Simulation of cochlear response by bone conducted tone
Yasuki MURAKAMI(1)(1) National Institute of Technology, Oshima College, Japan, murakami@oshima-k.ac.jp

Abstract
To investigate cochlear responses to bone-conducted (BC) tones, a two-dimensional nonlinear cochlear modelis proposed in this paper. The proposed model comprises macromechanical and micromechanical behaviors
that can be described by the Laplace equation and Neely and Kims model, respectively. An outer hair cells
model is also included as the source of cochlear nonlinearities. Cochlear responses to air-conducted (AC) tonescan generally be simulated through stapes vibration. However, in this study, sound pressure was employed
to affect the cochlear ducts outer wall for simulating cochlear responses to BC tones. Both frequency- andtime-domain solutions can solve the proposed model. Through simulation, a forward traveling wave in thecochlea can be observed for both AC and BC tones in the frequency-domain solution. The time-domain solution
indicates compressive nonlinearity for both AC and BC tones. However, the degree of compressive nonlinearity
is the primary differentiating factor for AC and BC tones. For lower and higher input levels, the degree ofcompressive nonlinearity for BC tones is respectively higher and lower than that for AC tones. The obtained
simulation results thus corroborate that compared to AC tones, BC tones generate a cochlear traveling wave
response with marginally different compressive nonlinearity.
Keywords: Sound, Insulation, Transmission

1INTRODUCTION

In general, sounds propagating in an external ﬁeld can be heard. In this situation, the sounds are collected by
the outer ear and transmitted into the cochlea. This path is known as air conduction (AC). However, a different
path exists in addition to this. Bone conduction (BC) hearing is an alternative to AC hearing, and refers to
the transmission of vibrations through the skull bone to the cochlea. However, the mechanism of BC hearing
is still unclear. Toward solving this problem, ﬁve pathways are suggested [1] as follows: sound pressure in
the ear-canal, compression and expansion of the cochlear shell, transmission of pressure from the cerebrospinal
ﬂuid, inertia of the middle ear ossicles, and inertia of the cochlear ﬂuid. Therefore, one can hear the BC toneby the combination of all these pathways.
The contribution of the BC propagating pathways depends on the frequency of the incoming sound. In a realsituation, however, it is impossible to consider them separately owing to technical difﬁculties in investigating acomplicated hearing system. Meanwhile, a modeling study can facilitate understanding the contribution of theBC pathways.
Linear cochlear models simulating BC excitation have been proposed and demonstrated different responses between AC and BC tones [2, 3]. These results may suggest different cochlear mechanics excited by AC and BC
pathways in a linear domain. However, it is well known that cochlear responses are nonlinear. A perception
study has indicated compressive curves in AC and BC loudness functions owing to nonlinear cochlear processes[4]. This process can be caused by nonlinear and active elements [5].
The aim of this study is to investigate the effects of nonlinear and active elements on the input–output (IO)
property of the cochlea, develop a two-dimensional (2D) cochlear model based on our previously proposedmodel [6], and estimate the BM responses for the vibrations of the stapes and cochlear shell caused by AC andBC tones, respectively.

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BC sound

Scala vestibuli

AC soundPressure difference p

Displacement w
Cochlear partition

Scala tympaniyzx

Scala vestibuli
Reissner’s membrane
Tectrial membrane

Outer hair cell

Inner hair cell

Basilar membrane
Scala tympani
Figure 1. Scheme of the cochlea.

2

COCHLEAR MODEL WITH BONE CONDUCTION

2.1 Macrocochlear model
The cochlea is a coiled duct containing ﬂuid. As shown in Fig. ??, in the longitudinal direction of this duct,
the cochlear partition (CP) divides two chambers that are the scala vestibuli and scala tympani. Incoming
wave through the middle ear transmits in the cochlear ﬂuid, and generates a pressure difference between twochambers. Particularly, this pressure difference vibrates the basilar membrane (BM), which is included in theCP, to face the scala tympani. The vibration pattern of the BM depends on frequency of the incoming sound. Atthe base and apex sides, higher and lower frequencies of the sounds vibrate the BM, separately. This resonantfrequency is called the characteristic frequency (CF). The BM contains numerous sensor cells such as innerand outer hair cells (IHCs and OHCs). The IHC transforms the incoming sound from mechanical to neural
information at different locations. Meanwhile, the OHC ampliﬁes the BM motion by 60 dB.
In this study, we develop a two-dimensional (2D) cochlear model based on our previously proposed model [6].
Fluid contained in the cochlear duct is assumed as an ideal ﬂuid. Hence, the pressure difference p affecting theCP can be expressed using the Laplace equation as follows [7]:∂ 2 p(t) ∂ 2 p(t)+= 0, 0 < x < L, 0 < y < H,
∂ x2
∂ y2(1)697

      
Figure 2. Schematic illustration of two-dimensional macromechanical cochlear model discretized by squaregrids. Filled and open circles represent boundary and inner points, respectively.
where t, x, and y represent time, and the location toward the length and height, respectively; L and H are thelength and height of the cochlear duct, respectively. In the proposed model, we assume that AC and BC tones
vibrate the stapes and cochlear shell, respectively, as shown in Fig. ??. Therefore, the boundary conditions inthis model are as follows:∂ p (t) (2)− 2ρ ẅSR (t) = 2ρ ẅSO (t) ,∂ x x=0
p (t)|x=L = 0,(3)∂ p (t) = 2ρ ẅ (t) ,(4)∂ y y=0∂ p (t) = 2ρ ẅCS ,(5)
∂y 
y=H

where ẅSR and ẅSO are the accelerations of the stapes caused by the incoming sound and reﬂection from thecochlea, respectively; ẅ and ẅCS are accelerations of the cochlear partition and cochlear shell, respectively; ρrepresents the ﬂuid density.
2.2 Microcochlear model
In this study, we use the Neely and Kim model [8] as the micromechanical model of the cochlea, as shownin Fig. 3. In the micromechanical system, the tectrial membrane (TM) and OHC exist on the BM. The displacement of the BM and TM are represented by ξ1 and ξ2 , respectively. The Displacement vector is denotedas ξ = [ξ1 ξ2 ]T . Neely and Kim assumed that the discrete CP can be modeled in two mass models as thefollowing equation:C ξ̇ξ + K ξ = P + P a ,(6)
M ξ̈ξ +C698








Figure 3. Micromechanical model of the cochlea [8].
where the matrices M , C , and K represent mass, resistance, and stiffness, respectively, and are deﬁned as
m1 0M =,0 m2

−c3
c1 + c3,C =−c3
c2 + c 3
−k3
k1 + k 3.
K =−k3
k2 + k 3

(7)
(8)
(9)

The external pressure P and P a affect the BM; the former and latter are the pressure difference between twochambers and the active pressure through the OHC, respectively.
p(x, 0,t)P =,(10)0pa (x,t)
Pa =.
(11)
0(12)
2.3 Simulation condition
To simulate human cochlear responses, we used the values of the parameters listed in Table 1.

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Table 1. List of parameters in the cochlear model to simulate the human’s cochlea
ParameterlN
M
ρ
Ab
m1
c1
k1
m2
c2
m3
c3
k3
c4
k4
γ

3Value0.035
300
10100010−610−33 × 10−2120 + 13400e150x
2.2 × 108 e300x
5 × 10−3
88e−165x
1.4 × 107 e−330x16e−60x2 × 106 e−3x
8800e−150x1.15 × 109 e−300x
0.5Unitmkg/m3
m2
mkg/m2N · s/m3N/m3kg/m2
N · s/m3
N/m3
N · s/m3
N/m3
N · s/m3
N/m3

SIMULATION RESULTS

3.1 Linear responses
To calculate the BM response in the frequency domain, in this simulation, we deﬁne the active pressure pa
according to the original Neely–Kim model as follows:pa (x,t) = γ c4 ξ̇c + k4 ξc ,(13)where γ , c4 , and k4 represent the gain factor of the feedback, and the damping and stiffness of the OHC,respectively; ξc denotes the gap between the BM and TM as follows:ξc (x,t) = ξ1 (x,t) − ξ2 .

(14)

Figure 4 shows the BM responses for BC and AC tones at three different frequencies. The amplitude in theBM responses exhibit peaks depending on the sound frequency. Additionally, the locations of these peaks matchwith those of both BC and AC tones. Furthermore, the angular in the BM responses is decayed from the baseto apex, and this trend is clear at the peak site of the amplitude in the BM responses. Therefore, these results
indicate that the BC tone can generate the traveling wave of the BM, as obtained by the AC tone.
3.2 Nonlinear responses
In the simulation above, we calculated the BM responses in the linear domain. However, the cochlear responses
demonstrated nonlinearities that were caused by nonlinear transductions in the OHC processing [5]. In thissimulation, we denote the active pressure pa containing the nonlinearity as follows:pa (x,t) = γ c4 ξ̇cnl + r4 ξcnl ,(15)where ξcnl is the saturating property that is calculated from the distance of the gap ξc as follows: ξcξcnl (t) = αtr tanh,
αtr700(16)

Figure 4. BM responses for 4-, 1-, and 0.25-kHz tones calculated in the frequency domain. Solid and dashedlines indicate the BM responses for BC and AC tones, respectively.
where al phatr is determined to be xic = xinl
c when xic is sufﬁciently small at 1 nm or less. The following
initial conditions are used to calculate the output from the model in the time domain.
ξ (x,t)|t=0 = ξ̇ξ (x,t) = 0,(17)t=0pa (x,t)|t=0 = 0.

(18)

Figure 5 shows the IO properties of the BM responses for BC and AC tones at frequencies of 0.25, 1, and 4
kHz. The amplitude in the BM responses is varied with the input level. For the lower input level, the IO curves
are linear for both the BC and AC tones. However, at moderate and higher input levels, the IO curves exhibit
different degrees of compressive nonlinearity. For the BC tone, this degree is higher than that for the AC tone
among the moderate input levels, but disappears at higher input levels.

4DISCUSSION

The vibrations of the stapes and cochlear shell are related the AC and BC pathways that generate the BM
traveling wave, respectively, as shown in Fig. 4. In Ref. ??, this trend was reported. Furthermore, at the CF
site, the proposed model demonstrated cochlear ampliﬁcation, i.e., a basic feature in cochlear mechanics, asreported in Ref. [5]. Thus, these results suggest that the cochlear responses are similar for both AC and BCtones.

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(a) 0.25 kHz tones(b) 1 kHz tones(c) 4 kHz tones
Figure 5. IO property of BM responses obtained from the nonlinear cochlear model. Solid and dashed lines
indicate the BM responses for BC and AC tones, respectively.
In comparison to the above consideration in the linear domain, however, real cochlear responses are nonlinear.
Hence, we proposed a nonlinear cochlear model excited by BC tones. As shown in Fig. 5, the IO propertyof the model shows nonlinear curves. Particularly, it is known that the growth of the IO on a moderate input
level is less than that at a lower input level [9]. This phenomenon is known as compression and is key forunderstanding cochlear mechanics [5]. Comparing the IO curves of AC and BC tones, at moderate soundpressure levels, the degree of compression caused by the BC tone is large. However, at higher sound pressurelevels, the degree of compression caused by the AC tone tends to increase. Hence, the IO curves derived fromthe loudness function demonstrated the similar result [4]. Therefore, these results indicate that the AC and BC
pathways can be divided by focusing on the difference in the degree of compression.

5CONCLUSIONS

Nonlinear and active elements that affect the IO property of the cochlea were investigated in this study. Additionally, a 2D cochlear model excited by both AC and BC tones was developed. The proposed model could702

exhibit the BM traveling waves for both tones. However, the degree of compression was different between thetwo pathways.

ACKNOWLEDGEMENTS
This work was supported by JSPS KAKENHI Grant Number 18K18081.

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