Sensory Systems/Vestibular System

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Introduction[edit]

The main function of the balance system, or vestibular system, is to sense head movements, especially involuntary ones, and counter them with reflexive eye movements and postural adjustments that keep the visual world stable and keep us from falling. An excellent, more extensive article on the vestibular system is available on Scholorpedia [1]. An extensive review of our current knowledge about the vestibular system can be found in "The Vestibular System: a Sixth Sense" by J Goldberg et al [2].


Anatomy of the Vestibular System[edit]

Labyrinth[edit]

Together with the cochlea, the vestibular system is carried by a system of tubes called the membranous labyrinth. These tubes are lodged within the cavities of the bony labyrinth located in the inner ear. A fluid called perilymph fills the space between the bone and the membranous labyrinth, while another one called endolymph fills the inside of the tubes spanned by the membranous labyrinth. These fluids have a unique ionic composition suited to their function in regulating the electrochemical potential of hair cells, which are as we will later see the transducers of the vestibular system. The electric potential of endolymph is of about 80 mV more positive than perilymph.

Since our movements consist of a combination of linear translations and rotations, the vestibular system is composed of two main parts: The otolith organs, which sense linear accelerations and thereby also give us information about the head’s position relative to gravity, and the semicircular canals, which sense angular accelerations.

Human bony labyrinth (Computed tomography 3D) Internal structure of the human labyrinth
Canaux osseux.png VestibularSystem.gif

Otoliths[edit]

The otolith organs of both ears are located in two membranous sacs called the utricle and the saccule which primary sense horizontal and vertical accelerations, respectively. Each utricle has about 30'000 hair cells, and each saccule about 16'000. The otoliths are located at the central part of the labyrinth, also called the vestibule of the ear. Both utricle and saccule have a thickened portion of the membrane called the macula. A gelatinous membrane called the otolthic membrane sits atop the macula, and microscopic stones made of calcium carbonate crystal, the otoliths, are embedded on the surface of this membrane. On the opposite side, hair cells embedded in supporting cells project into this membrane.

The otoliths are the human sensory organs for linear acceleration. The utricle (left) is approximately horizontally oriented; the saccule (center) lies approximately vertical. The arrows indicate the local on-directions of the hair cells; and the thick black lines indicate the location of the striola. On the right you see a cross-section through the otolith membrane. The graphs have been generated by Rudi Jaeger, while we cooperated on investigations of the otolith dynamics.

Semicircular Canals[edit]

Cross-section through ampulla. Top: The cupula spans the lumen of the ampulla from the crista to the membranous labyrinth. Bottom: Since head acceleration exceeds endolymph acceleration, the relative flow of endolymph in the canal is opposite to the direction of head acceleration. This flow produces a pressure across the elastic cupula, which deflects in response.

Each ear has three semicircular canals. They are half circular, interconnected membranous tubes filled with endolymph and can sense angular accelerations in the three orthogonal planes. The radius of curvature of the human horizontal semicircular canal is 3.2 mm [3].

The canals on each side are approximately orthogonal to each other. The orientation of the on-directions of the canals on the right side are [4]:

Canal X Y Z
Horizontal 0.32269 -0.03837 -0.94573
Anterior 0.58930 0.78839 0.17655
Posterior 0.69432 -0.66693 0.27042

(The axes are oriented such that the positive x-,y-,and z-axis point forward, left, and up, respectively. The horizontal plane is defined by Reid's line, the line connecting the lower rim of the orbita and the center of the external auditory canal. And the directions are such that a rotation about that vector, according to the right-hand-rule, excites the corresponding canal.) The anterior and posterior semicircular canals are approximately vertical, and the horizontal semicircular canals approximately horizontal.

Orientation of the semicircular canals in the vestibular system. "L / R" stand for "Left / Right", respectively, and "H / A / P" for "Horizontal / Anterior / Posterior". The arrows indicate the direction of head movement that stimulates the corresponding canal.

Each canal presents a dilatation at one end, called the ampulla. Each membranous ampulla contains a saddle-shaped ridge of tissue, the crista, which extends across it from side to side. It is covered by neuroepithelium, with hair cells and supporting cells. From this ridge rises a gelatinous structure, the cupula, which extends to the roof of the ampulla immediately above it, dividing the interior of the ampulla into two approximately equal parts.

Haircells[edit]

The sensors within both the otolith organs and the semicircular canals are the hair cells. They are responsible for the transduction of a mechanical force into an electrical signal and thereby build the interface between the world of accelerations and the brain.

Transduction mechanism in auditory or vestibular haircell. Tilting the haircell towards the kinocilium opens the potassium ion channels. This changes the receptor potential in the haircell. The resulting emission of neurotransmittors can elicit an action potential (AP) in the post-synaptic cell.

Hair cells have a tuft of stereocilia that project from their apical surface. The thickest and longest stereocilia is the kinocilium. Stereocilia deflection is the mechanism by which all hair cells transduce mechanical forces. Stereocilia within a bundle are linked to one another by protein strands, called tip links, which span from the side of a taller stereocilium to the tip of its shorter neighbor in the array. Under deflection of the bundle, the tip links act as gating springs to open and close mechanically sensitive ion channels. Afferent nerve excitation works basically the following way: when all cilia are deflected toward the kinocilium, the gates open and cations, including potassium ions from the potassium rich endolymph, flow in and the membrane potential of the hair cell becomes more positive (depolarization). The hair cell itself does not fire action potentials. The depolarization activates voltage-sensitive calcium channels at the basolateral aspect of the cell. Calcium ions then flow in and trigger the release of neurotransmitters, mainly glutamate, which in turn diffuse across the narrow space between the hair cell and a nerve terminal, where they then bind to receptors and thus trigger an increase of the action potentials firing rate in the nerve. On the other hand, afferent nerve inhibition is the process induced by the bending of the stereocilia away from the kinocilium (hyperpolarization) and by which the firing rate is decreased. Because the hair cells are chronically leaking calcium, the vestibular afferent nerve fires actively at rest and thereby allows the sensing of both directions (increase and decrease of firing rate). Hair cells are very sensitive and respond extremely quickly to stimuli. The quickness of hair cell response may in part be due to the fact that they must be able to release neurotransmitter reliably in response to a threshold receptor potential of only 100 µV or so.

Auditory haircells are very similar to those of the vestibular system. Here an electron microscopy image of a frog's sacculus haircell.

Regular and Irregular Haircells[edit]

While afferent haircells in the auditory system are fairly homogeneous,those in the vestibular system can be broadly separated into two groups: "regular units" and "irregular units". Regular haircells have approximately constant interspike intervals, and fire constantly proportional to their displacement. In contrast, the inter-spike interval of irregular haircells is much more variable, and their discharge rate increases with increasing frequency; they can thus act as event detectors at high frequencies. Regular and irregular haircells also differ in their location, morphology and innervation.

Signal Processing[edit]

Peripheral Signal Transduction[edit]

Transduction of Linear Acceleration[edit]

The hair cells of the otolith organs are responsible for the transduction of a mechanical force induced by linear acceleration into an electrical signal. Since this force is the product of gravity plus linear movements of the head

 \vec F = \vec F_g + \vec F_{inertial} = m(\vec g-\frac{d^2\vec x}{dt^2})

it is therefore sometimes referred to as gravito-inertial force. The mechanism of transduction works roughly as follows: The otoconia, calcium carbonate crystals in the top layer of the otoconia membrane, have a higher specific density than the surrounding materials. Thus a linear acceleration leads to a displacement of the otoconia layer relative to the connective tissue. The displacement is sensed by the hair cells. The bending of the hairs then polarizes the cell and induces afferent excitation or inhibition.


Excitation (red) and inhibition (blue) on utricle (left) and saccule (right), when the head is in a right-ear-down orientation. The displacement of the otoliths was calculated with the finite element technique, and the orientation of the haircells was taken from the literature.

While each of the three semicircular canals senses only one-dimensional component of rotational acceleration, linear acceleration may produce a complex pattern of inhibition and excitation across the maculae of both the utricle and saccule. The saccule is located on the medial wall of the vestibule of the labyrinth in the spherical recess and has its macula oriented vertically. The utricle is located above the saccule in the elliptical recess of the vestibule, and its macula is oriented roughly horizontally when the head is upright. Within each macula, the kinocilia of the hair cells are oriented in all possible directions.

Therefore, under linear acceleration with the head in the upright position, the saccular macula is sensing acceleration components in the vertical plane, while the utricular macula is encoding acceleration in all directions in the horizontal plane. The otolthic membrane is soft enough that each hair cell is deflected proportional to the local force direction. If denotes the direction of maximum sensitivity or on-direction of the hair cell, and the gravito-inertial force, the stimulation by static accelerations is given by

 stim_{otolith}= \vec F \cdot \vec n

The direction and magnitude of the total acceleration is then determined from the excitation pattern on the otolith maculae.

Transduction of Angular Acceleration[edit]

The three semicircular canals are responsible for the sensing of angular accelerations. When the head accelerates in the plane of a semicircular canal, inertia causes the endolymph in the canal to lag behind the motion of the membranous canal. Relative to the canal walls, the endolymph effectively moves in the opposite direction as the head, pushing and distorting the elastic cupula. Hair cells are arrayed beneath the cupula on the surface of the crista and have their stereocilia projecting into the cupula. They are therefore excited or inhibited depending on the direction of the acceleration.

The stimulation of a human semicircular canal is proportional to the scalar product between a vector n (which is perpendicular to the plane of the canal), and the vector omega indicating the angular velocity.

This facilitates the interpretation of canal signals: if the orientation of a semicircular canal is described by the unit vector  \vec n , the stimulation of the canal is proportional to the projection of the angular velocity  \vec \omega onto this canal

 stim_{canal}= \vec \omega \cdot \vec n

The horizontal semicircular canal is responsible for sensing accelerations around a vertical axis, i.e. the neck. The anterior and posterior semicircular canals detect rotations of the head in the sagittal plane, as when nodding, and in the frontal plane, as when cartwheeling.

In a given cupula, all the hair cells are oriented in the same direction. The semicircular canals of both sides also work as a push-pull system. For example, because the right and the left horizontal canal cristae are “mirror opposites” of each other, they always have opposing (push-pull principle) responses to horizontal rotations of the head. Rapid rotation of the head toward the left causes depolarization of hair cells in the left horizontal canal's ampulla and increased firing of action potentials in the neurons that innervate the left horizontal canal. That same leftward rotation of the head simultaneously causes a hyperpolarization of the hair cells in the right horizontal canal's ampulla and decreases the rate of firing of action potentials in the neurons that innervate the horizontal canal of the right ear. Because of this mirror configuration, not only the right and left horizontal canals form a push-pull pair but also the right anterior canal with the left posterior canal (RALP), and the left anterior with the right posterior (LARP).

Central Vestibular Pathways[edit]

The information resulting from the vestibular system is carried to the brain, together with the auditory information from the cochlea, by the vestibulocochlear nerve, which is the eighth of twelve cranial nerves. The cell bodies of the bipolar afferent neurons that innervate the hair cells in the maculae and cristae in the vestibular labyrinth reside near the internal auditory meatus in the vestibular ganglion (also called Scarpa's ganglion, Figure Figure 10.1). The centrally projecting axons from the vestibular ganglion come together with axons projecting from the auditory neurons to form the eighth nerve, which runs through the internal auditory meatus together with the facial nerve. The primary afferent vestibular neurons project to the four vestibular nuclei that constitute the vestibular nuclear complex in the brainstem.

Vestibulo-ocular reflex.

Vestibulo-Ocular Reflex (VOR)[edit]

An extensively studied example of function of the vestibular system is the vestibulo-ocular reflex (VOR). The function of the VOR is to stabilize the image during rotation of the head. This requires the maintenance of stable eye position during horizontal, vertical and torsional head rotations. When the head rotates with a certain speed and direction, the eyes rotate with the same speed but in the opposite direction. Since head movements are present all the time, the VOR is very important for stabilizing vision.

How does the VOR work? The vestibular system signals how fast the head is rotating and the oculomotor system uses this information to stabilize the eyes in order to keep the visual image motionless on the retina. The vestibular nerves project from the vestibular ganglion to the vestibular nuclear complex, where the vestibular nuclei integrate signals from the vestibular organs with those from the spinal cord, cerebellum, and the visual system. From these nuclei, fibers cross to the contralateral abducens nucleus. There they synapse with two additional pathways. One pathway projects directly to the lateral rectus muscle of eye via the abducens nerve. Another nerve tract projects from the abducens nucleus by the abducens interneurons to the oculomotor nuclei, which contain motor neurons that drive eye muscle activity, specifically activating the medial rectus muscles of the eye through the oculomotor nerve. This short latency connection is sometimes referred to as three-neuron-arc, and allows an eye movement within less than 10 ms after the onset of the head movement.

For example, when the head rotates rightward, the following occurs. The right horizontal canal hair cells depolarize and the left hyperpolarize. The right vestibular afferent activity therefore increases while the left decreases. The vestibulocochlear nerve then carries this information to the brainstem and the right vestibular nuclei activity increases while the left decreases. This makes in turn neurons of the left abducens nucleus and the right oculomotor nucleus fire at higher rate. Those in the left oculomotor nucleus and the right abducens nucleus fire at a lower rate. This results in the fact than the left lateral rectus extraocular muscle and the right medial rectus contract while the left medial rectus and the right lateral rectus relax. Thus, both eyes rotate leftward.

The gain of the VOR is defined as the change in the eye angle divided by the change in the head angle during the head turn

 gain = \frac{\Delta_{Eye}}{\Delta_{Head}}

If the gain of the VOR is wrong, that is, different than one, then head movements result in image motion on the retina, resulting in blurred vision. Under such conditions, motor learning adjusts the gain of the VOR to produce more accurate eye motion. Thereby the cerebellum plays an important role in motor learning.

The Cerebellum and the Vestibular System[edit]

It is known that postural control can be adapted to suit specific behavior. Patient experiments suggest that the cerebellum plays a key role in this form of motor learning. In particular, the role of the cerebellum has been extensively studied in the case of adaptation of vestibulo-ocular control. Indeed, it has been shown that the gain of the vestibulo-ocular reflex adapts to reach the value of one even if damage occur in a part of the VOR pathway or if it is voluntary modified through the use of magnifying lenses. Basically, there are two different hypotheses about how the cerebellum plays a necessary role in this adaptation. The first from (Ito 1972;Ito 1982) claims that the cerebellum itself is the site of learning, while the second from Miles and Lisberger (Miles and Lisberger 1981) claims that the vestibular nuclei are the site of adaptive learning while the cerebellum constructs the signal that drives this adaptation. Note that in addition to direct excitatory input to the vestibular nuclei, the sensory neurons of the vestibular labyrinth also provide input to the Purkinje cells in the flocculo-nodular lobes of the cerebellum via a pathway of mossy and parallel fibers. In turn, the Purkinje cells project an inhibitory influence back onto the vestibular nuclei. Ito argued that the gain of the VOR can be adaptively modulated by altering the relative strength of the direct excitatory and indirect inhibitory pathways. Ito also argued that a message of retinal image slip going through the inferior olivary nucleus carried by the climbing fiber plays the role of an error signal and thereby is the modulating influence of the Purkinje cells. On the other hand, Miles and Lisberger argued that the brainstem neurons targeted by the Purkinje cells are the site of adaptive learning and that the cerebellum constructs the error signal that drives this adaptation.

Vestibular Implants[edit]

Computer Simulation of the Vestibular System[edit]

Semicircular Canals[edit]

Model without Cupula[edit]

Simplified semicircular canal, without cupula.

Let us consider the mechanical description of the semi-circular canals (SCC). We will make very strong and reductive assumptions in the following description. The goal here is merely to understand the very basic mechanical principles underlying the semicircular canals.

The first strong simplification we make is that a semicircular canal can be modeled as a circular tube of “outer” radius R and “inner” radius r. (For proper hydro mechanical derivations see (Damiano and Rabbitt 1996) and Obrist (2005)). This tube is filled with endolymph.

The orientation of the semicircular canal can be described, in a given coordinate system, by a vector  \vec n that is perpendicular to the plane of the canal. We will also use the following notations:

 \theta Rotation angle of tube [rad]
 \dot{\theta} \equiv \frac{d \theta}{dt} Angular velocity of the tube [rad/s]
 \ddot{\theta} \equiv \frac{d^2 \theta}{dt^2} Angular acceleration of the tube [rad/s^2]
 \phi Rotation angle of the endolymph inside the tube [rad], and similar notation for the time derivatives
 \delta = \theta - \phi movement between the tube and the endolymph [rad].

Note that all these variables are scalar quantities. We use the fact that the angular velocity of the tube can be viewed as the projection of the actual angular velocity vector of the head  \vec \omega onto the plane of the semicircular canal described by  \vec n to go from the 3D environment of the head to our scalar description. That is,

 \dot{\theta} = \vec \omega \cdot \vec n

where the standard scalar product is meant with the dot.

To characterize the endolymph movement, consider a free floating piston, with the same density as the endolymph. Two forces are acting on the system:

  1. The inertial moment  I \ddot{\phi} , where I characterizes the inertia of the endolymph.
  2. The viscous moment  B \dot{\delta} , caused by the friction of the endolymph on the walls of the tube.

This gives the equation of motion

 I \ddot{\phi} = B \dot{\delta}

Substituting  \phi = \theta - \delta and integrating gives

 \dot{\theta} = \dot{\delta} + \frac{B}{I} \delta .

Let us now consider the example of a velocity step  \dot{\theta}(t) of constant amplitude  \omega . In this case, we obtain a displacement

 \delta = \frac{I}{B} \omega \cdot (1-e^{-\frac{B}{I}t})

and for  t \gg \frac{I}{B} , we obtain the constant displacement

 \delta \approx \frac{I}{B} \omega .

Now, let us derive the time constant  T_1 \equiv \frac{I}{B} . Fora thin tube,  r \ll R , the inertia is approximately given by

 I = m l^2 \approx 2 \rho \pi^2 r^2 R^3 .

From the Poiseuille-Hagen Equation, the force F from a laminar flow with velocity v in a thin tube is

 F = \frac{8 \bar{V} \eta l}{r^2}

where  \bar{V} = r^2 \pi v is the volume flow per second,  \eta the viscosity and  l = 2 \pi R the length of the tube.

With the torque  M = F \cdot R and the relative angular velocity  \Omega = \frac{v}{R} , substitution provides

 B = \frac{M}{\Omega} = 16 \eta \pi ^2 R^3

Finally, this gives the time constant  T_1

 T_1 = \frac{I}{B} = \frac{\delta r^2}{8 \eta}

For the human balance system, replacing the variables with experimentally obtained parameters yields a time constant  T_1 of about 0.01 s. This is brief enough that in equation (10.5) the  \approx can be replaced by " = ". This gives a system gain of

 G \equiv \frac{\delta}{\omega} = \frac{I}{B} = T_1

Model with Cupula[edit]

Effect of the cupula.

Our discussion until this point has not included the role of the cupula in the SCC: The cupula acts as an elastic membrane that gets displaced by angular accelerations. Through its elasticity the cupula returns the system to its resting position. The elasticity of the cupula adds an additional elastic term to the equation of movement. If it is taken into account, this equation becomes

 \ddot{\theta} = \ddot{\delta} + \frac{B}{I} \dot{\delta} + \frac{K}{I} \delta

An elegant way to solve such differential equations is the Laplace-Transformation. The Laplace transform turns differential equations into algebraic equations: if the Laplace transform of a signal x(t) is denoted by X(s), the Laplace transform of the time derivative is

 \frac{dx(t)}{dt} \xrightarrow{Laplace Transform} s \cdot X(s) - x(0)

The term x(0) details the starting condition, and can often be set to zero by an appropriate choice of the reference position. Thus, the Laplace transform is

 s^2 \tilde{\theta} = s^2 \tilde{\delta} + \frac{B}{I} s \tilde{\delta} + \frac{K}{I} \tilde{\delta}

where "~" indicates the Laplace transformed variable. With  T_1 from above, and  T_2 defined by

 T_2 = \frac{B}{K}

we get the

 \frac{ \tilde{\delta} }{ \tilde{\theta} } = \frac{T_1 s^2}{T_1 s^2 + s + \frac{1}{T_2}}

For humans, typical values for  T_2 = B/K are about 5 sec.

To find the poles of this transfer function, we have to determine for which values of s the denominator equals 0:

 s_{1,2} = \frac{1}{T_1} \Big(-1 \pm \sqrt{1-4\frac{T_1}{T_2}} \Big)

Since  T_2 \gg T_1 , and since

 \sqrt{1-x} \approx 1 - \frac{x}{2} for x \ll 1

we obtain

 s_1 \approx - \frac{1}{T_1}, and s_2 \approx - \frac{1}{T_2}

Typically we are interested in the cupula displacement  \delta as a function of head velocity  \dot{\theta} \equiv s \tilde{\theta} :

 \frac{\tilde{\delta}}{s \tilde{\theta}}(s) = \frac{T_1 T_2 s}{(T_1 s +1)(T_2 s + 1)}

For typical head movements (0.2 Hz < f < 20Hz), the system gain is approximately constant. In other words, for typical head movements the cupula displacement is proportional to the angular head velocity!

Bode plot of the cupula displacement of a function of head velocity, with T1 = 0.01 sec, T2 = 5 sec, and an amplification factor of (T1+ T2)/ (T1* T2) to obtain a gain of approximately 0 for the central frequencies.

Control Systems[edit]

For Linear, Time-Invariant systems (LTI systems), the input and output have a simple relationship in the frequency domain :

 Out(s) = G(s)*In(s)

where the transfer function G(s) can be expressed by the algebraic function

G(s)=\frac{num(s)}{den(s)}=\frac{n(0)*{{s}^{0}}+n(1)*{{s}^{1}}+n(2)*{{s}^{2}}+...}{d(0)*{{s}^{0}}+d(1)*{{s}^{1}}+d(2)*{{s}^{2}}+...}

In other words, specifying the coefficients of the numerator (n) and denominator (d) uniquely characterizes the transfer function. This notation is used by some computational tools to simulate the response of such a system to a given input.

Different tools can be used to simulate such a system. For example, the response of a low-pass filter with a time-constant of 7 sec to an input step at 1 sec has the following transfer function

G(s)=\frac{1}{7s+1}

and can be simulated as follows:

With Simulink[edit]
Step-response simulation of a lowpass filter with Simulink.
Commandline[edit]

If you work on the command line, you can use the Control System Toolbox of MATLAB or the module signal of the Python package SciPy:

MATLAB Control System Toolbox:

% Define the transfer function
num = [1];
tau = 7;
den = [tau, 1];
mySystem = tf(num,den)
 
% Generate an input step
t = 0:0.1:30;
inSignal = zeros(size(t));
inSignal(t>=1) = 1;
 
% Simulate and show the output
[outSignal, tSim] = lsim(mySystem, inSignal, t);
plot(t, inSignal, tSim, outSignal);

Python - SciPy:

# Import required packages
import numpy as np
import scipy.signal as ss
import matplotlib.pylab as mp
 
# Define transfer function
num = [1]
tau = 7
den = [tau, 1]
mySystem = ss.lti(num, den)
 
# Generate inSignal
t = np.arange(0,30,0.1)
inSignal = np.zeros(t.size)
inSignal[t>=1] = 1
 
# Simulate and plot outSignal
tout, outSignal, xout = ss.lsim(mySystem, inSignal, t)
mp.plot(t, inSignal, tout, outSignal)
mp.show()

Otoliths[edit]

Consider now the mechanics of the otolith organs. Since they are made up by complex, visco-elastic materials with a curved shape, their mechanics cannot be described with analytical tools. However, their movement can be simulated numerically with the finite element technique. Thereby the volume under consideration is divided into many small volume elements, and for each element the physical equations are approximated by analytical functions.

FE-Simulations: Small, finite elements are used to construct a mechanical model; here for example the saccule.

Here we will only show the physical equations for the visco-elastic otolith materials. The movement of each elastic material has to obey Cauchy’s equations of motion:

 \rho \frac{\partial^2 u_i}{\partial t^2} = \rho B_i + \sum_{j} \frac{\partial T_{ij}}{\partial x_j}

where  \rho is the effective density of the material,  u_i the displacements along the i-axis,  B_i the i-component of the volume force, and  T_{ij} the components of the Cauchy’s strain tensor.  x_j are the coordinates.

For linear elastic, isotropic material, Cauchy’s strain tensor is given by

 T_{ij} = \lambda e \delta_{ij} + 2 \mu E_{ij}

where  \lambda and  \mu are the Lamé constants;  \mu is identical with the shear modulus.  e = div(\vec u) , and  E_{ij} is the stress tensor

 E_{ij} = \frac{1}{2} \Big( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \Big).

This leads to Navier’s Equations of motion

 \rho \frac{\partial ^2 u_i}{\partial t^2} = \rho B_i + (\lambda + \mu) \frac{\partial e}{\partial x_i} + \mu \sum_{j} \frac{\partial ^2 u_i}{\partial x_j^2}

This equation holds for purely elastic, isotropic materials, and can be solved with the finite element technique. A typical procedure to find the mechanical parameters that appear in this equation is the following: when a cylindrical sample of the material is put under strain, the Young coefficient E characterizes the change in length, and the Poisson’s ratio  \nu the simultaneous decrease in diameter. The Lamé constants  \lambda and  \mu are related to E and  \nu by:

 E = \frac{\mu (3 \lambda + 2 \mu)}{\lambda + \mu}

and

 \nu = \frac{\lambda}{2(\lambda + \mu)}

Central Vestibular Processing[edit]

Central processing of vestibular information significantly affects the perceived orientation and movement in space. The corresponding information processing in the brainstem can often be modeled efficiently with control-system tools. As a specific example, we show how to model the effect of velocity storage.

Velocity Storage[edit]

The concept of velocity storage is based on the following experimental finding: when we abruptly stop from a sustained rotation about an earth-vertical axis, the cupula is deflected by the deceleration, but returns to its resting state with a time-constant of about 5 sec. However, the perceived rotation continues much longer, and decreases with a much longer time constant, typically somewhere between 15 and 20 sec.

Vestibular Modeling: The blue curve describes the deflection of the cupula as a response to a velocity step, modeled as a high-pass filter with a time-constant of 5 sec. The green curve represents the internal estimate of the angular velocity, obtained with an internal model of the cupula-response in a negative feedback look, and a feed-forward gain-factor of 2.

In the attached figure, the response of the canals to an angular velocity stimulus ω is modeled by the transfer function C, here a simple high-pass filter with a time constant of 5 sec. (The canal response is determined by the deflection of the cupula, and is approximately proportional to the neural firing rate.) To model the increase in time constant, we assume that the central vestibular system has an internal model of the transfer function of the canals, \hat{C}. Based on this internal model, the expected firing rate of the internal estimate of the angular velocity, \hat{\omega}, is compared to the actual firing rate. With a the gain-factor k set to 2, the output of the model nicely reproduces the increase in the time constant. The corresponding Python code can be found at [5].

It is worth noting that this feedback loop can be justified physiologically: we know that there are strong connections between the left and right vestibular nuclei. If those connections are severed, the time constant of the perceived rotation decreases to the peripheral time-constant of the semicircular canals.

Central Vestibular Processing can often be described with control-system models. Here "omega" is the head velocity, "C" the transfer function of the semicircular canals, and "k" a simple gain factor. The "hat"-ed variables indicate internal estimates.

Mathematically, negative feedback with a high gain has the interesting property that it can practically invert the transfer function in the negative feedback loop: if k>>1, and if the internal model of the canal transfer function is similar to the actual transfer function, the estimated angular velocity corresponds to the actual angular velocity.

\begin{align}
  & \hat{\omega }=(\omega C-\hat{\omega }\hat{C})k \\ 
 & \hat{\omega }(1+\hat{C}k)=\omega Ck \\ 
 & \frac{{\hat{\omega }}}{\omega }=\frac{C}{1/k+\hat{C}}\,\,\xrightarrow[if\,C\approx \hat{C}]{k>>1}1  
\end{align}

Alcohol and the Vestibular System[edit]

As you may or may not know from personal experience, consumption of alcohol can also induce a feeling of rotation. The explanation is quite straightforward, and basically relies on two factors: i) alcohol is lighter than the endolymph; and ii) once it is in the blood, alcohol gets relatively quickly into the cupula, as the cupula has a good blood supply. In contrast, it diffuses only slowly into the endolymph, over a period of a few hours. In combination, this leads to a buoyancy of the cupola soon after you have consumed (too much) alcohol. When you lie on your side, the deflection of the left and right horizontal cupulae add up, and induce a strong feeling of rotation. The proof: just roll on the other side - and the perceived direction of rotation will flip around!

Due to the position of the cupulae, you will experience the strongest effect when you lie on your side. When you lie on your back, the deflection of the left and right cupula compensate each other, and you don't feel any horizontal rotation. This explains why hanging one leg out of the bed slows down the perceived rotation.

The overall effect is minimized in the upright head position - so try to stay up(right) as long as possible during the party!

If you have drunk way too much, the endolymph will contain a significant amount of alcohol the next morning - more so than the cupula. This explains while at that point, a small amount of alcohol (e.g. a small beer) balances the difference, and reduces the feeling of spinning.

References[edit]

  1. Kathleen Cullen and Soroush Sadeghi (2008). "Vestibular System". Scholarpedia 3(1):3013. http://www.scholarpedia.org/article/Vestibular_system. 
  2. JM Goldberg, VJ Wilson, KE Cullen and DE Angelaki (2012). "The Vestibular System: a Sixth Sense"". Oxford University Press, USA. http://www.scholarpedia.org/article/Vestibular_system. 
  3. Curthoys IS and Oman CM (1987). "Dimensions of the horizontal semicircular duct, ampulla and utricle in the human.". Acta Otolaryngol 103: 254–261. 
  4. Della Santina CC, Potyagaylo V, Migliaccio A, Minor LB, Carey JB (2005). "Orientation of Human Semicircular Canals Measured by Three-Dimensional Multi-planar CT Reconstruction.". J Assoc Res Otolaryngol 6(3): 191-206. 
  5. Thomas Haslwanter (2013). "Vestibular Processing: Simulation of the Velocity Storage [Python"]. http://work.thaslwanter.at/CSS/Code/vestibular_feedback.py. 

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