Reconsidering tympanal-acoustic interactions leads to an improved model of auditory acuity in a parasitoid fly

Although most binaural organisms locate sound sources using neurological structures to amplify the sounds they hear, some animals use mechanically coupled hearing organs instead. One of these animals, the parasitoid fly Ormia ochracea (O. ochracea), has astoundingly accurate sound localization abilities. It can locate objects in the azimuthal plane with a precision of 2°, equal to that of humans, despite an intertympanal distance of only 0.5 mm, which is less than 1/100 th of the wavelength of the sound emitted by the crickets that it parasitizes. O. ochracea accomplishes this feat via mechanically coupled tympana that interact with incoming acoustic pressure waves to amplify differences in the signals received at the two ears. In 1995, Miles et al developed a model of hearing mechanics in O. ochracea that represents the tympana as flat, front-facing prosternal membranes, though they lie on a convex surface at an angle from the flies’ frontal and transverse planes. The model works well for incoming sound angles less than ±30∘ but suffers from reduced accuracy (up to 60% error) at higher angles compared to response data acquired from O. ochracea specimens. Despite this limitation, it has been the basis for bio-inspired microphone designs for decades. Here, we present critical improvements to this classic hearing model based on information from three-dimensional reconstructions of O. ochracea’s tympanal organ. We identified the orientation of the tympana with respect to a frontal plane and the azimuthal angle segment between the tympana as morphological features essential to the flies’ auditory acuity, and hypothesized a differentiated mechanical response to incoming sound on the ipsi- and contralateral sides that depend on these features. We incorporated spatially-varying model coefficients representing this asymmetric response, making a new quasi-two-dimensional (q2D) model. The q2D model has high accuracy (average errors of under 10%) for all incoming sound angles. This improved biomechanical model may inform the design of new microscale directional microphones and other small-scale acoustic sensor systems.


Introduction
The ability to localize sound allows animals to avoid predators and assists them in finding mates and capturing prey. Binaural organisms locate sound-emitting objects by comparing the intensity and timing of incident sound waves arriving at their two hearing organs ( figure 1(A)). Sound localization in binaural organisms is commonly described using two metrics: the ITD, the difference in the time it takes sound to reach the two hearing organs, and the IAD, the difference in sound amplitude between the two organs figure 1(A) [1]. Once a sound wave has reached an organism, IAD and ITD typically depend primarily on the animal's size. Larger-headed animals generally have larger ITD values due to the greater distance between their hearing organs.
Larger animals also have larger IAD values due to acoustic shading by their heads when the sound wavelength is comparable to or smaller than their head size, and to the lower amplitude of the acoustic signal at the ear farther from the sound source, since sound decays as 1/r 2 . When the animal is located at a great enough distance from the sound source, acoustic decay can reduce the amplitude of incoming sound below the threshold for acoustic sensing on one side or both.
The particulars of sound localization are complex and vary widely among animals, as the ITD and IAD ranges experienced by binaural animals demonstrate ( figure 1(B)). In vertebrates, ITD is calculated in the brain stem's superior olivary nucleus, and IAD is calculated in the inferior colliculus in the midbrain nucleus [2]. Many vertebrates have ears that function independently to receive pressure from incoming sound waves, determining directionality through neural processing and diffraction [3,4]. Other animals, especially smaller ones, can take advantage of tympanal membranes coupled through shared bodies of air, converting their ears into pressure gradient receivers [3,5]. However, as differences in received pressure and organismal length scales become smaller, this coupling strategy becomes less practical. And many invertebrates lack significant neural investment in central processing and rely heavily on mechanical structures to pre-process sensory signals [5][6][7][8].
The biophysics of sound localization for specific species is often too complex to be modeled easily. But a simple mathematical hearing model exists for the tachinid fly Ormia ochracea (O. ochracea). It can be used to extract key principles of binaural hearing without complex physiological modeling of neural processes.
For O. ochracea, the ability to hear its host plays a key role in its reproductive cycle and overall fitness. As a parasitoid, O. ochracea listens to chirping male crickets and follows the sound back to the source, where female O. ochracea then deposit their larvae [9]. Gravid O. ochracea females will remain in an area for extended periods in response to cricket chirping sounds, even if no cricket is present [10]. Given its small size, if O. ochracea relied exclusively on the distance between its prosternal tympanal membranes (figure 1(C)), the ITD it experienced would be at the nanosecond scale or below for the sound frequencies to which they are sensitive, and there would be no practical difference in sound amplitude between the two membranes (IAD). To solve this scaling problem without costly neurological investment, O. ochracea have two mechanically coupled membranous tympana directly beneath their head ( figure 1(C)). These coupled tympana are composed of a pair of prosternal membranes, joined together by an intertympanal bridge [11], and are significantly larger in female Ormia ( figure 1(C)). This distinctive mechanical coupling increases the ITD and IAD perceived by the fly. We will refer to the increased ITD and IAD as mITD and mIAD, respectively. These mechanically amplified values allow it to localize chirping crickets successfully. Since O. ochracea are active at or after dusk they also use their hearing to avoid predation by bats, exhibiting a startle response while in flight to bat sonar frequency sound, similar to praying mantises [12,13]. Given O. ochracea's ability to accurately and quickly locate a source of incoming sound at high levels of lateral angular resolution, even in potentially noisy environments [14], it may serve as a good source of bioinspiration to tackle the socalled cocktail party problem [15] (isolating sounds in a noisy environment) for directional microphones and hearing aids.
Accurate models of binaural hearing in animals are generally highly complex. Because of the mechanical nature of its acoustic sensing organ, O. ochracea is one of the few exceptions, and it has been the focus of numerous studies featuring its 'simple' hearing organs and how they function [11,12,[16][17][18][19][20][21][22]. To investigate the biomechanical mechanisms that underlie O. ochracea's unusual hearing abilities, Miles, Robert, and Hoy developed mechanical and mathematical models of the ormiine flies' coupled tympana in 1995 [18]. The authors validated their model against experimental data, recording tympanal membrane positions and velocities, and consequently mIAD and mITD, as a function of the incident sound pressure, intensity, and angle. The Miles model becomes analytically solvable under the assumptions of continuous sinusoidal input and symmetric model parameters, in addition to being numerically solvable without requiring the assumptions of symmetry or continuity. The model allowed Miles et al to demonstrate that O. ochracea's impressive sound localization abilities are due to the pre-processing performed by their structurally coupled tympana, which mechanically amplifies the ITD and IAD experienced by the fly.
In addition to providing a physiological explanation for O. ochracea's localization prowess, the Miles model also accurately predicted mITD for all incoming sound angles and mIAD for angles below ±30 • in a sample O. ochracea population. Both the measured ITD is the time delay between sound reaching one sensory organ relative to the other, defined by the equation τ = dsin(θ)/c, where d is the distance between the hearing organs and c is the speed of sound in air. IAD is the difference in response amplitude between the left and right sensory organs, usually due to acoustic shading or signal decay. (B) Estimated ITD values in binaural animals of increasing size (O. ochracea, rat, cat, human, and rhinoceros) for a sound source at 45 • from the midline, calculated using the formula in (A), with zygomatic breadth (cheekbone-to-cheekbone distance) used as a measure of interaural distance for mammals. The expected ITD for O. ochracea calculated based on intertympanal distance is indicated with a blue dot. The true value of ITD for O. ochracea is included as a red square ('mechanical ITD') to illustrate the degree of improvement in O. ochracea's hearing acuity provided by its mechanically coupled tympana. Data for mammals from [32][33][34][35]. (C) Female O. ochracea post-decapitation, showing the location of the prosternal membrane, tympanal pits, and intertympanal bridge, key physical features in modeling its binaural hearing. and predicted mITD indicated that O. ochracea possesses an mITD comparable to the ITD of an animal closer in size to a rat ( figure 1(B)). Later experiments successfully determined that O. ochracea has a sound localization precision in the azimuthal plane of 2 • [23,24], comparable to that of humans. This high precision, together with the relative simplicity of the model and the easily reproducible structure of the hearing mechanism used by O. ochracea, led to a new stream of research in O. ochracea-inspired designs for directional microphones and hearing aids [25][26][27][28][29][30][31]. Despite its utility, the model contains a number of simplifications that limit its biological accuracy.
The Miles model is a lumped-element model that considers the dynamics of the intertympanal bridge and the tympanal membranes (figure 1(C)), modeling each membrane as a flat plate with a purely onedimensional amplitude response. The tympana are represented as flat, front-facing membranes, though they lie on a convex surface at an angle from the flies' frontal and transverse planes. Miles et al adjusted the model's spring and damper coefficients until the model response approximated the experimental responses of the tympana in recently deceased O. ochracea specimens measured using laser vibrometry. Although the model is accurate for mITD in a narrow range of incident sound angles, it displays significant errors in mIAD for incident sound angles larger than approximately ±30 • from the midline of the fly, and mITD becomes increasingly inaccurate at angles above approximately ±40 • . This inaccuracy at large incident sound angles limits the model's power for explaining binaural hearing in O. ochracea and its potential for inspiring new hearing-based engineering.
In this work, we used three-dimensional (3D) reconstructions of the tympanal organs of two O. ochracea specimens, which we obtained via synchrotron x-ray imaging, together with the available behavioral hearing and tympanal response data, to identify candidate morphological features and possible additional mechanics not represented in the Miles model that could help explain the error in mITD predicted for large incident sound angles. While previous scanning electron microscopy images of O. ochracea tympana indicated that a degree of morphological complexity was present [11], the 3D reconstructions allowed us to examine different views of the tympanal organ and form a more complete understanding of its structure.
Using the reconstructions as a guide, we identified the tympanal bridge arm angle, ϕ, which quantifies the tympanal membranes' relative orientation, and the angular width of the frontal tympanal organ segment between the membranes, as important to the perception of sound. We then modified the Miles model to include a differentiated mechanical response to incoming sound on the hearing organ's ipsi-and contralateral sides. These asymmetric parameter adjustments were represented mathematically via modified spring and damper coefficients that depend on the tympanal bridge angle, the frontal tympanal organ segment angle, and the incoming sound angle. Our model modifications were based on the hypothesized existence of unknown hearing mechanics akin to acoustic shadowing (which is not a feature of O. ochracea's hearing mechanics at the host-cricket chirp frequencies that the flies' hearing system is tuned to perceive) that act to damp the response of the hearing organ on the contralateral side for certain incoming sound angles. We discuss the hypothesized mechanics further in this article's q2D model modifications, results, and discussion sections.

Synchrotron x-ray imaging of the ormiine tympanal organ
To examine the 3D nature of Ormia ochracea's tympanal morphology, we performed tomographic imaging of preserved O. ochracea specimens using synchrotron x-ray imaging at the Advanced Photon Source at Argonne National Laboratory. Two O. ochracea dried specimens were borrowed from the Virginia Tech Insect Collection. The specimens were placed in slender tubes made of polyimide, and the ventral thorax was imaged at beamline 2-BM using x-rays with an energy of 14 keV. Each specimen was imaged using the beamline's fast 2D phase-contrast imaging, giving stacks of images along the anteriorposterior-axis at intervals of 1.72 µm [43].
Reconstructed microtomographic images were cropped and down-sampled using FIJI [36], and segmented in SlicerMorph, [37] an imaging extension of 3D Slicer [38,39]. To segment, features of the tympana were highlighted and then rendered in 3D for applicable measurements. The three-dimensional scans are available upon request.

Previous model
The previous model of binaural hearing in O. ochracea includes two components: a mechanical model of the anatomy and its function and a corresponding mathematical model. The mechanical model [18] treats the tympanal structure as a pair of beams pinned at a central pivot, with lumped-mass approximations of the two sides of the hearing organ located at the ends of the beams (figures 2(A) and (B)). The beams are anchored to the substrate at their distal ends with a pair of symmetric spring-damper elements and to each other with a third spring-damper element (figure 2(B)). Pressure forces from incident sound waves are applied to the point masses via a forcing function composed of the product of the incident pressure magnitude, the inward-facing unit normal vector, and the tympanal surface area, A (see the supplemental material for numerical values used in this study). A time delay is applied between the left and right sides based on the angle θ, which is the angle of the incoming sound wave relative to the midline of the fly, with 0 • defined as straight ahead [18]. The time delay is calculated via the equation that appears in figure 1 The mathematical model is a set of coupled ordinary differential equations representing the mechanical model's dynamics. It treats the incident acoustic pressure acting on the tympanal membranes as two point forces, f 1 (t) and f 2 (t), acting on the point masses representing the tympanal membranes and associated structures. The dependent variable in the problem is x(t), which represents the one-dimensional response of the tympana. The model can be written as follows: [ is the unknown response vector containing the vertical displacement of the left and rightmost tips of the beams in figure 2(B), which represent the two sides of the cuticular intertympanal bridge, the applied force is f = (f 1 (t), f 2 (t)), and( ) represents differentiation with respect to time, t. The parameters k i and c i are spring stiffness and damper constants, respectively, and the parameter m is the effective mass of all the moving parts of the auditory system [18].

Q2D model modifications based on ormiine morphology
The Miles model (equations (1.1) and (1.2)) consists of two coupled one-dimensional ordinary differential equations with time as the sole independent variable. In Miles et al's analysis of their model, the ormiine hearing structure is assumed to be leftright symmetrical, and the spring and damper coefficients on the right and left sides are identical and constant for all incident sound angles, with To add the level of sensitivity to the incoming sound angle represented in the available behavioral data, we modified the spring and damper parameters to incorporate aspects of the 3D morphology of the fly's hearing organ. The resulting model is quasi-two-dimensional (q2D) since it depends on the sound source's location in the azimuthal plane. Specifically, we treated the magnitude of the spring and damper coefficients, k and c, on the contralateral side as functions of the incoming sound angle for sounds incident between azimuthal angles approximately corresponding to sound arriving perpendicular to the ipsilateral membrane (θ 1 ) and sound arriving parallel to the contralateral membrane (θ 2 ). This variable response region corresponds to the yellow-shaded angle segment in figure 4(D). The functions were structured such that for incident sound angles above θ 1 (which is equivalent to ϕ − 90 • , where ϕ is the angular width of the frontal tympanal organ segment between the membranes), the k and c values corresponding to the contralateral tympanum are increased compared to those for the ipsilateral tympanum. This represents an overall damping of the contralateral tympanal response, or a stiffening of the contralateral tympanum. Our modified model posits three distinct tympanal response regimes: a symmetric regime in which incoming sound is incident near the front of the fly between the two tympana, and the tympana respond identically (|θ| ⩽ θ 1 , the blue shaded region in figure 4(D)), a linear ramp regime in which the contralateral tympanum's response is damped and the damping increases as a linear function of the incoming sound angle (θ 1 ⩽ |θ| ⩽ θ 2 , the yellow shaded region in figure 4(D)), and a saturated regime in which the contralateral damping has reached a maximum level and no longer increases with incoming sound angle (|θ| ⩾ θ 2 , the red shaded region in figure 4(D)).  [18], as no other experimental data exists with which to compare our model's predictions. Data were obtained from [18] using the MATLAB program GRABIT [42] and the published PDF file, and the error for a given variable was calculated by finding the absolute value of the difference between the model results and the experimental data for that variable, dividing the difference by the experimental data value of the variable, and multiplying the result by 100. The significant divergence from behavioral data present in the Miles model outside ±30 • , particularly for mIAD, is rectified in the modified q2D model. In (C), the gray box indicates errors below ±30 • , which are not considered because the q2D and Miles models are identical for these ranges. The errors for the q2D model peak close to ±30 • , then decrease as the incident sound angle increases.
This stiffening of the contralateral tympanum is due to unknown mechanics related to the structure of the tympanal organ that causes an asymmetric response to laterally oriented incoming sounds ( figure 2(C)). This behavior would be consistent with the enhancement of directional hearing via a pressure gradient receiver mechanism resulting from internal acoustic coupling of the tympana, as in pyraloid moths [41] and many terrestrial vertebrates.
We provided the following q2D modification to the Miles model of ormiine hearing: where k i and k f and c i and c f are the minimum and maximum values that the spring stiffness and damping coefficients take on, respectively, and ϕ is the intertympanal bridge arm angle (see figures 2(A) and 4(D)). The form of the modified spring coefficient function, two constant segments with a linear ramp between ±θ 1 and ±θ 2 (figures 2(C) and 3(A)), was also informed by the lateralization behavior observed in O. ochracea [23] and the analysis of an O. ochracea-inspired sensor [40]. These works indicated the presence of two separate behavioral regimes, a localization regime from 0 • to 30 • and a lateralization regime at higher angles. This choice is further supported by the accuracy of the fit to experimental data for sound incident at ⩾ 30 • (figures 3(B) and (C)), and physically represents a variable mechanical response to sound waves incoming laterally. These equations can be represented in a fully generalized form as:  6) for the modified q2D model. A custom peak-finding algorithm was implemented to calculate mITD and mIAD, which was necessary because of the asymmetry in the model coefficients k and c. Further computational details and a link to representative code samples can be found in [44]. The colored panel reflects the approximate symmetric, ramp, and saturation azimuthal response ranges of the modified q2D model tuned to this specimen. The q2D model regime bounding angles, θ1 and θ2 are 22 • and 68 • , respectively, for this specimen. See also movie S1 in supplemental information.

Results
The tympanal organ of O. ochracea protrudes anteriorly from underneath the fly's neck (cervix), with distinct lateral faces (prosternal tympanal membranes) and sharply angled intertympanal bridge arms (figure 5). The tympanal organ is a modification of the prosternum and is composed of a pair of front-facing (cephalad) tympanal membranes that cover a hollow prosternal cavity. The cavity opens to the environment through a pair of tracheae connected to lateral mesothoracic spiracles [11]. Each bridge arm has a pit that connects to an auditory apodeme that runs longitudinally through the sternal cavity to an auditory sensory organ (a bulba acustica, a chordotonal organ composed of many sensory scolopidia). Vibrations are conducted via the apodemes to the bulbae acusticae, which are innervated to the frontal nerve of the thoracic ganglion. Figure 5 shows 3D surface renderings of O. ochracea tympanal membranes in teal, with the supporting sternal structures highlighted in peach. The organs are far from the simple two-dimensional surfaces most often depicted in the literature [17,18,31,40]. These new 3D models motivated our modifications to include aspects of actual morphology. The confirmation of significant bilateral faces of the tympanal organ led to the modifications present in the q2D model (equations (1.3)-(1.6)), which account for the lateral tympanal organs' response to acoustic stimuli. Because the specimens are preserved, some aspects of the morphology, especially the degree of bowl-like curvature of tympanal membranes visible in figure 5, may be due at least partially to the desiccation of the samples. However, the overall topography of the structures, especially the tympanal bridge, appears to remain essentially unchanged when compared to previous still images taken of recently sacrificed specimens [18].
Using the 3D renderings, we measured several features of Ormia tympanal morphology, as shown in figures 4(A)-(D). Because the rendering is produced using images sampled every 1.72 µm, features smaller than this may not be visible.
In figure 4(A), the prosternal inflation of the tympanal structure is traced in purple. The path length and curvature of the tracing were measured. The mean curvature is 0.004 µm −1 , and the maximum curvature is 0.018 µm −1 , highlighting the presence of extended flat portions of the supporting structure. The plane of the intertympanal bridge arms (shown in purple) is oriented upwards relative to the azimuthal (x-y) plane by 38 • . The vertical height of the connecting line between the horizontal span of the intertympanal arms and the point at which they join, in the y direction (also shown in purple), is   As discussed in the q2D model modifications section, careful inspection of the 3D renderings led us to hypothesize that including a representation of the orientation of the tympanal membranes and the angular distance between their front edges in the mathematical model would improve its performance. This improvement was hypothesized because the membranes' relative orientation appears to be related to the angles at which the flies' tympanal responses changed in experimental studies. In the updated q2D model, the contralateral tympanum's response is damped for incoming sound first incident at locations between the front edge and the center of the ipsilateral tympanal membrane. Because acoustic shading is not possible for the frequencies to which the flies are sensitive, one reasonable explanation is a previously unidentified mechanical feature causing angle dependence in the tympanal organ's response, as discussed in the model modifications section above.
The arms form an angle ϕ of approximately 136 • within the azimuthal plane, which creates a 112 • angle with the midline of the fly. Therefore, sound incoming at a 22 • angle (relative to the y-axis or midline in figure 3(D)) would be oriented perpendicular to the ipsilateral arm, indicating the beginning of an incoming sound angle regime in which the contralateral tympanal membrane exhibits increasingly damped responses. Similarly, incoming sound at a 68 • angle from the midline would be parallel to the contralateral arm, corresponding to maximal damping of the contralateral membrane.
Hence, 22 • and 68 • from this specimen's midline (y-axis) represent incoming sound angles that divide the three hypothesized tympanal response regimes introduced in the q2D model modifications section: symmetric, linear ramp, and saturated. In the symmetric regime (|θ| < 22 • ), the tympana will respond symmetrically. In this regime, the model coefficients k i and c i are constant. In the linear ramp regime (22 • < |θ| < 68 • ), both tympanal membranes will vibrate in response to incoming sound in a ratio that depends on θ, and the model coefficients vary linearly as described in equations (1.3)-(1.6). In the saturated regime, the contralateral tympanal membrane's damper and spring coefficients will have reached their maximum values, c f and k f . The coefficients will be constant in this regime. This trimodal behavior appears to be due to unidentified mechanics of the tympanal organ that are seemingly well-modeled by increasing damping of the contralateral membrane in response to increasing incoming sound angles in the specified range.
We used the values θ 1 = 30 • and θ 2 = 55 • in our modified q2D model to obtain the q2D model results shown in figures 3-7. These values are tuned to match the available behavioral data from [18], shown in figure 3(B) (mITD and mIAD values derived from laser-vibrometry measurements of tympanal membrane displacements in O. ochracea specimens in response to a 6 kHz sound source, as a function of incident sound angle). We assumed the total tympanal surface area, A, was fixed and used the value from [18] (A = 0.288 × 10 −6 m 2 ). The increases in the spring and damper coefficients, normalized relative to their nominal values (k i ), are visible in figure 4(A). The values for θ 1 and θ 2 used in our computations differ from the angles that bound the ramp regime represented on the fly in figure 4(D) (shaded yellow). This is because we used 3D imaging of the fly in figure 4 to identify the geometric features of O. ochracea's hearing organ that are important for sound localization. While the importance of these features generalizes to all O. ochracea individuals, the exact value of the intertympanal bridge arm angle, ϕ, is subject-specific. We expect that if we had behavioral data for the fly imaged in figure 4, the values of the modified model's ramp regime bounding angles that provided the best match to that data would be ±22 • and ±68 • , as shown in figure 4(D).
Values of mITD and mIAD, calculated from the q2D and Miles models, are shown in figure 3(B) as a function of incident sound angle and are compared to experimental measurements in recently sacrificed O. ochracea specimens [18]. Both models are identical for incident sound angles less than ±30 • , so the results are identical within that range ( figure 3(C), gray box). When we included the differentiated lateral response through the new k(θ) and c(θ) functions in equations (1.3)-(1.6), the gap between experimental measurements and model results in both mIAD and mITD narrowed significantly for 6 kHz signal input (figures 3(B) and (C)), with the q2D model having an average error of approximately 6% and a peak error of approximately 28% in mITD, and an average error of approximately 7% and a peak error of approximately 10% in mIAD. These results additionally confirm that aspects of mechanics in two dimensions are important elements of ormiine hearing.
The model behaviors outside of the regime associated with cricket calls (roughly 4-6 kHz) were also examined and are shown in figures 6 and 7. These results demonstrate that, as a general rule, mITD values from both models approach zero as the frequency of incoming sound increases beyond 20-30 kHz. The sensitivity in mITD to the angle of incoming sound also decreases. There are small asymmetric regions in our q2D model, one from approximately 45 • to 75 • and 20 to 30 kHz, and one from approximately −15 • to 0 • and 0 to 10 kHz ( figure 6(B)). Because the region where |θ| ⩽ 30 • should be identical between the Miles model and the q2D model, with no asymmetry present, these asymmetries are likely numerical artifacts related to the peak-finding algorithm we used to determine mITD. This asymmetry is not present in the mIAD results ( figure 7), nor in the divergence from the Miles model within the ±30 • range, further supporting its characterization as a numerical artifact.
By contrast, figure 7 demonstrates that mIAD values are significant for all frequencies and angles well into the range of expected bat ultrasonic calls, especially in the q2D model. The angular resolution of mIAD (that is, the change in mIAD as the incoming sound angle changes by a fixed amount) decreases with incoming sound angle for high frequencies, but this difference is consistent with observed Ormia behaviors that indicate that behavioral responses are tied more directly to successful bat identification, and less to precise localization [12] of the origin of ultrasonic bat vocalizations.
Additional figures showing independent left/right membrane response curves at specific incoming sound angles, as well as versions of figures 6 and 7, focused on the range from 1 to 20 kHz are available in the appendix. It should also be noted that above approximately 10 kHz in figure 6, the sign of the mITD values provided by both models shifts, indicating the contralateral response occurs prior to the ipsilateral response at high frequencies. This sign change and differences in mIAD signal values at high frequencies could provide an effective categorization mechanism for bat-signal and cricket-signal perception, as well as basic localization abilities into the ultrasonic frequencies for incoming sound, as was proposed in [12]. . mITD values are highest at high incoming sound angles. White spots on the plot represent elements that were measured incorrectly due to erroneous identification in the peak finding algorithm used to determine mITD values.

Discussion
This paper presents the results of 3D x-ray synchrotron imaging of the mechanically-coupled tympana in the parasitoid fly, O. ochracea, and our subsequent modification to the classic mathematical model of hearing in O. ochracea, which was informed by those imaging studies. Three geometric features of the tympanal organ were identified as particularly important for sound localization: the angle ϕ between the intertympanal bridge arms, the angle θ 1 , which is equivalent to 90 • − ϕ, and the angle θ 2 , which is half the azimuthal angle segment at the center of the fly, between the two tympanal membranes (see figure 4(D)). Thus, the convexity of the tympanal organ is represented in our modified model, in contrast to previous simplified representations of Ormia ochracea's tympana as being located in a common frontal plane of the fly, level with the peak of the tympanal bridge.
Detailed knowledge of the hearing organ's morphology allowed us to update the classic onedimensional mathematical model (the Miles model) into a quasi-two-dimensional (q2D) model of ormiine hearing that mimics the tympanal organ response in the lateral direction. Our updated q2D model has significantly improved agreement with available experimental data [18] compared to the Miles model, both in the mITD and in the mechanical mIAD (figures 3(B) and (C)). Compared to the Miles model, the new q2D model exhibits a reduction in maximum errors (relative to experimental values) of approximately 50% and 85% in mITD and mIAD, respectively. This result supports the premise that there are essential aspects of the mechanics of Ormia hearing aside Figure 7. Model comparison at extended frequencies: mIAD. The system response amplitude difference (IAD) between the tympanal membranes at incoming sound angles ranging from −90 • to 90 • , and frequencies ranging from 1 kHz to 60 kHz are shown for the Miles model (A) and the q2D model (B). Unlike for mITD, non-zero mIAD values persist up to 60 kHz, the highest incoming sound frequency simulated. In both models, mIAD values are the highest at high incoming sound angles, with steeper value gradients occurring at lower frequencies, closer to the values expected values for cricket calls. The range of peak mIAD values (dark blue/bright yellow) is taller for the q2D model and extends into the low ultrasonic range.
from the uniplanar response of the tympanal membranes and that the hearing organ structures are sensitive to the angle of incoming sound, a feature that was not included in the Miles model.
Prior to our study, the original Miles model was the only existing model of internally coupled earsbased hearing in ormiine flies [45]. This is one of the first attempts to update the foundational Miles model for hearing in O. ochracea. Our model may be further refined by incorporating additional mechanical behaviors of the tympana, such as tympanal deflection in the lateral direction or a representation of the tympanal response in the vertical direction. Simple analytic modifications could also improve and expand the model's capabilities without impacting its tractability, such as using functions that are more flexible than simple linear ramps for the spring and damper coefficients. For example, in our q2D model, the 'bump' visible near ±45 • in mIAD in figure 3(B) and the uptick at the same point in mITD may be a result of the values for either the springs, dampers, or the ratio between the two, being slightly too high at that point. It is also important to note that this work and the Miles model both rely on tuning the coefficients so that the model outputs better match the experimental response to sinusoidal input (2 kHz for the original 1995 work and 6 kHz for the work here). Although the q2D model's performance was not observed to degrade (compared to the original model) at other frequencies that we checked, the degree of improvement (relative to the 6 kHz experimental data) was far less significant for other frequencies. The model's reduced performance at frequencies other than those tuned specifically for crickets could potentially be resolved by introducing other morphological features in the form of frequency-dependent functions in a similar way as we have introduced spatially-dependent functions here.
In the larger context of bidirectional mechanically-coupled hearing, this model provides an alternate pathway for exploring possible frameworks for mechanically-coupled ears. Many existing models of coupled hearing focus on the coupling mechanism itself and do not consider the orientation of incoming sound when determining the properties of the mechanism [3,4,46]. While the particular mechanism used by Ormia is unlikely to be replicated in other systems without similar mechanical coupling, the work presented here demonstrates the importance of the sensitivity of some microscale mechanical structures to the direction of incoming acoustic forces. It also underscores some shortcomings inherent to model simplification and motivates the development of models that, while remaining tractable, employ slightly more sensitive approaches.
Our model demonstrates a potential role for detailed tympanal anatomy in the mechanics and modeling of directional hearing in O. ochracea, especially with respect to the values of mIAD in response to incoming sound waves at high angles. Including angle-dependent behavior in the spring and damper coefficients may provide a more accurate representation of how the insect receives sound. Previous work has demonstrated that O. ochracea engages in different behaviors depending on the relative angle of incoming sound [18,23,40,47], with two distinct response patterns. In the first, from 0 • to ±30 • , the fly makes relatively narrow adjustments to localize the origin of the sound (localization). In the other, at angles exceeding approximately ±30 • , the fly makes more significant positional adjustments, more akin to determining the side from which the sound originates (lateralization). Our results suggest that angle-dependent tympanal membrane mechanics could contribute to this observed behavioral sensitivity to incoming sound angle.
There is growing evidence that some O. ochracea are involved in an evolutionary arms race with their host species [48,49], and that they are capable of differentiating between different cricket host species based on their acoustic signaling, exhibiting preference towards local populations [50]. Consequently, the mechanical parameters for the model may depend not only on the geographic origin of O. ochracea samples, but also when collection occurred. The degree of tuning to host-searching behavior, as opposed to predator-avoidance behavior, also remains unaddressed experimentally, despite the startle responses when in flight and subjected to sound consistent with bat sonar frequencies [12]. O. ochracea exhibits a sorting behavior (being able to rapidly categorize sounds as belonging to a predator or not) in response to predator-consistent sound sources, as opposed to host or neutral sound sources [12]. O. ochracea is also only one of many Ormia species, which parasitize a diverse range of hosts, and display different behavioral responses to the acoustic signaling of their hosts [10]. Only O. ochracea has been examined in sufficient detail to develop a mechanical model with accurate parameters; consequently, it may be worth investigating the mechanics of other ormiine species [10,51] and developing mechanical models similar to the q2D model presented here. It may also be worth revisiting the hearing organs in Emblemasoma, another group of parasitoid flies representing a case of convergent evolution in a distantly related family, Sarcophagidae [52,53].
O. ochracea's hearing system has repeatedly been a source of inspiration for bio-inspired designs for directional microphones and hearing aids [25][26][27][28][29][30][31]40]. Including the angle-dependent behavior of the expanded q2D model in future Ormia-inspired device designs may also provide significant avenues for improvement in device performance or may expand the functionality of devices like acoustic sensors through miniaturization and tunable frequency sensitivities. Currently, work is being undertaken to explore the inclusion of lateral faces on a directional microphone to study these elements' role further and attempt to develop a novel practical application. However, there are numerous avenues for exploration remaining, both experimental and theoretical. These include the development of improved bio-inspired technology by incorporating higherdimensional features and parameter variations in the mechanical system, studying the model's behavior at frequencies commensurate with bat sonar, and investigating the role mechanical differences play in O. ochracea's hearing when addressing acoustic preferences.
Finally, our expanded q2D model is the first mathematical model of hearing in a binaural fly that is accurate for all measured incident sound angles. It demonstrates the importance of incorporating higher-dimensional model elements consistent with observed physiology to further our understanding of binaural and insect hearing.

Data availability statements
The data that support the findings of this study are openly available at the following URL/DOI: https:// github.com/aestaples/Ormia3D.