Physics of how the cochlea isolates frequencies along its length?

Question

Can anyone explain the separation of frequencies along the basilar membrane of the cochlea please? (equations would be nice)

I understand it being related to the resistance caused by fluid in the cochlear increasing in proportion to the rate of change at the ear drum versus the stiffness of the basilar membrane decreasing along it's length which somehow creates a condition where the membrane only vibrates in a local region. However all descriptions I have seen so far are always very general and never go into the actual physics of the system.

Answer 1

There are two primary factors that allow the cochlea to isolate frequencies. These are generally referred to as passive and active properties:

tl;dr version: The passive properties are due to the mechnical properties of one of the membranes in the cochlea, the basilar membrane, primarily the width and stiffness at a given point. The active properties are due to the vibration of a special class of cells within the membrane, called the outer hair cells, which change their shape in response to sound in such a way that it narrows the range of frequencies that this part of the cochlea responds to.

Passive

The cochlea is a complex structure, with multiple membranes and fluid-filled chambers.

Cochlea cross-section

However, the part that is primarily responsible for the frequency tuning of the cochlea is the basilar membrane. When a sound wave enters the cochlea (by way of the oval window), the sound wave travels down a fluid-filled chamber (scala vestibuli). This causes the basilar membrane to vibrate. Different points along the basilar membrane vibrate most strongly at a different frequencies (that point's resonant frequency), although there is a range of frequencies that it will respond to more weakly. Due to the fact that the sound travels at a finite speed down the cochlea, this induces what is called a traveling wave in the basilar membrane.

Basilar membrane traveling wave

The resonant frequency is determined primarily by two factors. First is the width of the basilar membrane at that point. The wider it is, the lower the resonant frequency (like the longer strings on a piano producing lower-frequency sound). The second is the stiffness. The stiffer the basilar membrane, the higher the resonant frequency.

These properties vary together across the the length of the cochlea, with the the narrowest, stiffest part (high-frequency) at the base, while the widest, most flexible part at the tip. In between these two extremes, there is a smooth, logarithmic gradient in frequency tuning along the basilar membrane.

Basilar membrane unrolled. In mammals in real life it is a spiral. In birds it is straight.

Plots of basilar membrane properties as a function of position

You can think of it sort of like strings on a harp, where each string vibrates at a particular frequency. However, instead a finite number of discrete strings, there are an infinite number of strings, all attached together to form a long sheet.

This is called the passive properties of the cochlea, since they are based on the effects of sound waves on the basic mechanical properties of the membrane.

For more detail see The Inner Ear

Active Properties

However, the earliest experiments on the basilar membrane were on dead tissue. Comparing these results with live tissue resulted in a surprise: a given point on the basilar membrane responds to a much smaller frequency in live tissue compared to dead tissue. This, it turns out, is due to what we call outer hair cells.

The cells in the cochlea that convert sound energy to neural activity are called hair cells. They are embedded in the basilar membrane in a structure called the organ of Corti, and have small protrusions (the "hairs", technically stereocilia, which are not at like hairs on your head). These hairs are embedded in another membrane, the tectorial membrane, which floats just above the basilar membrane. When the basilar membrane vibrates, the tectorial membrane moves relative to the basilar membrane, causing the hairs to bend, and this bending is converted into neural activity through a mechanism that I won't go into because it isn't relevant.

Organ of corti operation. Inner hair cells are the leftmost row, outer hair cells are the other three rows.

However, in mammals these hairs are divided into two distinct groups: the outer hair cells and the inner hair cells, so-called because of their position along the basilar membrane. It turns out they also have different specializations (although with some overlap). The inner hair cells are primarily signal transducers, they are main group responsible for converting the vibrations into neural activity.

The outer hair cells, however, have a very different role. They receive inputs from both the inner hair cells and higher brain areas. In response to these inputs, the outer hair cells change their length very quickly and very strongly.

Outer hair cell vibrating

The inputs from the inner hair cells cause a very fast feedback loop where, in response to sound, the outer hair cells change their length to amplify vibrations caused by sounds at around the frequency at the "proper" tuning of that part of the cochlea and attenuate those at different frequencies, which results in the membrane vibrating to a narrower range of frequencies and more strongly to the "correct" frequency than you would expect from the mechanical properties alone. The higher-level brain inputs allow for more complex dynamic changes in the cochlear response.

Passive vs. active responses

The mechanism by which outer hair cells change their length is a subject of some debate right now. Various people will tell you very confidently that some different factor is at play, but the truth is we really aren't sure yet. It isn't related to how muscles work, that much is clear, it is way too fast. It is know that a protein called prestin is required, but whether it is directly responsible for the change in length or are required for the proper operation of something else that causes the change in length is not certain. It may be related to changes in the capacitance of the cell, which result in changes in the cell membrane's surface area, but exactly how this occurs is not certain.

The vibrations caused by the outer hair cells are so strong, they produce sound that can be detected outside the ear (called otoacoustic emissions) using sensitive microphones. There was one dog that had this effect so strongly, you could hear the sound just by putting your ear next to its ear.

These are called active properties because they come from activity in the membrane.

References on outer hair cells: Two Kinds of Hair Cells in the Cochlea, Inner and outer hair cells

Answer 2

The cochlea has a complex physical structure, with multiple membranes and fluid-filled chambers.

Therefore to explain the separation of frequencies along the basilar membrane of the cochlea is complex to. Sure, there are a lot of very general descriptions (even the answer of theblackcat) and a lot never go into the actual physics of the system.

This second answer of my, again very long, covers in every detail the actual physics of the system.

More than 50 years after Georg von Békésy became a Nobel price laureate we still use a two compartment model, for our three compartment hearing sense:

The scala media is expected not to play a significant role and is therefore in models reduced to a 'thin baslilar membrane' (even in the answer of theblackcat).

And it is hypothesized that a traveling wave, running from base to apex and counting several wavelengths in its pathway in the cochlea to a place on the basilar membrane where resonance fits the best, is supposed to transport the sound energy:

Now most of his (Georg von Békésy) hypothese appeared to be incorrect. His traveling pressure wave is modified as well.

It was Georg von Békésy, who first observed and described a kind of „wave propagation‟ over the basilar membrane when the cochlea was stimulated with an acoustic vibration. A „wave‟ that always runs from base – the round window – to apex – the helicotrema.

He also noticed that the properties of the basilar membrane were quite remarkable. Smaller in size and mechanically more rigid near the base, broader and more flexible towards the apex. A logarithmic, frequency-location relation was observed along the basilar membrane known as the tonotopic frequency distribution. This observation has led to general agreement among cochlear experts, supported by an overwhelming amount of experimental evidence, that higher frequencies are detected near the base, whereas the lower frequencies evoke the most mechanical activity near the apex.

As a result of his experiments with cochleae which he stimulated with extreme intensity, von Békésy formulated the hypothesis that the observed „propagating wave‟ – which evidently moves faster near the base with a smaller wavelength, and thereafter gradually decreases its propagation velocity and increases in wavelength towards the apex, finally even stops there, (at the place) where detection of the characteristic resonance frequency is found – carries acoustic energy to the place of detection.

Von Békésy also executed experiments with a number of mechanical models. In these models he replaced the scala media, which is confined by the extremely thin Reissner membrane on one side and the much more rigid basilar membrane on the other, by means of a single flexible membrane dominated by the properties of the basilar membrane.

In order to justify this concept he formulated the hypothesis that the Reissner membrane must be so thin and flexible that it cannot possibly influence the hydrodynamic behaviour of the perilymph within the cochlea.

Even though this concept has been adapted by a few alterations and hypothesis that are better adapted to current thinking, the basic ideas of wave propagation and the replacement of the scala media by just one single, flexible membrane remain to be cornerstones for the hypothetical explanation of signal transfer within the cochlea.

Several attempts have been made by cochlear and mathematical experts to develop theoretical models that combine the acoustic stimulation of the ear by means of a travelling wave, (carrying acoustic energy) with the detection of signals in the organ of Corti.

All these attempts failed to lead to an ultimate solution that is free of anomalies.

This failure can be explained by one important reason.

Namely, both of these assumptions:

1. The existence of the Reissner membrane can be neglected while the hydrodynamic behaviour is both observed and explained, and

2. The propagating wave must, somehow, carry the frequency dependent sound energy to the corresponding location of resonance on the basilar membrane, are at variance with the fundamental rules of physics.

A correct application of physics leads to a totally different working model of our hearing sense.

Taking a lot of considerations and objections: From a physics point of view against the current hypothesis of basilar membrane stimulation, one can take a lot of considerations and objections.

With a reference to: http://www.researchgate.net/publication/48323925_Applying_physics_makes_auditory_sense__a_new_paradigm_in_hearing

In the manuscript "Applying Physics Makes Auditory Sense." the authors explain and here I substantiate their statements:

The enervation of the general assumption:

The existence of the Reissner membrane can be neglected while the hydrodynamic behaviour is both observed and explained.

In literature one finds many images of the cross section of the cochlear partition (the Cochlea cross-section image in the answer of thetheblackcat is such an image) clearly demonstrating that the Reissner membrane is not a loose piece of tissue. Although thin, it can be regarded as a flat structure in a normally functioning cochlea, which is an indication that some stretching must be present in it.

In patients suffering from Ménière's disease, endolymphatic hydrops causes the Reissner membrane to balloon, clearly showing pressure on the Reissner membrane.

This implies however, that if the movement of perilymph is parallel to the surface of the Reissner membrane, this membrane is not capable of lateral movement. Not, unless local stress forces are evoked in the Reissner membrane. For that purpose, the non-viscous perilymph fluid would have to develop shear stress forces in the interface of the Reissner membrane, which is not possible.

Within the science of physics we know of no such mechanism that would allow for the transfer of this lateral movement to the endolymph in the interior of the scala media, which is on the other side of the Reissner membrane, as if the Reissner membrane simply does not exist. Therefore, the hydrodynamic behaviour on either side of the Reissner membrane will always remain to be different.

It follows, that ignoring the existence of the Reissner membrane is fundamentally incorrect.

Any results of theoretical attempts to clarify cochlear functioning, based on a cochlear model that consists of two channels separated by a thin, flexible partition, ignoring the scala media as a third channel filled with endolymph in between both perilymph ducts, are therefore, also fundamentally incorrect.

The enervation of the general assumption:

The travelling wave must, somehow, carry the frequency dependent sound energy to the corresponding place of resonance on the basilar membrane,

Starting with the observations of Von Békésy, the „travelling wave‟ phenomenon has always been ascribed to the effects of sound energy transportation.

Even though from time to time, serious doubts with regard to this hypothesis have arisen – even Wever, Lawrence and Von Békésy himself – and other critical researchers have voiced the opinion that perhaps the „wavy character‟ of the basilar membrane must be found in a different cause, not one of them has ever developed an analytical calculation model accounting for all the remarkable properties of the basilar membrane. Cochlear scientists have always shared the expectation that developing such a model would be extremely complicated, and would therefore lead to unreliable results. [It is known that Von Békésy would disparagingly dispose of any such attempts as „armchair theories‟ that would not lead to anything of use.]

In a 1954 paper, Wever, Lawrence, and von Békésy reconciled some of their views on the nature of the traveling wave. They stated that when the cochlea is stimulated with a tone, a BM "displacement wave seems to be moving up the cochlea. Actually...each element of the membrane is executing sinusoidal vibrations...different elements...executing these vibrations in different phases. This action can be referred to as that of a traveling wave, provided that...nothing is implied about the underlying causes. It is in this sense that Békésy used the term ‘traveling wave’..." [pp. 511-513 of Wever et al. (1954)].

Unhindered by his disdain; as always following the curiosity that leads the way in science:

One can do the following math:

Start by calculating the sinusoidal pressure stimulation with frequency, which uniformly acts on the basilar membrane, while this membrane is infinitesimally divided into an array of individual resonators with a logarithmically decreasing resonance frequency from base to apex.

The reason for this uniform pressure stimulation is found in the fact that it has shown that the perilymph moves as a whole fluid column along the front side of the basilar membrane, thus resulting in uniform pressure effects on the basilar membrane as well.

Making use of complex function theory and conformal transformations this general vibrational transfer model of the basilar membrane, despite its complexity, offers an analytical solution.

What's more: this solution has led to a very useful result:

And it is in accordence with what Ren and his team observed with their direct laser interferometer measurements of basilar membrane movements.

Ren’s unintentional attack on Von Békésy’s “Traveling Wave Theory”

The paper of Ren is: Longitudinal pattern of basilar membrane vibration in the sensitive cochlea

Proceedings of the National Academy of Sciences - pnas.org PNAS | December 24, 2002 | vol. 99 | no. 26 | 17101-17106.

Experiment: Laser interferometrical measurements of the basilar membrane movement. In the 13,3 – 19 kHz area of the basilar membrane of a gerbil.

Results: The movement of the basilar membrane, from the higher frequency side towards the lower side, is restricted to 300 μm on both sides of the point of maximum activity. The shape of the movement was exactly symmetrical around this point.

The authors of the manuscript "Applying Physics Makes Auditory Sense." have actually paid rather a lot of attention to the form of displacement, which corresponds with the form that Ren et al. have actually measured.

So, there is a discrepancy between the assumed travelling wave from current theories and the experimental results by Ren et al.

In their experiments Ren et al. they observed a short „wave pattern‟, symmetrically divided on either side of the point of resonance. What's more, according to Ren et al, the movement of this observed wave pattern along the basilar membrane, running from base to apex, did not decrease in speed.

According the manuscript "Applying Physics Makes Auditory Sense.": Due to the peculiar basilar membrane resonance possibilities found in practice, a uniform sinusoidal pressure stimulus results in a mirror symmetrical phase wave pattern that shows a propagating wave running from base to apex. And this waveform on the basilar membrane is identical to that which Ren et al. observed in their laser interferometer experiments on gerbils.

The reason for this phase dependent behavior is explained in more general terms in the manuscript "Applying Physics Makes Auditory Sense.".

A detailed mathematical explanation and analytical calculation has been excluded from that manuscript – but is available.

The insight that the pressure stimulus on the basilar membrane is regarded to be uniform is part of the next:

The Bernoulli effect:

Bernoulli effect is mostly known under steady flow conditions. But that do not exist within the cochlea.

But the Bernoulli effect is not only known under steady flow conditions. And it is not so that the Bernoulli effect can only be valid under steady flow conditions.

In the cochlea: Therefore, the quasi-static approach, a quasi-stationary, in the concept: quasi-static relation, the quasi-static solution, the quasi-static Bernoulli, in quasi-static form.

It is non stationary within the cochlea.

Let me clarify (like the authors do in that manuscript) why and that indeed applying the Bernoulli effect is correct when in an oscillating column of fluid all conditions for a quasi-stationary potential flow are met. Such an quasi-stationary potential flow will not only lead to the valid use of the Bernoulli effect, but will also perfectly fulfil Laplace's equation, which, of course, is another prerequisite.

Consequently, in the case of the cochlea we do not have to make use of Navier-Stokes equations at all.

The basilar membrane moves in reaction to the sound energy signal, which is generated by the quasi-static Bernoulli effect.

According to Bernoulli's law, this pressure difference on either side of both the Reissner membrane and basilar membrane is represented by:

pressure difference = - 1/2 (the density in kg/m³ of the perilymph) times (the velocity of the perilymph in m/s)^2

An erroneous assumption and point of departure – encouraged by the advise of cochlear experts – is the commonly accepted two compartment concept, in essence, ignoring the existence of the endolymph filled scale media between the two with perilymph filled channels, the scala vestibuli and the scala tympani. The latter, unavoidably lead to an incorrect hydrodynamic concept.

The fact is then that the “physics” in these current theories are clearly at variance with the general laws of physics:

1. Ignoring the hydrodynamic role of the Reissner membrane in the cochlear mechanics is erroneous according to general physics.

2. Overlooking the fact that the cochlear potential changes only when the push-pull of the perilymph velocity is generated by a sound signal, which means that the incoming stimulus is differentiated, constitutes an omission in terms of general physics.

3. Not taking into account that the increase of these cochlear potentials with 6 dB in case of a doubled incoming stimulus signal, signifies that there is a quadratic relation between the sound stimulus and the cochlear potentials, is another such omission.

4. The common assumption that acoustic frequencies evoke waves in the perilymph channel, while the length of this channel is only a fraction of the wavelengths that can be generated, is erroneous. The existence of these waves is at variance with the general laws of physics.

regarding the Bernoulli effect: All conditions in the cochlear channel are such that the quasi-static solution, which is equal to the use of the Bernoulli equation, applies perfectly.

The authors make use of the – from literature, such as Von Békésy – known mechanical properties of the basilar membrane, which lead to a logarithmic distribution of the resonance frequencies over the basilar membrane. Nothing more. If subsequently, the deflection profile of the basilar membrane is calculated, the absolute deflection remains dependent on the size of the stimulus, and therefore remains arbitrary.

In the manuscript "Applying Physics Makes Auditory Sense." the authors clearly describe that the basilar membrane and the Reissner membrane both move outwards from the scala media and thus away from each other.

Indirectly (outside their manuscript) they argue the fact that apparently the scala media must expand, however that the incompressibility of the endolymph prevents this expansion. Which, by the way, would have been correct if there would not be a connection between the scala media and the saccus endolyphaticus within the cerebrospinal cavity.

In their model of the auditory sense: The saccus endolymphaticus – situated within the cerebrospinal cavity, an environment that is not subject to pressure variations such as the scala media is, but is still capable of adapting itself – functions as an expansion vessel and maintains the supplementation of the endolymph in the somewhat expanding scala media.

The fact is then that the scala media is capable of reacting to pressure variations, even though it's content is incompressible

Yes indeed, their proposed theory is completely different from the current theory. This is why, in an attempt to explain, they provided a new analysis. From this analysis that has been held against the light with regard to general physics, it becomes crystal clear that current theories are based upon interpretations of oversimplified and therefore erroneous working models, in turn leading to hypothesis and formulations that are at variance with general physics.

When one finds that an overwhelming part of literature departs from a model of two compartments, while it is clearly evident that it concerns a three compartments concept, this inevitably leads to an erroneous hypothesis. If then, finally, after much verification by erroneous interpretations of experimental results, one declares the theory departing from two compartments valid, one still cannot speak of a „large body of evidence‟.

Therefore: radically new ideas refreshing, I would like to emphasize that the authors have covered: all of these items. To mention just a few:

The claim that the auditory sense differentiates and squares and that we therefore receive the sound energy frequency signal in the organ of Corti, is completely based on the results published by Wever and Lawrence in 1950.

The „travelling wave‟ along the basilar membrane can be described as a passive reaction of a complex organised second order resonance system, and is therefore a phase wave in the cochlea, that will always run from base to apex, completely in accordance with the referred experimental observations by Ren et al.

An explanation that can satisfy the anomaly presented by the absence of the backwards travelling waves in the DPOAE experiments. Therefore, this anomaly that was acknowledged by cochlear experts like de Boer, still remains unsolved. It stands to reason; as sound energy transporting travelling waves simply cannot exist in the cochlea.

Whether the Reissner membrane is flexible or rigid in fact does not contribute much to its hydrodynamic behaviour. If the Reissner membrane were infinitely flexible –therefore deformable in the membrane surface itself - it would not exert a dominating influence on the hydrodynamic behaviour of either the perilymph moving along the membrane in the scala vestibuli nor on the immobile endolymph on the other side in the scala media. Precisely because of this lack of extreme deformability in the membrane surface the scala vestibuli and the scala media may not be considered as one hydrodynamic unit. To understand this better: one can perform physics at the kitchen table.

So, basic physics which can be performed at the kitchen table. You only have to dip a wire ring into a soap solution, create a soap film, and then softly blow air towards the surface of the soap film. You will see the soap film bending away from you, which is experienced as completely normal behaviour.

When you gently continue to blow air sideways along the soap film you will notice the extreme deformability of the surface of this soap film. The liquid in the soap film starts to move very easily, parallel to the direction of the airflow. This causes the liquid in the soap film to be pushed in an outward direction, while the soap film rapidly decreases in thickness and subsequently looses its structure and bursts.

When however, you blow in short puffs on one side of that soapy film, you at once realize that:

The soap film, which is many times more deformable and flexible than the Reissner membrane – and should therefore be better suited to the hypothesized behaviour that Von Békésy describes – each time bulges in the direction of that side where the pulsating air flow passes by, and will then be much more resistant to bursting.

This bulging of the soap film towards that side of the soap film where the airflow passes by is exactly what the quasi-static Bernoulli effect predicts.

Based on physics' principles, when correctly applied, it is entirely impossible that either a slow wave or even a fast wave exists in the cochlea.

I can wonder, when did calculating an erroneous model ever lead to correct results?

The reproache for the wavy behaviour within the cochlea, to introduce a three compartment model into the research.

In essence, new calculations are based on a three-compartment model, for which the authors assume as point of departure that the changing and location dependent pressure difference over the basilar membrane, between the scala tympani and the scala vestibuli, must be calculated on a three-compartment model.

Previously, it has been discussed and shown that the Bernoulli effect may be applied for all frequencies within the cochlea, be it in quasi-static form.

So one should not depart from the erroneous two compartment model, whereas in view it most definitely should be a three compartment model.

The basilar membrane moves in reaction to the sound energy signal, which is generated by the quasi-static Bernoulli effect.

How do we have to interpret that “wavy” movement of the basilar membrane? In this we have to observe the following facts in physics:

In a medium [ gas, liquid, solid material ] there exists a uniform relation between the propagation velocity v of sound or vibration, the frequency f and the wavelength λ of the sound or vibration wave:

v = f × λ

v is lowest in gasses: In air 330 m/s

v in water but also in perilymph 1500 m/s

v is highest in solid material to ca. 8000 m/s

Together with the lowest [ 20 Hz ] and highest [ 20.000 Hz ] sound frequencies that we are able to hear, the wavelength varies in the perilymph from 75 meter to 7.5 cm Always significantly larger than the size of the cochlea.

Consequences: In the much shorter perilymph duct there cannot run a “sound wave”.

The perilymph between oval and round windows is just able to move forwards and backwards as a whole. Tissue around the perilymph channel behaves more like a solid material than like a liquid. That tissue needs a larger size for a traveling wave.

Conclusion: There cannot propagate a traveling wave inside the cochlea.

But what kind of movement is observed then ? Therefore we must observe at first the way of movement of a singular resonator.

A resonator exist of a body connected to a spring, and is possessing in practice also damping. If the body is given a deflection in opposite direction to the spring influence and that body is released, it will move harmonically with descending amplitude around the equilibrium point. The frequency in that case is known as resonance frequency fr

Let us observe the reaction of a spring-mass-system on a periodic stimilus

If the resonator is brought into a vibrating movement, then three different situations can exist, dependent on the relationship between stimulus frequency f and resonance frequency fr :

f < fr : reduced in phase movement, with phase angle: 0

f = fr : increase due to resonance but also a phase retardation with phase angle: ½ π

f > fr : strongly reduced movement in opposite direction with phase angle: π

Followed by the remarkable mechanical setup of the basilar membrane: This basilar membrane [ BM ] exists of an array of small resonators, that have gradually decreasing resonance frequencies from the round window up to the helicotrema.

And then in case of an everywhere equal in phase stimulus on the entire BM, the following is happening:

All parts of the BM having fr > f : move in phase with the stimulus. That movement becomes larger if fr approaches f closer and will retard gradually in phase.

In case of resonance a large movement is and there exist a phase retardation of ½ π All parts of the BM with fr < f are more and more moving in opposite phase with the stimulus and with a growing decreasing in deflection.

And what phenomenon is comparable to this? The “wave” in the stadium!

And dependent on the quality factor in resonance, strongly coupled to the rate of damping, the moving area becomes smaller, while the maximum deflection becomes larger.

On theoretical grounds it is no mystery that this “wavy movement” of the BM is always running from the round window [base] towards the helicotrema [apex] of the cochlea.

It is a locally bound reaction behavior on a universally existing stimulus. Using the material specifications this behavior can be calculated in a perfect way.

If we calculate the phase relations of that same second order resonance system with the equation of phase, we find that for membrane resonance frequencies higher than the stimulus frequency, the phase of the membrane movements equals the phase of the stimulus frequency.

For membrane resonance frequencies that are lower than that of the stimulus frequency, the movements of the basilar membrane show a retarded phase shift of 180°.

The phase for the basilar membrane movement at center frequency is retarded over 90°.

This means that the auditory nerve receives the final signal, almost exclusively, from the contributions in the center frequency region. The contributions of the two flanks however, cancel each other due to their identical amplitude and opposite phase.

This mathematical calculation shows for the logarithmically distributed local resonance frequencies fc of the basilar membrane, the response characteristic that Ren observed in his experiments on gerbils: a very restricted symmetrical local movement phenomenon, which travels along the basilar membrane.

In this opinion this phenomenon is erroneously interpreted as evidence for a ‘traveling wave’ along the basilar membrane. That it is not a traveling wave, but a ‘phase wave’, that consists of coherent place dependent phase shifted local reactions to a stimulus that is simultaneously present throughout the basilar membrane.

Which results in the following animation:

http://www.a3ccm-apmas-eakoh.be/downloads/files/WMV-animation-local-wave-from-base-to-apex-phase-wave.wmv

Answer 3

UPDATE - With a reference to:

http://www.researchgate.net/publication/48323925_Applying_physics_makes_auditory_sense__a_new_paradigm_in_hearing

OP, user263399, COMMENT:

Can you explain the phase wave and its cause? On reading the linked paper I'm confused on how ther explanation involving the change in liquid volume velocity would create a localized effect that could be sensed as a different tone (seems the pressure on the cochlea duct would continue to decrease for the entire length as it tapers for all frequencies). – user263399

And now, a VERY DETAILED ANSWER on that 'COMMENT OF OP, user263399' -

So, with a reference to the full paper:

If one agree with the authors of the referred paper, that the perilymph inside the cochlear duct, existing of scala vestibuli and scala tympani, is just moving back and forth over distances not exceeding a few micrometer and if one admit this fact, one should also agree with the authors that all the known and involved physical quantities and parameters indicate that we are confronted here with the problem to find the hydrodynamic solution for the non-stationary small movements of an incompressible non-viscous fluid in a tiny narrow duct.

According to the rules of physics it is then permitted without any additional constraints to use the non-stationary Bernoulli equation. The result is exactly the mathematical expression the authors have used in the booklet: the pressure decrease in the perilymph duct in front of the basilar membrane is everywhere proportional to the perilymph velocity squared.

What leads to the overall result that the pressure stimulus on the basilar membrane is proportional to the sound energy stimulus offered to the ear.

Considered is thus the possibility that the observed (passive) movements of the basilar membrane could be caused by another phenomenon than a sound energy transporting traveling wave.

In case of flow in a tube under the material conditions incompressible and non-viscous and a rotation free flow condition, for stationary flow counts the Bernoulli equation. In a horizontal orientated tube gravity doesn’t play a role, what leads to the well known equation: the decrease of the overall existing internal pressure is proportional to fluid velocity squared.

And as you can see: in the case of a non-stationary flow with all other conditions the same as above, that overall pressure inside the tube – and thus also on its boundaries – is proportional to the time dependent fluid velocity squared.

In practice it also means that the movements of the incompressible viscous-free perilymph, in the direction perpendicular to the core of the perilymph duct, are negligibly small compared to the movement in the core direction.

Also: there doesn’t exist a pressure gradient inside that fluid column, as long as the area of the cross section remains the same. If the diameter of the cross section changes as function of the distance measured along the tube axis, while the shape of the cross section remains the same, the fluid velocity will be reciprocal to the squared diameter and consequently the local pressure decrease changes with the fourth power of that diameter. So if the duct has a tapered shape the pressure distribution along the duct shows a small pressure drop in the wider areas and an increasing pressure drop in narrower areas. However that isn’t the pressure gradient that acts as the driving force one may have in mind, because this pressure distribution isn’t the cause of the flow in the fluid but the result of it.

While for the flow, inside the perilymph duct, not just one single but all conditions for a potential flow and thus for the analytical solution according to Bernoulli’s relation for non-stationary flow are fulfilled. And it is that solution based on the sound and solid use of hydrodynamic rules and laws that is the straight forward outcome.

And that result can be summarized in the ultimate short statement that the changes in the internal pressure everywhere in the perilymph – perilymph, that moves, or better wiggles (along-its-length) (along/over his whole length), on the rhythm of the sound pressure in front of the eardrum – are proportional to the corresponding sound energy.

The change in the internal pressure is a decrease proportional to the time derivative of the sound pressure signal squared.

The non-stationary potential flow according to Bernoulli in the perilymph duct, like the authors have calculated, includes that everywhere inside this fluid there exist the balance between the kinetic energy represented by the expression ‘1/2 rho v^2’ – or for the total perilymph volume V ‘1/2 m v^2' and the decrease in potential energy, given by the expression : ‘– V delta p’.

Here rho is the density of the fluid; v the fluid velocity; delta p the pressure difference and m the mass of the fluid column.

So also here the sound energy signal is present inside the perilymph fluid. However not in the form of an assumed traveling wave, but as a uniform pressure stimulus all over the volume.

And therefore all the existing Fourier frequency components in the sound energy signal are present inside the perilymph to stimulate the basilar membrane including their relative amplitudes and their relative, but extremely precise, phase relations.

Regarding the compliance of the membranes, the authors have also used a frequency dependent ‘compliance’ of the basilar membrane in the description of the evoked movements in this membrane due to a sinusoidal sound stimulus.

It results in a DC deflection all over the basilar membrane due to the ‘time average of the sound energy signal’ and the locally evoked AC or frequency dependent deflection at the corresponding resonance locus with a doubled frequency.

And it results in an all over the Reissner membrane existing combination of a DC deflection towards the scala vestibuli and an AC deflection with a doubled frequency.

I repeat the part:

It results in a DC deflection all over the basilar membrane due to the ‘time average of the sound energy signal’ and the locally evoked AC or frequency dependent deflection at the corresponding resonance locus with a doubled frequency.

It isn’t a basilar membrane movement due to an ‘overpressure’ caused by an increase in pressure inside the perilymph. That is in essence the specific behavior of a potential flow – like this Bernoulli flow actually is – where the decrease in internal pressure [delta p] is proportional to the decrease in potential energy [E potential], while the kinetic energy [E kinetic] of the entire perilymph mass [m] in the flow tube increases proportionally to the fluid velocity [v] squared. Thereby potential energy and kinetic energy in the potential flow remain always in balance.

About the general mechanism of the potential flow:

I read (in Dutch) (1.5 Rotatievrije stromingen -- page 17 and page 18 -- )

Bernoulli equation for incompressible, non-stationary, rotation free streaming.

This equation in its general form is given by: ...

http://www.math.rug.nl/~veldman/Colleges/stromingsleer/Stromingsleer1011.pdf

(1.5 Rotatievrije stromingen -- page 17 and page 18 -- )

and then I read:

http://www.a3ccm-apmas-eakoh.be/downloads/index.php?file=NonstationaryBernoulli.pdf

And in off-wiki forum:

Explained in an off-wiki forum ( by the author: W. Chr. Heerens ) ( in an off-wiki forum ) ( lists.mcgill.ca/archives/auditory.html ) on at best with the example of a straight tube in which a (periodic) potential flow exists. For this flow condition the fluid in the tube is incompressible and nonviscous and the flow isn’t turbulent, which means ‘rotation free’.

Since there doesn’t exist internal laminar friction [the fluid isn’t viscous] it will ‘stream’ along the core direction of the tube everywhere with the same velocity.

In that case the (non-stationary) Bernoulli equation is valid and the internal pressure in the tube is everywhere the same and given by the well-known Bernoulli relation. The decrease in internal pressure in the fluid is equal to half the density of the fluid multiplied with the square of the fluid velocity. In case of the non-stationary Bernoulli flow, the involved velocity in this equation is then a function of time.

If we insert pressure sensors at two places along the tube in its wall, each of the pressure sensors will detect a decrease in pressure proportional to the square of the fluid velocity – in full accordance with the Bernoulli equation.

However, if we try to measure the pressure difference between the two locations, we will find zero as the result. That is logic because the fluid velocities in both cross sections are equal.

However, if we want to measure the fluid velocity in the tube, we can use the solution found by Venturi. Then we have to place in the tube an intersection in which the cross section along the length of that partition gradually and fluently decreases from the tube cross section to a minimum value and then fluently increases again to the size of the original tube cross section.

And let us place this Venturi tube in-between the two original pressure sensors. If we insert now in the wall of the narrowest cross section of that Venturi tube a third pressure sensor, we will measure there an extra decrease in pressure related to the other two pressure sensors.

The now measurable pressure difference between the Venturi pressure sensor and the pressure sensor either ‘up-streams’ or ‘down-streams’ is proportional to the fluid velocity in the tube multiplied with the total of the square of the ratio between tube cross section and Venturi cross section minus 1.

Hence there exists a lower pressure in the Venturi tube, but equal pressures on both sides of the Venturi tube.

And remark that in principle the Venturi tube in the potential flow isn’t forming an obstacle in that flow. Otherwise there would exist a pressure difference between locations on both sides of the Venturi tube.

Now we can make one further step: we can smoothly fold the tube in the Venturi partition in such a way that the narrowest cross section also forms the ‘elbow’ in the folded tube. [ let us name that the helicotrema].

Hence – contrary to what one may think – under the potential flow conditions inside the cochlea there doesn’t exist a longitudinal pressure gradient, which evokes a substantial pressure difference across the helicotrema.

Finally we can place in-between the two parts of the tube [scala vestibuli and scala tympani] a third one [the scala media] that forms an intersection of the two other ones. As long as the cross sections of both perilymph ducts at some place x away from the base [oval and round window] are identical, the evoked pressures on both sides of the scala media will be directed outwards and equal. Exactly as is shown in Fig. 3 on page 22 of the referred booklet ‘Applying Physics Makes Auditory Sense’.

Let one first calculate what pressure decrease will be evoked in front of the basilar membrane by a stimulus of 1000 Hz which let the oval window deflect with an amplitude of 2 micrometer.

With the density of perilymph [ 1000 kg/m3 ] the maximum pressure decrease will be 72 mPa.

Not really a low value.

The animation of such a phase wave can be seen in:

http://www.a3ccm-apmas-eakoh.be/aobmm/bm-movement.htm

( The movements shown in several animations on the Internet, where the stapes activation creates ‘waves’ of higher frequency stimulus contributions which leave the core flow in the scala vestibuli and let the Reissner membrane and the basilar membrane simultaneously vibrate at a location nearer to the base, while from that location in the scala tympani a reverse ‘wave’ propagates toward the round window, is based on a hypothesis for which one cannot find a sound physics principle. )

And it really must be reminded: the fact that a mechanical vibration – and the sound stimulus is such a vibration – in a fluid, or in this case water like perilymph, will always propagate with the speed of sound, which has typically here the value of 1500 m/s.

That is just one of those constraints dictated by general physics. And with the equation that counts for the relation between sound velocity, frequency and wavelength we simply can calculate that for a 1000 Hz stimulus the corresponding wavelength in the perilymph is 1.5 meter. So approximately equal to 50 times the length of the active partition of the basilar membrane.

That is the only reason why the round window is moving in opposite direction related to the oval window. A widely reported always observed phenomenon in experiments.

And under the existing conditions in the cochlea there is no physics ground for so-called ‘slow waves’ with wavelengths even in the order of fractions of a millimeter. With the same equation for the relation between wave propagation velocity, frequency and wavelength as is used for the ‘fast’ running waves here above. Just because such a slow wave demands the propagation of a row of successively higher and lower pressure areas with sizes in the order of those wavelengths and even smaller. And that is impossible in general physics. The incompressibility of the perilymph fluid makes this impossible.

It cannot be that a mathematical wish of researchers for explaining the existence of a hypothesized traveling wave with a small wavelength prescribes that physics has to offer the possibility for such a slow wave. Just because the general physics rule prescribes that wave propagation velocity equals frequency times wavelength.

And therefore the only possibility that remains is that under the incompressibility constraint the entire perilymph fluid column between oval window – helicotrema – round window is moving as a whole, while it is stimulated by a mechanical vibration of the stirrup.

If we look closer to the basilar membrane properties, we observe that there exists a frequency-place related distributed resonance capability. With a subdivision that has a logarithmical scale from apex to base. High resonance frequencies near the base and low resonance frequencies near the apex.

Actually this unique property is the cause that a stimulus, that is equally present all over the length of the basilar membrane, evokes phase related movements which appear as a ‘wave’ that is always running from base to apex.

And it is this ‘wave’ phenomenon that is erroneously interpreted as the ‘traveling wave’ that transfers the sound stimulus.

And of course the perilymph fluid can be stimulated from both sides. Wever and Lawrence have reported that already in 1950. They reported that stimulating either the oval window or the round window results in identical cochlear potentials.

But that doesn’t imply that there have to run traveling waves in both directions. We can only draw the conclusion that a perilymph push-pull caused by a sound stimulus isn’t dependent on the pathway that is chosen.

Indeed: A direct measurement of velocity inside the cochlea is known as extremely difficult. So far every attempt fails, mostly because of the intolerable disturbances of the properties in the location that has to be examined. This makes the experimental results unreliable. And non-invasive measurements still show not enough details of fluid movements.

But from what really happens there we nevertheless can still make a reliable imagination, which is simply based on physics and the physiological properties and parameters which exist in the cochlea.

Let us make an inventory of them ... Or let us also look at:

Is the change in cochlear potentials dependent on perilymph velocity?

Edit/comment:

( The system said: You must have 50 reputation to comment. So this last edit here is a comment, because I could not post it as a comment on the answer of 'users/76352/theblackcat' , so I post my comment here on top in my own answer with this edit, but it is a comment on the answer of 'users/76352/theblackcat' . I do not have that 50 eputation to comment on his or her answer. )

The answer of 'users/76352/theblackcat' shows a cochlea cross-section picture with two membranes (Reissner's membrane and basilar membrane) and fluid-filled chambers. And there is a picture 'Basilar membrane unrolled'. I see in that latter picture the replacement of the scala media by just one single, flexible membrane remain (basilar membrane). An assumption and point of departure: the commonly accepted two compartment concept, in essence, ignoring the existence of the endolymph filled scale media between the two with perilymph filled channels, the scala vestibuli and the scala tympani. The latter assumption and point of departure is unavoidably an other hydrodynamic concept than in the three compartment cochlear model of the authors reffered in my answer.

So, from the basic ideas of wave propagation in the answer of 'users/76352/theblackcat' : primarily basilar membrane: stiffest part (high-frequency) at the base, while the widest, most flexible part at the tip.

About the proposed nature of the travelling wave in that answer answer of 'users/76352/theblackcat': the assumption that this wavy behaviour actually is a consequence the interaction of two energy sources: kinetic energy of fluid and elastic of the basilar membrane. However: within the oscillating column of perilymph fluid in the three compartment cochlear model of my answer: the authors in the paper, with the Bernoulli effect, they find an interaction between kinetic and potential energy (similar to Newton's laws), as a stimulus for the basilar membrane. The hydrodynamic stimulus introduction in the cochlea in the form of the sound energy signal (the perilymph velocity distribution and the perilymph pressure distribution), activating the motion of that basilar membrane by that non-stationary Bernoulli effect.

In my answer: according to those authors: The travelling wave along the basilar membrane can be described as a passive reaction of a complex organised second order resonance system, and is therefore a phase wave in the cochlea, that will always run from base to apex, completely in accordance with the referred experimental observations by Ren et al. Longitudinal pattern of basilar membrane vibration in the sensitive cochlea Proceedings of the National Academy of Sciences - pnas.org PNAS | December 24, 2002 | vol. 99 | no. 26 | 17101-17106.

And: If one assumes an active energy source to be present in the basilar membrane, one thinks of the cochlear amplifier. It is precisely this matter, the presence of such a mechanism that is an issue of great uncertainty in literature for which explicit experimental evidence is lacking.

End Edit/comment.

EDIT 1:

With a reference to: http://www.researchgate.net/publication/48323925_Applying_physics_makes_auditory_sense__a_new_paradigm_in_hearing There is a possibility that the observed movements of the basilar membrane could be caused by a phenomenon of a phase wave, what contributes to the discriminating mechanism for frequency selectivity, originated out of the manner in which the resonators in the basilar membrane are grouped.

The observed ‘waves’ always run from base to apex. It is conform to the mechanics of the peculiar basilar membrane system that this phase wave behavior is prescribed.

Exactly a mathematical solution

for this mechanics problem of resonators with a from low frequencies near the apex to high frequencies near the base logarithmically divided frequency distribution, in case of a pure sinusoidal tone,

provides for a tonotopical exact symmetrical envelope of that running phase wave with center frequency equal to the corresponding resonance frequency.

And the running direction of the phase wave is always from base to apex.

Everywhere along the basilar membrane  very local resonance with a high quality factor takes place. However not on the primary sound pressure signal, but on the sound energy signal.

And for the explanation of our hearing sense, in case of more sinusoidal tones, from all the distinguishable frequencies next of course to their frequency also their individual amplitude and phase are transmitted to the auditory cortex.

Our brain can directly compare the entire frequency selected sound energy stimulus with patterns that are stored in our memory.

EDIT 2a:

To explain the phase wave and its cause:

First, that cochlear fluid dynamics concept of the authors is based on the following points:

· Both perilymph and endolymph are [equal to water] almost incompressible. · Both fluids have viscosity coefficients not more than 3 times that of water. · The maximum displacement of perilymph in the cochlear duct doesn’t exceed 0.1 mm, mostly it is much less than that. · Reynolds number for flow due to acoustic stimuli are very low and will never reach the value for which laminar flow conditions change into turbulent conditions.

All these conditions together result in the possibility of solving the Navier-Stokes equation in the scala tympani and scala vestibule if one – what due to the above conditions and constraints is allowed here – takes the viscosity coefficient as zero. In that case the solution is completely analytical and results in the non-stationary Bernoulli effect:

The non stationary Bernoulli effect details: in may 2011: See: http://www.a3ccm-apmas-eakoh.be/NonstationaryBernoulli/non-stat-bern.htm

see: http://www.a3ccm-apmas-eakoh.be/figures/figures.htm see: figure 3

In the cochlear duct in every ‘slice volume’ formed by two cross sections perpendicular to the core axis at close distance from each other the decrease in pressure is proportional to the kinetic energy of the moving perilymph. And therefore also proportional to the sound energy of the evoked acoustic stimulus.

It means that the BM is stimulated with the rhythm and frequency components of the sound energy signal.

They use a potential flow concept (resulting in the non-stationary Bernoulli effect).

The ‘there and back’ travel of the footplate creates a movement of perilymph with a volume that is equal to footplate area times travel length. The travel length in the scala tympani is given by the ratio of footplate area over scala tympani cross section. There seems to be no obstacle for a smooth shape of the stream pattern in the potential flow. The embodiment of the cochlea is extremely stiff and the orientation of the stapes footplate has no influence for the potential flow.

And if one compares the length of the travel of perilymph in the scala tympani with the radius of curvature in the coiling of the cochlea there is such a large difference in scale between the two that in every ‘slice’ the perilymph movements can be seen as straight.

And in the figure 5 they calculate the phase wave:

A ‘phase wave’, that consists of coherent place dependent phase shifted local reactions to a stimulus that is simultaneously present throughout the basilar membrane.

The mathematical calculation of the authors shows for the logarithmically distributed local resonance frequencies fc of the basilar membrane, the response characteristic: a very restricted symmetrical local movement phenomenon, which travels along the basilar membrane.

EDIT 2b :

comment "... for the entire length as it tapers for all frequencies"

answer: On page 26 of the paper I read:

Another result becomes clear when we observe that in similarly shaped cross sections, the evoked signal at location x1, where for example the characteristic diameter of the cochlear duct is 2 times smaller than at location x2, the cross section is actually 4 times smaller. This results locally in a perilymph velocity that is 4 times higher, whereas due to the squaring of this velocity, the stimulus on the basilar membrane is 16 times stronger in accordance with the Bernoulli effect. This offers an explanation for the fact that the tapered shape of the scala tympani, from the round window to the helicotrema, is beneficial to sensing lower frequencies.

Answer 4

For our three compartment hearing sense, from a physics point of view, there is a basilar membrane stimulation, from base to apex, in its pathway in the cochlea, to a place on the basilar membrane.

By periodic movement of perilymph, non viscous fluid, backwards and forewards, in the cochlear duct meet the conditions of a potential flow.

The basilar membrane reacts in a quadratic way on that movements, according to the non-stationary Bernoulli effect. A pressure difference over basilar membrane exist.

The human hearing sense transfers the incoming sound into the sound energy stimulus everywhere in the cochlear duct by means of the non-stationary Bernoulli effect.

The omnipresent alternating pressure in the cochlea, being proportional to the sound energy signal, is the evoking stimulus to the basilar membrane.

Resonance phenomena according to the laws of physics forms the basis for the basilar membrane movements.

Both stimulation of reissner's membrane and basilar membrane exists by Bernoulli's effect.

A local wave be developed on the basilar membrane (from base to apex), (over a very restricted area, not over the entire membrane in case of a one frequency tone stimulus).

For details:

Let us observe the reaction of a spring-mass system on a periodic stimulus.

'mass' ; 'spring' ; 'point of impact stimulus on a periodic stimulus',

'forced movements of a mass - spring system', 'stimulus frequency', 'resonance frequency'.

Now consider the basilar membrane as an array of weakly coupled spring-mass systems.

With resonance frequencies logarithmically descending from base to apex.

This is a mechanical problem that can be calculated completely analytically:

Reference: Willem Chr. Heerens: Georg Von Békésy's traveling wave is a deceptive appearance interpreted as a traveling wave ; PDF First version: January 13, 2003 ; Second version: December 28, 2009.

There are indications in hearing literature:

Indication 1:

for the squaring of stimulus, Ernest Glen Wever and Merle Lawrence: The acoustic pathways to the cochlea, JASA 1950 July, 22: 460-467

changes in cohlear potentials 4 times as strong (when doubling of stimulus) as for single sided stimulus

Indication 2:

for the short traveling wave: Tianying Ren: Longitudinal pattern of basilar mebrane vibrations in the sensitive cochlea, PNAS USA 2002, 99: 17101-6

The detectable basilar membrane response to a low-level 16-kHz tone occurs over a very restricted (600 μm) area. Not over the entire membrane.

So, to continu:

For the three-compartment hearing model, (for our three compartment hearing sense)

Conclusion 1:

From a physics point of view two fundamental prosess steps are taking place:

1. The sound pressure stimulus is transferred to perilymph movement.

2. The perilymph movement is transferred by the Bernoulli effect to pressure decrease in front of the basilar membrane in scala tympani.

Conclusion 2:

Mathematically this means:

1. Sound pressure -> perilymph movement:

Differentiation

2. Perilymph movement -> decrease in pressure in front of basilar membrane:

Squaring

Conclusion 3:

All together:

The performance of the Stimulus on the basilar membrane over it's entire length is proportional to the Sound energy signal.

Final Conclusions:

After the meta analysis, and the developed insight in the functioning of the cochlea:

Conclusion:

The mammalian hearing sense transfers the incoming sound into the sound energy stimulus everywhere in the cochlear duct by means of the non-stationary Bernoulli effect.

Due to the resonance frequency gradient in the basilar membrane the omnipresent sound energy stimulus evokes a phase wave on it.

Finnally due to this omnipresent sound energy stimulus all frequency dependent contributions keep their phase relations without distortion.

Consequences for the sound stimulus transfer.

In case the sound pressure stimulus is transferred inside the cochlea to the sound energy stimulus.

There is a relation between sound energy stimulus and what we hear. See: Positions and corresponding terminology of "acoustics energy"?

Final conclusions:

The signal transfer to the sound energy stimulus results for a pure tone - one frequency - in a pure tone with doubled frequency, which evokes local resonance, and a stationary stimulus which is active all over the basilar membrane. (here in case of a one frequency tone).

The differentiation step in that transfer makes that all frequency contributions in the sound pressure signal with 1/f amplitude ratio's have equal contributions in the sound energy signal. (here in case of more tones).

With the squaring step in that transfer all primary present frequency contributions and in their combinations they evoke sum and difference frequencies. (here in case of more tones).

If I consider the comment of fall-apart:

To explain the behavior: as an epiphenomenon caused by a stimulus everywhere on the BM in combination with the resonance frequency distribution along the BM. High near the RW and low near the helicotrema. And that epiphenomenon is nothing else but a phase wave. The crowd on the gallery in a stadium produces the same type of phenomenon if they initiate a ‘wave’ by standing up and sitting down in an along the rows propagating phase order.

There is a electrical analogon for that basilar membrane movements:

For the description of the correct analogon, we have to consider that voltage E exist on each of the junctions like L1;L2; C2, so we have to short-circuit all the junctions to that voltage E in common. Now we have the analogon in which we have individual resonators existing of in series connected L and C impedances. If we have to introduce damping we have to add a resistor in series as well. On the nodes marked A1, etc., we will find the signal to the auditory nerve (see also Figure).

Such an analogon can be found nowadays in the museum of scientific instruments. It is the vibrating tongue frequency analyzer. On one massive stiff bar a large array of elastic tongues is mounted, each of them acting as individual mass-spring system, tuned from left to right in descending frequencies. (The tuning is of course not bound to that.) Stimulating the whole massive bar with a certain frequency will cause the vibration of that tongue having the same resonance frequency. Without damping and coupling all the other ones remain at rest. This because the quality factor of each of the resonating tongues is extremely high.

And that mechanical analogue of such a combination of the electrical quantities L, C and R is represented by a mass connected to a spring. So if you mount on a bar, next to each other, a series of leaf springs, which in one direction along the common beam: in each next case of the leaf springs: 'be slightly shorter in length', but where it is fastened with the same mass at the ends: it then behaves exactly, as a whole, as that series of combinations of L, C and R.

And if that beam, as a whole, into the right rhythm, will move back and forth, you'll see that these masses at the ends of the leaf springs, exactly as spectators in the stadium, will perform the "wave", who always "runs" from the side of the stiffest springs towards the side of the weakest.

The comparable signals A1; A2. . . etc., from the Figure, they waves as such as 'wave' like those springs with masses.

So far the electrical and the mechanical analogons. Inside that perilymph duct of the cochlea, due to the non stationary Bernoulli effect, the resulting rhythm and frequency components, stimulating the BM, is however the rhythm and frequency components of the sound energy signal. That is what I read in the answer of fall-apart: In the cochlea: Everywhere along the basilar membrane very local resonance with a high quality factor takes place. However not on the primary sound pressure signal, but on the sound energy signal. It means that the BM is stimulated with the rhythm and frequency components of the sound energy signal. The details of the sound energy signal, one can find in the answer of fall-apart. Because: I read in the answer of fall-apart: the pressure decrease in the perilymph duct in front of the basilar membrane is everywhere proportional to the perilymph velocity squared. (non stationary Bernoulli effect). Because: the sound energy signal is present inside the perilymph fluid: as a uniform pressure stimulus all over the volume. Because: Because the perilymph duct actually functions as a stream channel and the BM (basilar membrane) is on the wall of that perilymph duct in the scala tympani.

If I consider the comment of Floris on this page:

Is this a good summary? "Inside the ear there is a resonant structure - and each frequency excites a different part of that structure. Each resonant part stimulates a different nerve - that is how pitch is perceived."

So, also: "... - that is how pitch is perceived." ?

If I also look at that one complicated pitch perception example, namely, the complicated pitch perception example given by De Cheveigné

De Cheveigné A. (2005) Pitch Perception Models. In: Plack CJ, Oxenham AJ, Fay RR, Popper AN, editors. Pitch: Neural Coding and Perception: 169 – 233. New York: Springer Science + Business Media, Inc. ISBN 10: 0-387-2347-1.

the corresponding and resulting sound energy frequency spectrum, according to that new paradigm, can be calculated:

This is also shown in this figures.

And again if I consider the comment of Floris on this page: Is this a good summary? "Inside the ear there is a resonant structure - and each frequency excites a different part of that structure. Each resonant part stimulates a different nerve - that is how pitch is perceived."

For myself I colored it as follows:

Answer 5

Perhaps I could share some idea for further research.

If we could make actual and correct pressure measurements in the cochlea to reveal wether the non-stationary Bernoulli effect is a good description of the actual physics-of-how-the-cochlea-isolates-frequencies-along-its-length?

I would consider:

I would propose to use a pitot tube, with sensor in the side wall [ B in the next figure, left side in that figure ] to actual have correct pressure measurements in the perilymph flow tube inside the cochlea.

Therefore I would propose to use a pitot tube, with sensor in the side wall [ B in the figure, left side in the figure ] to actual have correct pressure measurements in the perilymph flow tube inside the cochlea.

So, I would propose to use a pitot tube, with sensor in the side wall [ B in the figure, left side in the figure ]

Answer 6

There is one more aspect of the main principles of operation to mention: - middle ear - - inner ear - namely the combination of the ear as an externally activated spring-mass system where the system functions as a damped mass-spring system activated by an externally generated sound signal and can be regarded as a system with amplifier, mass, spring, damping and 'resonance frequency' and an increase in perilymph velocity for frequencies lower than this 'resonance frequency' and with decrease in perilymph velocity for frequencies higher than this 'resonance frequency'. This whole aspect can also be taken in account when making calculations of velocity: the perilymph velocity as a function of frequency can be calculated by means of a standard solution in physics: the second order differential equation.

Namely, the combination of eardrum, ossicular chain, oval window, perilymph in the cochlea and, finally, the round window is a perfect example of a highly damped second order system with a relatively low 'resonance frequency', which is adjustable in amplitude.

Details: The cooperative functioning of the eardrum and ossicular chain attenuation can be regarded as the amplifier. The perilymph fluid column equals the mass. The combined elastic behavior of the eardrum, oval window and round window are represented by the spring constant, while damping is indicated.

Year Article Author(s) Source 2010 1 Applying physics makes auditory sense : a new paradigm in hearing Abstract | Full-text Heerens, W.C., Ru, J.A. de Medicine (2010), pp: 16 and Fig 2. on page 17.

The other operating principles in the hearing organ I will leave out in this answer for this moment. The other operating principles in the hearing organ one can find in the answers of fall-apart.

Thus here: To start with:

Second-order system: A second-order system is a dynamic system which can be modeled as two integrators with some negative feedback. The state of a second-order system can be defined with two numbers. A well-known example of a second-order system is a mass on a spring: one state is the velocity of the mass, the other the stretch of the spring relative to the equilibrium state. The negative feedback are the spring for the position and the damper for the speed.

Yes, and in a more general and simpler way one can say: A damped second-order system is a system able to move internally. That movement is characterized by the fact that the movement takes place around a position of rest or equilibrium position. The moving part makes displacements, and it is drawn back by a restoring force to the idle position or balance. Due to internal friction during that movement the kinetic energy is gradually converted into heat, thus heating of the parts, in which deformations occur. Because there is less and less kinetic energy, the movements gradually continues to become lower, which is referred to as damping.

And then: A mass connected to a spring and a damper, therefore, acts as such.

Randomly I can give three simple examples:

And because rubber has a high internal damping is a weight hanging from a rubber band already such a strong damped second order system. A bungee jumper, together with that elastic cords exactly is such a thing. But also a string of a guitar, only there is less damping and vibration sounds longer. But hearing that tone means that all the string energy is transfered to the surrounding air. And so, the air in this case is the damper.

And then a first point to consider:

• What makes: the combination of eardrum, ossicular chain, oval window, perilymph in the cochlea and, finally, the round window

is a perfect example of a second order system?

The three examples mentioned above (weight hanging from a rubber band) (a bungee jumper) (a string of a guitar) are that, in principle, also, only the combination in our hearing is formed in a natural way, while the three other examples have been created artificially.

And then a second point,

• What makes: the combination of eardrum, ossicular chain, oval window, perilymph in the cochlea and, finally, the round window

is a perfect example of a highly damped second order system?

That answer is obviously a lot more complicated.

Studies show that the inner ear can only selective distinguish good closely adjacent frequencies because there on the basilar membrane also composite resonance systems are arranged, very much parallel arranged, and at specific frequencies tuned, so on the basilar membrane also second order systems exist, but here they are aranged with very low damping. Because the stronger the damping, the worse the frequency selectivity. So on the basilar membrane there it is a sharp resonance responsive system.

But now: If, between ' 'the sound source' ' and ' 'the of a between 20 Hz and 20,000 Hz presented frequencies sharp resonance responsive system' ', somewhere between no strong damping would be present, then we would have a permanently heavily resonance troubled inner mechanism, to become with a lot of distraction and then on unbearable severe tinnitus to become deaf.

So nature has chosen to greatly highly damping here of the total system ( = in the combination of 'eardrum, ossicular chain, oval window, perilymph in the cochlea and, finally, the round window'). Then it will succeed.

And then a third point:

• What makes: the combination of the eardrum, ossicular chain, oval window, perilymph in the cochlea and, finally, the round window

is a perfect example of a highly damped second order system with a relatively low 'resonance frequency'?

Normally one makes microphones so that their lowest natural frequency, the lowest frequency at which the microphone - also a second-order system - can resonate, is clearly a lot higher than the highest frequency to be recorded and transmitted as a signal.

Then you certainly do not get in trouble with the sensitivity of the microphone to cover frequency range. However if you choose that frequency in the middle of that area, you know in advance that around and at that point resonance by the then much higher sensitivities there occurs with according problems with too high sensitivity.

But nature has - as is apparent from measurements - choosen for a selected 'resonant frequency' in the here above mentioned combination of that hearing system between 1000 and 2000 Hz.

Why (will you ask)? Because with this system than it is for frequencies in that area extremely sensitive. And then it holds for that particular system that all displacements of frequency 0 [very low] to the 'resonance frequency' of the combination system it can be raised up 'one on one'.

But this means that the velocity of movement increases proportionally to the frequency. So for perilymph velocity this goes up by 3 dB / octave.

Because of the strong damping of the total system, there arises no resonance peak, and the displacement at signals at frequencies higher than the 'resonant frequency' goes down by precisely 6 dB / octave.

However, pay attention that then the velocity which is associated with the latter displacement than decreases 3 dB / octave. This is because here exists inverse proportion of velocity with frequency.

In other words, for the velocity of the perilymph, for frequencies below the 'resonant frequency' it is true that the velocity increases in proportion to the frequency, therefore, increases by 3 dB / octave, but for frequencies higher than the 'resonance frequency' that velocity strength decreases, and then with 3 dB / octave.

After squaring, such as a particular hearing model indicates, the picture is as follows:

Around the 3000 Hz [doubling in frequency from 1500 to 3000 by squaring] delivers this system then to the signal on the basilar membrane a band [frequency] filter of the second order around 3000 Hz. This is because a band-pass filter of which the sensitivity in the flanks with 6 dB / octave rise, respectively, decreases by 6 dB / octave has the ideal characteristic for such a filter.

And to make this filter in its entirely wider and flatter, relatively little has to be adapted by nature in the cochlea in the resonance frequency arrangements and placement of its haircells.

Repeat: It's like this:

The piece of text:

--- But nature has - as is apparent from measurements - choosen for a selected 'resonant frequency' of that system between 1000 and 2000 Hz.

Why (will you ask)? Because with this system than it is for frequencies in that area extremely sensitive. And then it holds for that particular system that all displacements of frequency 0 [very low] to the 'resonance frequency' of the system it can be raised up 'one on one'.

But this means that the velocity of movement increases proportionally to the frequency. So for perilymph velocity this goes up by 3 dB / octave.

Because of the strong damping of the total system, there arises no resonance peak, and the displacement at signals at frequencies higher than the 'resonant frequency' goes down by precisely 6 dB / octave. ---

The explanation is as follows:

The displacement will remain from 0 to 'resonance frequency' [Hz 1000/2000] constant, independent of the frequency for a signal with constant amplitude, and sound pressure increasing frequency. But will the change of the displacement [so the perilymph velocity v] increase precisely proportional to the frequency.

In the logarithm scale it means - velocity proportionately increase with the frequency - which that velocity is increased by 3 dB per octave.

In a strongly damped second order system takes the displacement for the same signal with constant amplitude and still further increasing frequency higher than the 'resonant frequency', however, by 6 dB / octave down.

This means that the displacement inversely proportional to the square of the frequency decreases.

However, since the velocity with increasing frequency and constant signal amplitude still continues to increase in proportion to the frequency, this means thus that the perilymph velocity at frequencies higher than the 'resonance frequency' goes down just inversely proportional to the frequency.

[Instead of down by 1 / f^2 it is down by 1 / f]

And that means:

From 0 Hz to 1000/2000 Hz assumes a signal with constant amplitude, the perilymph velocity increases proportional to the frequency. [= 3 dB / octave]

Beyond that 1000/2000 Hz assumes that signal the perilymph velocity decreases inversely proportional to the frequency. [= -3 DB / octave]

Takes the square than in dB's, respectively, than with 6 dB / octave increase to point to the resonance, and then with 6 dB / octave decrease from above the resonance point.

And one final point:

- What makes: That is, the combination of the eardrum, ossicular chain, oval window, perilymph in the cochlea and, finally, the round window

is a perfect example of a highly damped second order system with a relatively low 'resonance frequency', which is adjustable in amplitude?

Yes firstly because the eardrum tension can be altered finally with about a factor of 30 if much sound is presented and then secondly the lever, which can be found in the ossicular chain structure, even again once such a factor of 30 it can change, you have to this system a total volume adjustment with a factor of 1000.

But by squaring in the cochlea this eventually becomes a factor 1,000,000 over the entire frequency spectrum. So a signal adjustment of 60 dB over the entire frequency range from 20 Hz to 20,000 Hz.

And it accomplishes all this with the characteristics of those three membranes (eardrum, oval window, round window), three tiny bones and two muscles.

[ Furthermore with perilymph fluid ( and a basilar mebrane ( also second order systems exist ) ). ]

That is a perfect example of a highly damped second order system with a relatively low 'resonance frequency', which is adjustable in amplitude.

To go into the actual physics of the system answers become very long and answers get comments, so I would just like to add: about the organ of Corti as a highly selective frequency analyser, about inner hair cells, about the arrays of outer hair cells and how they all are embedded in the basilar membrane and looking at the places where they are embedded, ... (of shear forces ) compared to ( stress forces when the basilar membrane moves away from the tectorial membrane, as a result of the under- pressure – caused by the Bernoulli effect – that is evoked in the scala tympani through the back and forth movement of the perilymph ), so: instead of shear forces that affect the hair bundles of the inner hair cells, stress forces are generated that affect the outer hair cells.

And about the comments here: Physics of how the cochlea isolates frequencies along its length?

and even about the Physics of how the cochlea isolates frequencies along its length? so, also even about the missing fundamental there are indications that indicates it can be mechanical.

I am affiliated with this book. I am co-author of the Appendices.

Year Article Author(s) Source 2010 1 Applying physics makes auditory sense : a new paradigm in hearing Abstract | Full-text Heerens, W.C., Ru, J.A. de Medicine (2010), pp: 1-74

"Mathematically, this signifies that the mammalian cochlea differentiates and squares the incoming sound pressure signal. In terms of physics, it means that a sound energy signal is offered to the organ of Corti. Functioning as a Fourier analyzer, the organ of Corti subsequently converts these incoming signals into the sound energy frequency spectrum that is transferred to the auditory cortex in a frequency selective way." relates to: Applying physics makes auditory sense : a new paradigm in hearing - ResearchGate. Available from: http://www.researchgate.net/publication/48323925_Applying_physics_makes_auditory_sense__a_new_paradigm_in_hearing [accessed May 29, 2015]. Page 40

5. The organ of Corti as a highly selective frequency analyser.

The common hypothesis is that only the inner hair cells are expected to generate signals that reach the auditory nerve. This hypothesis is based on the idea that the basilar membrane moves towards the tectorial membrane in reaction to the occurring waves of pressure, higher than the ambient pressure that normally exists in the cochlear duct (scala tympani and scala vestibuli). The long-held assumption is that this movement of the basilar membrane causes the development of shear forces that stimulate the inner hair cells. Whereas, the inner hair cells are very sensitive to these shear forces, according to Hudspeth et al. [ ]. In our functional concept however, the pressure that is evoked in the cochlear duct (scala tympani and scala vestibuli) is not a wave, but an overall pressure that is lower than the ambient pressure. This merely causes the basilar membrane to bend away from the tectorial membrane. Therefore, the presumed development of shear forces that could stimulate the inner hair cells is clearly out of the question.

The tectorial membrane lies completely enclosed in the scala media and apart from its connection to the stiff bony center axis of the cochlea; it is surrounded by endolymph fluid at rest. When its morphology and ultra structure are taken into consideration, the tectorial membrane is more likely to function as a relatively non- deformable position reference. Consequently, the tectorial membrane cannot be a moving object. The arrays of outer hair cells are embedded in the basilar membrane at those places along the core spiral of the cochlear duct, where due to pressure stimuli the largest local displacements of the basilar membrane are to be expected.

Moreover, the top of a hair bundle that is part of the outer hair cell is anchored in the tectorial membrane. Therefore, these hair bundles undergo stress forces when the basilar membrane moves away from the tectorial membrane, as a result of the under- pressure – caused by the Bernoulli effect – that is evoked in the scala tympani through the back and forth movement of the perilymph.

Because the electrical current inside the hair cells is responsible for evoking the signal contribution in the auditory nerve, the direct interconnection of every ten afferent axons of nerve cells that are connected to the outer hair cell is such that a parallel switching of these electrical current sources is attained. Careful experiments may prove that indeed, the outer hair cells cooperate in order to evoke a much stronger combined signal to the auditory cortex. Moreover, this parallel switching leaves the organ of Corti less vulnerable to individual hair cell collapse or local damage. 40

Page 41

As stated, the Reissner membrane and the basilar membrane bend away from the tectorial membrane and therefore, instead of shear forces that affect the hair bundles of the inner hair cells, stress forces are generated that affect the outer hair cells. The inner hair cells, however, are situated along a strip close to the edge where the basilar membrane is attached to the solid wall. This means that the inner hair cells are located in places where they can hardly be affected by vibrations or deflections of the basilar membrane. Thus, apart from the relatively small displacements that are mainly induced by unwanted mechanical distortions, the inner hair cells probably only evoke signals to compensate for the influence of these distortions of the signal. Therefore, the number of inner hair cells can be much smaller than that of the outer hair cells, reduced by a factor 10, actually.

Moreover, we would like to introduce the following contribution; an article by Leibbrandt [ ] that was first published in 1966, which only very recently came to our attention. During our contact with the author he informed us that this article received very little acknowledgement or response from the scientific community at that time. In our opinion, a significant injustice to this paper and the findings presented therein. We would, therefore, like to share the entire abstract from this publication with you.

--- In guinea pigs cochlear microphonic responses were studied during stimulation with harmonic high tone complexes. In the apical portion of the cochlea a sine wave with the frequency of the "missing fundamental" could be recorded. The amplitude of this low frequency microphonic potential (CM) in the third turn of the cochlea appears to be about equal to the amplitude of the strongest component of the high tone complex recorded in the basal turn. The "missing fundamental" appears to stimulate the apical portion of the cochlea, which indicates cochlear analysis according to a certain place principle, although apparently not the Fourier principle.---

In this publication Leibbrandt reports that aside from the presence of the missing fundamental, he also found the performance of the sum frequencies of the sound pressure stimuli.

The experimental results that were reported in this publication clearly confirm our assertion. The electronic signals that are evoked in the organ of Corti closely resemble the in frequency components split sound energy signal. 41