Basilar membrane motion in a spiral-shaped cochlea.

To examine the influence of the spiral coiling of the cochlea upon the motion of the basilar membrane, a mathematical model of the cochlea is constructed. The formulation of the problem leads to Laplace's equation in three dimensions in a curvilinear coordinate system plus corresponding boundary conditions. By basing the choice of the coordinate system upon the form of the helix representing the centerline of the basilar membrane, a relatively simple formulation is obtained. The helix parameters appear only in the Laplacian, not in the boundary conditions. From experimental data the equations of the basilar membrane's centerline are derived for a human cochlea, both in intrinsic form and in regular form. The relative simplicity of the formation permits the tentative conclusion that, in spite of the large curvature near the apex, the spiral shape of the cochlea has only a small influence upon the motion of the basilar membrane.