Increasing Computational Efficiency of Cochlear Models Using Boundary Layers

Abstract

Our goal is to develop methods to improve the efficiency of computational models of the cochlea for applications that require the solution accurately only within a basal region of interest, specifically by decreasing the number of spatial sections needed for simulation of the problem with good accuracy. We design algebraic spatial and parametric transformations to computational models of the cochlea. These transformations are applied after the basal region of interest and allow for spatial preservation, driven by the natural characteristics of approximate spatial causality of cochlear models. The project is of foundational nature and hence the goal is to design, characterize and develop an understanding and framework rather than optimization and globalization. Our scope is as follows: designing the transformations; understanding the mechanisms by which computational load is decreased for each transformation; development of performance criteria; characterization of the results of applying each transformation to a specific physical model and discretization and solution schemes. In this manuscript, we introduce one of the proposed methods (complex spatial transformation) for a case study physical model that is a linear, passive, transmission line model in which the various abstraction layers (electric parameters, filter parameters, wave parameters) are clearer than other models. This is conducted in the frequency domain for multiple frequencies using a second order finite difference scheme for discretization and direct elimination for solving the discrete system of equations. The performance is evaluated using two developed simulative criteria for each of the transformations. In conclusion, the developed methods serve to increase efficiency of a computational traveling wave cochlear model when spatial preservation can hold, while maintaining good correspondence with the solution of interest and good accuracy, for applications in which the interest is in the solution to a model in the basal region.

FIGURE 1. Schematic representation of two transformation mechanisms

Top: Mechanism A (more basal absorption). Bottom: Mechanism B (reduced spatial gradients). Solid line: no transformation; Dashed line: with transformation. P is the differential pressure across the basilar membrane

FIGURE 2. Complex spatial transformation

Left top: Wavenumber as a function of the simulated variable before transformation. Left bottom: Wavenumber as a function of the simulated variable after transformation. Right top: spatial extension into the complex domain occurs after the beginning of the boundary layer (at 0.5 cm) as indicated by the dashed vertical line. Right middle: |P(u)| for original and transformed models. Right bottom: Relative magnitude of difference vector between original and transformed model solutions. The input frequency was 2.4 kHz; the step size was 2.9 ·10⁻³ cm.