Close-to-critical Hopf oscillators boost cochlear gain in realistic models with fluid focusing

Critical Hopf oscillators have been proposed for modeling the cochlear micromechanics, and the generation of spontaneous otoacoustic emissions. In such top-down models, the cochlear micromechanical elements are abstractly represented through equations written in normal form. There is a gap to be filled between this abstract formulation and the complexity of the motion of the Organ of Corti as revealed by recent optical coherent tomography measurements. The aim of the present work is bridging this gap using a bottom-up approach: starting from realistic equations schematizing the actual physical elements of cochlear micromechanics, we show explicitly how these equations admit a normal form representation. To do that, the Organ of Corti is schematized as a system of two oscillators coupled by an internal active low-passed nonlinear force, and to an incompressible 2D fluid. We generalize the classical two-equation normal form to a three-equation system, demonstrating that the normal form approach may be used to compute the relative motion of the main elements of the Organ of Corti, while the center of mass motion depends on the fluid forces only. This way, the parameters of the critical oscillator are no longer abstract mathematical quantities, but acquire a direct physical interpretation in terms of the dynamics of the underlying physical elements. In particular, the approach to the critical behavior, and the consequent dramatic variation of the cochlear response gain and bandwidth, are quantified in terms of a few dimensionless parameters of obvious physiological meaning.