662 
A PASSIVE SHARED ELEMENT ANALOG 
ELECTRICAL COCHLEA 
Joe Eisenberg 
Bioeng. Group 
U.C. Berkeley 
David Feld 
Dept. Elect. Eng. 
207-30 Cory Hall 
U.C. Berkeley 
Berkeley, CA. 94720 
ABSTRACT 
Edwin Lewis 
Dept. Elect. Eng. 
U.C. Berkeley 
We present a simplified model of the micromechanics of the human 
cochlea, realized with electrical elements. Simulation of the model 
shows that it retains four signal processing features whose importance 
we argue on the basis of engineering logic and evolutionary evidence. 
Furthermore, just as the cochlea does, the model achieves massively 
parallel signal processing in a stmcturally economic way, by means of 
shared elements. By extracting what we believe are the five essential 
features of the cochlea, we hope to design a useful front-end filter to 
process acoustic images and to obtain a better understanding of the 
auditory system. 
INTRODUCTION 
Results of psychoacoustical and physiological experiments in humans indicate that 
the auditory system creates acoustic images via massively parallel neural computations. 
These computations enable the brain to perform voice detection, sound localization, and 
many other complex taskS. For example, by recording a random signal with a wide range 
of frequency components, and playing this signal simultaneously through both channels 
of a stereo headset, one causes the brain to create an acoustical image of a "shsh" sound in 
the center of the head. Delaying the presentation of just one frequency component in the 
random signal going to one ear and simultaneously playing the original signal to the 
other ear, one would still have the image of a "shsh" in the center of the head; however if 
one mentally searches the acoustical image space carefully, a clear tone can be found far 
off to one side of the head. The frequency of this tone will be that of the component with 
the time delay to one ear. Both ears still are receiving wide-band random signals. The 
isolated tone will not be perceptible from the signal to either ear alone; but with both 
signals together, the brain has enough data to isolate the delayed tone in an acoustical 
image. The brain achieves this by massively parallel neural computation. 
Because the acoustic front-end filter for the brain is the cochlea, people have proposed 
that analogs of the cochlea might serve well as front-end filters for man-made processors 
of acoustical images (Lyon, Mead, 1988). If we were to base a cochlear analog on 
current biophysical models of this structure, it would be extraordinarily complicated and 
extremely difficult to realize with hardware. Because of this, we want to start with a 
cochlear model that incorporates a minimum set of essential ingredients. The ears of 
lower vertebrates, such as alligators and frogs, provide some clues to help identify those 
ingredients. These animals presumably have to compute acoustic images similar to ours, 
A Passive Shared Element Analog Electrical Cochlea 663 
but they do not have cochleas. The acoustic front-end filters in the ears of these animals 
evolved independently and in parallel to the evolution of the cochlea. Nevertheless, those 
front-end filters share four functional properties with the part of the cochlea which 
responds to the lower 7 out of 10 octaves of hearing (20 Hz. to 2560 Hz.): 
e 
e 
They are multichannel filters with each channel 
covering a different part of the frequency spectrum. 
Each channel is a relatively broad-band frequency 
filter. 
Each filter has an extremely steep high-frequency 
band edge (typically 60 to 200 db/oct). 
Each filter has nearly linear phase shift as a function 
of frequency, within its passband. 
The front-end acoustical filters of lower vertebrates also have at least one structural feature 
in common with the cochlea: namely, the various filter channels share their dynamic 
components. This is the fifth property we choose to include. Properties 1 and 3 provide 
good resolution in frequency; properties 2 and 4 are what filter designers would add to 
provide good resolution in time. 
In order to compute acoustical images with the neural networks in our brain, we need 
both kinds of resolution: time and frequency. Shared elements, a structural feature, has 
obvious advantages for economy of construction. The fact that evolution has come to 
these same front-end filter properties repeatedly suggests that these properties have 
compelling advantages with respect to an animal's survival. We submit that we can 
realize all of these properties very well with the simplest of the modem cochlear models, 
namely that of Joseph Zwislocki (1965). This is a transmission line model made 
entirely of passive elements. 
Figure 1 - Drawing of the ear with the cochlea represented by an 
electrical analog. 
664 Feld, Eisenberg and Lewis 
A COCHLEAR MODEL 
In order to illustrate Zwislocki's model, a quick review of the mechanics of the 
cochlea is useful. Figure 1 depicts the ear with the cochlea represented by an electrical 
analog. A sound pressure wave enters the outer ear and strikes the ear drum which, in 
turn, causes the three bones of the middle ear to vibrate. The last bone, known as the 
stapes, is connected to the oval window (represented in figure 1 by the voltage source at 
the beginning of the electrical analog), where the acoustic energy enters the cochlea. 
As the acoustic energy is transferred to the oval window, a fluid-mechanical wave is 
formed along a structure known as the basilar membrane (This membrane and the 
surrounding fluid is represented by the series and shunt circuit elements of figure 1). As 
the basilar membrane vibrates, the acoustical signal is transduced to neural impulses 
which travel along the auditory nerve, carrying the data used by the nervous system to 
compute auditory images. Figure 2 is taken from a paper by Zweig et al. (1976), and 
depicts an uncoiled cochlea. As the fluid-mechanical wave travels through the cochlea: 1) 
The wave gradually slows down, and 2) The higher-frequency components of the wave are 
absorbed, leaving an increasingly narrower band of low-frequency components proceeding 
on toward the far end of the cochlea. If we were to uncoil and enlarge the basilar 
membrane it would look like a swim fin (figure 3). If we now were to push on the 
basilar membrane, it would push back like a spring. It is most compliant at the wide, 
thin end of the fin. Thus as one moves along the basilar membrane from its basal to 
apical end, its compliance increases. Zwislocki's wansmission-line model was tapered in 
this same way. 
ci{'cular 
canal 
Fund 
window 
Scol a 
tympani 
membrane 
vestibuli 
Cochlea 
stretched out 
I"'lelicot rema 
Figure 2 - Uncoiled cochlea (Zweig, 1976). 
A Passive Shared Element Analog Electrical Cochlea 665 
Basal End 
Figure 3 - Simplified uncoiled and enlarged drawing of the basilar 
membrane. 
Zwislocki's model of the cochlea is a distributed parameter transmission line. 
Figure 4 shows a lumped electrical analog of the model. The series elements (L1,...Ln) 
represent the local inertia of the cochlear fluid. The shunt capacitive elements (C 1,...Cn) 
represent the local compliance of the basilar membrane. The shunt resistive elements 
(R 1,...Rn) represent the local viscous resistance of the basilar membrane and associated 
fluid. The model has one input and a huge number of outputs. The input, sound 
pressure at the oval window, is represented here as a voltage source. The outputs are 
either the displacements or the velocities of the various regions of the basilar membrane. 
L 1 I-, 2 L n 
Figure 4 - 
R2 n 
  
c 
Transmission line model of the cochlea represented as an 
electrical circuit. 
In the electrical analog, shown in figure 4, we have selected velocities as the outputs (in 
order to compare our data to real neural tuning curves) and we have represented those 
velocities as the currents (I1,...In). The original analysis of Zwislocki's tapered 
transmission line model did not produce the steep high frequency band edges that are 
observed in real cochleas. This deficiency was a major driving force behind the early 
development of more complex cochlear models. Recently, it was found that the original 
analysis placed the Zwislocki model in an inappropriate mode of operation (Lewis, 1984). 
In this mode,determined by the relative parameter values, the high frequency band edges 
had very gentle slopes. Solving the partial differential equations for the model with the 
help of a commonly used simplification (the WKB approximation), one finds a second 
mode of operation. In this mode, the model produces all five of the properties that we 
desire, including extraordinarily steep high-frequency band edges. 
666 Feld, Eisenberg and Lewis 
RESULTS 
We were curious to know whether or not the newly-found mode of operation, with 
its very steep high-frequency band edges, could be found in a finite-element version of the 
model. If so, we should be able to realize a lumped, analog version of the Zwislocki 
model for use as a practical front-end filter for acoustical image formation and processing. 
We decided to implement the finite element model in SPICE. SPICE is a software 
package that is used for electrical circuit simulation. Our SPICE model showed the 
following: As long as the model was made up of enough segments, and as long as the 
elements had appropriate parameter values, the second mode of operation indeed was 
available. Furthermore, it was the predominant mode of operation when the parmeter 
values of the model were matched to biophysical data for the cochlea. 
 2o t 
o  
FREQUENCY (kHz) 
Figure 5 - Frequency response of 
the basilar membrane velocity. 
Figure 6 - Inverted neural tuning curves 
from three afferent fibers of a cat cochlea 
(Kiang and Moxon, 1974). 
Figure 5 shows the magnitude of the electrical analog's response plotted against 
frequency on log-log coordinates. The five curves correspond to five different locations 
along our model. The cutoff frequencies span approximately seven octaves. Further 
adjustments of the parameters will be needed in order to shift these curves to span the 
lower seven octaves of human audition. For comparison, figure 6 shows threshold 
response curves of a cat cochlea from a paper by Kiang and Moxon (1974). These curves 
are inverted intentionally because Kiang and Moxon plotted stimulus threshold vs. 
frequency rather than response amplitude vs. frequency. We use these neural tuning 
curves for comparison because direct observations of cochlear mechanics have been 
limited to the basal end. Furthermore, in the realm of single frequencies and small 
signals, Evans has produced compelling evidence that this is a valid comparison (Evans, 
in press). These three curves are typical of the lower seven octaves of hearing. One 
obvious discrepancy between Kiang and Moxon's data and our results is that our model 
does not exhibit the sharp comers occurring at the band edges. The term sharp corner 
denotes the fact that the transition between the shallow rising edge and steep falling edge 
of a given curve is abrupt i.e. the comer is not rounded. 
A Passive Shared Element Analog Electrical Cochlea 667 
Figure 7 shows what happens to the response curve at a single location along our 
model as the number of stages is increased. The curve on the right, in figure 7, was 
derived with 500 stages and does not change much as we increase the number of stages 
indefinitely. Thus the curve represents a convergence of the solution of the lumped 
parameter Zwislocki model to the distributed parameter model. The middle curve was 
derived with 100 stages and the left-hand curve was derived with 50. In any lumped- 
element transmission line, there occurs an artifactual cutoff which occurs roughly at the 
point where the given input wavelength exceeds the dimensions of the lumped elements. 
If we do not lump the stages in our model finely enough, we observe this artifactual 
cutoff as opposed to the true cutoff of Zwislocki's distributed parameter model. This 
phenomena is clearly observed in the curve derived from 50 stages and may account for 
the sharper comers in response curves from real cochleas. However, in order to make our 
finite element model operate in a manner analogous to that of the distributed parameter 
Zwislocki model we need approximately 500 stages. 
I I I II1111 I [ I III1[ I I I I Illll I I I I lITIT 
LOG (FREQUENCY) 
100000 
I I IllIll] [ I IllIll] I f IllIll I I I IllIll I t I liTIll 
Apcal End 
LOG (FEQUENCY) (HZ) 
Figure 7 - Convergence of cut- 
toff points as the number of 
branches increase. 
Figure 8 - Frequency response of the 
basilar membrane velocity without 
the Heliocotrema. 
Basal End 
100000 
A critical element in the Zwislocki model is a terminating resistor, representing the 
heliocotrema (see Rh in figure 3). The heliocotrema is a small hole at the end of the 
basilar membrane. Figure 8, shows the effects of removing that resistor. The irregular 
frequency characteristics are quite different from the experimental data and represent 
possibly wild excursions of the basilar membrane. 
Figure 9, shows phase data for the Zwislocki model, which is linear as a function of 
frequency. Anderson et al (1971), show similar results in the squirrel monkey. 
With lumped-element analysis we are able to obtain temporal as well as spectral 
responses. For a temporal waveform such as an acoustic pulse, the linear relationship 
between phase and frequency guarantees that those Fourier components which pass 
through the spectral filter will be reassembled with proper phase relationships at the 
output of the filter. As it travels down the basilar membrane, the temporal waveform 
will simply be smoothed, due to loss of its higher-frequency components and delayed, due 
to the linear phase shift. Figure 10 shows the response of our electrical analog to a 1 
msec wide square pulse at the input. The curves represent the time courses of basilar 
membrane displacement at four equally spaced locations along the cochlea. The curve on 
the right represents the response at the apical end of the cochlea (the end farthest from the 
input). The curve on the left represents the response at a point 25 percent of the distance 
8 Feld, Eisenberg and Lewis 
input end. The impulse responses of mammalian cochleas and of the auditory filters of 
lower vertebrates all show a slight ringing, again indicating a deficiency in our model. 
2000 
Basal End 
TIME (SECONDS) 
Apical End 
Figure 9 - Phase response of the 
basilar membrane velocity. 
Figure 10 - Traveling square wave 
pulse along the membrane from the 
basal to apical end. 
CONCLUSION 
Research activity studying the function of higher level brain processing is in its 
infancy and little is known about how the various features of the cochlea, such as linear 
phase, sharp band edges, as well as nonlinear features, such ,as two-tone suppression and 
cubic difference tone excitation, are used by the brain. Therefore, our approach, in 
developing a cochlear model, is to incorporate only the most essential ingredients. We 
have incorporated the five properties mentioned in the introduction which provide 
simplicity of analysis, economy of hardware consauction, and the preservation of both 
temporal and spectral resolution. The inclusion of these properties is also consistent with 
the fact that they are found in numerous species. 
We have found that in the correct mode of operation a tapered transmission line 
model can exhibit these five important cochlear properties. A lumped-element 
approximation can be used to simulate this model as long as at least 500 stages are used. 
As observed in figure 7, by decreasing the number of stages below 500, the solution to 
the lumped-element model no longer adheres to the Zwislocki model. In fact, the output 
of the coursely lumped model more closely resembles the neural tuning data of the 
cochlea in that it produces very sharp comers. There is some evidence that indicates the 
cochlea is constructed of discrete components. Indeed, the hair cells themselves are 
discretized. If this idea is valid, a model constructed of as little as 50 branches may more 
accurately represent the cochlear mechanics then the Zwislocki model. 
Our simple model has some drawbacks in that it does not replicate various properties 
of the cochlea. For example, it does not span the full ten octaves of human audition, nor 
does it explain any of the experimentally observed nonlinear aspects seen in the cochlea. 
However, we take this approach because it provides us with a powerful analysis tool that 
will enable us to study the behavior of lumped-element cochlcar models. This tool will 
allow us to proceed to the next step; the building of a hardware analog of the cochlea. 
A Passive Shared Element Analog Electrical Cochlea 669 
allow us to proceed to the next step; the building of a hardware analog of the cochlea. 
RESEARCH DIRECTIONS 
In and of itself, the tapered shared element travelling wave structure we have chosen is 
interesting to analyze. In order to get even further insight into how this filter works and 
to aid in the building of a hardware version of such a filter, we plan to study the 
placement of the poles and zeroes of the transfer function at each tap along the structure. 
In a travelling wave transmission line we expect that the transfer function at each tap will 
have the same denominator. Therefore, it must be the numerators of the transfer 
functions which will change, i.e. the zeroes will change from tap to tap. It will be of 
interest to see what role the zeroes play in such a ladder structure. Furthermore, it will be 
of great interest to us to study what happens to the poles and zeroes of the transfer 
function at each tap as the number of stag.qs is increased (approaching the distributed 
parameter filter), or decreased (approaching the lumped-element cutoff version of the filter 
with sharper comers). We should emphasize that our circuit is bi-directional, i.e. there is 
loading from the stages before and after each tap, as in the real cochlea. For this reason, 
we must consider carefully the options for hardware realization of our circuit. We might 
choose to make a mechanical structure on silicon or some other medium, or we could 
convert our structure into a uni-directional circuit and build it as a digital or analog 
circuit. 
Using this design we plan to build an acoustic imaging device that will enable us to 
explore various signal processing tasks. One such task would be to extract acoustic 
signals from noise. 
All species need to cope with two types of noise, internal sensor and amplifier noise, 
and external noise such as that generated by wind. Spectral decomposition is on effective 
way to deal with internal noise. For example, the amplitudes of the spectral components 
in the passband of a filter are largely undiminished, whereas the broadband noise, passed 
by the filter, is proportional to the square root of the bandwidth. External noise reduction 
can be accomplished by spatial decomposition. When temporal resolution is preserved in 
signals, spatial decomposition can be achieved by cross correlation of the signals from 
two ears. Therefore, from these two properties, spectral and temporal resolution, one can 
construct an acoustic imaging system in which signals buffed in a sea of noise can be 
extracted. 
Acknowledgments 
We would like to thank Thuan Nguyen for figure 1, Eva Poinar who helped with the 
figures, Michael Sneary for valuable discussion, and Bruce Parhas for help with 
programming. 
670 Feld, Eisenberg and Lewis 
References 
Anderson, D.J., Rose, J.E., Hind, J.E., Brugge J.F., Temporal Position of Discharges in 
Single Auditory Nerve Fibers Within the Cycle of a Sine-Wave Stimulus: Frequency and 
Intensity Effects, J. Acoust. Soc. Am.., 49, 1131-1139, 1971. 
Evans, E.F., Cochlear Filtering: A View Seen Through the Temporal Discharge Patterns 
of Single Cochlear Nerve Fibers. A talk given at the 1988 NATO advanced workshop, to 
be published as (J.P. Wilson, D.T. Kemp, eds.) Mechanics of Hearing, Plenum Press, 
N.Y. 
Kiang, N.Y.S., Moxon, E.C., Tails of Tuning Curves of Auditory-Nerve Fibers, J. 
Acoust. Soc. Am., 55, 620-630, 1974. 
Lewis, E.R., High Frequency Rolloff in a Cochlear Model Without critical-layer 
resonance, J. Acoust. Soc. Am., 76 (3) September, 1984. 
Lyon, R.F., Mead, C.A., An Analog Electronic Cochlea, IEEE Trans.-ASSP, 36, 1119- 
1134, 1988. 
Zweig, G., Lipes, R., Pierce, J.R., The Cochlear Compromise, J. Acoust. Soc. Am., 
59, 975-982, 1976. 
Zwislocki, J., Analysis of Some Auditory Characteristics, in Handbook of Mathematical 
Psychology, Vol. 3, (Wiley, New York), pp. 1-97, 1965. 
