Internally coupled ears: mathematical structures and mechanisms underlying ICE

Abstract

In internally coupled ears (ICE), the displacement of one eardrum creates pressure waves that propagate through air-filled passages in the skull, causing a displacement of the opposing eardrum and vice versa. In this review, a thorough mathematical analysis of the membranes, passages, and propagating pressure waves reveals how internally coupled ears generate unique amplitude and temporal cues for sound localization. The magnitudes of both of these cues are directionally dependent. On the basis of the geometry of the interaural cavity and the elastic properties of the two eardrums confining it at both ends, the present paper reviews the mathematical theory underlying hearing through ICE and derives analytical expressions for eardrum vibrations as well as the pressures inside the internal passages, which ultimately lead to the emergence of highly directional hearing cues. The derived expressions enable one to explicitly see the influence of different parts of the system, e.g., the interaural cavity and the eardrum, on the internal coupling, and the frequency dependence of the coupling. The tympanic fundamental frequency segregates a low-frequency regime with constant time-difference magnification (time dilation factor) from a high-frequency domain with considerable amplitude magnification. By exploiting the physical properties of the coupling, we describe a concrete method to numerically estimate the eardrum's fundamental frequency and damping solely through measurements taken from a live animal.

Geometric perturbation theory and Acoustic Boundary Condition Dynamics

“…Together with the pressure-difference driven membranes, the ICE model can be formulated in terms of three wave equations with homogeneous initial and boundary conditions for the acoustic pressure p and the membrane displacements u 0/L . That is, equations (40) and (41). Applying the technique of Picard iterations and setting p (0) = 0 in agreement with the perturbative argument before, the three partial differential equations decouple.…”

“…Further domain questions will be discussed at the beginning of section 2.3. The full ICE equations [40,41] read in the present notation…”

Animals and ICE: meaning, origin, and diversity

“…The directionality of d shows the response of the left tympanum and is in general strongly frequency-dependent. The diagram shows the most directional response; see also Vedurmudi et al (2016b, Fig. 15).…”

“…Based on the 2- and 3-dimensional wave equation that describe waves in the 3-dimensional air-filled cavity and with damping, which is important, in the 2-dimensional eardrums, Vossen et al (2010) and Vedurmudi et al (2016a) were the first to present a complete mathematical theory, the former focusing on lizards and the latter encompassing the characteristics of all terrestrial vertebrates with ICE. It turns out that the 3-dimensional geometry plays a remarkable, unforeseen, role; for details including the key role played by the tympanic fundamental frequency, we refer to Vedurmudi et al (2016b).…”

“…These coefficients are called transfer functions. Their mathematical derivation can be found in Vedurmudi et al (2016b, Section 3) where, for the sake of mathematical convenience, u IL is rather expressed in the form u IL (ω) = ½ [H IL (ω)+H CL (ω)] (p IL +p CL ) + ½ [H IL (ω)−H CL (ω)] (p IL −p CL ).…”