Hair Cell Biophysics, Semicircular canal mechanics and VIII nerve responses
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Reading
Hair Cells:
A.J. Hudspeth:
"The hair cells of the inner ear", Scientific American, Jan 1983: 54-64
"The cellular basis of hearing: The biophysics of hair cells", Science 230: 745-752 (1985)

Nature 428: 901 (2004) review by Hudspeth's student David Corey, now at Mass General, "Tightrope Act."

Semicircular Canal (SCC) Mechanics: Victor J. Wilson & Geoffrey Melvill-Jones, Mammalian Vestibular Physiology, Plenum Press, 1979.
on reserve at Sci. Li., [QP471/W54]. Chapter 3, "Biophysics of peripheral end organs."

RHSC2 chapter 2: "Vestibular eye movements".

Outline:
Anatomy & physiology hair cells & semicircular canals

Anatomy & structure of hair cells

Second order mechanics of hair cell deflection in the SCC

Hair cells can be found in the vertebrate cochlea, otolith, lateral line, and the vestibular apparatus. In the semicircular canals (SCC) of the vestibular system the hair cells are embedded in the cristae and protrude into the cupula, which spans the canal at the enlargement of the ampula. In the canal is endolymph fluid. The basal side of the hair cells contact by synapses fibers of the VIII nerve. VIII projects into the brainstem, to the vestibular nuclei. About 20,000 of the VIII fibers are due to the vestibular apparatus.

Viewgraphs from Hudspeth 1983:
anatomy of canal
ampulla widened region of canal
cupula the gelatinous mass + hairs
cristae "special region," like the maculae. it's the rise in the ampula
3 canals on each side: six all total!

For the vestibuloocular reflex we will concern ourselves with the two horizontal canals

Cilia on apical side of hair cell
A close-up of a hair cell shows that it extends from the endolymph to the basal membrane. The set of hair cells and support cells forms a complete barrier between the two realms (endolymph and synaptic contact as base of hair cells). On the endolymph side each hair cell has about 50 cilia in a hexagonal array, forming a staircase assembly projecting up into the cupula. The array of cilia has an orientation, because of one special stereocilium on one end. Of the thousands of hair cells in any one cupula, all are oriented in the same direction with respect to their stereocilia. It is the deflection of the hairs on the apical membrane that is the mechanical event which causes the inside of the hair cell to change voltage, and this changed voltage in turn causes more or less neurotransmitter to be released onto the awaiting VIII fibers at the other end of the cell.

Hudspeth was able to isolate single hair cells and impale them with glass pipette recording electrodes. He measured intracellular voltage change as a function of mechanical deflection.

Endolymph and hair cell resting potential
The endolymph is a special fluid with a high potassium ion concentration, similar to the intracellular potassium ion [K+] in the cytoplasm of the hair cell itself. The other side of the hair cell, where synaptic contact is made with VIII nerve fibers, has a membrane encountering a more standard low-K extracellular fluid. Because of this basal membrane concentration gradient, there is a Nernst potential of (say) -90mV internal.

[See en123 website, http://www.engin.brown.edu/courses/en123/Lectures/electrodes.htm lecture notes on Electrodes, for more details. Qualitatively, imagine that K+ ions diffuse from the inside to the outside, through membrane channels, leaving a deficit of + charge on the inside of the cell].

In order to have a resting potential of about -60mV, there would need to be about twice the conductance at the basal membrane as there is at the endolymph membrane.


The mechanism of transduction
Only to the extent the central cilia are deflected along their axis with the lone stereocilia does the haircell voltage change. If the cilia are deflected at right angles to that axis, the hair cell intracellular voltage does not change at all. This result is a clue about how the transduction takes place!


image below clipped from www.du.edu/~kinnamon/3640/ hearing/hearing.html
in turn copied from Levitan and Kaczmarek, The Neuron, Figure 13-4...

As you can see from other diagrams in Hudspeth's papers, current flow during transduction is largest when sensed near the tips of the cilia, not when recorded at the base of the cilia where it contacts the hair cell "body". The hypothesis is: Protein filaments stretch from a longer cilia to it's "downstream" shorter cilia neighbor. When the bundle is deflected in the direction of the stereocilia, the filaments extend further and mechanically open channels that allow K+ to flow. The hair cell potential becomes more positive (depolarized). See diagram above. Such filaments have been seen at the electron microscope level.

DEMO:
[It is a change in the viscosity of endolymph that is a contributing factor in the dizziness of inebriation. Enough EtOH mixes with the endolymph to lower its viscosity.]

SCC mechanics: Time constants
First consider the endolymph fluid in the canal, as the canal begins to rotate about its axis. Because of its rotational inertia, the fluid will attempt to remain in place, but will be dragged along by viscous resistance of the canal wall. Without for now worrying about the hair cells and cupula being in the path of flow, what will be the time constant of the fluid reaction? We have the following equation:

where h(t) is the position in time of head, and θ is the position of the fluid w.r.t. the canal reference frame. In Laplace transform terms we have

where an integration reduces the order of the equation. I/B is the time constant, as could be concluded from prior work on first order diff. equations...remember T=B/K in the derivation of the mechanical load of the eye.

We can calculate the rotational inertia of the fluid in the canal, assuming a density of ρ=1, with r the inner radius and R the diameter of the canal as a whole. Reasonable values for r and R are 0.16 mm and 3 mm.

On p. 47 Wilson & Melvill-Jones(1979) derive drag coefficient B
of the whole canal length 2*pi*R, to be

ending up, with no dependence on the internal diameter r of the canal: see details below:

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Details of Wilson p47 derivation of fluid drag coefficient:
Start with the Poiseuille-Hagen equation, which calculates the average "volumetric flow" V (units are cm^3/sec) of a fluid with viscosity η in a pipe of radius r, and length L, which experiences a pressure drop ΔP,

Wilson recasts this equation in terms of force = pressure*area, coming up with

then Wilson wants the "turning moment" (torque) of this force:

B, the viscous drag coefficient, will end up in an equation T = B*Ω, so we must extract Ω:
First, convert discharge J to velocity, with units cm/sec.

where dividing by the area of the canal leaves average "linear" velocity of the fluid.
Since there are 2π radians per cycle, then one radian is R length and that R will covered at a rate
Ω = v/R
radians per seconds. Therefore

Now we can compute B:

and (curiously) small r has disappeared from the formula...
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Given the viscosity η of endolymph as 0.01 poise
(poise units are dyne-sec/cm^2, again, from Wilson & Melvill-Jones, Table 2.1), we find

Net rotation of endolymph in canal, due to head rotation.
If the head rotation in radians, as a function of time, is described by h(t), and the lagging rotation of the endolymph in the canal is θ(t), then the net rotation of the endolymph in the skull is h(t) - θ(t). Considering the viscous resistance of the endolymph in the canal, B, and the resisting stiffness K of the cupula in the ampula of the canal, we have this differential equation, and its "solution" in Laplace transform terms:

You can see one of the time constants, I/B, appears as the coefficient of s in one form of the denominator.

Torsional spring stiffness of cupula and two time constants
How do we account for the value of the torsional spring stiffness of the cupula (wherein lie the hair cells) in a second order equation? We know that the cupula reacts to and resists the endolymph pressing against it during rotation. But we have no straightforward way to calculate or measure directly the cupula stiffness. What we can do is observe the response of the canal (by recording activity on the VIII nerve) to a stimulus that isolates the time constants. What stimulus h(t) has a Laplace transform that is 1/s^2? Recall the Laplace transform of the impulse is 1, the Laplace transform of the integral of the impulse, the step, is 1/s and the integral of the step: the ramp has Laplace transform 1/s^2. So we drive the canal with a ramp function of rotation.

In the example below we illustrate what happens if s1 = 10 and s2 = 1, resulting in a denominator of

Let's use s1=10, s2=1. In such a case s1+s2 ≈ s1, approx. Try the following Simulink model:

The scope shows:

Where you can see the fast time constant (1/10) creates the rising edge and the slow time constant (1 sec) dominates the falling edge of the waveform response to the ramp. Notice the response returns to zero, since the ramp has a zero second order derivative. What we have in the SCC is an accelerometer: responding to rotational acceleration of the head.

Experimental results from VIII recordings suggest the other time constant is much greater than 3msec, maybe about 3 sec, or 1000x larger. We can derive K by the following, assuming s1 >> s2:

But really all we care about for purposes of simulation are the two time constants, 3 msec and 3 seconds...

Bode plot
What is the frequency response of the VIII nerve transfer function? As suggested in Assgn 1B, you can type into the command line of Matlab, or into a Matlab script, a frequency range and the exact complex number formulation of the transfer function and ask Matlab to plot. Below is a version of such a plot where the time constants used were 3 msec and 5 sec. (script Bode00.m)

Notes: overall the SCC is a high pass filter. Script Bode00 has arranged that the
high frequency gain = 0 dB = gain of 1.00
At v. low freq's the slope of the curve is +40dB/decade, indicating a second-order system (accelerometer) while in the "physiological range" of rotation frequencies the slope is +20dB/dec: the range of the "velocity sensitive accelerometer". It is the remaining s in the numerator of the transfer function in the physiological range that the neural integrator in the final common path (OMN circuit) is designed (evolved) to compensate.

Question: can you add to the Bode00 script to generate the phase vs frequency plot?

Summary
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