Back
 OJBIPHY  Vol.9 No.1 , January 2019
Effective-Spring Model of Tympanic Response in Archosaurs
Abstract: Whereas for smaller animals the eardrums are well-characterized as excitable membranes or drums, some animals such as several archosaurs feature, as a first approximation, a rather stiff elastic shell supported by an elastic ring. Mathematically, the theory of plates and shells is applicable but its governing equations overly complicate the modeling. Here the notion of tympanic structure is introduced as a generalization of “ordinary” tympanic membranes so as to account for sound perception as it occurs in archosaurs, such as birds and crocodilians. A mathematical model for the tympanic structure in many archosaurs called two-spring model implements this notion. The model is exactly soluble and solutions are presented in closed form and as a series expansion. Special emphasis is put onto offering an easy-to-apply model for describing experiments and performing numerical studies. The analytic treatment is supplemented by a discussion of the applicability of the two-spring model in auditory research. An elasticity-theoretic perspective of the two-spring model is given in the Appendix.
Cite this paper: Heider, D. and van Hemmen, J. (2019) Effective-Spring Model of Tympanic Response in Archosaurs. Open Journal of Biophysics, 9, 21-50. doi: 10.4236/ojbiphy.2019.91003.
References

[1]   van Hemmen, J.L., Christensen-Dalsgaard, J., Carr, C.E. and Narins, P. (2016) Animals and ICE: Meaning, Origin and Diversity. Biological Cybernetics, 110, 237-246.
https://doi.org/10.1007/s00422-016-0702-x

[2]   Schnupp, J. and Carr, C.E. (2009) One Hearing with More than One Ear: Lessons from Evolution. Nature Neuroscience, 12, 692-697.
https://doi.org/10.1038/nn.2325

[3]   Vedumurdi, A.P. (2018) General Aspects of Sound Localization through Intenally Coupled Ears. PhD Thesis, Technische Universität München, München.

[4]   Vossen, C. (2010) Auditory Information Processing in Systems with Internally Coupled Ears. PhD thesis, Technische Universität München, München.

[5]   Vedurmudi, A.P., Goulet, J., Christensen-Dalsgaard, J., Young, B.A., Williams, R. and van Hemmen, J.L. (2016) How Internally Coupled Ears Generate Temporal and Amplitude Cues for Sound Localization. Physical Review Letters, 116, 028101.
https://doi.org/10.1103/PhysRevLett.116.028101

[6]   C. Vossen, J. Christensen-Dalsgaard, and J. L van Hemmen. Analytical model of internally coupled ears. Journal of the Acoustical Society of America, 128(2):909-918, 2010.
https://doi.org/10.1121/1.3455853

[7]   Howe, M.S. (2007) Hydrodynamics and Sound. Cambridge University Press, Cambridge, UK.

[8]   Howe, M.S. (2014) Acoustics and Aerodynamic Sound. Cambridge University Press, Cambridge, UK.

[9]   Autrum, H. (1940) über Lautäusserungen und Schallwahrnehmungen bei Anthropoden II. Das Richtungshören von Locusta und Versuch einer Hörtheorie für Tympanalorgane vom Locustidentyp. Zeitschrift für Vergleichende Physiologie, 28, 326-352.

[10]   Autrum, H. (1942) Schallempfang bei Tier und Mensch. Naturwissenschaften, 30, 69-85.
https://doi.org/10.1007/BF01475622

[11]   Vedurmudi, A.P., Christensen-Dalsgaard, J. and van Hemmen, J.L. (2018) Modeling Underwater Hearing and Sound Localization in the Aquatic Frog Xenopus laevis. Journal of the Acoustical Society of America, Submitted.

[12]   Vedurmudi, A.P., Young, B.A. and van Hemmen, J.L. (2016) Internally Coupled Ears: Mathematical Structures and Mechanisms Underlying ICE. Biological Cybernetics, 110, 237-246.
https://doi.org/10.1007/s00422-016-0696-4

[13]   Fletcher, N.H. (1992) Acoustic Systems in Biology. Oxford University Press, Oxford, UK.

[14]   Saunders, J.C., Duncan, R.K., Doan, D.E. and Werner, Y.L. (2000) The Middle Ear of Reptiles and Birds. In: Dooling, R.J., Fay, R.R. and Popper, A.N., Eds., Comparative Hearing: Birds and Reptiles, Springer, New York, Chapter 2.
https://doi.org/10.1007/978-1-4612-1182-2_2

[15]   Timoshenko, S.P. and Goodier, J. (1969) Theory of Elasticity. 3rd Edition, McGraw-Hill, New York.

[16]   Timoshenko, S.P. and Woinowsky-Krieger, S. (1959) Theory of Plates and Shells. 2nd Edition, McGraw-Hill, New York.

[17]   Van Hemmen, J.L. and Leibold, C. (2007) Elementary Excitations of Biomembranes: Differential Geometry of Undulations in Elastic Surfaces. Physics Reports, 444, 51-99.
https://doi.org/10.1016/j.physrep.2006.12.007

[18]   Hassani, S. (2013) Mathematical Physics: An Introduction to Its Foundations. 2nd Edition, Springer, Berlin.
https://doi.org/10.1007/978-3-319-01195-0

[19]   Van der Waerden, B.L. (1993) Algebra I. 9th Edition, Springer, Berlin.
https://doi.org/10.1007/978-3-642-85527-6

[20]   Van der Waerden, B.L. (1993) Algebra II. 6th Edition, Springer, Berlin.
https://doi.org/10.1007/978-3-642-58038-3

[21]   Colbert, E. (1946) The Eustachian Tubes in the Crocodilia. Copeia, 12, 12-14.
https://doi.org/10.2307/1438813

[22]   Spencer, P. and Moon, D. (1971) Field Theory Handbook: Including Coordinate Systems, Differential Equations and their Solutions. Springer, Berlin.

[23]   Lautrup, B. (2004) Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World. IoP Publishing, Bristol.

[24]   Skudrzyk, E. (1971) Foundations of Acoustics: Basic Mathematics and Basic Acoustics. Springer, Berlin.
https://doi.org/10.1007/978-3-7091-8255-0

[25]   Falloon, P., Abbott, P. and Wang, J. (2002) Theory and Computation of the Spheroidal Wave Functions.
https://arxiv.org/ftp/math-ph/papers/0212/0212051.pdf

[26]   Meixner, J. and Schäfke, R. (1954) Mathieusche Funktionen und Sphäroidfunktionen. Springer, Berlin.
https://doi.org/10.1007/978-3-662-00941-3

[27]   Meixner, J., Schäfke, R. and Wolf, G. (1980) Mathieu Functions and Spheroidal Functions and Their Mathematical Foundations (Further Studies). Lecture Notes in Mathematics 837. Springer, Berlin.
https://doi.org/10.1007/BFb0096194

[28]   Zhao, L. (2017) Spherical and Spheroidal Harmonics: Examples and Computations. Master’s Thesis, Ohio State University, Columbus.

 
 
Top