Anatomical set of mechanical models for the organ of Corti​
​by Jorge Berger
This notebook makes available the core of the code used in the article cited above.

Synopsis of the article

A key role in the hearing process is played by the cochlea, located in the inner ear. Sound waves are converted in the middle ear into elastic vibrations. Two transductions occur in the cochlea: The elastic vibrations are converted into motion of the basilar membrane, and then this motion is converted into signals in the auditory nerve. The second transduction is performed by the organ of Corti (OoC). The OoC has amazing sensitivity, being able to detect motions smaller than thermal noise. We present a set of simple mechanical models for the OoC components that reproduces this sensitivity.
Nunerous studies of the OoC can be found in the literature. What characterizes this article is that each anatomical component of the OoC is described by a simple model that translates into a simple differential equation. It is possible to change the model of any of the components by changing just one of the differential equations in the code presented below.
The scope of this article differs from typical articles on the subject. It is more speculative and, rather than studying the OoC of a particular species, raises qualitative questions such as what is the advantage of various features of the OoC, how it could have evolved, and how an artificial OoC could be designed.
With appropriate parameters, the OoC can become a critical oscillator. This feature confers the OoC impressive frequency selectivity, signal to noise improvement and amplitude compressibility. Figures 5-7 of the article illustrate this behavior.
​Fig, 5 shows the gain (output divided by input) as a function of the frequency, for several amplitudes. The inset compares these results with those expected from a critical oscillator
​​
​Figures 6 and 7 illustrate the filtering capability of the OoC. The upper figure shows inputs as functions of time that contain a signal with noise added to it and the lower figure shows outputs. In particular, the lower panel B is the output that corresponds to the input in upper panel C, where the signal was the same in the three cases.
The article contains four appendices that include (1) generalization of the phase-lag concept to non-sinusoidal functions, (2) extension of Poiseuille’s equation (flow-pressure relation) to non-stationary boundaries, (3) input-output dependence near a Hopf bifurcation and (4) differentiation between amplification in the cochlea along distance or along time.
The cells below provide the most common calculations required and they closely follow the definitions in the article. If you run into any difficulty, you are welcome to contact the authors.

Geometric parameters

L=10;(*lengthofRL*)​​LH=10;(*lengthofHC*)​​LT=L+0.5LH^2/(L+LH);(*lengthofTM*)​​x1=3;(*centerof1stOHC*)​​x2=7;(*centerof2ndOHC*)​​ell=2;(*extensionofOHCinradialdirection*)

Dynamic parameters

m=10;(*massateachendofanOHC*)​​mH=120;(*massofHC*)​​IRL=2.*^3;(*momentofinertiaofRL*)​​KRL=1.*^3;(*torsionalspringconstantonRL*)​​KCP=1.*^2;(*springconstantbetweenRLandCP*)​​betaCP=3;(*dampingconstantbetweenRLandCP*)​​KC=50;(*springconstantbetweenendsofOHC*)​​betaC=43;(*dampingconstantbetweenendsofOHC*)​​DEL=0.25375;(*maximumcontraction*)​​KD1=4.*^2;(*springconstantbetween1stDCandBM*)​​KD2=4.*^2;(*springconstantbetween2ndDCandBM*)​​beta1=3;(*damping1stDC*)​​beta2=3;(*damping2ndDC*)​​beta12=3;(*frictionbetweenDC1-2*)​​KB=10;(*springconstantouterbundles*)​​H1=6.5*^-3;(*extensionunstablerangeOHB1*)​​H2=5.*^-3;(*extensionunstablerangeOHB2*)​​KIHC=10;(*torsionalspringconstantinnerbundle*)​​HIN=5.*^-3;(*extensionsoftrangeIHB*)​​cs=2;(*innersulcusresistancetocompression*)​​mu=10;(*dragcoefficientIHB*)​​amp=310^-5;(*inputamplitude*)​​wbm=5.3294;(*inputangularfrequency*)​​noise=2.*^-6;(*representativevaluefor2k_BT/thickness*)

Variables

q: Q(0, t); tet: θ; b1:
b
1
; b2:
b
2
; s1:
s
1
; s2:
s
2
; tin:
θ
in
; pin:
p
in
​
wn1, wn2, wn3, wn4: noise frequencies
ph1, ph2, ph3, ph4: noise phases

Initial values

(*Letter"p"standsfortimederivativeand"0"forinitialvalue*)tet0=RandomVariate[NormalDistribution[0,Sqrt[noise/KRL]]];​​tetp0=RandomVariate[NormalDistribution[0,Sqrt[noise/IRL]]];​​b10=RandomVariate[NormalDistribution[0,Sqrt[noise/KCP]]];​​b1p0=RandomVariate[NormalDistribution[0,Sqrt[noise/m]]];​​b20=RandomVariate[NormalDistribution[0,Sqrt[noise/KCP]]];​​b2p0=RandomVariate[NormalDistribution[0,Sqrt[noise/m]]];​​s10=RandomVariate[NormalDistribution[0,Sqrt[noise/KD1]]];​​s1p0=RandomVariate[NormalDistribution[0,Sqrt[noise/m]]];​​s20=RandomVariate[NormalDistribution[0,Sqrt[noise/KD1]]];​​s2p0=RandomVariate[NormalDistribution[0,Sqrt[noise/m]]];​​pin0=RandomVariate[NormalDistribution[0,Sqrt[noisecs]]];​​q0=0;tin0=0;

Input

x-dependence, forces and torques

Dynamic equations

Time integration

Re-initialization for extension of follow up time

Amplitude and phase