This image is adapted from Bioengineering: Biomedical, Medical and Clinical Engineering by A. Terry Bahill, Prentice Hall.
Dynamics of the inner ear
or
Why you get dizzy
The Vestibular Organ
This image is adapted from Bioengineering: Biomedical, Medical and
Clinical Engineering by A. Terry Bahill, Prentice Hall.
The inner ear has three vestibular organs, one of which is shown. These organs have a radius of a little less than a millimeter. They are used by the brain to sense movement of your head. You can demonstrate this for yourself by movieng your head from side to side. As you move your head the output of the vestibular organ (from the afferent nerve) is processed by the brain, and the eyes reflexively move in a compensatory way such that your gaze remains fixed. This even happens when yours eyes are closed, so it is not visual clues that are being used by the brain.
The inner ear also lets your brain compensate for movements of the body in visual processing. As you move your head (and your eyes with it - for example when walking), even though the image on your retina is moving, your mind correctly perceives that you are moving, and that the world is fixed. A large part of the information about your movement is taken from the vestibular organ. To see this in play, try moving your eye gently with your finger, as your eye moves (with no compensating signal from the vestibular organ), it seems as if the world is shaking. Without feedback from the vestibular organ, your visual perceptions fail.
The Model
In our model of the inner ear, we will define y as the position of the head (and thus the vestibular organ), q as the position of the fluid within the organ, and f as the deflection of the cupula. As the head turns (y), the fluid will tend to remain motionless (q), and thus the cupula will be forced to bend (f). We will call the deflection of the cupula the difference between the position of the head, and the position of the fluid, f=q-y, or q=f+y.
We can write equations of motion.
or, making substitutions we can write,
If we call the acceleration term
then we can write
The restoring force of the spring is very weak, and the fluid is very viscous, so the system is highly overdamped, which means it has two real roots that are quite far apart.
The Solution
Consider a typical turn of the head which lasts for one half second, we can plot the velocity and acceleration of the head:
Since the system is linear, we are not concerned with the exact magnitude of the input, so let's assume that the acceleration is a pair of impulses:
We want to plot the position of the cupula as a function of time. If we assume we start at rest, we have a zero state problem, and we can start by taking the Laplace Transform of the differential equation (with zero initial conditions).
If the system is overdamped we can factor the characteristic equation into two real roots:
Since the system is highly overdamped, one of the roots of the characteristic equation is much longer than the other
Experiments have shown that:
We can now begin the solution:
We can do a partial fraction expansion of both terms (which will have identical coefficients)
so
and
The graph below shows cupula deflection versus time for the example described (impulse and delayed impulse for head acceleration). Note that the deflection of the cupula is at first very rapid, this is the fast time constant. The deflection then slowly goes towards zero, this is due to the (small) restoring force and is caused by the slow time constant. In this case the deflection of the cupula is very nearly proportional to velocity - we can say that the cupula is acting as a velocity transducer. Note: don't worry about the vertical scale, it is totally arbitrary at this point.
Why we get dizzy
Now consider the case where you spin at a constant velocity for one minute, and lets plot the deflection of the cupula. Note the much larger time scale of this image.
For the first sixty seconds, you would be spinning at a constant velocity. You spin for long enough that the small restoring force of the cupula is enough to bring it back to its rest position. For the next sixty seconds, you are still, but your cupula tells you that you are spinning in the opposite direction - you are dizzy. If you look at a person's eyes when they are dizzy, you will see that they are going back and forth. Their inner ear is trying to keep them focussed in one direction, but the inner ear thinks the person is still spinning and hence must move the eyes to keep them focussed in one direction.
Another explanation of the velocity transduction.
I stated earlier that the inner ear acts as a velocity transducer, and we showed how this works if the head only spins for a short period. Let's see how the math works out.
Consider only short time scales. Over these time scales, we can assume the spring has no effect, or k=0. Therefore we can rewrite the differential equation:
Recall that multiplication by 1/s in the Laplace domain is equivalent to integration in the time domain, thus we can rewrite the right hand side of the equation as a velocity term:
This makes the system a first order system with a velocity input:
From this we conclude that for short time scales the inner ear should act like a first order system that responds to the velocity of the head. This behavior can be seen from the first graph of cupula deflection which looks a lot like a first order response to the velocity input. The input and output graphs are shown below for convenience.
For a more complete discussion see Biological Control System Analysis, by J.H. Milsum, McGraw Hill, 1966.
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