\documentclass[twoside]{article}
\usepackage{psfig} % for importing PostScript figures.
\pagestyle{myheadings}
\markboth{\hfil Mathematical model for the basilar membrane \hfil}%
{\hfil H. Y. Alkahby, M. A. Mahrous, \& B. Mamo \hfil}
\begin{document} \setcounter{page}{115}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc 15th Annual Conference of Applied Mathematics,
Univ. of Central Oklahoma}, \newline
Electronic Journal of Differential Equations, Conference~02, 1999,
pp. 115--124. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu (login: ftp)}
\vspace{\bigskipamount} \\
Mathematical model for the basilar membrane as a two dimensional plate
\thanks{ {\em 1991 Mathematics Subject Classifications:} 92C05, 92610, 35G15, 34B10.
\hfil\break\indent
{\em Key words and phrases:} Basilar membrane, eigenvalue,
hearing frequencies. \hfil\break\indent
\copyright 2000 Southwest Texas
State University and University of North Texas.
\hfil\break\indent Published January 21, 2000.} }
\date{}
\author{H. Y. Alkahby, M. A. Mahrous, \& B. Mamo}
\maketitle
\begin{abstract}
In this paper we present two mathematical models for the
basilar membrane. In the first model the membrane is represented as an
annular region. In the second model the basilar membrane is treated as
a rectangular region. Comparison of the two models allows us to study
the effect of the curvature of the basilar membrane on the range of
the frequencies of hearing. The differential equation of both models
is a fourth order partial differential equation derived from the
classical plate theory. Boundary conditions are defined as a region
with four sides. The conditions are different on each side and
together form an interesting physiological combination, relative to
standard engineering problems. Eigenvalues of the differential
equations of the two models are obtained numerically. A comparison of
the eigenvalues of the two models clearly shows that the range of the
hearing frequencies of the first model is larger than that of the
second model. The results indicate strongly that the curvature of the
basilar membrane plays an important role in the hearing process.
Curvature and measurement of curvature should be allowed in future
models and experiments of the inner ear.
\end{abstract}
\def\D{\mbox{\boldmath {$D$}}}
\def\h{\mbox{\boldmath {$h$}}}
\def\u{\mbox{\boldmath {$u$}}}
\section{Introduction}
Before the mathematical models of the basilar membrane are presented,
it is necessary to briefly describe the components of the inner ear.
This gives a better understanding of the role of the basilar
membrane's curvature in the hearing process. The inner ear is the
location in the auditory system where mechanical and
electrophysiological mechanisms are combined. The cochlea of the inner
ear is a small bony structure with a small coiled tube in its
interior. The walls of this tube are composed of special hard bone
(the hardest in the body).
\begin{center}
\begin{figure}[t]
\psfig{file=fig1.ps}
\caption{Schematic diagram of the cross section of the cochlear duct.
Perylimph space (a), scala vestibuli (b), modiolus (c), cochlear nerve (d),
scala tympani (e), basilar memebrane (f), hair celss (g), supporting cell (h),
organ of Corti (i), cochlear duct (j), endolymph (within membrane) (k),
tectotial membrane (l), vestibular (Reissner's) membrane (m)}
\end{figure}
\end{center}
In a cross section of the coiled tube (Fig. 1), there are three
distinct chambers, namely, the scala vestibuli, the scala media and
the scala tympani. The scala media is bounded by the Reissner's
membrane and the basilar membrane. Vibrations of the oval window are
transferred to the perilymph in the scala vestibuli, which transfers
them to the basilar membrane, triggering the electrical impulses in
the orgin of Corti, where the terminals of the acoustic nerve reside.
The organ of Corti rests on top of the basilar membrane. Therefore,
the bsilar membrane is considered to be of primary importance in the
stimulation of the hair cells and the transmission of signals to the
brain. The basilar membrane is a three-dimensional structure. It forms
the helical spiral ramp. The edges, described as being from the base
to the apex, from a diminishing spiral with radii of curvature
becoming increasingly shorter. Interestingly, the basilar membrane is
coiled, with no exception, in any species. The length of the basilar
membrane varies from a short as 7 mm in laboratory mice, 20 mm in
cats, 32 - 35 mm in humans and sheep to 60 mm in elephants (see [2]
for references and Fig.2). The number of coils or ``turns'' ranges
from 2 to 4.25 spiral turns. The number of coils in man is 2.25, 3
coils in cats and dogs, and 4 coils in the guinea-pig. The width of
the basilar membrane (in man) is 0.1 mm at the basal end and increases
to 0.5 mm at the apical end. From the size and location of the basilar
membrane, direct experimental procedures are almost impossible.
Nevertheless, Bekesy [1] pioneered an important experimental work on
the inner ear for which he received a Nobel Laureat in 1961.
Presently, there are several theories of hearing and all assume a
place principle; that is, different frequencies are triggered at
different locations on the membrane. In general, this principle
assumes that hearing frequency is a function of the mechanical
properties of the basilar membrane and also a functioin of the
location on the basilar membrane. In the most recent experiments, with
the aid of advanced technological tools, better information is
available about the basilar membrane [5]. Yet, data for an important
parameter ``curvature at the edges'' is still missing. The work of
Bekesy [2] supports a place principle where the basilar membrane is
more sensitive to successively low frequencies progressively toward
the apical end and to successively higher frequencies toward the basal
end. It is pointed out that ``the place principle predicts that apical
or mid-apical regions of the cochlea are the first to mature and that
basal regions are last, and just the opposite results are consistently
found [4].'' Thus, a more reliable place principle is needed.
From the above discussion, it is clear that the curvature of the edges
of the basilar membrane must play some role in the hearing mechanism,
in addition to the accepted evolution's explanation as a space saving
feature. Notice that curvature information embodies information about
the height of the cochlea, its base diameter, the number of coils and
the diameter of each coil, and also the diminishing rate of the
spiral. Thus, any attempt to model the mechanism of the basilar
membrane in the hearing process must allow for curvature. In this
paper two models are presented to illustrate the effect of the
curvature on the vibration response of the membrane. In the following
sections we present results that do, indeed, indicate that the
curvature of the basilar membrane is an important property that cannot
be ignored.
In the first model, the basilar membrane is considered as an
incomplete annular region (see Fig. 3). The radii of the curvature of
the inner and outer circles of the annular region simulate the
curvature of the edges of the basilar membrane. The mathematical model
is derived from the classical plate theory. It is a linearized fourth
order partial differential equation. Eigenvalues for this differential
equation are obtained numerically and they are dependent on the radii
of the curvature. We believe that the dependence of the eigenvalues on
the radii of the curvature has an important influence on the hearing
process. To emphasize the importance of this conclusion the above
model is compared with a second model. In the second model the
membrane on a rectangular region is considered. The rectangular and
the annular models have the same properties, and the same boundary
conditions. We also have chosen the regions that have equal areas. Two
sets of eigenvalues, for both models, are compared. Comparison clearly
indicates the importance of the curvature and its effect on hearing
frequencies.
\begin{center}
\begin{figure}
\psfig{file=fig2.ps}
\caption{Schematic diagram of uncoiled cochlea and basilar membrane.
Round window (a), Oval window connected to the stapes of the middle eat (b),
Scala vestibuli (c), Scal tympani (d), Basilar membrane (f),
Helicotrema (apical end) (g)}
\end{figure}
\end{center}
\begin{center}
\begin{figure}
\def\fiverm{}
\input prepictex
\input pictex
\beginpicture
\setcoordinatesystem units <8mm, 8mm>
\setplotarea x from -6 to 6, y from -1 to 6
\putrule from -5.5 0 to 5.5 0
\circulararc 180 degrees from 5 0 center at 0 0
\circulararc 180 degrees from 4 0 center at 0 0
\plot 0 0 4 3 /
\put{$r=0.95$} at 5 4.3
\plot 0 0 2.4 3.2 /
\setdashes
\plot 2.4 3.2 4 4 /
\put{$r=1.00$} at 5 3.2
\put{$S_4$} at -2.5 2.5
\put{$S_3$} at -3.2 4.5
\put{$S_2$} at -4.5 -0.5
\put{$S_1$} at 4.5 -0.5
\endpicture
\input postpictex
\caption{Semiannular plate configuration}
\end{figure}
\end{center}
Finally, in this work the membrane is represented by two dimensional
models, where most of the previous ``membrane specific'' models are
one dimensions. For example, [3] the basilar membrane was modeled as a
one dimensional beam, which implies that the vibrations are dependent
only on the longitudinal direction along the membrane. On the other
extreme, the fibers of the basilar membrane are examined in the radial
direction [5].
\section{Basic equations and boundary conditions}
The basilar membrane is considered as a plate according to classical
plate theory. The equation of motion for the traverse displacement,
$u$, is given as:
\begin{equation}
\D\nabla^4\u+2c{\partial u\over{\partial
t}}+\zeta\h{\partial^2u\over{\partial t^2}}=0~,
\end{equation}
where $D$ is flexural rigidity and defined by
$$D={Eh^3\over{12(1-\nu^2)}}$$
$h$ is plate thickness, $E$ is Young's modules of plate material, $v$
is Poisson's ratio for plate material, $\zeta$ is density of plate
material, $c$ is damping coefficient, and $\nabla$ is the gradient
operator.
Thickness, density and flexural rigidity are considered constant. The
basilar membrane is assumed to be a region with the following four
sides:
\begin{list}{}{}
\item[{``$S_1$''}] corresponding to the basal end of the membrane,
\vskip-10pt
\item[``$S_2$''] corresponding to the apical end of the membrane (at the
helicotrema),
\vskip-10pt
\item[``$S_3$''] corresponding to the outer wall of the cochlea, and
\vskip-10pt
\item[``$S_4$''] corresponding to the inner wall (see Fig. 3).
\end{list}
Since the side ``$S_1$'' is clamped we have:
\begin{equation}
\u|_{t=0}=0~,\qquad {\partial u\over{\partial n}}~,
\end{equation}
where $n$ is the normal direction to $S_1$.
The basilar membrane is not attached to anything at the helicotrema,
so the condition on $S_2$ is:
\begin{eqnarray}
u_{ss}+vu_{nn}&=&0~,\\ u_{sss}-2(1-v)u_{snn}&=&0~.
\end{eqnarray}
The side ``$S_3$'' is attached to the outside cochlear wall and simply
supported. On this side, usually called the Spiral Ligament (SL),the
fibers present little resistance to moment. The condition on $S_3$ is:
\begin{equation}
u_{ss}+u_{nn}=0~.
\end{equation}
At the primary osseous spiral lamina (SPL), which is represented by
$S_4$, the filaments are held between an upper and lower bony layers
and enter the support with zero slope. Therefore, this side is
clamped, and the condition on $S_4$ is:
\begin{equation}
\u|_{t=0}=0~,\qquad {\partial u\over{\partial n}}=0~.
\end{equation}
In this section the basilar membrane is modeled as an annular plate.
Moreover, $a$ is the radius of the outer circle and $b$ is the radius
of the inner circle.
\section{Solution of the model as an annular region}
In this section the basilar membrane is modeled as an annular plate.
The radii of curvature are constants. In spite of the absence of
information about the real curvature of the membrane edges, we still
get results for the radial waves as well as the longitudinal waves.
Equation (1) is considered in polar coordinates where
\begin{equation}
\nabla^2={\partial^2\over{\partial r^2}}+{1\over
r}{\partial\over{\partial
r}}+{1\over{r^2}}{\partial^2\over{\partial\theta^2}}~.
\end{equation}
Let $u=W(r,\theta)f(t)$ in (1), and assume that
\begin{equation}
f(t)=\exp(-ct/\omega h)\sin(i\omega)~,
\end{equation}
and $f(0)=0$, then equation (1) becomes
\begin{equation}
\nabla^4 W-k^4W=0~,
\end{equation}
where
\begin{eqnarray}
\omega^2&=&{\zeta hDk^4-c^2\over{\zeta^2
h^2}}~,~~\mbox{or}\nonumber\\ k^4&=&{\zeta^2
h^2\omega^2+c^2\over{\zeta hD}}~.
\end{eqnarray}
The boundary conditions (2-6) are now as follows:
\medskip
\noindent On $S_1:\theta=0$ and
\begin{equation}
W(r,0)=0~,\qquad{\partial W\over{\partial r}}(r,0)=0~.
\end{equation}
\noindent On $S_2:\theta=\pi$ and
\begin{eqnarray}
W_{\theta\theta}(r,\pi)=0&,~~\qquad W_{rr}(r,\pi)=0~,\nonumber\\
W_{\theta\theta\theta}(r,\pi)=0&,\qquad W_{\theta\pi\pi}(r,\pi)=0~.
\end{eqnarray}
\noindent On $S_3:r=a$ and
\begin{equation}
W_{rr}(a,\theta)=0~,\qquad W_{\theta}(a,\theta)=0~.
\end{equation}
\noindent On $S_4:r=b$ and
\begin{equation}
W(b,\theta)=0~,\qquad W_{\theta}(b,\theta)=0~.
\end{equation}
The differential equation (9) has a solution of the following form:
\begin{equation}
W(r,\theta)=\sum^{\infty}_{m=1}g_m(r)\cos
m\theta+\sum^{\infty}_{m=1}\bar{g}_m(r)\sin m\theta~,
\end{equation}
where
\begin{equation}
g_m(r)=A_m J_m(kr)+B_m Y_m(kr)+C_m I_m(kr)+D_mK_m(kr)~,
\end{equation}
and
\begin{equation}
\bar{g}_m(r)={\bar A}_m J_m(kr)+{\bar B}_m Y_m(kr)+{\bar C}_m
I_m(kr)+{\bar D}_m K_m(kr)~.
\end{equation}
The coefficients $A_m$, $B_m$, $C_m$,${\bar A}_m$, ${\bar B}_m$,
${\bar C}_m$, ${\bar D}_m$ are constant to be evaluated using the
boundary conditions, and $J_m$, $Y_m$, $I_m$, $K_m$-- Bessel functions
and modified Bessel functions of the first and second kind. Using the
boundary conditions (11 - 14), and after some simplification, we
obtain the following frequency determinant $T$. A nontrivial solution
is obtained with $T=0$.
\begin{equation}
T=\left|\begin{array}{cccc}
J_m(\lambda_1)&Y_m(\lambda_1)&I_m(\lambda_1)&K_m(\lambda_1)\\
T_{21}&T_{22}&T_{23}&T_{24}\\
J_m(\lambda_2)&Y_m(\lambda_2)&I_m(\lambda_2)&K_m(\lambda_2)\\
J_{m+1}(\lambda_2)&Y_{m+1}(\lambda_2)&-I_{m+1}(\lambda_2)&K_{m+1}(\lambda_2)
\end{array}\right|
\end{equation}
where
\begin{eqnarray}
T_{21}&=&(1-v)J_{m+1}(\lambda_1)-2\lambda_1 J_m(\lambda_1)~,\nonumber\\
T_{22}&=&(1-v)Y_{m+1}(\lambda_1)-2\lambda_1 Y_m(\lambda_1)~,\nonumber\\
T_{23}&=&-(1-v)I_{m+1}(\lambda_1)~,\nonumber\\
T_{24}&=&(1-v)K_{m+1}(\lambda_1)~,\nonumber\\
\lambda_{1}&=&ka,\qquad
\lambda_2=kb~.
\end{eqnarray}
Moreover, $a$ is the radius of the outer circle, $b$ is the radius of
the inner circle and $k$ is defined in (10).
\section{Solution of the model as a rectangular plate}
The above model of the basilar membrane as an annular plate is closer
to the real model than a rectangular plate model. Our interest in a
rectangular plate in this section is to compare the eigenvalues from
both models for plates with the same characteristics and essentially
the same area. The literature is wealthy in results for rectangular
plates but results for the specific boundary condition we selected for
the basilar membrane are scarce. Therefore two solutions that lead to
some estimates of eigenvalues are given here.
\noindent The boundary conditions are as follows:
\noindent On $S_1:x=0$ and
\begin{equation}
W(0,y)=W_x(0,y)=0~.
\end{equation}
\noindent On $S_2:x=a$ and
\begin{eqnarray}
W_{xx}(a,y)+(2-v)W_{yyx}(a,y)&=&0~,\nonumber\\
W_{xx}(a,)+vW_{yy}(a,y)&=&0~.
\end{eqnarray}
\noindent On $S_3:y=0$ and
\begin{equation}
W(x,0)=W_{yy}(x,0)=0~.
\end{equation}
\noindent On $S_4:y=b$ and
\begin{equation}
W_{yy}(x,b)+vW_{xx}(x,b)=0~.
\end{equation}
The operator $\nabla^2$ has the standard form:
$$\nabla^2={\partial^2\over{\partial x^2}}+{\partial^2\over{\partial y^2}}~.$$
Let
\begin{eqnarray}
W=\sum^{\infty}_{m=1}\Bigl[\sum^{4}_{n=1}(A_{mn}\sin cy&+&B_{mn}\cos
cy\nonumber\\ C_{mn}\sinh dy&+&D_{mn}\cosh dy)F_n(x)\Bigr]
\end{eqnarray}
where $A_{mn}$, $B_{mn}$, $C_{mn}$, $D_{mn}$ are constants and
\begin{equation}
F_1(x)=\sin ax, F_2(x)=\cos ax, F_3(x)=\sinh ax. F_4(x)=\cosh
ax~,
\end{equation}
where
\begin{equation}
c=\sqrt{k^2-a^2}~,~~d=\sqrt{k^2+a^2}~.
\end{equation}
Using the boundary conditions (20-23) we obtain
\begin{equation}
\tan cb={c\over d}\tan db~,~~\mbox{and}~~\tan aa=\tan aa~.
\end{equation}
\begin{center}
{\bf Table 1. Eigenvalues for annular plate and rectangular\\ plate
models}
\bigskip
\begin{tabular}{|{l}{c}{c}{c}{c}{c}|}\hline
$a/b$ & Harmonics & Annular & Plate & Rectangular & Plate\\ \hline
&&$\lambda_1$ & $\lambda_2$ & $\lambda_1$ & $\lambda_2$\\ \hline 0.8 &1 &$15.5611$
& $12.4489$ & & \\ \hline 0.8 &2 & $31.5640$ & $25.2512$ & &\\ \hline
0.8 &3 & $47.0748$ & $37.6598$ & &\\ \hline 0.8 &4 & $62.906$ &
$50.3252$ & &\\ \hline 0.8 &5 & $78.5104$ & $62.8083$ & &\\ \hline 0.8
&6 & $94.2975$ & $75.438$ & &\\ \hline 0.8 &7 & $109.9347$ & $87.9478$
& &\\ \hline 0.8 &8 & $125.7010$ & $100.5608$ & &\\ \hline 0.8 &9 &
$141.3555$ & $13.0844$ & &\\ \hline 0.8 &10 & $157.1095$ & $125.6876$
& &\\ \hline 0.8 &11 & $172.7740$ & $138.2190$ & &\\ \hline 0.9 &1 &
$31.1005$ & $27.990$ & &\\ \hline 0.9 &2 & $62.8981$ & $56.6083$ & &\\
\hline
0.9 &3 & $94.143$ & $84.7280$ & &\\ \hline 0.9 &4 & $125.697$ &
$13.1270$ & &\\ \hline 0.9 &5 & $175.017$ & $41.3153$ & &\\ \hline
0.95 &1 & $62.4469$ & $59.3245$ &$44.8387$ &$42.5968$\\ \hline 0.95 &2
& $125.695$ & $119.4100$ &$93.8786$ &$89.1847$\\ \hline
\end{tabular}
\end{center}
\section{Computations and conclusions}
The eigenvalues of both models are computed numerically. It is shown
that if the curvature of the basilar membrane is taken into account
the range of the hearing frequencies is significantly wider. On the
other hand, for the rectangular model, the hearing frequencies
appeared to be significant only when the rectangular region tends to a
secured region. This, of course, contradicts the real shape of the
basilar membrane. As a result, any experimental or theoretical work
should take the curvature of the basilar membrane into account.
Finally, these numerical computations for the eigenvalues were
obtained on a VAX-8600 system with IMSL standard mathematical
routines.
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\end{thebibliography}
\noindent{\sc H. Y. Alkahby \& B. Mamo}\\
Division of the Natural Sciences, Dillard University \\
New Orleans, LA 70122 USA \\
Telephone: 504-286-4731 e-mail: halkahby@aol.com \smallskip
\noindent{\sc M. A. Mahrous}\\
Department of Mathematics, University of New Orleans \\
New Orleans, LA 70148, USA
\end{document}