Delay differential equations in a nonlinear cochlear model.

The human auditory system performs its primary function in the cochlea, the main organ of the inner ear, where the spectral analysis of a sound signal and its transduction into a neural signal occur. It is filled with liquid and divided in two cavities by the basilar membrane (BM). A sound stimulus propagates in air as an acoustic pressure wave through the outer and the middle ear. The pressure of the stapes on the oval window (boundary between the middle and the inner ear) causes the cochlear fluid to flow between the two cavities through a hole at the end of the BM. A spatial partial differential equation of fluid-dynamics describes this physical process. As a consequence of the differential pressure between the two cavities, each micro-element of the BM oscillates as a forced damped harmonic oscillator. The BM displacement is amplified by the overlying outer hair cells (OHCs) through a nonlinear nonlocal active feedback mechanism. The latter can be modeled by means of various representations. Among them, the delayed stiffness model of Talmadge et al. (J. Acoust. Soc. Am. 104, 1998) has been considered in this thesis. Specifically, the cochlear nonlinearity is introduced as a quadratic function of the BM displacement in the passive linear damping function. Moreover, the active mechanism is described by two additional forces, each one proportional to the BM displacement delayed by a slow and a fast feedback constant time, respectively. According to this model, a time delay differential equation (DDE) of the second order describes the oscillating dynamics of the BM. A different formulation of the nonlinear active mechanism, driven by the OHCs, is expressed as a nonlinear function of the BM velocity by the anti-damping model of Moleti et al. (J. Acoust. Soc. Am. 133, 2013). In this case the model equations do not contain time delays. The numerical integration of the above mentioned models has been obtained by finite differencing with respect to the space variable in the state space, as introduced by Elliott et al. (J. Acoust. Soc. Am. 122, 2007), and then integrating in time with the adaptive package introduced by Bertaccini and Sisto as a modification of the popular Matlab ode15s package (J. Comput. Phys. 230, 2011). The semidiscrete formulation of the delayed stiffness model and the anti-damping model has a non trivial mass matrix, and eigenvalues of the system matrix with large negative real part and imaginary part. That is why an implicit solver with an infinite region of absolute stability should be used. Therefore, the customized Matlab ode15s package by Bertaccini and Sisto seems to be the convenient choice to integrate the problem at hand numerically. In particular, for the delayed stiffness model, an integrator for constant DDEs (the method of steps; Bellen and Zennaro, Oxford University Press 2003) has been formulated and based on the customized ode15s. All these topics have been discussed in this doctoral thesis, which is subdivided in the following chapters. Chapter 1 describes the anatomy of the human ear, with special regard to the cochlea. Some experimental evidences about the cochlear mechanisms are discussed, in order to support the cochlear modeling. Two physical models with one degree of freedom are shown: the anti-damping model of Sisto et al. (J. Acoust. Soc. Am. 128, 2010) and Moleti et al. (J. Acoust. Soc. Am. 133, 2013), and the delayed stiffness model of Talmadge et al. (J. Acoust. Soc. Am. 104, 1998). Chapter 2 discusses the general theory of DDEs, with greater reference to constant and time dependent DDEs from Bellen and Zennaro (Oxford University Press 2003). Existence and uniqueness of time dependent DDEs are briefly analyzed, while the method of steps is shown as a basic approach to find a numerical approximation of the DDEs solution. According to this method, IVPs of constant DDEs (as for the semidiscrete delayed stiffness model) are turned into IVPs of ODEs in a subinterval (of length less than or equal to the time delay) of the whole integration interval. Each IVP of ODEs can be integrated by means of any ODEs numerical method, and its convergence is then discussed. Chapter 3 describes the main tools used to find an approximate solution of the considered models. In particular, the discretization for spatial partial derivatives by means of finite differences is shown. Such a representation turns a model, which is continuous in the space-time domain, into a semidiscrete model to be integrated in time. The models considered in this thesis are stiff, so the phenomenon of stiffness is discussed and the ode15s package of Matlab for integrating stiff ODEs is described. Nevertheless, greater benefits can be obtained by using the ode15s package customized by Bertaccini and Sisto as a hybrid direct-iterative solver which exploits Krylov subspace methods. Chapter 4 shows the semidiscrete formulation of the continuous models (anti-damping model and delayed stiffness model) in the state space with respect to the spatial variable, as introduced by Elliott et al. (J. Acoust. Soc. Am. 122, 2007). The algebraic properties of the semidiscrete models are discussed in order to show why the customized ode15s package may perform a faster numerical integration of the semidiscrete models and how this solver can be used in an integration numerical technique for constant DDEs (the method of steps). Chapter 5 shows the results produced by the numerical experiments of the delayed stiffness model by supplying a sinusoidal tone, and compares them with the numerical results produced by the anti-damping model. Some considerations about the numerical approach of the time integration are also discussed, and a part of the simplified code used for integrating the semidiscrete delayed stiffness model, is reported. The results are comparable with those obtained by the anti-damping model, and then the numerical experimental evidences seem to justify the proposed integration technique for constant DDEs. Delayed model properties of tonotopicity, anti-damping and nonlinearity are verified, as well as the dependence of the approximate solution on some free parameters of the model. The cochlear response described by the delayed stiffness model shows a typical tall and broad BM activity pattern. This behavior is also found in the numerical results of a model with two degree of freedom produced by Neely and Kim (J. Acoust. Soc. Am. 79, 1986) and Elliott et al. (J. Acoust. Soc. Am. 122, 2007).