Re: A new paradigm?(On pitch and periodicity (was "correction to post"))

[Willem Christiaan Heerens <heerens1@xxxxxxxxx>]

Peter van Hengel, Dick Lyon and other contributors, my apologies for the 
late response to your contributions to this topic. 

However in my opinion this subject is still far away from a satisfactory 
result. And I prefer to give you a clear and well overthought reaction.

Therefore let me first give you a comment on Peter?s following reaction to 
Matt Flax:

** One small comment to your statement that there are only forward 
traveling waves. I think much of the confusion stems from the fact that 
the observed motion of the cochlear partition is often referred to as the 
traveling wave. In actual fact this is only a 'reflection' of the actual 
wave which is traveling in the fluid. The fluid supports waves traveling 
in both directions, but the unique properties of the cochlea cause only 
the appearance of a forward traveling on the cochlear partition.**

Yes indeed that is the explanation for the ?traveling wave hypothesis? 
within the common paradigm. And if you want to explain a reported 
apparently observed ?traveling wave? over the basilar membrane, running 
from the base ? near the round window ? to the apex ? near the 
helicotrema, with a mathematical model designed by you, of course it is 
legitimate to try this.  

However in the background of all the exercises there remains to be 
incorporated a constraint that cannot be ignored or disobeyed: the basics 
of such mathematical models must be in full agreement with the general 
rules and laws of physics.

I seriously hope that the AUDITORY List members agree with me that there 
doesn?t exist ? beside general physics ? a special cochlear physics theory 
with its own laws and rules, which are different from and occasionally 
even at variance with the general ones. If not, we can end our discussion 
right away, because we then can only agree that we fundamentally disagree.

But let me comment on Peter?s statement about the observed basilar 
membrane traveling wave: 

**In actual fact this is only a 'reflection' of the actual wave which is 
traveling in the fluid.**

I really must remind you to 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 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.   

Peter you further stated:
**The fluid supports waves traveling in both directions, but the unique 
properties of the cochlea cause only the appearance of a forward traveling 
on the cochlear partition.**

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.

Peter, you made the remark:
** If one wants to observe the reverse traveling waves in the cochlea it 
is necessary to measure fluid velocity, which I believe is not yet 
possible.**

No 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.

The physiological structure data:

?	The walls of the cochlear envelope are extremely rigid. Hardly 
compliant to not compliant at all. So bone conduction based on deformation 
of that envelope is not possible.
?	The cochlea shaped cavity is subdivided into the perilymph filled 
duct, which is folded at the apex and which parts are the scala vestibuli 
between oval window and apex and the scala tympani between apex and round 
window.
?	In between these two scalae the third one ? scala media ? is 
located, filled with endolymph.
?	The partition between scala vestibuli and scala media is formed by 
the Reissner membrane. This membrane is extremely thin, but on all 
available electron microscope pictures it is observed as straight, except 
for Ménière cochlea, where it is curved into the direction of the scala 
vestibuli. This membrane has compliance.
?	The partition between the scala tympani and the scala media is 
formed by the basilar membrane. This membrane has substantially more 
volume, while both the  inner and the outer hair cells are embedded in it. 
This membrane has a place located frequency dependent compliance. It can 
be observed as a resonance device.
?	The hair bundles of the outer hair cells are at their top 
connected with the tectorial membrane, a rim structure that is completely 
located in the scala media and connected with the bony envelope of the 
cochlea.
?	At every location along the cochlear partition the cross sections 
of scala tympani and scala vestibuli are practically equal in size. In 
average the channel diameter is 0.3 mm.
?	There exists a tapered shape in the perilymph duct, larger at the 
base to smaller at the apex.
?	The maximal deflections of the oval window and the round window 
are estimated to be in the order of a few micrometers.
?	Deflections of the basilar membrane also do not exceed a few 
micrometers. Otherwise the hair bundles of the outer hair cells would be 
damaged due to overstressing.   
  
The involved material quantities:

?	The perilymph fluid in the scala vestibuli and scala tympani as 
well as the endolymph fluid in the scala media has a density equal to that 
of water. So 1000 kg/m3.
?	Both fluids are incompressible and have a low viscosity, 
comparable with water. This can be considered in practice as ?viscous 
free?.
?	The propagation velocity of acoustic vibrations in the perilymph 
is 1500 m/s. 

Cardinal fluid dynamics numbers:

?	The main criterion for non-turbulent fluid flow in the cochlea is 
the Reynolds number. Calculation in case of the highest hearable frequency 
stimulus of a 20 kHz vibration with an amplitude of 1 micrometer with a 
dynamic viscosity coefficient equal to that of water: 0.001 Ns/m2 gives 
for this Reynolds number a value of 36. For vibrations with lower 
frequencies and with  similar amplitudes this Reynolds number is 
proportionally lower. 

Hence the Reynolds number is for all situations far below the boundary of 
2000, the value that counts for upper boundary of the laminar flow 
conditions. And therefore the perilymph flow inside the cochlea is 
definitely non-turbulent. 

All these aspects together result in the fact that it is allowed to 
consider the perilymph movement inside the cochlea as a periodic movement 
that can be theoretically expressed by the non-stationary Bernoulli 
equation. Just as I have done in the attached PDF in my message to the 
AUDITORY List of Saturday, October 1, 2011.

For those of you who think that I misuse the Navier-Stokes theory in the 
case of the cochlear fluid dynamics I can point to the following 
elucidating introductory explanation placed on-line on the Internet by the 
Academic Medical Center of Amsterdam that I cite here:

-- The Navier-Stokes equations are a set of equations that describe the 
motion of fluids (liquids and gases, and even solids of geological sizes 
and time-scales). These equations establish that changes in momentum 
(acceleration) of the particles of a fluid are simply the product of 
changes in pressure and dissipative viscous forces (friction) acting 
inside the fluid. These viscous forces originate in molecular interactions 
and dictate how sticky (viscous) a fluid is. Thus, the Navier-Stokes 
equations are a dynamical statement of the balance of forces acting at any 
given region of the fluid.

They are one of the most useful sets of equations because they describe 
the physics of a large number of phenomena of academic and economic 
interest. They are useful to model weather, ocean currents (climate), 
water flow in a pipe, motion of stars inside a galaxy, flow around a wing 
of an aircraft. They are also used in the design of aircraft and cars, the 
study of blood flow, the design of power stations, the analysis of the 
effects of pollution, etc.

The Navier-Stokes equations are partial differential equations which 
describe the motion of a fluid, so they focus on the rates of change or 
fluxes of these quantities. In mathematical terms these rates correspond 
to their derivatives. Thus, the Navier-Stokes for the simplest case of an 
ideal fluid (i.e. incompressible) with zero viscosity states that 
acceleration (the rate of change of velocity) is proportional to the 
derivative of internal pressure. Poiseuille?s Law and Bernoulli?s equation 
are special cases of 1D Navier-Stokes.

The fluid motion is described in 3-D space, and densities and viscosities 
may be different for the 3 dimensions, may vary in space and time. Since 
the flow can be laminar as well as turbulent, the mathematics to describe 
the system is highly complex. 

In practice only the simplest cases can be solved and their exact solution 
is known. These cases often involve non turbulent flow in steady state 
(flow does not change with time) in which the viscosity of the fluid is 
large or its velocity is small (small Reynolds number). 

For more complex situations, solution of the Navier-Stokes equations must 
be found with the help of numeric computer models, here called 
computational fluid dynamics. 
Even though turbulence is an everyday experience it is extremely hard to 
find solutions for this class of problems. Often analytic solutions cannot 
be found.
Reducing the models to 1D, as is often done in fluid dynamics of blood 
vessels, makes the problem handsome. --

[See also:  http://onderwijs1.amc.nl/medfysica/compendiumDT.htm edited by 
N. A. M. Schellart 2005]

After studying my PDF that I sent to the List on October 1 2011, you can 
see for yourself that I have derived the analytical solution for the non-
stationary non-viscous incompressible time dependent wiggle-waggle 
movements directed along the core of the perilymph duct. Because in that 
case the reduction of the complex set of Navier-Stokes equations to the 
non-stationary Bernoulli equation is fully permitted.

And this finally results in the fact that everywhere inside the perilymph 
duct the evoked pressure variations are proportional to the sound energy 
stimulus.

This means that by resonance in the basilar membrane, i.e. the frequency-
place related distributed resonance capability, the stimulus can evoke 
simultaneously all the frequency contributions of the sound energy signal, 
including the exact phase relation for each contribution, which will be 
sent to the auditory cortex.
Further details I will give in my answers to the comments of Dick Lyon 
here below.

Finally Peter you reacted to Matt Flax with:

** Model calculations clearly show the reverse traveling wave and produce 
results in accordance with data on OAEs (see e.g. the work of Mauermann et 
al or Epp et al). **

Yes that is based on the ?transmission line model?. And of course that 
will present a reverse traveling wave. The entire model that is used is 
based on the hypothesis that there must exist and consequently must be 
explained a traveling wave whatsoever. 

However that is the cardinal subject of difference between your statements 
and mine.
In my concept, based on the Bernoulli solution, there still exists a ?wavy 
motion? on the basilar membrane, but that is the result of phase dependent 
behavior. 
Locations with higher resonance frequencies react in phase with the 
frequency stimulus; at resonance locus this motion is 90 degrees behind in 
phase; locations with lower resonance frequencies react 180 degrees 
retarded in phase.
And because higher resonance frequencies are found near the round window, 
while lower resonance frequencies are found near the helicotrema 
the ?wave? always runs from base to apex.  And as Tianying Ren et al. also 
experimentally detected: there doesn?t exist a reversed traveling wave.


And now my reactions to the comments of Dick Lyon:

Dick you remarked:
** Willem, your approach in which "the flow behaves as a parallel 
streaming oriented along the core of the perilymph duct" and in 
which "there exists only a contribution in the x-direction" is what might 
be called a "non-compliant membrane" approximation.**

In strict theoretical terms you could name it as such, but in practice it 
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.

You commented also with:
** Generally, the BM is interpreted as being variably compliant (and the 
RM very compliant), such that there is some velocity (and pressure 
variation) ortogonal to the x dimension, which corresponds to BM 
displacement. **

Regarding the first part of it, the compliance of the membranes, I agree 
with you. And I have also used this frequency dependent ?compliance? of 
the basilar membrane in my 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.

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.

You commented:
** If you assume no BM displacement, then of course you have no traveling 
wave. **

Nowhere in my explanation have I stated that basilar membrane mobility 
doesn?t exist.

Of course there exist basilar membrane displacements. But their influences 
on the local cross section of the perilymph duct are rudimentary.

And for clarity let us make an indicative calculation: 
The average diameter of the perilymph duct is 0.3 mm. And let us assume 
the cross section to be circular. Then the size of the surface can be 
calculated as 0.0706 square mm.  The local deflection of the basilar 
membrane cannot be much larger than a few micrometers. Otherwise the hair 
bundles of the outer hair cells would be overstretched or even disrupted.
The width of the basilar membrane is approximately 0.1 mm. If that is 
displaced 1 micrometer over its entire width, the corresponding surface of 
that cross section is 0.0001 square mm. 

This means that the cross section at the place of the membrane deflection 
is relatively 1.4 pro mille decreased. So you cannot maintain that this 
will have a serious influence on the main fluid movements in the core 
direction of the perilymph duct.

You commented further:
** The BM (or the whole of scala media in your approximation) separates 
the two parts of the folded duct in which you have a longitudinal pressure 
gradient, so there will be a substantial pressure difference across it, 
from the far-apart x locations (except near the apex where it folds). **

I am sorry that I have to tell you, but with this statement you show that 
you do not really understand the general mechanism of the potential flow.  

And I can explain this 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 non-viscous 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 you suggested in your comment ? 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 our booklet ?Applying Physics 
Makes Auditory Sense?.

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 I cannot find a sound physics principle. 

[See for instance: http://www.blackwellpublishing.com/matthews/ear.html  ]

Dick, you commented also:
** If you allow the pressure across the BM to deflect it, as we usually do 
with membrane compliance, you get a very different analysis, based on the 
same physics but different mechanical approximations.  In this analysis, 
the v-squared pressure differences due to Bernoulli's law are generally 
very small compared to the pressure differences accelerating the fluid 
within the short wavelength of the traveling wave, so are neglected. **

Let me 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/m^3 ] the maximum pressure 
decrease will be 72 mPa. Not really a low value.

Another fact is that we also have to cope with the problem that the 
pressure differences accelerating the fluid within the short wavelength of 
the traveling wave ? that would exist indeed in the ?transmission line? 
hypothesis and in the companying mathematical model, which calculates 
these hypothesized effects ? have no physics ground to exist in the wiggle-
waggle movements of the perilymph.

Finally you commented:
** Which approximation is better?  Probably the one that yields a 
traveling wave like the one seen in direct mechanical measurements, I 
think.**

Already years ago I corresponded with Tianying Ren, the man who did this 
kind of mechanical measurements. On his request because those measurements 
did not fit well within the ruling traveling wave paradigm. Actually they 
were flawed in that  time by the auditory peers. 
In 2008 De Boer et al. also reported that there were serious doubts about 
the hypothesized existence of reversed traveling waves. Only 
forward ?waves? on the basilar membrane could be observed. He also 
suggested that in the first place there must be found an explanation that 
can dispose this newly arisen anomaly. 

The observed ?wave phenomena? on the basilar membrane resemble the forms 
that are shown in the following animations, which can be downloaded from 
the Internet:

	http://lab.rockefeller.edu/hudspeth/graphicalSimulations

or:

	http://www.youtube.com/watch?v=dyenMluFaUw

These simulated animations also resemble very well with the phase wave 
phenomena which I have calculated and which I have presented in brief in 
our booklet for one sinusoidal stimulus. 
The traveling wave you refer to in your comment here above is by far not 
an accomplished fact.

Therefore I hope you will agree with me that the best theoretical 
description of the functioning of our hearing sense is the one that is in 
the first place in agreement with physics, describes and explains the 
experimental findings very well even in detail, can cope with the existing 
anomalies, and can predict correctly in detail new and until now unknown 
hearing and auditory perception phenomena.

And I am convinced that under these requirements our hearing paradigm is a 
very serious candidate. 


Kind regards,

Pim Heerens