Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 49–52___________________________________________________________________________________________________________An Improved Parameter Value Optimization Technique
for the Reflectionless Transmission-Line Model of the Cochlea
Takemori Orima and Yoshihiko Horio

Research Institute of Electrical Communication, Tohoku University
2-1-1 Katahira, Aoba-ku, Sendai, Miyagi, 980-8577, Japan
E-mail: takemori.orima.q2@dc.tohoku.ac.jp

Abstract
The passive reflectionless transmission-line model can reproduce physiological cochlear properties. We haveapplied an optimization technique to obtain good parameter values that give a desired cochlear property. However,
we found that, in some case, it is difficult to find the sub-optimal parameter values because of complex
dependencies among parameters. In this paper, we explicitly formulate some of the cochlear properties in thereflectionless transmission-line model, and improve the optimization technique using these formulae.
Keywords: cochlea, passive model, reflectionless transmission-line model, optimization technique.

optimization methods, and chose the best one for thisdesign technique. Finally, we demonstrate some design
examples to confirm the validity and efficiency of theproposed method.

1. IntroductionThe physiological transmission-line model of thecochlea can reproduce the mechanical vibration of thebasilar membrane well. To approximate individual
human cochlear characteristics to experimental data, it
is necessary to tune parameter values of the model.
However, because of the large number of parameters, it
is difficult to determine these values.
The passive reflectionless transmission-line model1(passive model) has been proposed to alleviate thisproblem. Although this model has less number ofparameters, it can reproduce the characteristics of thecochlea well. Therefore, we proposed a designtechnique for this model using parameter optimization2.
However, this method requires a long calculation time
to design.
In this paper, we explicitly give approximate
formulae for some key characteristics of the passivemodel to accelerate the calculation. In addition, anoptimization method is applied in the design techniquewith proposed formulae. We compare three

2. The passive reflectionless transmission-line
model
A minute part between x and x + dx of the passivereflectionless transmission-line model1 is shown in Fig.
1, where x is a distance from the input terminal. Theparallel impedance Zp(x, ω) in Fig. 1 is given by

Z p (x, ω ) = jωL p (x ) + R p (x ) +1,jωC p (x )(1)Fig. 1. A minute part between x and x + dx of the passivereflectionless transmission-line model1.

Copyright © 2017, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 49–52___________________________________________________________________________________________________________where ω is the angular frequency of the input signal.
Assuming that the value of each of the circuit
elements in Zp(x,ω) changes exponentially with distance,
they can be given as

R p (x ) = R0 e − ax , L p ( x) = L0 e ax , C p (x ) = C0 e ax , (2)where R0, L0, and C0 are values at x = 0, and a is aconstant.
In Fig. 1, we further assume that the characteristic
impedance Z0(x, ω) is independent of the distance, sothat

Z 0 ( x, ω ) = r ,Fig. 2. The solid line: the frequency vs. gain characteristics ofthe original passive model obtained from Eq. (7). The dashedline: the approximation with Eqs. (13) and (14).

(3)

where r is a constant. As a result, the series impedanceZs(x,ω) in Fig. 1 can be defined asr2
Z s ( x, ω ) =.
Z p ( x, ω )Q(x ) =

(4)

1
L0 C0

e −ax .

ω1 (x ) =(5)Taking the integral of the propagation constant withrespect to x yieldsΓ ( x, ω ) = ∫x01exp(− Γ (x, ω )).
Z p ( x, ω )

(6)

(7)

(11)
(12)

ln g max (x ) − ln g z (x ),ωc (x ) − ω z (x )(14)g z (x ) = F (x, ω z (x )) ,(15)ω z (x ) = µ z (x )ω res (x ),(16)µ z (x ) =− v z (x ) − v z (x ) − 4
22,2
 10  C0 − ax 2
v z (x ) =− 2.
(R0 e ) −  
L0 
 g max (x )  We explicitly formulate some of the key characteristicsof the passive model, the center angular frequency
ωc(x), the maximum gain gmax(x), and the quality factorQ(x) shown in Fig. 2 as follows:

g max (x ) = F (x, ωc (x )) , 1 g max (x ) 1,ln⋅A(x )  B(x )10 B(x ) =3. Explicit formulation of key characteristics ofthe passive modelωc (x ) = µ c (x )ω res (x ),1 ω res (x ),10  ω (x )ln g z (x ) − ω z (x )ln g max (x ) A(x ) = exp c
, (13)ωc (x ) − ω z (x )Finally, the transfer function is given by

F ( x, ω ) =10

ωc (x ) + 1 −where  1 + jωR C + ω L C e ax 0 00 0× lnax  1 + jωR0C0 − ω L0C0 e 

− ln





1

ω 2 (x ) =rjrdx =Z p ( x, ω )2a L0 C0 + jωR0 L0

1 + jωR0C0 + ω L0C0 
.
1 + jωR0C0 − ω L0C0 (10)where ω1(x) and ω2(x) are the angular frequencies at
g max (x ) 10 given by

On the other hand, the resonance angular frequency
ωres(x) is given byω res (x ) =ωc (x ),
ω1 (x ) − ω 2 (x )

(17)
(18)

In addition, µc(x) is given byµ c (x ) =(8)where(9)50− vc ( x ) + vc ( x ) + 4
22,

(19)

Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 49–52___________________________________________________________________________________________________________vc ( x ) =C0  r− ax − ax
 − R0 e  R0 e .
L0  a(20)

Comparisons between the original characteristics
calculated numerically from Eq. (7) and approximatedvalues from the above formulae are shown in Figs. 3 to5, where a = 0.288, R0 = 1.5, L0 = 2.385 × 10-7, C0 =2.132 × 10-7, and r = 5. Equation (7) took about 0.26
sec. for calculation, in contrast, the formulae about 2.5µsec. From these results, we can confirm validities andefficiency of the proposed formulae.
4. Design method using parameter optimization

Fig. 4. The original (circles) and approximated (solid line)
maximum gains gmax(x) according to the distance x obtained
from Eq. (7) and Eq. (9), respectively. The crosses: the percent
errors between them.

We design the passive model using the proposed
formulae to reproduce the human cochlear
characteristics. We employ an optimization technique toobtain the model parameters.
First, we define the object functions as follows:ωc (x ) − cf (x )E1 (x ) =E2 (x ) =,(23)

g max (x ) − mg (x ),mg (x )(24)

Q(x ) − q (x ),q (x )(25)

cf (x )

E3 ( x ) =where cf(x), mg(x), and q(x) are the target values for thecenter frequency, maximum gain, and quality factor at a
distance x, respectively. Then, we optimize the weighted
sum of these object functions E(x).

Fig. 5. The original (circles) and approximated (solid line)
quality factors Q(x) according to the distance x obtained from
Eq. (7) and Eq. (10), respectively. The crosses: the percent
errors between them.

E (x ) =33∑ w E (x ) and ∑ w = 1,i =1ii

i =1i(21)where wi ≥ 0 is a weight. In the following section, we
design the passive model with the target parametervalues shown in Table 1, where cf(x), mg(x), and q(x)were estimated based on the physiologicalexperiments3,4.
4.1. Optimization techniques
We compare the downhill simplex (DHS), thegenetic algorithm (GA), and the particle swarm
Table 1. The target design specifications.
Fig. 3. The original (circles) and approximated (solid line)
center frequencies ωc(x) according to the distance x obtained
from Eq. (7) and Eq. (8), respectively. The crosses: the percent
errors between them.

51x

15 mm

20 mm

25 mm

cf(x) [Hz]

2500

1200

500mg(x) [A/V]

9.4

9.4

6.1

q(x) [Hz/Hz]3.6

2.6

1.8

Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 1 (June 2017) 49–52___________________________________________________________________________________________________________optimization (PSO), using the design target at x = 25mm in Table 1. The conditions of each optimization
methods are summarized in Table 2. In the table, N is
the number of simplexes, individuals, and particles in
DHS, GA, and PSO, respectively. Note that, in the caseof PSO, one iteration is counted each time as the global
best is updated.
The results of the comparisons are shown in Table 3.
In the table, DHS gives the smallest optimization
function value.
Therefore, we use DHS as theoptimization method in our design.

Table 3. The obtained parameters and object function values
with DHS, GA, and PSO for x = 25 mm.

aRp
Lp
Cp
r
E (Eq. (21))

DHS

GA

PSO

0.19490.22990.21412.315 × 10-3 2.596 × 10-3 2.816 × 10-3
4.963 × 10-9 8.227 × 10-10 2.122 × 10-9
5.875 × 10-8 1.068 × 10-8 2.071 × 10-8
2.5
2.5
2.5
1.47 × 10-6

1.34 × 10-4

6.38 × 10-2

Table 4. The obtained center frequencies ωc(x), maximum
gains gmax(x), and quality factors Q(x) by the proposed
design technique with DHS.

4.2. Design examples
We designed the passive model specified in Table 1
using DHS. The results are shown in Table 4 and Fig. 6.
From Table 4, we confirm that the designed values arealmost the same as the target ones, although some errors
remain due to approximation errors in the proposed
formulae, in particular, Eq. (10).

x15 mm

20 mm

25 mm

ωc(x) [Hz]

2496

1196

495

gmax(x) [A/V]

9.408

9.417

6.133

Q(x) [Hz/Hz]

3.858

2.733

1.8345. Conclusion
We have explicitly formulated some key characteristicsof the passive reflectionless transmission-line model ofthe cochlea to alleviate the complex calculation loads.
These expressions efficiently reproduced the originalcharacteristics.
In addition, we defined the object functions forparameter optimizations. We compared DHS, GA, and
PSO as the optimization methods. As a result, we foundthat DHS is the best method.
We finally designed three cochlear characteristics at
different distances successfully using the proposed
formulae with DHS.
As a future problem, we will improve the formulaefor Q(x) for more precise designs.

Fig. 6. The obtained frequency vs. gain characteristics at x =15 mm, 20 mm, and 25 mm through the proposed optimization
technique using downhill simplex method.

References

Table 2. The simulation conditions.
NTrials
Iterations

DHS

GA

PSO

3030
5000

30
30
5000

30305000

1. T. Kohda, T. Une, and K. Aihara, “An active,reflectionless transmission-line model of the cochlea:Revisited,” in AIP Conf. Proc., vol. 1403, no. 1, pp. 578–583, 2011; DOI:10.1063/1.3658152.
2. T. Orima, Y. Horio, and T. Kohda, “A parameter value
optimization technique for a reflectionless transmissionline model of the cochlea,” in Proc. of Int. Symp. on
NOLTA2016, pp. 503–506, 2015.
3. G. V. Bekesy, “Experiments in Hearing,” McGraw-Hill
Book Co., New York, 1960.
4. M. A. Ruggero and A. N. Temchin, “Unexceptional
sharpness of frequency tuning in the human cochlea,”
PNAS, vol. 102, no. 51, pp. 18614–18619, 2005; DOI:10.
1073/PNAS.0509323102.

Acknowledgments
This work was supported, in part, by JSPS KAKENHI
Grant Number 16K00340, and the Cooperative
Research Program of the Research Institute of ElectricalCommunication, Tohoku University. The authors are
grateful to Emeritus Professor Tohru Kohda of Kyushu
University for his valuable discussions.

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