Influence of discretization error on instability of cochlear model

Abstract

The instability derived from the cochlear model is a phenomenon used to characterize its active processing, which is one of the fundamentals of hearing. This processing relates to the activities of outer hair cells (OHCs) lining in the cochlea. The decline of this processing function is accounted for by the spatially irregular activities of OHCs. Current cochlear models show excessive instability when cochlea containing the irregular activity of the OHCs is simulated. To explain this phenomenon, there are two possibilities. First, excessive instability may be essential to the model. Second, discretized errors may affect the instability of the model in numerical calculation. To investigate these possibilities, a state space model of two-dimensional cochlear mechanics is developed and applied to a high-accuracy scheme of numerical calculation. As a result of this simulation, the basilar membrane responses include the discretization error, which depends on the modeling accuracy. Furthermore, the transfer function and the temporal response show more instability with higher modeling accuracy, with the irregularity of the feedback gain corresponding to the activities of the OHCs. Therefore, when the discretization error is large, the instability is weakened. Consequently, this paper concludes that excessive instability is an essential feature of the cochlear model.

Efficiency limit of nonuniform grid setting in two-dimensional cochlear model

“…Nonuniform Grid In this section, we apply the finite difference method with a nonuniform grid [14] to the 2D cochlear model represented by the state space formulation [13]. To discretize the 2D cochlear model, we introduce the central difference operator Á 0 , which acts on individual elements of a sequence z:…”

“…Equation 7is divided into N and M segments with varying step widths, Áx n and Áy m , respectively. A procedure for constructing a state space model of 2D cochlear mechanics using a finite difference scheme with a uniform grid has been proposed [13]. In this study, we develop a state space model of 2D cochlear mechanics with a nonuniform grid [14].…”

“…(11) of Ref. [13]. Therefore, the expansion of the following expressions is equivalent to their method [13], and the following state space equations are derived:…”

“…For this problem, we introduce a statespace formulation that is commonly used for stability analysis of models. A 2D cochlear model represented by a state space formulation has shown that errors in the discretized process have no influence on the stability of the model if the degree of discretization is sufficient [13]. Thus, the problem here is determining the sufficient degree of discretization.…”

“…The structure of this paper is as follows: in Sect. 2, we introduce a 2D cochlear model, apply a nonuniform grid [14] to the state space represented model [13], and investigate the correct step width for the model. In Sect.…”