Cochlear Mechanics: Analysis and Analog VLSI

This dissertation describes a simplified model of a cochlear mechanics problem and a technique to solve the problem. A silicone slug modeling the behavior of a passive slug was manufactured and tested.

Review of Previous Work

Rhode showed that the sharp tuning was dependent on the animal's health and experimental condition. In the 1980s and early 1990s, research shifted towards an understanding of the active outer hair cells.

Overview

The model is compared with the lter cascade analog VLSI model of Lyon and Mead [66], and with the classical transmission line model [85]. The circuit is shown to generate the appropriate delayed signal as required by the conceptual model; however, the method for feeding the signal back to the basilar membrane is still under development.

Original Contributions of the Present Work

The circuit is able to mimic the features of the two-dimensional conceptual model, including the transition from long-wave to short-wave propagation and the appearance of the second wave mode predicted by the LG mode-coupling solution of Chapter 3. chapter describes the anatomy and basic function of the cochlea and cites groundbreaking measurements that shaped the modern understanding of cochlear function.

Anatomy

Dieter's cells have processes that extend upward to support the tips of the outer hair cells; to. Erent connections with outer hair cells are made by nerve fibers from the olivo-cochlear bundle in the superior olivary region of the brainstem.

Figure 2.1 Anatomy of the human auditory periphery. Adapted from Kessel and Kardon [50].

Function

The effect of basilar membrane displacement on the stereocilia of the hair cells is shown in Figure 2.11. Other circumstantial evidence implicates the outer hair cells as the force-generating active elements of the cochlea.

Figure 2.8 Propagation of a wave down the cochlea, for a xed input fre- fre-quency, viewed at one moment

Measurements

Ruggero and Rich [95] have provided compelling evidence that the amplitude nonlinearity is due to mechanically active cells in the organ of Corti, most likely the outer hair cells. Figure 2.15 compares the isovelocity curve of a point on the guinea pig cochlea with the neural isoresponse curve of a spiral ganglion cell in the guinea pig.

Figure 2.12 Rhode's data, taken from a live squirrel monkey using the Mossbauer tech- tech-nique

Abstraction

Steele and Zais have shown that the winding of the biological slug is indifferent to the wave propagation [113]. In the short-wave range, just before the amplitude of the membrane displacement peaks, the outer hair cells affect the signal and preferentially amplify soft sounds that would otherwise be too weak to hear.

Figure 2.16 Summary of the basic ideas in cochlear wave propagation. At the basal end of the cochlea, waves travel with long wavelength and high speed.

Formulation of the Passive Two-Dimensional Problem

Hydrodynamics

The boundary conditions of the hard walls at the right and bottom of the model imply that there is no fluid flow in a direction perpendicular to the boundary. At x = 0 the movement of the fluid is determined by the movement of the stirrups, i.e.

Basilar-Membrane Boundary Condition

Some authors use a 'tension' term, which corresponds to longitudinal coupling between the beam-like laments of the basilar membrane. Finally, there is the issue of the active behavior of the outer hair cells, which influences the separation boundary condition.

Figure 3.2 The mathematical two-dimensional model of passive cochlear mechanics. The contribution of the active outer hair cells is not included.

Review of Established Solution Techniques

Numerical Solutions

Waves are naturally expected in hyperbolic systems, but Whitham points out that elliptic systems, such as those governed by Laplace's equation, are also capable of propagating waves due to the interesting eect of the boundary conditions [126, p.

Exact Solution for a Uniform Cochlea

S M!2)(kitanhkrh krtankih) !(krtanhkrh+kitankih) 2!2tanhkrhtankih= 0; which are merely the real and imaginary parts of the complex dispersion relation. M!2 for Re[kh]1: (3:18) Note that the wave number is independent of the channel height h in the short wave case.

Figure 3.3 The complex function tanh(kh), as a function of its complex argument kh. (a) The real part

Approximate LG Solution for a Nonuniform Cochlea

We will develop the condition for the invincible box, which is quite simple; the validity of the result in the damped case has been shown by de Boer and Viergever [22]. The amplitude and phase of the basilar-membrane displacement ratios are shown in Figure 3.5 for the two methods. However, the quantitative agreement is poor for lower frequencies, and in the cut-o region, the amplitude of the LG solution decreases very quickly and the phase behavior is inaccurate.

New Solution Techniques

Higher-Order Calculation of Stapes Displacement

The final difference solutions are shown as solid lines; the LG solutions are shown as dotted lines. The agreement becomes increasingly poor for low frequencies, and there is a sharp difference between the numerical solutions and the LG solutions after the peak. The improved stapes displacement calculation has been incorporated into the LG calculation and the results are shown in Figure 3.6.

Figure 3.5 Comparison of LG and nite-dierence solutions for the basilar- basilar-membrane displacement ratio, for various frequencies

The Mode-Coupling LG Solution

Note that the off mode has the largest amplitude near the best site. The traveling-wave and cut-off mode phases are shown in Fig. 3.14(b) for the case of destructive interference. The RLE of the LG solution with mode coupling is compared with the RLE of the simple LG solution in Figure 3.16.

Figure 3.6 LG solution with improved calculation of stapes displacement (dashed lines) compared with nite-dierence solution (solid lines), using Neely's parameters

Discussion

A plain three-dimensional rectangular box model showing the extension of the basilar membrane into the supporting bony shelf. We have seen that the increasing membrane mass and participating uid mass can be attributed to the expansion of the basilar membrane. So it is likely that the expansion of the basilar membrane can be a dominant factor in the variation of all the physical parameters.

Figure 3.17 Three-dimensional models. (a) A common three-dimensional rectangular-box model, showing the widening of the basilar membrane within the supporting bony shelf

Development of the Circuit Elements

The Fluid Subcircuit

The mesh interior is modeled by an incompressible uid; It is analogous to the mass uid 2 in the upper and lower chambers of the original model. Boundary edges with double size resistors are analogous to the hard wall boundary conditions on the right and bottom sides of the physical model. The boundary conditions of the rigid wall are represented by the edges on the right and bottom sides of the grid.

Figure 4.1 Finite-dierence approximation of a resistive sheet by a resistive network. (a) Resistive sheet

The Membrane Subcircuit

The basil membrane is represented by the group of boxes marked \B" along the top edge of the network. The boundary between the basil and the membrane is represented by a series of circuits along the top edge of the network, as described in section 4.1.2 The flow I flowing between the terminals is proportional to the rate at which the voltage across the two plates changes.

Figure 4.3 Idealized analog VLSI circuit elements for implementing linear systems. (a) Transconductance amplier

Variation of Parameters

The resistive-network cochlear model would not propagate waves using the constant-mass scaling conguration. In the basic membrane circuit of Figure 4.4(a), the attenuation term is the one parameter that cannot be adjusted by a transconductance. Similarly, in the resistive network model, the exponentially varying membrane stiffness, membrane mass and participating fluid mass are all controlled by a single tilted bias wire.

Figure 4.5 Tilted bias lines. An exponentially varying membrane pa- pa-rameter is achieved through the use of a dierential voltage applied across a resistive bias wire

Characterization of the Cochlear Model

The One-Dimensional Cochlear Model

The one-dimensional cochlear model consists of a chain of cochlear stages, as shown in Figure 4.12. The measured and simulated frequency responses for every fifth voltage tap of a 64-stage cochlea are shown in Figure 4.13. The phase response for the voltage signals at every fifth tap is shown in Figure 4.14.

Figure 4.13 Magnitude of frequency response of every fth voltage tap, from tap 10 to tap 60, in a 64-stage cochlea

The Two-Dimensional Cochlear Model

A notch is evident in the response near the bottom of the uid, corresponding to the cut-o conditions where the lowest wavenumber mode begins to dominate. To allow visualization of the complex two-dimensional behavior of the uid pressure, the data in Figure 4.19 are plotted as an intensity grid in Figure 4.20. Finally, the magnitude data is shown as a surface plot in Figure 4.21 to illustrate the notch associated with destructive interference of the traveling wave and cut-o modes.

Figure 4.15 Magnitude of frequency response of every fth current tap, from tap 10 to tap 60, in a 64-stage cochlea

Comparison to Other Circuit Models

The data are plotted as a surface to illustrate the notch associated with destructive interference from the traveling wave and cut-o modes. For ease of comparison, the two models are shown in Figure 4.23, where the resistive network model is drawn to emphasize the series resistanceR and the equivalent impedanceZm of the basilar membrane circuit. The bias circuitry allows the resistance to be relatively independent of the DC operating point.

Figure 4.20 Chip data, showing the real part of uid pressure at a xed frequency, at eight phases to allow visualization of the traveling wave

Analog VLSI Implementation

• Resistor Circuit
• Basilar-Membrane Circuit
• Reduction of Parasitic Capacitance
• DC Operating Point
• Instrumentation, Fabrication, and Testing

The parameter is related to the efficiency of the bias voltage in controlling the current coming into the bias circuit. The basic element of the circuit is the transconductance amplifier, described in detail by Mead [71]. A single transistor makes a copy IOUT of the current forced into the circuit by the resistive network.

Figure 4.25 Detailed transistor-level schematic diagram of the basilar- basilar-membrane circuit of Figure 4.4(a)

Summary

In two-dimensional cochlear chips, there are 305 voltage signals and 61 current signals to be observed externally, but there are only or 84 pins on standard packages. All data in this dissertation was taken from chips fabricated on the Orbit 2 dual-pole dual-metal p-well process via the MOSIS service. Single section data is from circuits placed on a TinyChip frame (2.22mm 2.25mm die).

Review of Previous Active Models

The outer hair cells are the active (energy producing) elements in the organ of Corti. Outer hair cells respond to the displacement of their stereocilia by exerting a force on the basilar membrane. Outer hair cells are limited in how much force they can exert on the basilar membrane.

The Outer Hair Cell Model

Mathematical Description

The parameters of the low-pass or band-pass filter are H(x) and QH, where we have implicitly assumed second-order filters. The conceptual model requires saturating nonlinearity, as there must be an upper limit to the magnitude of the force that can be produced by the outer hair cell. For this reason, we cannot talk about the impedance of the organ of Corti," since the concept of impedance is a concept of a linear system.

Analysis and Simulation

Meanwhile, we need an experimental medium to investigate the behavior of the model. 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 real time) required 8.5 hours of CPU time on an HP9050 computer, using a very ecient integral-equation method [24]. Hoping to exploit these advantages, we continue to develop the circuit model of the outer hair cell.

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

The Circuit Model

The second block calculates Vd2, a filtered and delayed version of Vd, corresponding to the delayed motile response of the outer hair cells to stereocilia bending. The second-order lter stage is a variation on the cochlea section of Lyon and Mead [66, 71] and has been extensively analyzed by Kerns [49]. The transconductance g4 controls the amount of current injected, i.e. the strength of the feedback.

Characterization of the Outer Hair Cell Circuit

By designing the chip with the terminals of the g4 amplifier reversed, we can choose the opposite sign of the feedback. The correct operation of the outer hair cell circuit can be checked by comparing the Ivel and Vd2 signals. Currently, the correct behavior of the outer hair cell circuit has not been verified at the system level, so we have to leave the project as is.

Figure 5.3 Response from a single basilar-membrane and outer-hair-cell circuit. (a) Magnitude and (b) phase of V out , normalized to V in

Analog VLSI Implementation

The cutoff frequency of the second-order lter is set to about 400 Hz, slightly lower than the peak frequency of the Ivelsignal, and the second-order lter is tuned to be slightly resonant, so that a significant bump appears around 400 Hz on the Vd2 signal. There is a large phase delay (about 0.3 cycle) in the Vd2 signal at 400 Hz, as required in the active stability model with negative feedback. Since there is still confusion in the hearing community about the form and sign of mechanical feedback from motile outer hair cells, we must remember that the model is intended to allow the investigation of hypotheses about the biological system and considerable experimentation with this model and with other models before reaching a consensus on the mechanisms underlying gain control and frequency tuning in the real cochlea.

Summary

In this work, we have seen that the traditional LG method does not satisfy the constraints of the problem and that a different wave mode is required for a consistent solution. An analog VLSI model of passive cochlear uid mechanics has been fabricated and tested; it is able to model the primary three-dimensional effects of the cochlea in considerable detail. The active and nonlinear behavior of the cochlea is a topic of intense research interest at present, and many questions remain unresolved.

Figure 5.4 Transistor-level outer-hair-cell circuit.

Finite-Dierence Method

In the nite-dierence method, the two-dimensional channel is conceptually divided into a grid of points NxNy, where Nx and Ny are the number of points in the x and y directions, respectively. Additional terms appear in the equations for points located at the boundary of the membrane or at the edge of the column. The equations can be written in the following block form, given here for the example Nx = 6,Ny = 5:.

LG Method

Mode-Coupling LG Method

Other Programs

Measurement of the amplitude and vibration phase of the basilar membrane using the Mossbauer Eect. Systemic injection of furosemide alters the mechanical response to sound of the basilar membrane. On the physical background of the point impedance characterization of the basilar membrane in cochlear mechanics.