Flexible solver for 1-D cochlear partition simulations

Transcription

1 Psychological and Physiological Acoustics (others): Paper ICA Flexible solver for 1-D cochlear partition simulations Pablo E. Riera (a), Manuel C. Eguía (b) (a) Laboratorio de Dinámica Sensomotora, Departamento de Ciencia y Tecnología, CONICET, Universidad Nacional de Quilmes, B1876BXD, Bernal, Argentina, pablo.riera@unq.edu.ar (b) Laboratorio de Acústica y Percepción Sonora, Escuela Universitaria de Artes, CONICET, Universidad Nacional de Quilmes, B1876BXD, Bernal, Argentina, meguia@unq.edu.ar Abstract There is a vast literature on cochlear modelling, much of it based on theoretical and numerical analysis of the hydromechanics of the canals and the physiology and micromechanics of the organ of Corti. During the past decades, many models have been developed from common theoretical grounds but with differences in the cochlear partition impedance, mainly because of the active mechanism adopted. Here we present a module for the Python language that allows to simulate and compare many different models in a simple manner, with the only need of writing the partition impedance expression. The outcome is a highly optimized C++ code that carries on the numerical simulation. The module can simulate models that fit in the long wave approximation of the cochlear fluid mechanics or, equivalently, a one dimensional transmission line. It allows an arbitrary number of variables for the partition impedance, including longitudinal coupling. Keywords: cochlear mechanics, nonlinear, numerical simulation

2 Flexible Solver For 1-D Cochlear Partition Simulations 1 Introduction The cochlea is the main organ responsible of transducing sound into the electrical impulses conveyed by the auditory nerve to the brain. It is a complex structure with many mechanical, chemical and electrical mechanisms interweaved, that are not yet completely understood. It is generally agreed that the cochlea presents two distinctive phenomena: a passive behavior, namely the propagation of sound as a traveling wave through the spatial extension of the cochlea, and an active amplification mechanism. The former is responsible for segregating frequencies: the wave travels from high to low frequencies and two different frequencies resonate at different locations. The later gives us the ability of hearing faint sounds, it contributes only at low intensities and is overridden by the passive mechanism at high intensities. Passive and active mechanism are due to the hydromechanics and physiology of the cochlear partition, a flexible structure surrounded by fluid that supports the basilar membrane (BM), the tectorial membrane and the organ of Corti, where the inner and outer hair cells perform the transduction and amplification of the signal respectively. The passive travelling wave is a consequence of the stiffness gradient in the BM and the fluid pressure coupling. The active amplification mechanism is generated by the outer hair cells somatic electromotility, which is intrinsically nonlinear. Thus it is also responsible of many other phenomena like compression, two tone suppression and distortion products, among others [1]. The passive mechanism could be model discretizing the continuous BM through an array of oscillators with varying damping and stiffness that are coupled each other by the hydrodynamics of an incompressible fluid. This discretization is depicted in Figure 1, each oscillator corresponds to a section or site in the membrane. Figure 1: Array of oscillators immersed in an incompressible fluid The active mechanism has its origin in the physiology and electro-mechanics of the outer hair cells, but its contribution to the dynamics of the oscillators could be modeled in several ways. For example, using lumped models with several degrees of freedom [9], tuned nonlinear oscillators [11], or time delayed or lateral forces [4], [5], among others. In this work we present a simple software tool (a Python module), which allows to simulate custom one dimensional cochlear models in a flexible fashion and makes comparison of different models straightforward. There are works in the literature with similar efforts [2]. In our module we adopt the approach proposed in [3]. 2

3 The module will be used to compare two types of active amplification mechanisms. The first one is made of two arrays of oscillators interconnected generating a positive feedback. This approach is common in many works, while in some cases the second array represents tuned non-linear oscillators [11] or the tectorial membrane and the outer hair cells [9], [10]. The second active mechanism relies on feed-forward and feed-backward forces. These forces can arise, for example, if the outer hair cells sense the vibration in one site of the partition and act on a neighbour site [13]. Both models include a saturating non-linearity that generates a compression in the amplitude responses and other nonlinear phenomena. In the following section we introduce the formulation of passive one dimensional cochlear mechanics and the discretization employed by the software module, then we present its features and finally the results obtained from the simulations of the two models D Cochlear Mechanics For many applications, it is sufficient to model the cochlea completely uncoiled and simplifying the fluid hydrodynamics to only one dimension. This is called the long wave approximation [7], and is valid primarily away from the resonance site. The laws of motion acting on the BM for the continuous one dimensional case are [6]: m 2 h(x,t) t 2 + µ(x) h(x,t) t + s(x)h(x,t) = p(x,t) (1) where h represents the displacement in the y direction of the BM, p the fluid pressure, m the mass, µ the damping factor and s the stiffness. The last three magnitudes are expressed per unit surface. All variables depend on position x, except the mass that could be considered as a constant [6]. In the long wave approximation, the hydrodynamic coupling between the different sections of the BM is given by the following expression: 2 p(x,t) x 2 = 2ρ H 2 h(x,t) t 2 (2) where H is the height of the cavity and ρ is the fluid density. This expression is derived from the fluid momentum and fluid mass conservation equations for a one dimensional cavity [6]. The corresponding boundary conditions are: p(x,t) x = 2ρ 2 h(0,t) x=0 t 2 2ρ 2 w s (t) t 2 (3) p(x,t) x=l = 0 (4) where w s (t) represents the displacement of the oval window or directly the input stimulus. 3

4 The complete system is given by equations (1-4). This system has to be solved first for the pressure p. For the sake of clarity, in what follows we will suppress the dependency on x and t and rewriting equation (1) as: g = µ h t + kh 2 h t 2 = p + g m (5a) (5b) Now, this last equation can be merged with equation (2): ( mh 2ρ 2 p x 2 2ρ 2 ) x p x 2 = 2ρ 2 h H t 2 2 p x 2 = 2ρ (p + g) mh (6b) mh p = 2ρ mh g p = g (6a) (6c) (6d) This last expression shows how to compute p knowing g, by inverting the left hand side operator. We continue with the discretization of equations (5b, 6d) using finite differences: mh 2ρ And, equivalently, for the boundary conditions (3, 4): 2 h i (t) t 2 = p i(t) g i (t) m (7a) p i 1 (t) 2p i (t) + p i+1 (t) dx 2 p i (t) = g i (t) (7b) (7c) m 2ρ ( p2 (t) p 1 (t) dx ) p 1 (t) = g 1 (t) m 2 w s (t) t 2 p N (t) = 0 (8a) (8b) where 1 i N, y dx = L/N. We then can express the discretized system in a matrix form: 2 h t 2 = p g m (9) (D I) p = g + q (10) 4

5 where D represents the matrix of the operator 2 and q is the vector of sources (oval window x 2 movement). The first step for obtaining the temporal evolution of the system is to solve the linear system of equations on p from the values of g and q (Equation 10). For the case of the one dimensional problem this becomes a tridiagonal system and a gaussian elimination method is used, which is linear in the number of oscillators. The second step is to integrate the temporal derivatives (Equation 9) and compute the next time values for g. 2 Flexible solver The software module was designed to be flexible and fast. Flexibility is provided by the ability of writing custom cochlear partition impedance expressions and dynamical systems with arbitrary degrees of freedom. Once the equations are set, the program builds an optimized C++ code that runs the simulations. The main features are the following. The module uses the odeint [8] library to integrate the dynamic equations which allows to select different integrators like fixed step Runge-Kutta 4 and variable step Dormand Prince or Cash-Karp. The equations should be written as an ordinary differential equation and using C++ syntax with functions from the common math C++ library or writing extra functions (Figure 2). The equation syntax accepts non local variables useful to feed-forward and feed-backward forces (figure 3) Figure 2: Code example for a model with four dynamic variables corresponding to two arrays of oscillators. The equations should use C++ syntax and functions should be in the common math C++ library or added as extra functions. In future versions we will add noise sources, time dependent parameters and a dynamic middle ear model coupled to the oval window. The module works with numpy arrays as input sound data and output data. Also, the module comes with an internal function to run threaded simulations for fast exploration of different 5

6 Figure 3: Code example for a model with bilateral forces. The syntax shortcut h[s] represents the variable h at position x + s, where x is the current position sound inputs. The work-flow starts by selecting a cochlear model and writing it down as set of ordinary differential equations and explicitly defining the variable g ( needed for the computation of the pressure p ). Then the parameters of the model must be specified. There are two classes of parameters allowed by the model, fixed and spatial. The fixed ones are the same for all the sites in the cochlea, and the spatial vary with the position along the BM. The initial conditions values are given for each dynamic variable, and are all zero by default. The output of the module is the time evolution of each dynamic variable. In order to avoid large memory usage when high sampling frequencies are used, the module allows saving only decimated samples of the time evolution. The complete set of parameters needed to run a simulation can be modified, but the module has defaults values for the cavity height, BM mass, fluid density, sampling frequency and number of oscillators. The module is available in In Figures 2 and 3 we show the sample codes used to run the simulations of Figures 4 and 5. The codes are not complete and displayed only for demonstrative purposes. Complete examples are available at the repository. 3 Simulations In this section we present some results from simulations of the Python module using two models that present different approaches to the active amplification system. The results are analyzed with reference to classical cochlear mechanics experiments [14] [15], where cochlear response curves are either measured for different amplitudes and frequencies or different cochlear positions for one frequency (our case). The first model of active amplification consists in two arrays of oscillators that are connected each other, generating a closed-loop feedback. The code corresponding to this model is depicted in Figure 2. The equations are the following: 6

7 h t = v (11) v = p g t (12) g = ω1 2 (h + y) + ω 1 v Q 1 (13) y t = w (14) w t = ω 2 2 (y αh) ω 2 Q 2 v(1 + β f (γw)) (15) f (x) = 4/π 2 arctan 2 x (16) The model accounts for the BM oscillators array in the variables h and v (displacement and velocity), fluid coupling in the pressure p, and an extra oscillator array (variables y and w) with a different natural frequency and damping. The natural frequencies ω 1 for the BM are spaced logarithmically from Hz in the base to 100 Hz in the apex, and the frequencies ω 2 = 0.75ω 1. This frequency scaling (or shift in terms of the site in the cochlea) is a key factor to generate the proper feedback and amplification. The f function is a nonlinear saturating function that provokes a limited increase in the damping factor as the displacement increases and generates the compression in the curves of Figure 4.b. In the second model the active mechanism is applied with feed-forward and feed-backward forces from the outer hair cells. These kind of models are sometimes called non-local or with bi-lateral coupling [12] [13]. The corresponding code is displayed in Figure 3 and the equations are the following: h(x) = v(x) t (17) v(x) = p(x) g(x) t (18) g = ω 2 (h(x) + αγ(arctan(h(x + s)/γ) β arctan(h(x s)/γ))) + ω v(x) (19) Q In this case we make the spatial dependency on x explicit in order to write the bi-lateral coupling as forces that depend on the displacement from the positions x + s and x s, where s is a fixed value. Here again, there is a saturating function (arctan) to limit these lateral forces. Finally, we want to evaluate the response of model with a more rich input. In Figure 6 we display the obtained cochleogram of a sinusoidally modulated tone. The parameters of the model, sampling rate and number of oscillators were adjusted to give a reasonably output in the shortest computational time possible, which was approximately 4 times slower than real time in an Intel(R) Core(TM) i GHz. 7

8 RMS displacement (nm) db SPL (a) Distance from base (mm) RMS displacement (nm) (b) Cochlea site Peak position 19.1 mm 16.3 mm 4.8 mm Stimulus intensity (db) Figure 4: Two array of oscillators model (a) RMS displacements of the BM for a pure tone of 1000 Hz and different amplitudes. (b) RMS displacements for four sites in the cochlea (gray vertical lines in (a) ) as function of the stimulus intensity. Parameters: The natural frequencies ω 1 for the BM are spaced logarithmically from Hz in the base to 100 Hz in the apex, ω 2 = 0.75ω 1, Q 1 = 4, Q 2 = 10, γ = m, β = 40, α = Conclusions In this work we introduced a Python module capable of simulating one dimensional hydrodynamic cochlear models with an arbitrary cochlear partition impedance. The main advantages of the module are performance and flexibility. The module provides a just-in-time C++ compilation of the model, allows to process numpy arrays with arbitrary inputs like sample sound files, and can incorporate different active mechanisms. Two active mechanisms were tested in order to show different types of input formulas. Both models yield reasonable outputs, when compared with experimental results, despite being simplified versions of more complex models from the literature. For the two cases there is a noticeable amplification in the region of resonance. A simple explanation for the underling amplification mechanism in both models can be found in the existence of a cycle-by-cycle positive feedback. In the first model the feedback arises from the connection between the oscillators, while in the second model the feedback loop is close through the fluid coupling. Furthermore, in the first model the second array of oscillators tuning is scaled down, so their responses will be similar to a feed-forward force. Acknowledgements This work was partially funded by CONICET, Argentina and Universidad Nacional de Quilmes, Argentina. References 8

9 RMS displacement (nm) db SPL (a) Distance from base (mm) RMS displacement (nm) (b) Cochlea site Peak position 19.1 mm 16.3 mm 4.8 mm Stimulus intensity (db) Figure 5: Bi-lateral outer hair cells forces model (a) RMS displacements of the BM for a pure tone of 1000 Hz and different amplitudes. (b) RMS displacements for four sites in the cochlea (gray vertical lines in (a) ) as function of the stimulus intensity. Parameters: The natural frequencies ω 1 for the BM are spaced logarithmically from Hz in the base to 100 Hz in the apex, Q = 4,γ = 10nm,β = 0.3,α = 0.2,s = 140µm Distance from base (mm) Time (ms) BM displacements (nm) Figure 6: Cochleogram for a sinusoidally frequency modulated tone with carrier frequency of 1000 Hz, modulation frequency of 5 Hz and modulation depth of 500 Hz at 40 db SPL Parameters are the same as Figure 4 9

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