The "inverse problem" solved for a three-dimensional model of the cochlea. I. Analysis.

With nonactive cochlear models of the "classical" type, it is impossible to stimulate the characteristic type of frequency selectivity that is revealed by modern mechanical measurements of the motion of the basilar membrane (BM). Locally active models of the cochlea have been proposed to alleviate this problem but it remains uncertain whether the real cochlea is active in this sense. The present study was undertaken to investigate this subject in a more general and systematic way than has hitherto been done. The "inverse" problem is solved for a three-dimensional (3-D) model and a procedure is developed for recovering the BM impedance needed to stimulate the given BM response function. In the present paper the theoretical basis of the procedure is presented, and an analysis is given of the validity of the method and the errors involved. It is shown why the inverse problem is "ill-posed" and why the results of our procedure are more accurate in the region of the response peak than in the more basal region of the model. The latter finding serves to delimit the domain within which results of an inverse procedure (for instance, in the short- or long-wave approximation) are to be judged. In a subsequent paper the technique developed will be applied to actual data sets from the literature on mechanical BM measurements.