Biological Cybernetics

, Volume 110, Issue 4–5, pp 359–382 | Cite as

Internally coupled ears: mathematical structures and mechanisms underlying ICE

  • Anupam P. Vedurmudi
  • Bruce A. Young
  • J. Leo van HemmenEmail author


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.


Internally coupled ears Bioacoustics Acoustic coupling Sound localization Tympanic vibrations Fundamental frequency 



The authors gratefully acknowlege André Longtin’s (uOttawa) constructive criticism. They also thank David Heider for an enjoyable collaboration on the mathematical foundations of the perturbation theory used here.


  1. Asmar NH (2005) Partial differential equations with Fourier series and boundary value problems. Pearson Education, New YorkGoogle Scholar
  2. Autrum H (1942) Schallempfang bei Tier und Mensch. Naturwissenschaften 30:69–85CrossRefGoogle Scholar
  3. Christensen-Dalsgaard J (2005) Directional hearing in non-mammalian tetrapods. In: Popper AN, Fay RR (eds) Sound source localization. Springer handbook in auditory research. Springer, New York, pp 67–123Google Scholar
  4. Christensen-Dalsgaard J, Manley GA (2005) Evolution of a sensory novelty: tympanic ears and the associated neural processing. J Exp Biol 208:1209–1217CrossRefPubMedGoogle Scholar
  5. Christensen-Dalsgaard J, Manley GA (2008) Acoustical coupling of lizard eardrums. J. Assoc Res Otolaryngol 9:407–416CrossRefPubMedPubMedCentralGoogle Scholar
  6. Christensen-Dalsgaard J, Carr CE (2009) Evolution of a sensory novelty: tympanic ears and the associated neural processing. Brain Res Bull 75:365–370CrossRefGoogle Scholar
  7. Christensen-Dalsgaard J, Tang Y, Carr CE (2011) Binaural processing by the Gecko auditory periphery. J. Neurophysiol 105:1992–2004CrossRefPubMedGoogle Scholar
  8. Copson ET (1973) Introduction to the theory of functions of a complex variable. Clarendon Press, OxfordGoogle Scholar
  9. Fletcher NH (1992) Acoustic systems in biology. Oxford University Press, OxfordGoogle Scholar
  10. Heider D, Vedurmudi AP, van Hemmen JL (2016) Acoustic boundary condition dynamics and internally coupled ears (in preparation). This paper gives a mathematical justification of the perturbative approach presented here. See also: van Hemmen JL (2016) Acoustic boundary-condition dynamics and internally coupled ears. SIAM News 49/8:1–3Google Scholar
  11. Jørgensen MB (1993) Vibrational patterns of the anuran eardrum. In: Elsner N, Heisenberg M (eds) Gene–brain–behaviour. Proceedings of the 21st Göttingen neurobiology conference. Georg Thieme Verlag, Stuttgart, p 231Google Scholar
  12. Jørgensen MB, Kanneworff M (1998) Middle ear transmission in the grass frog Rana temporaria. J Comp Physiol A 182:59–64PubMedGoogle Scholar
  13. Jørgensen MB, Schmitz B, Christensen-Dalsgaard J (1991) Biophysics of directional hearing in the frog Eleutherodactylus coqui. J Comp Physiol A 168:223–232CrossRefGoogle Scholar
  14. Koeppl C, Carr CE (2008) Maps of interaural time difference in the chicken’s brainstem nucleus laminaris. Biol Cybern 98:541–559CrossRefGoogle Scholar
  15. Manley GA (1972a) The middle ear of the Tokay gecko. J Comp Physiol 81:239–250CrossRefGoogle Scholar
  16. Manley GA (1972b) Frequency response of the middle ear of Geckos. J Comp Physiol 81:251–258CrossRefGoogle Scholar
  17. Manley GA (2000) Cochlear mechanisms from a phylogenetic viewpoint. Proc Natl Acad Sci USA 97:11736–11743CrossRefPubMedPubMedCentralGoogle Scholar
  18. Michelsen A, Larsen ON (2008) Pressure difference receiving ears. Bioinspir Biomim 3:011001CrossRefPubMedGoogle Scholar
  19. Pozrikidis C (2009) Fluid dynamics: theory, computation, and numerical simulation. Springer, New YorkCrossRefGoogle Scholar
  20. Rajalingham C, Bhat RB (1998) Vibration of circular membrane backed by cylindrical cavity. Int J Mech Sci 40:723–734CrossRefGoogle Scholar
  21. Rschevkin SN (1963) A course of lectures on the theory of sound. Pergamon, Oxford, p 111Google Scholar
  22. Schnupp JW, Carr CE (2009) On hearing with more than one ear: lessons from evolution. Nat Neurosci 12:692–697CrossRefPubMedPubMedCentralGoogle Scholar
  23. Stoer J, Bulirsch R (2002) Introduction to numerical analysis, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
  24. Szpir MR, Sento S, Ryugo DK (1990) Central projections of cochlear nerve fibers in the alligator lizard. J Comp Neurol 295:530–547CrossRefPubMedGoogle Scholar
  25. Temkin S (1981) Elements of acoustics. Wiley, New YorkGoogle Scholar
  26. van Hemmen JL, Christensen-Dalsgaard J, Carr CE, Narins P (2016) Animals & ICE: meaning, origin, and diversity. Biol Cybern 110(4–5). doi: 10.1007/s00422-016-0702-x
  27. Vedurmudi AP, Goulet J, Christensen-Dalsgaard J, Young BA, Williams R, van Hemmen JL (2016) How internally coupled ears generate temporal and amplitude cues for sound localization. Phys Rev Lett 116:028101CrossRefPubMedGoogle Scholar
  28. Vossen C (2010) Auditory information processing in systems with internally coupled ears. Doctoral dissertation, Technische Universität MünchenGoogle Scholar
  29. Vossen C, Christensen-Dalsgaard J, van Hemmen JL (2010) An analytical model of internally coupled ears. J Acoust Soc Am 128:909–918CrossRefPubMedGoogle Scholar
  30. Young BA, Vedurmudi AP, van Hemmen JL (2016) Auditory balance (in preparation)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Physik Department T35 and BCCN–MunichTechnische Universität MünchenGarching bei MünchenGermany
  2. 2.Kirksville College of Osteopathic MedicineA.T. Still UniversityKirksvilleUSA

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