A cylindrical cochlea model: the bridge between two and three dimensions

In this paper the response is studied of a linear two-channel of the cochlea with a circular cross-section. In conformity with anatomical data the basilar membrane occupies only a small fraction of the width of the cochlear partition separating the channels. Fluid displaced by the membrane will move mainly in the longitudinal (x) direction, the remainder is assumed to occur only in the radial (r) direction. In this way the major effect of three-dimensional fluid movement is built in, yet the mathematical treatment can proceed as in two-dimensional theory. The problem can be solved by analytical means when the membrane is taken as completely described by the mechanical impedance z(x) and the functional form of z(x) is simplified to a sufficient degree. Near the locus of resonance z(x) is nearly a linear function of x. For this case - the straight-line approximation of z(x) - the solution of the problem is straightforward. The response of this three-dimensional model is found to have characteristics about midway between those of one- and two-dimensional models. The same general properties are found when a better approximation - the hyperbolic approximation - of z(x) is introduced. In the resonance region the computed response for this model agrees excellently with experimental data, much better than for a two-dimensional model. A noteworthy feature of the three-dimensional response is that in the resonance region the response is well elevated above its course in the remainder of the cochlea (where wave propagation can be regarded as one-dimensional model). In this respect the three-dimensional response differs fundamentally from that of the two-dimensional model. This property is also found in certain experimental data, and from the present results this feature can be interpreted as specific for a three-dimensional structure. Further work should be directed at other three-dimensional structures in which the basilar membrane also occupies a small fraction of the width of the cochlear partition, and at more elaborate types of model.