Signal processing in the cochlea: the structure equations

The main result is a linear system of partial differential equations (referred to as the structure equations) that describe the result of signal processing in the cochlea. It is formulated for phase and for the logarithm of the amplitude. The changes of these quantities are the essential physiological observables in the description of signal processing in the auditory pathway.The structure equations display in a quantitative way the subtle balance for processing information on the basis of phase versus amplitude. From a mathematical point of view, the linear system of equations is classified as an inhomogeneous ？ ˉ -equation. In suitable variables the solutions can be represented as the superposition of a particular solution (determined by the system) and a holomorphic function (determined by the incoming signal). In this way, a global picture of signal processing in the cochlea emerges.At the outset of this work is the quest to understand signal processing in the cochlea.It has been known since 1992 that cochlear signal processing can be described by a wavelet transform (Daubechies 1992 [1], Yang, Wang and Shamma, 1992 [2]). There are two basic principles that lie at the core of this description: Linearity and scaling.In the cochlea, an incoming acoustical signal f ( t ) in the form of a pressure fluctuation (t is the time variable) induces a movement u ( x , t ) of the basilar membrane at position x along the cochlea. At a fixed level of sound intensity, the relation between incoming signal and movement of the basilar membrane is surprisingly linear. However as a whole this process is highly compressive with respect to levels of sound - and thus cannot be linear.In the present setting this is taken care of by a ‘quasilinear model’. This is a model that depends on parameters, for example, in the present situation the level of sound intensity. For fixed parameters the model is linear. It is interpreted as a linear approximation to the process at these fixed para