Order reduction and efficient implementation of nonlinear nonlocal cochlear response models

Abstract

The cochlea is an indispensable preliminary processing stage in auditory perception that employs mechanical frequency-tuning and electrical transduction of incoming sound waves. Cochlear mechanical responses are shown to exhibit active nonlinear spatiotemporal response dynamics (e.g., otoacoustic emission). To model such phenomena, it is often necessary to incorporate cochlear fluid–membrane interactions. This results in both excessively high-order model formulations and computationally intensive solutions that limit their practical use in simulating the model and analyzing its response even for simple single-tone inputs. In order to address these limitations, the current work employs a control-theoretic framework to reformulate a nonlinear two-dimensional cochlear model into discrete state space models that are of considerably lower order (factor of 8) and are computationally much simpler (factor of 25). It is shown that the reformulated models enjoy sparse matrix structures which permit efficient numerical manipulations. Furthermore, the spatially discretized models are linearized and simplified using balanced transformation techniques to result in lower-order (nonlinear) realizations derived from the dominant Hankel singular values of the system dynamics. Accuracy and efficiency of the reduced-order reformulations are demonstrated under the response to two fixed tones, sweeping tones and, more generally, a brief speech signal. The corresponding responses are compared to those produced by the original model in both frequency and spatiotemporal domains. Although carried out on a specific instance of cochlear models, the introduced framework of control-theoretic model reduction could be applied to a wide class of models that address the micro- and macro-mechanical properties of the cochlea.

Appendices

Appendix 1: Efficiently computing $\hbox {A}_{\mathrm{R2}}$

Let

$$\begin{aligned} A_{B2} =\mu A_N \tau \end{aligned}$$

where:

$$\begin{aligned} \mu =T.\hbox {Mass}^{-1}\qquad \tau =A_{LA} .T^{-1} \end{aligned}$$

Let’s divide $\mu$, $\tau$ and $A_N$ into block matrices:

$$\begin{aligned} A_N= & {} \left[ {{\begin{array}{c@{\quad }c} A_{N1} &{} 0_{k,N-k} \\ 0_{N-k,k}&{} A_{N2} \\ \end{array} }} \right] ;\\ \mu= & {} \left[ {{\begin{array}{c@{\quad }c} \mu _{11} &{} \mu _{12}\\ \mu _{21} &{} \mu _{22} \\ \end{array} }} \right] ; \tau =\left[ {{\begin{array}{c@{\quad }c} \tau _{11} &{} \tau _{12} \\ \tau _{21} &{} \tau _{22} \\ \end{array}}} \right] \end{aligned}$$

where:

$$\begin{aligned} \begin{array}{cc} A_{N1} , \mu _{11} \,\, \hbox {and} \,\, \tau _{11} \epsilon \mathbb {R}^{k\cdot k} &{} \qquad A_{N2} , \mu _{22} \,\,\hbox {and} \,\,\tau _{22} \mathbb {R}^{k\cdot k} \\ \mu _{12} \,\, \hbox {and} \,\, \tau _{12} \epsilon \mathbb {R}^{k\cdot \left( {N-k} \right) } &{} \qquad \mu _{21} \,\, \hbox {and} \,\, \tau _{21} \epsilon \mathbb {R}^{\left( {N-k} \right) \cdot k} \end{array} \end{aligned}$$

Then:

$$\begin{aligned} A_{R2} =\mu _{11} \cdot A_{N1} \cdot \tau _{11} +\mu _{12} \cdot A_{N2} \cdot \tau _{21} \end{aligned}$$

To reduce the simulation time, the calculation of $A_{R2}$ should be done according to the value of k. As a matter of fact, there are 3 cases: $k<2N,\,2N<k<3N$ and $3N<k<4N$. The first case is shown here and the other cases follow similar reasoning.

$$\begin{aligned} A_{N1}= & {} 0_k \\ A_{{N2}}= & {} \left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 0_{{2N - k}} &{} 0 &{} 0 \\ 0 &{} \hbox {diag}\left( {\vec {\gamma }\left( t \right) } \right) &{} 0 \\ 0 &{} 0 &{} 0_{N} \\ \end{array}} \right] \end{aligned}$$

Hence,

$$\begin{aligned} A_{{R2}} = \mu _{{122}} \cdot \left( {\vec {\gamma } \otimes \tau _{{212}}} \right) \end{aligned}$$

where,

$$\begin{aligned} \mu _{122}= & {} \mu _{12} \left( {1:k,2N-k+1:3N-k} \right) \\ \tau _{212}= & {} \tau _{21} \left( {2N-k+1:3N-k,1:k} \right) \end{aligned}$$

Appendix 2: List of operators

• $\cdot *:$ Element by element matrix multiplication

• $\cdot /$: Element by element matrix division

• $\cdot 2$: Element by element matrix square

Let

$$\begin{aligned} v=\left[ \begin{array}{c} v_1 \\ v_2 \\ \vdots \\ v_{n} \\ \end{array}\right] \qquad \qquad A=\left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} a_{11}&{} a_{12} &{} \ldots &{} a_{1n} \\ a_{21} &{} a_{22} &{} \ldots &{} a_{2n} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ a_{n1} &{} a_{n2} &{} \ldots &{} a_{nn} \\ \end{array} \right] \end{aligned}$$

Then, the operators: $\otimes$ and d(.) are defined as follows:

$$\begin{aligned} v\otimes A= & {} \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} v_1 a_{11}&{} v_1 a_{12} &{} \ldots &{} v_1 a_{1n} \\ v_2 a_{21} &{} v_2 a_{22} &{} \ldots &{} v_2 a_{2n} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ v_n a_{n1} &{} v_n a_{n2} &{} \ldots &{} v_n a_{nn} \\ \end{array} \right] \\ d\left( v \right)= & {} \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} v_{1} &{} 0&{}\ldots &{} 0 \\ 0&{} v_{2} &{} \ldots &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0&{} 0 &{} \ldots &{} v_{n} \\ \end{array}\right] \\ d\left( A \right)= & {} \left[ \begin{array}{c} a_{11} \\ a_{22} \\ \vdots \\ a_{nn} \\ \end{array} \right] \end{aligned}$$

Appendix 3: Terminologies and numerical values

Table 3 gives the terminologies and the values of the parameters referred to throughout the paper. The values are based on LaMar et al. (2006) in cgs units.

Table 3 List of parameters, symbols and numerical values