The Functional Anatomy of the Semicircular Canals in Man

Anatomy of the Semicircular Canals

The semicircular canals are generally described as a set ofmutually perpendicular membranous rings that meet in a dilated expansion calledthe utricle. The anterior andposterior canals are approximately vertical with each canal at an approximately45° angle to the midsagittal plane and the lateral canal approximatelyhorizontal, which is why it is also called the horizontal canal.  The planes of the canals form an arraysimilar to the corner of a box that is rotated 45° out of the sagittal plane.

The vestibular part of the membranous labyrinth iscomposed of the utricle and saccule and three roughly mutually perpendicularsemicircular canals. Thesemicircular canals arise from and end in the utricle.  One end of each canal has a dilationcalled an ampulla, which contains the sensory epithelia for rotationalacceleration. The utricle andsaccule each contain a sensory epithelium for linear acceleration.

The semicircular canals lie in a similarly shaped bonychamber called the bony labyrinth, which also has a cochlea and threesemicircular canals. The canals,along with the utricle and saccule and the scala media of the cochlea and theendolymphatic duct form a complex membranous structure called the membranouslabyrinth. The scala media isattached to the walls of the bony cochlea and the membranous semicircularcanals lie within the bony semicircular canals. The utricle and saccule lie in the vestibule of the bonylabyrinth.

The membranous labyrinth is filled with an ionic solutioncalled endolymph and surrounded by another ionic fluid called perilymph.  The different ionic compositions of thetwo fluids are important for the correct operation of the hair cells.  Both fluids are approximately the samedensity and viscosity as water.

The vestibular parts of the membranous labyrinth, thesaccule and utricle and the three semicircular canals are suspended in the bonylabyrinth by thin connective tissue trabeculae, so that the membranouslabyrinth is floating in the bony labyrinth, but retrained from large movementswithin the bony chamber. Thearrangement is like a water-filled balloon floating in a substantially largerpool of water, held in place by many thin elastic bands.

The cross-sectional radius of the membranous semicircularcanal is about one-third that of the bony labyrinth and it is modestlyflattened so that the canal has an elliptical cross-section.  The elliptical cross-section means thatthe canal can expand in volume readily, simply by assuming a more circularcross-section. A circle has agreater area than an ellipse with the same circumference.

At one end of each canal, just before it meets the utricle,the membranous semicircular canal dilates into a bulbous ending called theampulla of the canal. The ampullahas about three times the radial cross-section of the remainder of thecanal. The ampulla has a transverseridge, called the crista ampullae, that contains sensory hair cells oriented soas to respond head movements in the plane of the canal.  Surmounting the crista ampullae andseparating the ampulla into two spaces is a gelatinous membrane called thecupula.

Nerves and blood vessels that pass into the base of thecrista supply the sensory epithelium of the crista ampullae.  Consequently each canal has its ownnerve and the different nerves combine with each other and the nerves from themaculae in the utricle and saccule to form first a superior and inferior branchand then the vestibular nerve. Thevestibular nerve runs in the internal auditory meatus with the cochlear nerveand the facial-intermediate nerve. The proximity of these nerves in the narrow passage of the internalauditory meatus means that they are often compromised together by mass lesionssuch as edema and tumors.

Rotations of the Head Stimulate the Semicircular Canals

Rotation of the head in the plane of a canal causes a slightlag in the flow of the endolymphatic fluid within the canals because of itsinertia. The forces involved aremuch the same as you experience when you are in a car that accelerates ordecelerated or turns a corner. Your body would continue in the same direction at the same speed exceptthat it is compelled to change its velocity by your seat pressing on yourbody. In a semicircular canal, itis the walls of the membranous canal and ultimately the bony canal that compelmovement. The fluid can also movearound the canal if there is not a wall to force it in the new direction or atthe new speed. Lagging fluid moving within the membranous canal ultimatelydisplaces cupula of the ampulla, causing it to billow like a sail in abreeze. As the cupula billows inthe displaced endolymphatic fluid, the cilia of the embedded hair cells aredisplaced.

Displacement of the many stereocilia of a hair cell towardsthe single kinocilium at the margin of the cluster of stereocilia will causechannels to open, leading to a depolarization of the hair cell.  Displacement in the opposite directionwill hyperpolarize the hair cell. In any given crista ampullae all the hair cells are oriented so thattheir kinocilia lie to the same side of their cluster of stereocilia.  In the ampulla of the horizontal canal,the stereocilia are arranged so that displacement of the cupula towards theutricle will cause excitation. Inthe anterior and posterior canals it is displacement away from the utricle thatis excitatory.

Each canal has a dilation at one end where it meets theutricle. That dilation, called theampulla has a transverse ridge, called the crista ampullae, that contains apopulation of embedded hair cells, all oriented so that displacement of thehairs in one direction elicits depolarization of the hair cells and movement inthe opposite direction causes hyperpolarization. The crista ampullae is surmounted by a gelatinous membrane,called the cupula, which seals off the ampulla. Movements of the fluid in asemicircular canal causes the cupula to billow, like a sail in a breeze, andthe shearing force exerted upon the embedded stereocilia and kinocila of thehair cells causes more or less neurotransmitter to be released on to theterminal processes of the sensory neurons of the vestibular nerve.  The result is a fluctuating dischargein the vestibular sensory neurons that is proportional to the displacements ofthe cupula.

Rotation in one direction will cause the discharge rates toincrease and rotation in the opposite direction will cause them todecrease. Rotation in planes thatare oblique to the canal plane will have less and less effect, until rotationsin a plane perpendicular to the canal plane will not modify activity in thecanal’s hair cells. Since thereare three canals that are approximately perpendicular to each other, one wouldexpect that rotations of the head in any direction will stimulate at least onecanal and most often multiple canals.

The mirror symmetrical arrangement of the canals on the twosides of the head means that any rotation will stimulate canals on both sidesof the head. The two sets ofcanals are set so that the two horizontal canals respond to rotations inopposite directions. The anteriorcanal on one side is approximately aligned with the posterior canal on theopposite side so rotations in certain oblique planes will affect a canal on oneside and the other canal on the opposite side. For instance, rotating the head so that the nose moves downand to the side will increase activity in the opposite anterior canal anddecrease activity in the posterior canal on the same side.

The Value of Normal Vectors

We will consider the sensitivities of the semicircularcanals in terms of vectors that are orthogonal to the planes of thecanals. This approach is generallyconsidered more difficult that using the planes of the canal themselves, but,with a little practice, one soon realizes that normal vectors are easier tounderstand, because they lack the ambiguity intrinsic to the planar approachand the magnitude and direction of a vector is easier to picture than theorientation of a plane. Inpractice, it will be convenient to move back and forth between the twodescriptions of the canals. Afterall, the planes are what we see when we look at the canals and rotation in theplane of the canal is intuitively clear, if difficult to quantify.  Oriented surfaces are about asdifficult for most people to understand as vectors, but that is what one mustuse to model the semicircular canals. Oriented surfaces are formally equivalent to quaternion vectors.

If we consider the sensitivity of the anterior canal, acanal at a roughly 45° angle to a sagittal plane, movements that cause theampulla to move anteriorly will excite hair cells in the crista ampullae of the canal.  The optimal axis of rotation for thecanal will be perpendicular to the plane of the canal.  For the right anterior canal, the axisor rotation is directed towards the opposite side at a roughly 45° angle to thesagittal plane, nearly confined to a horizontal plane.  Consequently, rotations of the headabout that axis will excite the canal.

Briefly consider the movements of the nose and ear duringoptimal movements for exciting the anterior canal, where the axis passesthrough or near the semicircular canal. The nose moves down and towards the opposite side from the canal, while theear moves down and towards the same side. This disparity is due to the geometry of the head and it is only anapparent difference. If the axisof rotation is set higher, then both the nose and the ear move to the oppositeside and, if it is set lower, they both move to the same side.  Clearly referencing the optimalmovement to the plane of the canal and landmarks of the head is intrinsicallyambiguous. The situation issimilar to the difference between swing and spin. What you see depends upon your point of reference.  That ambiguity is removed if one usesthe axis of rotation and each of the above outcomes flows directly from anunderstanding of the axis of rotation and the anatomy of the head relative tothe axis of rotation.

Normal Vectors for the Planes of the Canals Lead to their Optimal Axes ofRotation

Each canal has two, oppositely directed, normal vectorswhich are perpendicular to the plane of the canal and which can be an optimalaxis of rotation. Semicircularcanals are stimulated by angular momentum and angular momentum is expressed bya vector perpendicular to the plane of rotation, that is, the plane defined bythe movement vector and a vector that extends perpendicularly from the axis ofrotation to the moving mass. Forthe most effective stimulation of a semicircular canal the center of rotationis on an axis of rotation perpendicular to the plane of the canal where itpenetrates the plane of the canal. Therefore, either normal vector to a canal is potentially the optimalaxis of rotation for the canal. However, rotation in one direction will depolarize the hair cells in thecanal’s ampulla and rotation in the opposite direction will hyperpolarize them.  If we choose the optimal axis ofrotation to be the canal that depolarized the canal’s hair cells, then there isa unique axis.

Planes to Normal Vectors

If one has a plane that includes the origin, it may bedescribed by an expression like the following.

The are constantsand the x, y, and z are variables. Normally, the plane is written in the form of the second expression, butthe basis vectors in the first expression are implied.  They are made explicit here, becausethey are relevant to the calculation of the normal vector to the plane.

If we express the plane in this format, then it is apparentthat a plane is the set of vectors for which the sum of their coefficients iszero. The orientation of the planeis determined by the relative values of the constants.  The ratios of the constants are thecritical factor. The absolutevalues are largely irrelevant as long as the ratios are maintained.  For that reason one can always writethe expression for a plane so that the sum of the constant coefficients isone. The reason that one mightwish to do so arises from the following argument.

We can readily calculate two vectors that lie in the planeby setting the coefficient of k equalto zero and setting the coefficient of j equal to zero.

Then the normal vector to the plane is the vector of theratio of these two vectors in the plane. We can take the ratio of the second to the first or the first to thesecond.

If we compute the ratio of the second to the first, then weget a vector that extends in one direction relative to the plane and if wecompute the other ratio, then the vector points in the opposite direction.  These are the two normal vectors to theplane. Also note that thecoefficients of the orthogonal vectors are the same as for the plane, give ortake a negative sign and the fact that the normal vectors are expressed as unitvectors. If the coefficients ofthe plane are normalized so that the sum of the squares of the coefficients is1.0 then the normal to the plane has the same coefficients as the plane.

Because of this relationship between the expressions for aplane and its unit normal vectors, we can take descriptions of the canal planesand immediately write down the expressions for the axes of rotation for thecanals.

The Actual Optimal Axes of Rotation for Human Semicircular Canals

Blanks et al.(1975) measured the planes of the semicircular canals in ten skulls of amixture of races and they expressed the planes of the canals as the regressionsof all of the canals of a given type. We can directly convert their expression for the regression plane intothe axes of rotation for each canal type. For the canals on the right side the values were as follows.

Horizontal canal	CHR	0.365 i + 0.158 j - 0.905 k
Anterior canal	CAR	0.652 i + 0.753 j + 0.17 k
Posterior canal	CPR	0.757 i – 0.561 j + 0.320 k

To get the values for the canals on the left side, we needonly reverse the values for the j terms.

Horizontal canal	CHR	0.365 i - 0.158 j - 0.905 k
Anterior canal	CAR	0.652 i - 0.753 j + 0.17 k
Posterior canal	CPR	0.757 i + 0.561 j + 0.320 k

However, a little thought and experimentation will convinceyou that the values as written are for movements relative to a left-handedcoordinate system. To render theminto a right handed coordinate system one must reverse the values for all thecomponents of the left-handed expressions. So the final values for the left-handed canals are as follows.

Horizontal canal	CHR	- 0.365 i + 0.158 j + 0.905 k
Anterior canal	CAR	- 0.652 i + 0.753 j - 0.17 k
Posterior canal	CPR	- 0.757 i - 0.561 j - 0.320 k

Visual Representations of the Canal Axes

It is much easier to think about the canal axes when one hasa graphical representation. Tothat end, this section considers a number of images that represent the canalaxes as vectors. In the firstfigure the canal axes for the semicircular canals on the right side of the headare represented as viewed from two viewpoints. The left panel is an oblique view from the left and abovethe horizontal plane and the right panel is the same axes viewed from directlyin front of the head and slightly above the horizontal plane.  Three mutually orthogonal blue vectorsindicate the axes of the head. Thei axis points directlyanteriorly. The j and k axes point directly laterally and directlyvertically, respectively.

One can see that the horizontal canal axis is verticallydirected with a modest anterior and lateral tilt. Its total tilt from vertical is 23.7°; it tilts 10°laterally and 22° anteriorly. Theanterior canal has an axis that crosses the midline at a roughly 45° angle to asagittal plane, actually 49.11° to the contralateral side a sagittalplane. It is nearly confined to ahorizontal plane, being 1° below the horizontal plane.  The posterior canal axis is directedanteriorly and laterally at about 45° to a sagittal plane and it risesvertically at a modest angle. Itsis 40.47° from the anterior axis (i),36.54° from a sagittal plane, and 18.76° above a horizontal plane through itsorigin.

All these measurements are relative to the Reid stereotaxiccoordinates in which the horizontal plane passes through the centers of theexternal auditory canals and the inferior margins of the orbits.  They would be slightly different in theFrankfurt coordinates, which use the superior margins of the external auditorycanals (Blanks, et al., ’75).

The right semicircular canal axes viewed from theleft and anterior (left panel) and from in front and slightly above thehorizontal plane (right panel). The basisvectors are aligned with the anterior axis (i), the left lateral axis (j), and the vertical axis (k) of the Reid stereotaxic system (Blanks et al,’75). The canals axes are labeledwith H for the horizontalcanal, A for the anteriorcanal, and P for theposterior canal. These conventionsapply to all of the following figures unless stated otherwise.

The next task is to represent the relationships between thecanal axes on the two sides of the head. In the next figure the right canal axes have been reflected in asagittal plane, because the canals on the two sides of the head areapproximately mirror reflections of each other. That representation is quite intuitive in that one set ofcanal axes is simply the reflection of the other set. Unfortunately, reflection also flips the coordinatesystem. A bit of thought andexperimentation will reveal that the axes on the left side are appropriate onlyif one uses one’s left hand to express the rotation. They are in a left-handed coordinate system, while the axeson the right side are in a right-handed coordinate system.  Consequently, we need to convert one orthe other set into the other coordinate system in order to compute with them.

We will consider the reflection of the right canal axes in abit, but there are some points that can be drawn from this representation thatare not as easily appreciate in a consistent single-handed representation.  The two anterior canal axes cross inthe midline and they lie in a plane that is nearly parallel with the horizontalplane. They are nearly orthogonalto each other, which means that the canals lie at approximately a 45° angle toa sagittal plane and nearly vertical. The posterior canal axes are at approximately a right angle to theanterior canal axes, actually 89.58° in the horizontal plane and 86.20° in theplane that contains both axes, which means that the posterior canal planes arenearly perpendicular to the anterior canal planes. The posterior canal axes are tilted up, meaning that thecanals are tilted posteriorly at their dorsal margin relative to their ventralmargin. The horizontal canals areapproximately horizontal, because their axes are predominantly verticallydirected, but they are tilted up and anteriorly along their anterior marginrelative to their posterior margin. That means that they do not have a common axis, so there is no rotationthat is optimal for both canals, but the best compromise is obtained by tiltingthe head forward until both axes lie in a coronal plane.  That inclination is generally said tobe about 30°, nose down from standard position, but the data considered herewould suggest that it should be more like 22°.

The right and left canal axes with the rightcanal axes reflected across the midline. This is anintuitive representation, but a little experimentation will reveal that theleft side axes work only if one uses one’s left hand to indicate the movement,that is they are in a left-handed coordinate system while the ones on the rightare in a right-handed coordinate system. The right canal axes have an Ras a subscript and the left canal axes have an L.

The next figure is similar to the last figure except that thecanal planes have been represented by circular discs that are perpendicular tothe axes of rotation. The centersof the canal axes have also been moved a bit further apart so that the anteriorcanal axes do not overlap and the basis vectors are clearly visible.  There is no proper separation for thetwo sets of canal axes and there is no need for the canal axes to have a commoncenter. They are vectors and soare free to move as is convenient as long as they maintain their direction andmagnitude. Actually, they arequaternion vectors, so it is important to keep their orientation or sense aswell, but they have no location, so their placement is largely arbitrary.  In this case, the elements have beengrouped and placed so as to facilitate an appreciation of the orientation ofthe canal axes and planes.

The reflected canal axes are drawn with theplanes of the canals indicated by circular discs and the canal centers moved further apart, so thatthe anterior canal axes do not overlap. The same reservations apply to this representation as for the previousfigure. One can see that the canalaxes do not leave through the intersections of the discs, which indicates thatthe canals are not actually orthogonal, although they are approximately so.

As stated above, the canal axes in the previous two figuresare in two different coordinate systems, which aids in understanding therelationships between the canals on the two sides of the head, but they willrapidly get one in trouble if one tries to compute with them.  A bit of experimentation will lead oneto the canal axes in a right-handed coordinate system that correspond to thoseillustrated in the last two figures. The correct reflection of the right canal axes is indicated in thefollowing figure and the three that follow it. The reflected axes do not look much like a mirrorreflection, but if one plays with the image by rotating it and by making theindicated gestures, it becomes apparent that it is a true representation of thereflected right canal axes.

The canal axes expressed in right-handedcoordinate system. The left canal axes are rendered interms of a right handed coordinate system. Consequently, the left canal axes are the reflection of theright canal axes across the midline. This representation is not as intuitive as a simple mirror reflection,but it is more appropriate for calculations.

As with the reflection that changed the handedness of thecanal axes, one can see that the three canals on one side are approximatelymutually perpendicular, but deviate from perpendicular by a readily detectedamount. The vertical canals onlyapproximately align with the matching canal on the opposite side (rightposterior with left anterior and right anterior with left posterior).  The horizontal canals are onlyapproximately aligned and each is tilted anteriorly and laterally to the sameside. Because the canal axes arequaternion vectors they have magnitude, direction, and orientation.  Because they are quaternion vectors,their mirror reflections are distinctly different appearing vectors.

The next three figures show the same canal axes from anumber of special viewpoints. Thefirst is from a point directly anterior to the canals.  The canal axes look very asymmetrical,but it you take the negative of each of the canal axes, then you will have aset of axes that are a mirror reflection of the axes for the oppositesemicircular canals. When vieweddirectly from the side, the axes collapse into a single center, because the twocenters are aligned. There is muchmore symmetry in that view, since each axis points in the opposite direction asthe same canal axis on the opposite side. When viewed from directly above there is also more apparent symmetry,but it is more like symmetry through the center point of the basis vectors,rather than mirror symmetry. Closer inspection reveals that the symmetry is only approximate and itis broken in many ways.

The symmetry of the two sets of canal axes is a more subtlesymmetry than one normally perceives, but they are truly symmetrical becausethey are transformed into each other with only a reversal of signs for thedirectional components. We alsoknow that the representations are reflections of anatomical structures that aresymmetrical across the midsagittal plane.

The same canal axes viewed from directlyanterior. This view makes it easier to appreciatethat the anterior canal axes are almost in a horizontal plane.  The horizontal canal axes are laterallydirected and the posterior canal axes have a substantial inclination to thehorizontal plane. The verticalcanals are clearly not aligned in the coronal plane and the horizontal canalshave opposite tilts.

The same canal axes viewed from directlylaterally. In this view, one can see that the rightland left canal axes are oppositely directed in the sagittal plane.  That is not true of the coronal plane(see previous figure). Theunderlying symmetry is most obvious from this viewpoint.

The same canal axes viewed from directly above. One can see that the right anterior and the left posteriorcanals are not actually aligned in the horizontal plane although they areapproximately aligned.

Angles Between the Canals

Given the plane and normal vectors of the canals, one canreadily compute the relative orientations of the canals.  The ratio of two planes is theirintersection, which is also the ratio of their normal vectors.  We can simply compute the ratios of thecanal axes and obtain the angular excursions and the axes of theintersection. The following tablesummarizes the results of a number of such calculations.

Canals	Ratio	Angle	Intersection	Blanks et al. ‘75
Horizontal to Anterior		112.216°
Horizontal to Posterior		95.948°
Anterior to Posterior		86.2003°
Right Posterior to Left Anterior		23.2833°
Right Anterior to Left Posterior		23.2833°
Right Horizontal to Left Horizontal		18.3942°

Non-optimal Rotations

Most movements will not be about an optimal canal axis forany canal and no rotation can be optimal for all the canals, because they havedifferent canal axes. That isfundamentally why there are multiple canals, so that any head rotation can besensed by at least one canal. Wenow consider how the different canals sense a given rotation.

The magnitude of response in a canal is proportional to theprojection of the axis of the rotation on the canal axis.  If both axes point in the samedirection, then the response to the rotation will be maximal for that rotation.  If they are perpendicular, then theresponse in the canal will be nil. If they point in opposite directions, the response will be minimal.  In brief, the response is a canal willbe proportional to the dot or scalar product of the canal axis with the axis ofrotation. Each vestibular afferentwas a substantial resting discharge rate () which may decrease as well as increase in response to headrotation. Also, theproportionality factor for response versus movement amplitude () is different for different afferents and possibly fordifferent canals. If the axis ofthe n’th canal is and the axis ofrotation for the rotation is , then the response for the canal is something like thefollowing.

The canal axis is a unit vector, so, the response is avector in the direction of the canal axis that varies about the resting rate toan extent proportional to the magnitude of the movement, the proportionalityfactor, and the projection of the axis of rotation upon the canal axis.  Each canal extracts a different imageof the movement and it is the consensus of the canals that determines thebrains image of the movement. Theconsensus is not necessarily the same at all central locations, because a differentarray of afferents may end in different targets, but one presumes that thedifferences are compensated for by internal circuits so that the effectiveimage of the movement is much the same at all targets.  Therefore, there are grounds forassuming a summation of the canal inputs is a fairly useful indication of theinformation flowing into the brainstem in the vestibular nerves.

In this expression the summation is over canals, althoughone could view it as a summation over canal afferents.  Note that information is a vector.  It has magnitude and direction.  The rate of change of information maybe a quaternion, because it may be expressed as a ratio of vectors.

If the nervous system responds to the movement, then itneeds to know the current direction of the movement and its rate of change withrespect to time.

All the canal axes are drawn from a common centerand viewed from the left front, above the horizontal plane. It is readily seen that the alignments that are generallyassumed when sketching the actions of the canals individually and collectivelyare not actually valid, but this discrepancy is not a particular problem whenmodeling the semicircular canals.

The Vestibular Signal

The Hypodermic/Hydraulic Model of the Semicircular Canal Dynamics

Cupula as a diaphragm between a large, low resistance,chamber and a small, high resistance, chamber

The conformational change of the flattened semicircular ductas a short time constant capacitance/spring.

Short duration signals affect primarily the capacitance

Long duration signals affect the resistive elements

The Model of the Dynamic Vestibular Response

The dynamic vestibular response behaves like a second degreedifferential equation with a very short time constant (~1/100 sec) and a longtime constant (20-30 sec).

The stimulus is almost certainly rotational head movements,which cause angular acceleration.

The short-term response mimics the angular velocity, whichis taken to indicate a mechanical integration.

The long-term response is characterized by an adaptivereturn to baseline in the presence of a sustained constant angular velocity.

The model is generally characterized by a driving functionthat is a torque, therefore proportional to the second derivative of an angulardisplacement. The semicircularduct responds with an inertial drag from the fluid within the duct, resistanceto flow of the fluid because of viscosity (proportional to first derivative ofangular displacement), and distortion of the cupular membrane as it resists therelative displacement of the fluid (proportional to angular displacement).  To this one might add the distortion ofthe duct itself as it tries to accommodate the back flow, which is proportionalto fluid displacement or angular displacement.

The semicircular duct is quite narrow, which results in asubstantial resistance to flow along its axis, and it is flattened, which meansthat has a cross-sectional area that is less than maximal for itscircumference. If it is necessaryfor fluid to flow through the duct, there will be a resistance that will bepartially offset by outwardly directed pressure on the duct walls, which willcause the duct to become more circular in cross-section.  This conformational change will act asa capacitive or spring element.

A pressure difference across the cupula will cause thecupula to billow like a sail in a breeze. Since the cupula is elastic, it will provide a restorative force thatresists fluid flow and returns the fluid to status quo if it is displaced. Most head movements are apt to be of short duration andmoderate speed. Under thoseconditions there is probably very little movement of the cupula and it acts asa linear capacitive element.

As the cupula billows it produces a transient pressure changesin the ampulla and the proximal duct. These are partially absorbed by fluid flow through the duct andpartially by changing the duct’s cross-sectional shape to accommodate more orless fluid. The resistance tofluid flow through the duct is proportional to the velocity of flow.  The distension/compression of the ductis proportional to the amount of fluid flow.

Because of the much greater resistance of the duct to fluidflow, one can open the circuit and to the first approximation model thecupular-duct dynamics as two chambers with an elastic diaphragm between.  On one side there is low resistance toflow and a large reservoir of fluid. On the other side is a high resistance to flow and a small volume offluid. The driving force is thepressure at the utriculo-ampullary junction, F_u.  The magnitude of the force isproportional to the rate of angular displacement in a manner that depends onthe axis and direction of rotation, the location, orientation, andcross-section of the junction. The amount of fluid flow will be proportional tothe driving force, V = F_u * R. The cupular membrane is distorted by the fluid displacement, V, and itgenerates a restoring force proportional to the amount of distortion.  In addition, the compression orrarefaction of the fluid on the duct side of the cupula will causeconformational changes in the duct and will cause fluid flow through theduct. The conformational changeswill be small and therefore linearly related to the volume displaced.  The flow through the duct will belaminar, therefore proportional to the rate of change of the volume displaced.

The forces on the cupula will be balanced so we can write anumber of expressions

Let represent thedisplaced volume. On the utricularside of the cupula there is a force that causes aflow through a resistance .

The elastic force, , from distortion of the cupula is linearly related to thedisplaced volume

The duct force is composed of the conformational distortion,, and the resistance to flow, . Theconformational distortion is proportional to the displaced volume for smallvolumes.

The flow resistance is proportional to the rate of change ofthe displaced volume or the flow.

If the duct is fed back to the large reservoir then therewill be a different force applied to its distal end, .

The two forces that impinge upon the ends of thesemicircular ducts, , are dependent upon the geometry of the system and themanner in which it moves.

The Alignment of the Canals with the Foramen Magnum

The torsion pendulum model assumes that rotations occuraround the center of the ring, but they almost never do so.  If they did for one semicircular duct,then they would not for any other duct.

An axis drawn from the foramen magnum to the semicircularcanals would be roughly orthogonal to the posterior canal, roughly in the planeof the anterior canal and roughly in the plane of the horizontal canal.

The positioning of the cupula at the utriculo-ampullaryjunction may important to the changes that occur in the cupula.

More Detail of Anatomy of Semicircular Canal With Functional Implications

Schematic of a Semicircular Canal with Frame of Reference

The sensory apparatus for a semicircular canal is located inthe ampulla, so, it is movements of the ampulla that are most relevant to thestimulation of a canal. For thatreason it makes sense to place the framework for describing the canal at theampulla. The framework will be astandard frame of reference. Thelocation of the ampulla, , is the first frame vector. It will be defined relative to a central origin, , and frame of reference, . The extensionvector, , is reserved for a convenient measurement and it may bemultiple vectors if need be. Theorientation for the canal will be the three vectors . The firstorientation vector is the axis of rotation for the canal.  Rotation about that axis will optimallystimulate the canal. The secondorientation vector is in the direction that will in which cupular movement willmaximally stimulate the hair cells for that canal. It is opposite the direction in which ampullary movementwill maximally depolarize the hair cells in the crista ampullae.  The third axis is the product of thefirst orientation vector times the second orientation vector.  It completes a right-handed coordinatesystem.

The framed vector for a canal is anchored at theampulla and its orientation reflects the movements that maximally stimulate theampulla.

Movements in the direction of or will not changethe discharge rate in the ampullary nerve. Rotations about either of those axes will maximallystimulate the canals as will rotations about any axis that may be expressed asa combination of those two vectors. Movements in the direction of will maximallystimulate the canal and rotations about that vector will not change themembrane potentials of the hair cells. Movement in the opposite direction will maximally reduce the dischargerate in the canals nerve. The nextstep is to define the consequences of rotations that are not in the abovecategories, that is, to solve for the general case.

The Pattern of Excitation in a Canal Hair Cell as a Function of MovementDirection

Canal afferents have a rest discharge rate that may increaseor decrease, depending on the direction of head movement.  Let that rate be . Then theincrease or decrease with movement is dependent upon how the movement lies withrespect to the axis from the stereocilia to the kinocilium of the haircells. In semicircular canals, allthe hair cells have a common axis although the saddle shape of the cristaampullae means that the actual direction is variable. Presumably, it is the shearing force of the cupula on thehair cell cilia that determines the displacement of the hairs.  The cupula is normally caused to billowby the pressure of the endolymph pressing on the broad surfaces of the cupula,so the shear is across the crista. Consequently, we would expect the activation of the hair cells toreflect the pressure differential across the cupula, so that we can choose thedirection orthogonal to the cupula and crista as the direction of movement thatwill optimally stimulate hair cells. That pressure differential will depend on the angle between the movementdirection and the optimal direction in a way that is close to the cosine of theangle. If the movement directionis and the optimaldirection is , then the effective stimulus magnitude is , where is the magnitude of the acceleration, is a unit vectorin the direction of the movement, and is the unitvector in the direction of optimal stimulation. Consequently, the response would have the following form.

The magnitude of the movement has been separated from thedirection to illustrate that they enter into the response in different ways,but there is no reason one might not have used a vector of variable magnitude,such as . Then theexpression would be simply . The form ofthis expression is an epicycloid that resembles a cardioid.

If we plot the responses for a series of accelerationsranging from no movement to a movement that completely suppresses thevestibular afferents activity. Then the response curves look like the followingfigure. The red curve is withminimal movement and the blue curve is with sufficient acceleration tocompletely suppress the activity when it is in the direction opposite thedirection of optimal responsiveness.

The responses to movements of an ampulla will be maximalwhen it is in the direction of , minimal when in the opposite direction and nil when in theplane perpendicular to the axis. So, each canal has a response surface that is like those drawn above,but rotated about the horizontal axis. If we assume similar response curves for each canal, then we can readoff the magnitudes of the responses in the different canals by computing theangle between the acceleration vector and the canal axis and plotting theradial vector at that angle. Wherethe vector intersects the surface of the appropriate acceleration the length ofthe vector will be the magnitude of the response. Because to the canals are approximately perpendicular,rotations about the axis of one canal will give a maximal response for thatcanal a near minimal response for its opposite on the other side of the headand near baseline responses in the other canals. The near baseline responses will be antagonistic in thecentral brainstem circuits therefore will approximately cancel each other whilethe maximal and minimal responses will show a near maximal difference.