Short waves in three-dimensional cochlea models: Solution for a ‘block’ model

Abstract In a previous paper the response of a three-dimensional cylindrical cochlea model was computed with a method that included a severe simplification. In the present paper the response of a different type of three-dimensional model is studied, a model that has the form of a rectangular block. The structure contains two fluid-filled channels, and the basilar membrane (BM) occupies only a fraction of the width of the separating partition. The only simplification introduced to make computation of the response feasible concerns the dynamics of the BM. It is assumed that the transmembrane pressure and BM velocity, both averaged over the width of the BM, are related according to the (given) mechanical impedance Z ( x ) of the BM. This assumption allows formulation of the problem in such a way that it can be solved with earlier developed methods. ‘Exact’ solutions are obtained for two approximations of the impedance function Z ( x ) that are valid in the region of resonance of the BM. The response of the three-dimensional model in this region is found to deviate considerably from that of a one- as well as a two-dimensional model. The response of the rectangular block model agrees excellently with that of the cylindrical model studied earlier, provided the BM occupies only a small fraction (smaller than approx. 0.1) of the width of the partition. This result indicates that the cylindrical model satisfactorily represents details of physical events in the vicinity of a narrow ribbon-like BM despite the simplification introduced. The results confirm also that a three-dimensional model can explain details of physical measurements on the cochlear response that are associated with BM resonance. The differences between the responses of the three types of models give rise to discussion on several points. First, it is established once more that in the resonance region short waves (the wavelength not being large with respect to the cross-section) dominate the response. Second, it is illustrated in which respects a three-dimensional and a two-dimensional model are fundamentally different and to which degree the latter model can simulate the former. Third, it is discussed how the approximation used to represent Z ( x ) in the resonance region, can be improved upon. Finally, the scope of models of the type considered here appears to be more or less exhausted. Further developments in cochlear mechanics should be sought in the realm of more complicated models.