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A numerical approach to
investigating the mechanisms
behind tonotopy in the bush-
cricket inner-ear
Emine Celiker
1
*
†
, Charlie Woodrow
1
, Natasha Mhatre
2,3
and Fernando Montealegre-Z
1
*
1
University of Lincoln, School of Life and Environmental Sciences, Joseph Banks Laboratories,
Lincoln, United Kingdom,
2
Department of Biology, Western University, London, ON, Canada,
3
Brain and Mind Institute, Western University, London, ON, Canada
Bush-crickets (or katydids) have sophisticated and ultrasonic ears located in the
tibia of their forelegs, with a working mechanism analogous to the mammalian
auditory system. Their inner-ears are endowed with an easily accessible
hearing organ, the crista acustica (CA), possessing a spatial organisation that
allows for different frequencies to be processed at specific graded locations
within the structure. Similar to the basilar membrane in the mammalian ear, the
CA contains mechanosensory receptors which are activated through the
frequency dependent displacement of the CA. While this tonotopical
arrangement is generally attributed to the gradual stiffness and mass changes
along the hearing organ, the mechanisms behind it have not been analysed in
detail. In this study, we take a numerical approach to investigate this
mechanism in the Copiphora gorgonensis ear. In addition, we propose and
test the effect of the different vibration transmission mechanisms on the
displacement of the CA. The investigation was carried out by conducting
finite-element analysis on a three-dimensional, idealised geometry of the C.
gorgonensis inner-ear, which was based on precise measurements. The
numerical results suggested that (i) even the mildest assumptions about
stiffness and mass gradients allow for tonotopy to emerge, and (ii) the
loading area and location for the transmission of the acoustic vibrations play
a major role in the formation of tonotopy.
KEYWORDS
insect hearing, crista acustica, frequency mapping, numerical modelling,
bioacoustics, acoustic vibration transmission
Frontiers in Insect Science frontiersin.org01
OPEN ACCESS
EDITED BY
Daniel Robert,
University of Bristol, United Kingdom
REVIEWED BY
Ionut Stefan Iorgu,
Grigore Antipa National Museum of
Natural History, Romania
Ryan Palmer,
University of Bristol, United Kingdom
*CORRESPONDENCE
Emine Celiker
eceliker@lincoln.ac.uk;
eceliker@dundee.ac.uk
Fernando Montealegre-Z
fmontealegrez@lincoln.ac.uk
†
PRESENT ADDRESS
Emine Celiker,
Division of Mathematics, University of
Dundee, Dundee, United Kingdom
SPECIALTY SECTION
This article was submitted to
Insect Neurobiology,
a section of the journal
Frontiers in Insect Science
RECEIVED 31 May 2022
ACCEPTED 21 July 2022
PUBLISHED 15 August 2022
CITATION
Celiker E, Woodrow C, Mhatre N and
Montealegre-Z F (2022) A numerical
approach to investigating the
mechanisms behind tonotopy in the
bush-cricket inner-ear.
Front. Insect Sci. 2:957385.
doi: 10.3389/finsc.2022.957385
COPYRIGHT
© 2022 Celiker, Woodrow, Mhatre and
Montealegre-Z. This is an open-access
article distributed under the terms of
the Creative Commons Attribution
License (CC BY). The use, distribution
or reproduction in other forums is
permitted, provided the original
author(s) and the copyright owner(s)
are credited and that the original
publication in this journal is cited, in
accordance with accepted academic
practice. No use, distribution or
reproduction is permitted which does
not comply with these terms.
TYPE Original Research
PUBLISHED 15 August 2022
DOI 10.3389/finsc.2022.957385
1 Introduction
Tonotopic organisation, or frequency maps, arise in the
auditory systems of many species (1). As is well known, the
main purpose of this phenomenon is to facilitate frequency
analysis of the acoustic vibrations entering the hearing chamber.
First discovered in the mammalian ear (2), a tonotopic hearing
organ has recently also been observed in the ears of bush-crickets
(3–5)], which have been shown to have a hearing system
analogous to mammals (6). For bush-crickets, whose ears are
located in their forelegs, a non-invasive investigation of the
biomechanism governing the inner-ear processes has been
possible through the transparent cuticle some species are
endowed with (4). In contrast, the structure and location of
the mammalian inner-ear makes it experimentally challenging to
investigate this mechanism non-invasively (7,8). Hence, the
convergent evolution between the bush-cricket and the
mammalian ears provides a unique opportunity to enhance
our understanding of these analogous hearing systems, making
the investigation of the mechanism behind the workings of the
bush-cricket inner-ear a timely and worthy pursuit.
Through his Nobel prize winning work, Georg von Bekesy
showed that the fluid-immersed basilar membrane in the
mammalian inner-ear (the cochlea) acted as a biological Fourier
transform, performing frequency analysis on the mechanical
travelling wave (2,9). As the travelling wave moved down the
organ, it was observed that the cochlear hair cells (or auditory
sensory cells) lying along the membrane would receive mechanical
input at specific frequencies, due to an amplitude maxima
response dependent on the stiffness and mass gradients of the
basilar membrane (10). Some properties of the basilar membrane
leading to such gradients include a tapering in width, and a
gradual increase in thickness and elasticity (1). Since then,
travelling waves have also been observed directly in the basilar
papilla of birds (11), and indirectly via the timing of responses of
auditory-nerve fibres in the auditory organs of some reptiles and
frogs (12,13). Analogous travelling waves have also been
measured invasively and non-invasively in the ears of bush-
crickets (3,4,14,15). The underlying mechanism is likely more
ancient since it has been observed in grigs, suggesting it was
shared by a common ancestor (16).
Bush-crickets have ultrasonic ears that are located in the
tibia of their forelegs, and each ear is endowed with two
tympanic membranes (TMs). The TMs are backed by an air-
filled tube (the acoustic trachea or ear canal), and their outer-ear
allows for sound to be received on both sides of the tympana:
directly to the external side, and through the acoustic trachea to
the internal side (17). The acoustic trachea bifurcates near the
TM, with the anterior tracheal branch backing the anterior
tympanic membrane, and similarly the posterior tympanic
branch lies behind the posterior tympanic membrane. After
travelling through the tracheal tube, sound arrives at the TM
with a phase difference and a pressure differential compared to
the external input (17), leading to the TM acting as a pressure-
time difference receiver (18). Acoustic vibrations are then
transmitted to the bush-cricket inner-ear, which is a separate,
fluid-filled chamber above the TM end of the tracheal tube (see
Figure 1). The mechanosensory organ of the bush-cricket, the
crista acustica (CA), is located inside this chamber and lies on
the dorsal wall of the anterior tracheal branch. While in other
insect hearing systems such as locusts and moths the
mechanoreceptors are in direct contact with the TMs (19), this
is not the case for bush-crickets, demonstrating another likeness
to the mammalian ear (20,21).
The CA has been observed to resemble an uncoiled basilar
membrane (4,22). Similar to mammals, in the bush-cricket
inner-ear the transmitted acoustic vibrations activated the
hearing organ, leading to the formation of travelling waves
moving along this structure (6). These waves were observed to
travel from the narrowest part (distal part) of the sensory organ
(the CA) containing the high frequency sensory cells, to the
broader (proximal) region containing the low frequency cells (3,
9,22). As the travelling wave moved along the hearing organ, the
CA maximum displacement was seen to occur in a frequency-
dependent fashion, after which the wave dissipated, showing
clear characteristics of a tonotopically ordered structure (4,6).
Consequently, the frequency dependent movement of the CA
activated the relevant mechanosensory cells. Even though the
tonotopy observed in the bush-cricket ear is generally attributed
to the stiffness and mass gradients of the CA (3), a more detailed
analysis, combining experimental and numerical approaches, is
required for more conclusive evidence on the mechanism behind
this phenomenon.
As is well known, for tonotopical vibrations to emerge along
the hearing organ, first the acoustic vibrations have to be
transmitted into this chamber through a mechanical
phenomenon. For the mammalian ear, the chain of
transmission by which sound is captured by the eardrum and
delivered, via the ossicles to the inner ear is well-studied (23,24).
However, in bush-cricket auditory mechanics, this process is
more controversial. To understand the workings of the bush-
cricket auditory system, it becomes important to discern the
process of vibration transmission into the bush-cricket inner-
ear. There are two main arguments in relation to the air-to-
liquid conversion of acoustic vibrations in the bush-cricket ear,
which are based on a lever-like system (25), where a higher
output force is generated through mechanical advantage (26).
The first mechanism was proposed by Bangert et al. (27)by
using the species Polysarcus denticauda and Tettigonia
viridissima as model systems, who likened the impedance
conversion in the bush-cricket ear to the TM acting as a type
2 lever. A type 2 lever is described as having a fulcrum located at
one end with the force applied at the other end. The resulting
force is then sensed at the middle of the lever (26). Hence,
according to Bangert et al. (27), the force load was sensed at the
area of intersection between the TM and the dorsal wall,
Celiker et al. 10.3389/finsc.2022.957385
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transferring the airborne acoustic energy to the fluid medium. It
was also suggested that there was a contribution to this
mechanism from the sound pressure acting on the dorsal wall,
emanating from the acoustic trachea (27). A similar mechanism
was also put forward by Nowotny et al. (28) for the species
Mecopoda elongata. Palghat Udayashankar et al. (22), however,
proposed that the dorsal wall played a more central role in
activating the CA in the M. elongata ear, through the pressure of
the sound wave on the dorsal wall as it travelled in the acoustic
trachea. Hence, it was suggested that the CA obtained a
displacement magnitude proportional to its local resonance
while stimulated through a pressure parallel to the hearing
organ, rather than a travelling wave. A similar idea was also
considered for the mammalian inner-ear (29).
A second transmission model was proposed by Montealegre-
Zetal.(6), using the species Copiphora gorgonensis as a model
system. For the neotropical bush-cricket C. gorgonensis,theCAis
located in the auditory vesicle (see Figure 1) which is believed to be
derived from the hemolymph channel and is filled with fluid, also
bathing the CA [(6)(25)]. The proposed transmission mechanism
in (6) was centered around the tympanal plate (TP), a cuticular
patch attached to the TM in contact with the distal end of the
auditory vesicle (25). The TP was recorded to have an out-of-
phase response to its hosting TM, hence performing an air-to-
liquid impedance conversion by transmitting the vibrations from
the air backed TM to the fluid-filled auditory vesicle. It is
suggested that the TP governs this transmission process by
acting as a type 1 lever: a lever with the fulcrum in the middle,
force applied at one end and the resulting force in the other end.
Therefore, while the role of TPs are not considered by Bangert
et al. (27), Montealegre-Z and colleagues (6) suggested that TPs play
a central role in the impedance conversion of acoustic vibrations to
the fluid medium. Hence, in (6)[seealso(25)], the TPs were posited
to have an analogous function to the mammalian middle-ear.
However, a middle-ear or the auditory vesicle were not observed
in the ears of the bush-cricket M. elongata (22). There is also a
considerable difference in the load areas between the species M.
elongala and C. gorgonensis. For both the species a part of the TM
(22,5)orTP(6) is in contact with the inner-ear fluid, however, the
TM has contact along the length of the CA (28), whereas the TP is
in contact with only the distal end of the auditory vesicle (6). Thus,
in addition to a different lever system, the mechanisms suggested
also propose differing sizes of load areas to the inner-ear. While
these different models were proposed, it has never been tested
whether such mechanical configurations actually give rise to the
observed tonotopic behaviour.
In this study, using C. gorgonensis as our model system we
investigated the underlying mechanism of the bush-cricket
inner-ear. By incorporating simple observable properties, such
as mass gradient and tapering (width and height) in the
geometry of the CA morphology, we tested mechanical
features that are crucial to the development of tonotopy. We
also used the constructed models to numerically investigate the
effect of the transmission load of acoustic vibrations to the
formation of tonotopy. Based on micro-computed tomography
(m-CT) measurements of the C. gorgonensis inner-ear an
idealised geometry was constructed, on which numerical
simulations were carried out by manipulating the “middle-ear”
conditions. Figure 2 demonstrates the idealised auditory vesicle
and CA geometry. We used this 3D model to test the hypothesis
that for C. gorgonensis, the existence of a separate chamber, the
auditory vesicle, makes it necessary to have a load area with a
smaller region as offered by the TP. Further, the role of the dorsal
wall in sound transmission was investigated. A comparison with
FIGURE 1
The m-CT image of the Copiphora gorgonensis ear and its components.
Celiker et al. 10.3389/finsc.2022.957385
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the experimental results in the literature suggested that our
numerical results gave the closest match to experimental data
when the impedance conversion took place with the influence of
both the TP and the dorsal wall.
2 Materials and methods
2.1 Model geometry
2.1.1 Morphological measurements
To produce an idealised geometry of the C. gorgonensis inner-
ear, a female specimen was scanned using (i) a SkyScan 1172 X-
ray m-CT scanner (Bruker Corporation, Billerica, MA, USA) with
aresolutionof1.6mm (40 kV source voltage, 165 mAsource
current, 2200 ms exposure and 0.1° rotation steps) and (ii)
synchrotron X-ray CT imaging at the Diamond Manchester
Imaging Branchline (I13-2, Diamond Light Source, Oxford). We
used monochromatic light and a 4× objective with a pco.edge 5.5
detector, providing a voxel size of 0.8125 mm. The obtained
images were then reconstructed with NRecon (v.1.6.9.18, Bruker
Corporation, Billerica, MA, USA) for a series of orthogonal slices.
The 3D segmentation of the inner-ear was performed with
the software Amira-Aviso 6.7 (Thermo Fisher Scientific,
Waltham, Massachusetts, USA), and were used for obtaining
the dorsal wall thickness and the dimensions of the auditory
vesicle, through the Center Line Tree module in AMIRA. For the
2D measurement of cap cell surface area, scolopale cell radius
and dendrite length, an Alicona InfiniteFocus microscope (G5,
Bruker Alicona Imaging, Graz, Austria) at ×5 objective
magnification was used, with a resolution of about 100 nm.
2.1.2 Idealised geometry
The idealised geometry was constructed in the Geometry
node of COMSOL Multiphysics, v. 5.6 (30), with parameter
dimensions based on the measurements obtained as described in
Section 2.1.1. The actual shape of the geometry can be seen in
Figure 1 and Figure 3. The acquired geometry is given in
Figure 2, and the used dimensions are presented in Figure 3
and Table 1. The auditory vesicle itself was represented as an
oblong hexahedron. The geometry included 28 individual cap
cells and corresponding dendrites of varying dimensions (see
Table 1), located inside the auditory vesicle. For cap cell
geometry, we assumed that they were shaped as cuboids (see
Figure 2B). Conjoined to the cap cells from the bottom were the
scolopale cells which were modelled as spheres, and were also
attached to the dendrite. On the other end, it was assumed that
dendrites were directly connected to the dorsal wall. The cap
cells, scolopale cells and dendrites made up the structure of the
modelled CA. The CA was covered by a surface representing the
tectorial membrane. Near the distal end of the constructed CA,
the TP was modelled to intersect with the auditory vesicle wall. A
second geometry with the TM intersecting the acoustic trachea
along the length of the CA was also constructed for comparison
(see Supplementary Materials, Figure S1). A block, intersecting
at the dorsal wall, was added representing the acoustic trachea.
2.2 Mathematical model
The mathematical models were developed using the Acoustics
and Structural Mechanics modules of COMSOL Multiphysics (v.
5.6) (30), and were set-up as an acoustic-structure interaction
A
B
C
FIGURE 2
An idealised geometry of (Copiphora gorgonensis inner-ear. (A) The geometry parameters (B) A mechanosensory cell and its components (C)
The travelling wave direction observed in the bush-cricket inner-ear.
Celiker et al. 10.3389/finsc.2022.957385
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problem. All the calculations were carried out in the frequency
domain so that the system of equations was not time dependent.
The auditory vesicle wall and the tectorial membrane were
represented with a shell formulation (31), and were coupled with
the fluid inside the auditory vesicle using the arbitrary Lagrangian-
Eulerian method (32). The fluid was assumed to be water, and the
pressure in the fluid was represented by the solution to the
linearised Navier-Stokes equations (33). Hence, any possible
viscous effects due to the fluid were accounted for. The CA
components (cap cells, scolopale cells and dendrites) were
assumed to be linear elastic and were represented by the elastic
Helmholtzequation (32).The CA was in turn coupled withthe fluid
inside the auditory vesicle, as well as the auditory vesicle boundary
(the shellformulation).In addition, we modelledthe propagationof
sound in the block representing the acoustic trachea. The block was
assumed to be filled with air, and the sound pressure was
represented by the solution to the acoustic Helmholtz equation
(34). The block was also coupled with the dorsal wall to reflect the
influence of the sound pressure in the auditory vesicle due to the
propagation of sound in the trachea.
A
B
FIGURE 3
Crista acustica (CA) measurements. (A) an Alicona InfiniteFocus microscope image of Copiphora gorgonensis CA. (B) The measurements of Copiphora
gorgonensis CA componentsusing the Alicona InfiniteFocus microscope. Cap cellarea refers to the surface area of the cap cell top surface.
TABLE 1 The parameter dimensions applied in constructing the idealised geometry.
Parameter Dimensions
Auditory vesicle volume 2.22×10
7
mm
3
Dorsal wall thickness Varying linearly in the interval
3.25-6.5 µm (proximal to distal)
Auditory vesicle wall thickness 15 mm
Cap cell dimensions Largest 64×40×40 µm
3
(proximal),
Smallest 4×9×4 µm
3
(distal)
Scolopale cell radius 9 mm
Tectorial membrane thickness 2 mm
Dendrite length 59-11 mm (proximal to distal)
Tympanal plate area 5493 mm
2
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The boundary of the auditory vesicle facing the proximal end
of the CA was assumed to act as a pressure release facilitator (6)
(see Supplementary Materials, Figure S2), as a result of a Free
boundary condition (34)defined there. This condition ensures that
the boundary moves based on its material properties and applied
forces, and is not constrained except at the edges. A free boundary
condition was also defined on the dorsal wall. The remainder of
the auditory vesicle walls were fixed. At the proximal end face of
the idealised acoustic trachea, the sound wave travelling through
the tracheal tube was modelled as a harmonic wave of frequency f
and amplitude 1 Pa (see Supplementary Materials, Figure S2). On
this face, we also defined a radiation boundary condition so that
there were no reflections there (34). The transmission of
vibrations through the TP were represented with an acceleration
condition with a magnitude of w
2
mm/s
2
,wherew=2pfis the
angular frequency (34). The frequency was considered in the
interval 10-90 kHz with a resolution of 10 kHz.
Table 2 summarises the material properties employed in the
mathematical models. These values were selected through
parametric sweeps, as a result of giving the closest numerical
results to the experimental data (see Supplementary Materials,
Section 1 for more details). The material properties were
assumed to be isotropic and homogeneous.
The model described above was also adapted to represent the
condition of a TM transmission of acoustic vibrations to the
auditory vesicle rather than a TP transmission, where a larger
area along the length of the auditory vesicle received the force.
This was achieved by a manipulation of the idealised geometry
(see Supplementary Materials, Figure S1). TP and TM entrances
without the influence of the dorsal wall were also considered by
removing the idealised acoustic trachea from the geometry.
2.3 Numerical simulations
The variational form of the developed mathematical models
were solved using the finite-element method. Linear Lagrange
elements were used for the solution so that a second order
accuracywasobtainedintheL
2
–norm (35). For the constructed
finite-element mesh, the tetrahedral mesh radii were between 1-20
mm. This mesh-size ensured that there were at least 10 tetrahedral
elements per wavelength for the considered frequency range of 10-
90 kHz. The mesh size was also based on the thickness of the
potential viscous boundary layers forming near the boundaries.
From the linearized Navier-Stokes equations, the thickness of the
viscous boundary layer can be obtained as
d=ffiffiffiffiffiffiffiffiffi
2m
wr0
s,
where m= dynamic viscosity, w=2pfis the angular
frequency with ffrequency, and r
0
the background density
(36). It is clear that the thickness ddepends on the properties
of the fluid considered, and that it becomes thinner with the
increase of frequency. Since the auditory vesicle is assumed to be
filled with water, at 20°C and 90 kHz the viscous layer thickness
d=1.8853mm. Hence, the adopted mesh size allows for
capturing the solution even in the thinnest boundary layer.
A mesh stability analysis is presented in Supplementary
Materials, Section 3. The COMSOL Multiphysics (30) inbuilt
feature of mixed interpolation of tensorial components (MITC)
shell elements were used for the meshing of the auditory vesicle
boundary. MITC shell elements have been established as
effectively capturing different shell behaviours with varied and
complex stress conditions (37). The obtained mesh was a
conforming finite-element mesh. The constructed mesh is
demonstrated in Supplementary Materials, Figure S3.
3 Results
In this study, we undertook a numerical investigation of the
mechanism behind the workings of the C. gorgonensis inner-ear,
the auditory vesicle. We found that a mass gradient generated by
the changing size of the cap cells, and the geometry of the CA
and dorsal wall were sufficient to generate tonotopic vibrations.
In addition, the two main hypotheses of vibration transmission,
(i) through the TP and dorsal wall, and (ii) through the TM and
dorsal wall, were tested numerically to determine their influence
on the formation of tonotopy. To investigate the individual
contributions of the dorsal wall, TP and TM, their effects on the
system were also simulated separately.
TABLE 2 The parameter material properties applied in the mathematical models.
Parameter Young’s Modulus
Dorsal wall 0.5 GPa
Auditory vesicle wall 1 GPa
Cap cell, scolopale cell 50 MPa
Dendrite 1 GPa
Tectorial membrane 10 MPa
Tympanal plate 1 GPa
Tympanic membrane 1 GPa
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Throughout this section, distance measurements refer to the
distance from the proximal end, as defined in Figure 2C.
3.1 Combined methods of
sound transmission
The mathematical models were set-up to represent a
combined method of sound transmission: through the (i) TP
and dorsal wall, and the (ii) TM and dorsal wall. For an analysis
of tonotopy along the CA, we checked for monotonicity in the
relationship between frequency and distance, in the range of
frequencies and positions sampled. Figure 4 demonstrates the
results obtained for vertical displacement maxima location along
the CA, for vibrations in the frequency range 10-90 kHz. As can
be observed, the tonotopical arrangement, or a frequency
dependent displacement maxima, was most pronounced when
the TP and dorsal wall transmissions were considered together.
For the TP and dorsal wall input, the three-dimensional
image of CA vertical displacement at various frequencies is
presented in Figure 5. For this input system, an increase in
frequency in the interval 20-90 kHz lead to the movement of the
displacement maxima location from 246 mm to 471 mm (see
Figure 4A). However, between 10 kHz and 20 kHz a
discontinuity was observed in the tonotopic vibrations, and
the displacement maxima location was at 408 mm for both 50
kHz and 60 kHz. The displacement magnitude with respect to
distance is given in Figure 6A. The results demonstrate a non-
decreasing displacement with the increase of frequency. From
Figure 5 and Figure 6A it can also be observed that the
displacement maxima are smooth peaks, dissipating a short
distance from the maxima.
A tonotopical pattern of vibration was not observed when
both the TM and dorsal wall served as vibrational inputs to the
CA (see Figure 4A and Supplementary Materials, Figure S4). In
the frequency range 30-90 kHz, the displacement maxima lied
within the short interval 369 - 408 mm from the proximal end,
showing the spatial frequency gradient to be less differentiated.
Further, displacement maxima at 30, 40 kHz were located at 360
mm, and similarly for 60-80 kHz they were located at 390 mm.
Tonotopy was also not present in the frequency range 10-30
kHz. The distance versus vertical displacement at the CA, for the
TM and dorsal wall input is given in Figure 6B. A correlation
between frequency and displacement magnitudes was
not present.
3.2 Single method of sound transmission
To determine the individual effects of the dorsal wall, TP and
TM input on the development of tonotopic vibrations along the
CA, the mathematical models were solved with these three
A
B
FIGURE 4
Location of displacement maxima. Distance, from the proximal end, of the vertical displacement maxima of crista acustica when vibrations are
transmitted (A) through the tympanal plate and dorsal wall (TP and DW) or the tympanic membrane and dorsal wall (TP and DW), (B) through
the tympanal plate (TP), tympanic membrane (TM) or the dorsal wall (DW).
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parameters as the separate excitation methods of the system.
Figure 4B demonstrates the displacement maxima location along
the CA with respect to vibration frequency. As can be observed
from Figure 4B, when these input methods are considered
separately, there is a lack of monotonicity between frequency
and distance. More precisely, some displacement maxima
locations were obtained as:
(i) For TP input 390 mm at 50 kHz; 369 mm at 60 kHz; 443
mm at 70 kHz; 408 mm at 80 kHz;
(ii) For DW input 299 mm at 20 kHz; 272 mm at 40 kHz; 427
mm at 50 kHz; 408 mm at 60 kHz; 443 mm at 70 kHz;
(iii) For TM input 299 mm at 40 kHz; 408 mm at 50 kHz; 369
mm at 60 kHz; 427 mm at 70 kHz; 408 mm at 80 kHz;
While a TP input leads to displacement maxima closest to
tonotopy, a fluctuation of the displacement location is present
throughout the considered frequency range. These fluctuations
become smaller in magnitude as the frequency increases. Similar
fluctuations also resulted from TM and dorsal wall inputs in the
frequency range 40-90 kHz. For the dorsal wall and TM
excitations a correlation between frequency and displacement
maxima location can not be discerned in the range 10-40 kHz.
The displacement magnitude along the CA obtained through
the three inputs is given in Supplementary Materials, Figure S5.
The dorsal wall excitation led to displacement magnitudes that
were orders of magnitude smaller than those generated by the
TP or TM excitation of the system.
4 Discussion
In this study, we numerically investigated the mechanism
behind the formation of tonotopical vibrations along the CA.
Further, we analysed how the transmission load area of acoustic
vibrations influenced this formation in the inner-ear.
4.1 Factors sufficient for
developing tonotopy
The local resonance frequency (f) of an acoustic structure
can be determined by the ratio of its stiffness (s) to its mass (m)
in the form
f=1
2pffiffiffiffiffi
s
m
r, (1)
where the stiffness is dependent on the elasticity (Young’s
modulus) and the dimensions of the structure, and mass is the
product of the structure’s density and volume. From equation
(1) it is easy to see that a change in stiffness or mass leads to the
change of the resonant frequency of the structure.
Due to a decrease in width and increase in thickness and
elasticity from the base to the apex, stiffness and mass gradients
are also present along the mammalian basilar membrane (9). As a
result, these stiffness and mass gradients contribute to the
A
B
DC
FIGURE 5
The three-dimensional crista acustica (CA) vertical displacement facilitated by the tympanal plate (TP) and dorsal wall (DW) transmission of
acoustic vibrations at (A) 20 kHz, (B) 40 kHz, (C) 60 kHz, (D) 90 kHz.
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formationof the observed tonotopical vibrations there(12). Using a
combined experimental and numerical approach, Olson and
Nowotny (14) demonstrated that the bush-cricket CA also had a
similar stiffness magnitude to the basilar membrane, and that the
stiffness decreased from the high-frequency region to the low-
frequency region. Nevertheless, when such a structure is located
inside a complex system, there are several factors that can modify
the local behaviour. In particular, for the bush-cricket inner-ear
these changes can arise from the oscillators in the system being
coupled to each other, the standing wave cavity resonance of the
auditory vesicle, and the standing wave resonances of the TM and
the dorsal wall. Hence, while equation (1) is generally true for a
single degree of freedom, it may not hold when the degrees of
freedom increase. Applying a three-dimensional numerical
analysis, we aimed to determine the extent to which the CA and
dorsal wall geometry contributed to the emergence of the
experimentally observed tonotopy there.
To incorporate the dimensions of the CA into the models, we
used the precise m–CT measurements of C. gorgonensis inner-ear
parts (see Figure 3). While it was not possible to directly use the
scanned geometry in our models due to the difficulties offorming a
finite-element mesh on the detailed features of the CA, we
constructed an idealised geometry based on the obtained
measurements (Figure 2). As a result, the numerical models
contained a CA tapering in width and height. A similar property
was also present for the dorsal wall, due to the narrowing of the
auditory vesicle towards the distal end of the chamber. Hence, the
model included the geometrical factors considered to be necessary
for the frequency dependent displacement of the CA. The elasticity
of these parts, however, were assumed to be homogeneous and
isotropic in the simulations. In addition, the geometry did not
includethe curvature presentalong the bush-cricket dorsal walland
CA, which could potentially add another gradient through its effect
on the stiffness (38).
Since the geometry of the TP and dorsal wall input system
matches the C. gorgonensis ear morphology, and TP input is the
mechanism identified for vibration transmission to the C.
gorgonensis inner-ear experimentally (6), in this section we
refer to this set of data when talking about our results. This
data set demonstrates a frequency based displacement along the
CA (see Figure 4A). Further, the maximum displacement peak is
a smooth peak dissipating a short distance after passing through
the point of resonance (Figure 6), a tonotopical property also
observed experimentally (6). Hence, we can see that tonotopy
emerge as a result of changes in the CA and dorsal wall geometry
that produce simplified stiffness and mass gradients.
Nevertheless, a discontinuity can be observed in the
tonotopic organisation around 10 kHz (see Figure 4), implying
AB
FIGURE 6
Crista acustica vertical displacement magnitude due to transmission of acoustic vibrations at various frequencies. Displacement magnitude due
to transmission of vibrations facilitated by the (A) tympanal plate and dorsal wall (TP and DW) shown in the left column, (B) tympanic membrane
and dorsal wall (TM and DW) shown in the right column.
Celiker et al. 10.3389/finsc.2022.957385
Frontiers in Insect Science frontiersin.org09
the requirement of additional features to the model for more
definitive tonotopy. Such a discontinuity implies that certain
properties have more significant effects at smaller frequencies,
for instance anisotropic elasticity, curvature or damping
properties. Another discrepancy between the experimental
data [(3)(6)] and our results can also be seen at the tonotopy
placement. While Montealegre-Z et al. (6)recordedthe
displacement maxima for 30 kHz at a distance of about 200
mm from the proximal end, for the numerical results it was
located at a distance of 320 mm, showing a shift in tonotopy at
the numerical results. Based on formula (1), a shift in the
maxima location points at a difference in stiffness and mass
between the actual and numerical geometry, which once again
suggests the requirement of more realistic material and
geometric properties in the model. Hence, for the investigation
of these discrepancies with the experimental data, as well as for
the further understanding of the inner-ear mechanism, an
enhancement of the model is certainly worth pursuing. Still, it
is worth noting that the distances presented in (6)were
measured through the insect cuticle rather than directly on the
CA, which was the case with the numerical results.
We believe our numerical approach exhibits the utility of
employing three-dimensional numerical models for investigating
a complex system alongside an experimental approach. For the
bush-cricket hearing system, while there are many studies
successfully explaining the general workings of the inner-ear [(3,
6,14,25,27) and references therein], it is not experimentally
possible to pick inner-ear parts apart to determine their individual
functions, without compromising the underlying mechanism of
the system to a certain extent. Our numerical technique provides
an alternative approach to such an analysis, which can be a
powerful tool in obtaining reliable predictions on the inner-ear
mechanism. Our approach is a first attempt to represent the
system with a simpler structure in order to understand the
contribution of specific properties, namely the basic morphology
of the chamber and its components.
Another simplifying assumption we applied for setting up the
mathematical models was that the TM and TP were comprised of
homogeneous and isotropic materials. Through experimental
investigations it has been observed that the tympana boost
frequencies relevant for the communication of conspecificbush-
crickets (5), indicating a more complex material structure. In the
mathematical models, properties like mass, stiffness and damping
of the TM and the TP, essential for capturing the impedance of the
system were also not based on measured values. Hence, the
magnitude of the displacement maxima presented in Figure 6
and Supplementary Materials, Figure S5 are not necessarily a true
reflection of the displacement magnitude. While the model
predictions are not directly comparable to the observed data, the
constructed model provides a simplified adaptation of the
biomechanics of the inner-ear, from which it is still possible to
obtain qualitative information related to tonotopy. Primarily the
model suggests that it is possible to develop a tonotopic vibrational
map, basedon themass and spatial gradientsthat result purelyfrom
geometrical changes.
4.2 Differentiation in CA morphology and
the role of TP, TM and DW inputs
In the mammalian hearing system,the transmission of acoustic
vibrations into the inner-ear is well-studied [see (23)(24)(39), and
references therein]. In particular, the middle ear is comprised of
three tiny bones (the ossicles) which through a lever action pass the
eardrum vibrations to the fluid filled cochlea, performing an air-to-
liquid impedance conversion (39). For bush-crickets, however, there
are still multiple untested ideas in the literature with regards to the
sound transmission to the bush-cricket inner-ear. Nevertheless, the
two mainpropositions for transmission mechanismsare both based
on lever systems. For instance, for the species M. elongata,itis
believed that the whole TM acts as the main input for sound
transmission (3)(5), through functioning as a type 2 lever. Hence,
the M. elongata ear will receive the maximum load at the intersection
of the TM with the dorsal wall. It has been observed that a large
section of the TM is in contact with the air-filled acoustic trachea,
and a smaller section is attached to the fluid-filled hemolymph
channel, thus allowing for impedance conversion. For M. elongata,
TheTMisincontactwiththehemolymphchannelalongthelength
of the CA (28). In contrast, for C. gorgonensis, vibration transmission
was identified to be facilitated by the type 1 lever action of the TP (6).
Thus, the large deflections of the airborne TM are transformed into
smaller deflections of the fluid-bound TP, demonstrating clear
impedance conversion characteristics in the ear. Moreover, the TP
is observed to be in contact with the fluid only towards the distal end
of the CA. Our numerical results suggest that the load area, or the
dimensions and location of the TM or TP in contact with the inner-
ear, also play a significant role in the formation of tonotopy inside
the inner-ear.
As can be observed from Figure 5, the frequency dependent
change in displacement maxima along the CA is evident when the
vibrations enter through the TP, with the maxima moving from
the proximal end to the distal end of the CA as frequency
increases. This is less pronounced for a TM entrance to the C.
gorgonensis inner-ear (Supplementary Materials, Figure S4).
Nonetheless, some differences between the bush-cricket C.
gorgonensis [established TP input (6)] and M. elongata
[established TM input (5,28)] are worth pointing out. The
species C. gorgonensis have a separate inner-ear chamber, the
auditory vesicle. Whereas for M. elongata, the CA is located in the
hemolymph channel, hence the mechanics driving the formation
of tonotopy may be different. In our numerical simulations, the
geometry only reflected the dimensions of the C. gorgonensis
auditory vesicle. Interestingly, Bangert et al. noted that for the
species Polysarcus denticauda and Tettigonia viridissima, while the
CA was activated through the TM, the tympana was in contact
with the hemolymph channel only where the high frequency
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Frontiers in Insect Science frontiersin.org10
receptor cells of the middle and distal crista acustica were located
(27). In this study, the outer surface of the TM for the considered
specieswereidentified to have an area called the ‘inner plate’,
which was a dark oval and stiff area that was surrounded by a paler
area, assumed to be more elastic. However, no out-of-phase
response between the inner-plate and the hosting tympana was
recorded, and the tympana was identified as a uniformly vibrating
membrane. Hence, there may be greater variation in auditory
mechanics even within bush-crickets than previously assumed.
Accordingly, we appreciate the differences in morphologies
between bush-cricket species and from this point on limit our
conjectures to the species C. gorgonensis.
The model also elucidates the transmission properties of the
TM, TP, andthe role of the dorsalwall which has alsobeen a topic of
interest [(22,27)]. While vibrational input from the wall was not
reportedin (6), the influence of sound pressurefrom the dorsal wall
was suggested in (5)and(27). In (22), it was further proposed that
the dorsal wall activated the CA before vibrations even reached the
TM. To analyse the dorsal wall influence numer ically, we simul ated
the transmission of sound by removing the acoustic trachea from
underneath the dorsal wall in the geometry, and activated the
system only through the TMor the TP (see Figure 4B). In addition,
we also considered a simulation where the transmission was only
through thedorsal wall. As can be observed in Figure4, the removal
of the dorsalwall pressure for a TP and TM excitation didnot lead
to a tonotopical arrangement, highlighting the role of the wall
displacement itself on the movements of the CA. A solitary dorsal
wall (Figure 4) excitation of the system did not lead to the
experimentally observed tonotopy at the CA either. Hence, we
conjecture that in addition to the effect of the stretching and
squeezing of the dorsal wall through the TM or the TP entrance,
the sound pressure from the acoustic trachea below also plays a
significant role in activating the mechanoreceptors. However, the
small displacement magnitude from the dorsal wall input
(Supplementary Materials, Figure S5C) leads us to believe that a
stronger mechanism than a solitary dorsal wall stimulation is
needed for the excitation of the system. These results suggest that
the resonance approach proposed in (22) is not a probable
mechanism for the stimulation of the C. gorgonensis CA.
Our attempt to simulate the biomechanical processes in the bush-
cricket ear, using a 3-dimensional idealised geometry, has resulted in a
reasonable qualitative match to the experimental results in the
literature. This suggests that for the investigation of such processes,
a numerical approach can provide a cost efficient alternative and
validation method to empirical studies. Reliable numerical models,
validated through experimental data, also provide a new platform for
analysing the individual effects of the parameters comprising the
bush-cricket inner-ear, which is not possible through
experimentation. Therefore, the further numerical investigation of
the bush-cricket inner-ear, through improving the incorporated
quantitative (material properties, fluid viscosity, potential non-
Newtonian fluid properties) and geometrical (curvature of the
dorsal wall and CA) properties is certainly worth pursuing.
Data availability statement
The datasets presented in this study can be found in online
repositories. The names of the repository/repositories and accession
number(s) can be found below: https://drive.google.com/file/d/
1xC7xfYp9896oEItITREDaAfGn5zLx-ZK/view?usp=sharing.
Author contributions
EC, NM and FM-Z contributed to conception and design of
study. CW took morphological measurements. EC developed
numerical models, ran simulations and analyzed obtained
results. EC and CW developed figures and images. EC wrote
the first draft of the manuscript. All authors contributed to
manuscript revision, read, and approved the submitted version.
Funding
EC, CW and FM-Z are funded by the European Research
Council Grant ERCCoG-2017-773067 (to FMZ for the project
“The Insect Cochlea”). NM is funded by an NSERC Discovery
grant and supplement (687216, 675248) and Canada Research
Chair (693206).
Acknowledgments
We thank the Orthopterists’Society for aiding the funding of
the micro-CT work of CW, and the University of Lincoln for
CW’s PhD studentship.
Conflict of interest
The authors declare that the research was conducted in the
absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the
authors and do not necessarily represent those of their affiliated
organizations, or those of the publisher, the editors and the
reviewers. Any product that may be evaluated in this article, or
claim that may be made by its manufacturer, is not guaranteed
or endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found
online at: https://www.frontiersin.org/articles/10.3389/
finsc.2022.957385/full#supplementary-material
Celiker et al. 10.3389/finsc.2022.957385
Frontiers in Insect Science frontiersin.org11
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