We study a class of nonlinear nonlocal cochlear models of the transmission line type, describing the motion of basilar membrane (BM) in the cochlea. They are damped dispersive partial differential equations (PDEs) driven by time dependent boundary forcing due to the input sounds. The global well-posedness in time follows from energy estimates. Uniform bounds of solutions hold in the case of bounded nonlinear damping. When the input sounds are multi-frequency tones, and the nonlinearity in the PDEs is cubic, we construct smooth quasi-periodic solutions (multi-tone solutions) in the weakly nonlinear regime, where new frequencies are generated due to nonlinear interaction. When the input consists of two tones at frequencies f 1, f 2 (f 1< f 2), and high enough intensities, numerical results illustrate the formation of combination tones at 2f 1 f 2 and 2f 2 f 1, in agreement with hearing experiments. We visualize
A two-space dimensional active nonlinear nonlocal cochlear model is formulated in the time domain to capture nonlinear hearing effects such as compression, multi-tone suppression and difference tones. The micromechanics of the basilar membrane (BM) are incorporated to model active cochlear properties. An active gain parameter is constructed in the form of a nonlinear nonlocal functional of BM displacement. The model is discretized with a boundary integral method and numerically solved using an iterative second-order accurate finite difference scheme. A block matrix structure of the discrete system is exploited to simplify the numerics with no loss of accuracy. Model responses to multiple frequency stimuli are shown in agreement with hearing experiments. A nonlinear spectrum is computed from the model, and compared with FFT spectrum for noisy tonal inputs. The discretized model is efficient and accurate
Dispersive instability appears in time-domain solutions of classical cochlear models. In this letter, a derivation of optimal initial data is presented to minimize the effect of instability. A second-order accurate implicit boundary integral method is introduced. Numerical solutions of two-dimensional models show that the optimal initial conditions work successfully in time-domain steady-state computations for both the zero Neumann and zero Dirichlet fluid pressure boundary conditions at the helicotrema.
A nonlinear, nonlocal cochlear model of the transmission line type is studied in order to capture the multitone interactions and resulting tonal suppression effects. The model can serve as a module for voice signal processing, and is a one-dimensional (in space) damped dispersive nonlinear PDE based on the mechanics and phenomenology of hearing. It describes the motion of the basilar membrane (BM0 in the cochlea driven by input pressure waves. Both elastic dampling and selective longitudinal fluid damping are present. The forner is nonlinear and nonlocal in BM displacement, and plays a kep role in capturing tonal interactions. The latter is active only near the exit boundary (helicotrema), and is built in to damp out the remaining long waves. The initial boundary value problem is numerically solved with a semi-implicit second order finite difference method. Solutions reach a multi-frequency quai-steady state
Feedback modules, which appear ubiquitously in biological regulations, are often subject to disturbances from the input, leading to fluctuations in the output. Thus, the question becomes how a feedback system can produce a faithful response with a noisy input. We employed multiple time scale analysis, Fluctuation Dissipation Theorem, linear stability, and numerical simulations to investigate a module with one positive feedback loop driven by an external stimulus, and we obtained a critical quantity in noise attenuation, termed as signed activation time. We then studied the signed activation time for a system of two positive feedback loops, a system of one positive feedback loop and one negative feedback loop, and six other existing biological models consisting of multiple components along with positive and negative feedback loops. An inverse relationship is found between the noise amplification rate and the signed activation time, defined as the difference between the deactivation and activation time scales of the noise-free system, normalized by the frequency of noises presented in the input. Thus, the combination of fast activation and slow deactivation provides the best noise attenuation, and it can be attained in a single positive feedback loop system. An additional positive feedback loop often leads to a marked decrease in activation time, decrease or slight increase of deactivation time and allows larger kinetic rate variations for slow deactivation and fast activation. On the other hand, a negative feedback loop may increase the activation and deactivation times. The negative relationship between the noise amplification rate and the signed