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A state space model for cochlear mechanics
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Image of FIG. 1.
FIG. 1.

Equivalent mechanical system for Neely and Kim’s active micromechanical model.

Image of FIG. 2.
FIG. 2.

Magnitude of the partition admittance of an isolated micromechanical element for feedback gains of and 1.1.

Image of FIG. 3.
FIG. 3.

Positions of the poles of the transfer function for the micromechanical element with feedback gains of and , plotted: (a) using the conventional control representation with positive and negative frequencies plotted vertically; and (b) using the representation used for the cochlea model described here, with frequency plotted horizontally.

Image of FIG. 4.
FIG. 4.

The discrete model of the cochlea including the micromechanical models of the cochlear partition for elements 2 to , the model of the middle ear dynamics at element 1 and of the helicotrema at element .

Image of FIG. 5.
FIG. 5.

Distribution of BM displacement along the cochlea calculated from the state space model for the parameters of Neely and Kim (1986) at with feedback gain (dashed) and (solid).

Image of FIG. 6.
FIG. 6.

Enhancement of the peak response in the BM velocity due to the cochlear amplifier for feedback gains of and .

Image of FIG. 7.
FIG. 7.

Distribution of poles in the coupled cochlear model for various values of feedback gain in the micromechanical model, . Any pole with a positive real part denotes an unstable system. The cochlear model is clearly stable when it is passive, (a), and also for feedback gains of 0.85 and 1 (b) and (c). The cochlear model is unstable if the feedback gain is increased to 1.03 (d).

Image of FIG. 8.
FIG. 8.

Poles of the coupled cochlea for a gain distribution of feedback gains with: (a) a 0.1% step decrease at ; system is unstable; (b) a step gain distribution of magnitude 0.85 at ; system is stable; and (c) a smoothly varying gain distribution (quarter sin-squared wave dip) of magnitude 2% change at ; system is stable. Note that the sketches of feedback gain with position along the cochlea have differing scales.

Image of FIG. 9.
FIG. 9.

Poles of the coupled cochlea for a random gain distribution of feedback gains: (a) with small magnitude (0.1%) and rough spatial variations; system is stable; (b) intermediate magnitude and rough spatial variations; system is unstable; (c) intermediate magnitude and smooth spatial variations; system is stable; and (d) large magnitude variations and smooth spatial variations; system is unstable.

Image of FIG. 10.
FIG. 10.

The number of unstable poles in coupled cochlea models having different amplitudes of cochlear amplifier gain variation with either smooth or rough distributions along its length. The bars denote the range of 40 simulations with different random gain distributions and the circles denote the average value.

Image of FIG. 11.
FIG. 11.

Block diagram of the micromechanical model of Neely and Kim (1986) with a saturating nonlinearity in the active force.

Image of FIG. 12.
FIG. 12.

The results of a time domain simulation of a single isolated nonlinear micromechanical element with , which is linearly unstable. The amplitude of the oscillation initially diverges exponentially, but is then limited by the saturation function so that it settles into a periodic limit cycle.

Image of FIG. 13.
FIG. 13.

BM velocity responses at different positions along the coupled cochlea as a result of a time domain simulation with saturating cochlear amplifiers and a distribution of feedback gains corresponding to Fig. 8(a).

Image of FIG. 14.
FIG. 14.

Grey scale representation of the results of the time domain simulation described in Fig. 13 at all positions along the cochlea. The grey scale corresponds to the amplitude of the pressure (positive white, negative black) at each position at each time. An animation of this response is available online (Ku, 2007).


Generic image for table

Parameters of the micromechanical model of Neely and Kim (1986) with corrections for original typographical errors and converted into SI units.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A state space model for cochlear mechanics