On the Analytic Origin of Two Species of Cochlear Eigenmodes

Abstract

After entering the ear, sound waves propagate as surface waves along the cochlea's basilar membrane. In recent work, we showed numerically that the system supports two types of modes: localized resonant modes, which underpin the modern understanding of cochlear mechanics, and a novel class of spatially extended modes. Here, we develop an analytic framework that explains the emergence of this mode structure. We show that extended modes arise from globally continuous standing-wave solutions, whereas localized modes result from internal resonance requiring matching across a singular point. These results clarify the generic structure of cochlear wave equations.

Figures (5)

• Figure 1: A linearly decreasing stiffness is qualitatively similar to an exponential BM. (A) The eigenvalue structure of cochlear modes elliott2007state, with an exponentially decreasing BM stiffness. An eigenvalue's imaginary part determines the oscillation frequency, and the real part determines stability. We define localized modes (red) as those that have a resonant position within the cochlea. Extended modes (blue) are those with frequencies lower than any resonant on the BM. Analogous to MyPaper Fig. 2 without active processes. (B) Eigenvectors corresponding to the circled eigenvalues in A. (C) The eigenvalue structure of cochlear modes, with a linearly decreasing BM stiffness. Note that the eigenvalue structure is qualitatively the same, though localized modes are now linearly spread instead of exponentially. The impedance used is from Eq. \ref{['lin_imped']} with $x_0=1$ (D) Eigenvectors corresponding to the circled eigenvalues in C.
• Figure 2: Analytic Solution for Extended Modes. (A) Eigenvalue structure for a cochlea with linearly decreasing BM stiffness. Yellow stars denote eigenvalues predicted by the analytic solution (Eqs. \ref{['solution']}-\ref{['RHS']}), showing excellent agreement with the matrix method. Though the numerical solution is identical to Fig. \ref{['fig:exp/lin']} the y-axis is truncated to highlight the extended mode structure. (B) Eigenvector corresponding to the marked eigenvalue in (A), demonstrating close agreement between analytic (solid line) and numerical (dashed lines) results.
• Figure 3: Analytic Solution for Localized Modes (A) ) Localized eigenvalues determined by the resonance condition $Z(x,\lambda)=0$, (yellow line), showing agreement with the numerically obtained localized modes.. (B) Eigenfunction for a mode resonant at $x=0.75$. The analytic solution obtained by matching the left and right solutions across the resonant point reproduces the spatial structure of the localized mode. The solution is truncated shortly beyond resonance where the amplitude rapidly vanishes.
• Figure 4: Extended and Localized Modes in the WKB Approximation. (A) The eigenvalue structure of cochlear modes, with a exponentially decreasing BM stiffness. Note that the extended modes predicted by the WKB approximation are qualitatively similar to those obtained numerically. (B) Eigenvector corresponding to the eigenvalues in A with a matching color. Both localized modes are indicated with a black circle.
• Figure 5: Discontinuous Additional Extended Modes. (A) This has the same data as Fig. \ref{['fig:Extended']} with a large range of x values so we see some additional extended modes predicted left of the red line. (B) A demonstration that these additional solutions are invalid as they have discontinuous eigenfunctions.