin
Vout
Outer-Hair-CellCircuit Ifb Figure5.2Outerhaircellcircuit,shownwiththeoriginalbasilar-membranecircuit.The currentIvelfromthemembranecircuitisconvertedtoavoltageVdwhichisanalogousto membranedisplacement.Thesecond-orderlow-passltercircuitcomputesVd2,adelayed versionofVd.Finally,thenarrow-input-rangeamplierconvertsVd2backtoacurrent,Ifb
; whichisfedbackintotheoriginalcurrentIvel.Toallowinvestigationofbothpositive-and negative-feedbackmodels,thepolarityofthenalampliermaybereversed.
The second block computes Vd2, a ltered and delayed version of Vd, corresponding to the delayed outer-hair-cell motile response to bending of the stereocilia. The second-order lter stage is a variation of the Lyon and Mead cochlea section [66, 71], and has been analyzed extensively by Kerns [49]. The transfer function is given by
H(s) = Vd2
Vd
= 1
1 +s=Q+2s2; where =p23,Q=p2=3,2=C=g2, and 3 =C=g3.
The third block uses a transconductance amplier to feed back a currentIfb that satu- rates at large values ofVd, corresponding to a saturating nonlinear force by the outer hair cells. The transconductanceg4 controls the amount of current that is injected|that is, the strength of the feedback. We use a narrow-input-range amplier for g4, to force saturation at low signal amplitudes, at which all the other ampliers (wide-range-input) in the circuit are still linear. The detailed transistor-level implementation is given later, in Section 5.4.
By designing the chip with the terminals of the g4 amplier reversed, we may choose the opposite sign of the feedback.
The reference level Vref is used in the conversions between current and voltage. Of course, osets within any given stage may cause inaccuracies in the computation, but since the DC level of the resistive network as a whole is determined by the voltages applied at the two ends of the resistive network, and is absolutely constant everywhere, any small error eects will be local|Vref does not have to be adjusted to track a globally drifting DC level.