A model of the nystagmus induced by off vertical axis rotation.

Timothy C. Hain, MD Page last modified: April 12, 2020

Hain, T. C. (1986). "A model of the nystagmus induced by off vertical axis rotation." Biol Cybern 54(4-5): 337-350.

In this page, we reprise the results of our 1986 paper, and have added some thoughts of subsequent workers on this interesting phenomenon.


A three-dimensional model is proposed that accounts for a number of phenomena attributed to the otoliths. It is constructed by extending and modifying a model of vestibular velocity storage. It is proposed that the otolith information about the orientation of the head to gravity changes the time constant of vestibular responses by modulating the gain of the velocity storage feedback loop. It is further proposed that the otolith signals, such as those that generate L-nystagmus (linear acceleration induced nystagmus), are partially coupled to the vestibular system via the velocity storage integrator. The combination of these two hypotheses suggests that a vestibular neural mechanism exists that performs correlation in the mathematical sense which is multiplication followed by integration. The multiplication is performed by the otolith modulation of the velocity storage feedback loop gain and the integration is performed by the velocity storage mechanism itself. Correlation allows calculation of the degree to which two signals are related and in this context provides a simple method of determining head angular velocity from the components of linear acceleration induced by off-vertical axis rotation. Correlation accounts for the otolith supplementation of the VOR and the sustained nystagmus generated by off-vertical axis rotation. The model also predicts the cross-coupling of horizontal and vertical optokinetic after nystagmus that occurs in head-lateral positions and the reported effects of tilt on vestibular responses.


A stepwise change in the velocity of head rotation about earth vertical, delivered with the head oriented in the normal upright position, induces a compensatory nystagmus (the vestibulo-ocular reflex or VOR) that dies away with a time constant of approximately 20 seconds. In contrast, rotation at a constant speed about an axis other than the earth vertical results in a sustained nystagmus.Rotation about the head-foot axis when the body is parallel to the earth, called "barbecue rotation" , evokes a sustained horizontal nystagmus (Guedry, 1965; Benson and Bodin, 1966a; Correia and Guedry, 1966; Correia and Money, 1970; Young and Henn, 1975; Raphan et al., 1981, Goldberg and Fernandez, 1982) (see Fig. 1). Rotation about the inter-aural axis (head-over-heels rotation) evokes a sustained vertical nystagmus (Young and Henn, 1975; Correia and Guedry, 1966; Bodin, 1978; Goldberg and Fernandez, 1982).
--------------------- figure 1 ---------------------
The nystagmus evoked by off-vertical axis rotation, called OVAN, consists of a sustained bias combined with a superimposed modulation that has the same frequency as the rotation.Typical values for a 60 deg./sec. barbecue rotation in man are about 15 deg./sec. bias and 5 deg./sec. modulation (Benson and Bodin, 1966a; Correia and Guedry, 1966).Slow-phase velocity maxima of modulation occur slightly prior to the time when the side of the ear that isdown coincides with the slow-phase direction of the OVAN bias, specifically, it lags the acceleration due to gravity by about 140 degrees.The amplitude of modulation is approximately proportional to rotational velocity from 10 deg./sec. through 180 deg./sec. (Benson and Bodin,1966 ;Correia and Guedry, 1966; Raphan et al., 1981; Wall and Black, 1984).OVAN similar to that found in humans can be recorded in monkeys though bias velocities are as large as 50-60 deg./sec. (e.g.Raphan et al., 1981).

The vestibular apparatus is necessary for OVAN since humans with no vestibular function do not exhibit OVAN (Guedry, 1965).Furthermore, OVAN requires otolith input as cutting the utricular nerve and destroying the saccule abolishes OVAN in rabbits (Janecke 1970).On the other hand, both bias and modulation persist in monkeys after plugging of the semicircular canals (Cohen et al., 1983).Because of these observations it is accepted that OVAN is generated through otolith activity.The discharge of individual otolith afferents shows a sinusoidal waveform during off-vertical axis rotation and the modulation component could derive from their output.However, neither the otolith nor canal afferents of the vestibular nerve show a bias during off-vertical axis rotation (Goldberg and Fernandez, 1982, Raphan et al., 1983).Therefore,the bias must be generated centrally. Presumably this is accomplished by reconstruction of rotational velocity from the sinusoidally modulated otolith signals.

The mechanism by which sinusoidally varying otolith component signals are transformed into a sustained bias that is appropriate tothe speed and direction of rotation is unknown.Much is known about the effects of the otoliths on the VOR in other experimental contexts and the question of whether OVAN can be understood through these actions arises.It is clear that otolith signals modulate the time constant of vestibular responses.Responses obtained by rotating about the earth vertical axis are shortened if the head is repositioned during the post-rotatory response (Benson and Bodin, 1966b). Likewise the time constant of vestibular nystagmus can be altered if the rotation is performed in a centrifuge (Benson and Whiteside, 1961; Lansberg et al., 1965; Crampton, 1966; Benson, 1974).

These effects on the VOR time constant presumably arise from otolith interaction with a neural network called the "velocity storage mechanism".This central neural circuit converts the cupula time constant, 5.7 seconds in monkeys (Fernandez and Goldberg, 1971), to the 16 second time constant of the VOR (Raphan et al., 1979).We propose in particular that otolith information about the orientation of the head to gravity modulates the degree to which velocity signals are connected to the velocity storage mechanism. It will be shown that a model incorporating this principle can simulate OVAN.

2 A Three-dimensional Model of Otolith-vestibular Interactions

--------------------- figure 2 ---------------------
The model is a three-dimensional extension of a one-dimensional kinematic model of the VOR (see Fig. 2) proposed by Robinson(1977).Velocity storage in Robinson's model is provided through positive feedback. The peripheral signal is supplemented with a stored central copy that perseverates the response.The effectiveness of velocity storage, as reflected in the time constant of the VOR, depends on a gain, k , that determines the amount of positive feedback, and upon a lag (storage element) with time constant To (Robinson, 1977; Robinson, 1981).When To is equal to Tc, the time constant of the cupula of the semicircular canals, the behavior of the model simulates that of normal subjects.Equation 1 relates eye velocity, e., to head velocity, h. (s denotes the Laplace operator) :

e. 1 1 (1)
- = sTc --- ----------------
h. 1-k sTc(1/(1-k)) + 1

The new model also takes as a starting point a three dimensional model of the VOR, without kinetics, again proposed by Robinson (1982).Here the VOR is formulated as being the result of an operation upon a input head angular velocity vector, h., by successive matrices representing the semicircular canals, brainstem, and extra-ocular muscles to produce an output eye velocity vector.Using capital letters to denote the matrices described above, the VOR eye velocity vector, e., is simply :
e. = M B C h. (2)
Before describing the combination of these two models we will first define the subsequent nomenclature :matrices will be denoted by bold capital letters, as the canal input matrix, CI.Matrixelements will be denoted numerically as MATRIX[row, col]. Vectors will be denoted by bold lower case letters as h. for the head angular velocity vector or e. for the eye velocity vector.To be consistent with previous workers (Robinson, 1982), a left handed Cartesian coordinate system is employed to describe angular velocities and linear accelerations relative to the head (see Fig.3A).The axes are X (+ = left), Y (+ = anterior) and Z (+ = superior).Vector components in this Cartesian frame will be denoted by _x, _y, or _z suffixes.Angular velocity sense is determined by the right hand rule -- positive rotations about the X, Y, and Z axes when the head is upright are forward pitch, counter-clockwise roll and leftward yaw.For example, the vector of head velocity that describes rotation counter-clockwise at 100 deg./sec. is {0, 0, 100}.
--------------------- figure 3 ---------------------
The otoliths, including the utricle and saccule, respond to linear acceleration.We will consider otolith output to be a Cartesian vector of linear acceleration, la, relative to the head (see Fig. 3B).Linear acceleration is positive when it could be due to an acceleration of the head along the positive sense of an axis.The la vectors that describe the force of gravity, with the head upright, supine, and right side dependent are respectively {0, 0, 1g}, {0, 1g, 0} and {1g, 0, 0}.
--------------------- figure 4 ---------------------
The operation of the three-dimensional portion of the model (Fig. 4) is as outlined by Robinson (1982) :the head velocity vector, h., is multiplied by a canal matrix, CI, to compute itsprojection on the canal axes, c (3).CI expresses the geometry of the canals as found in the human (Blanks et al., 1975) and converts the Cartesian head velocity vector into a conjugate canal coordinate reference frame (see Fig. 3A).The conjugate canal pairs are left and right lateral (lrh), right-posterior and left-anterior (rpla), and right-anterior and left-posterior (ralp).Vectors formulated in this frame, as for instance the c vector, are in "canal coordinates" and their components will be denoted by _lrh, _rpla, and _ralp suffixes respectively.Vectorc is combined with stored activity from velocity storage and retinal and otolith inputs, to form vector vn.
c CI h.
| c_lrh | | .000 -.374 .927 | | h._x | (3)
| c_rpla | = | -.673 .723 .156 | | h._y |
| c_ralp | | .673 .723 .156 | | h._z |
The output limb of the model converts central vestibular signals encoded in canal coordinates into Cartesian eye angular velocity.Vector vn is multiplied by the brainstem output matrix, BO to produce an ocular muscle innervation vector, m.The muscle innervations are converted to the eye velocity vector, e., through the muscle output matrix, MO which expresses the pulling directions of the muscles.These matrices are unchanged from those of Robinson (1982).

The new model also contains additions that provide three-dimensional dynamics and otolith interactions that will be summarized briefly at this point.All extensions except thosedealing with otolith inputs are simple substitutions of vectors for the scalar quantities of the original model of Robinson (1977).First, interposed between the canal input (CI) and brainstem output (BO) matrices, three velocity storage mechanisms have been included, one for each conjugate canal plane.A multidimensional implementation of velocity storage has been previously suggested by Cohen and others (1983).Equation (1), developed for the one-dimensional model, is still correct for the three-dimensional form but it applies only to activity confined to a single canal plane.

A retinal input matrix, RI, numerically identical to CI (3), transforms the Cartesian retinal image velocity vector, rslip, into canal coordinates and allows the model to produce optokinetic nystagmus and afternystagmus.Otolith input, denoted as linear acceleration vector la, causes gaze deviation, supplements the canal signals, and modulates the time constant of the VOR.The gaze deviation vector, gdev, is constructed from the product of la and matrix GI.The resulting vector is operated upon by an expression incorporating a delay and lag that simulates reported dynamics andthe derivative is then added to the eye velocity developed by the canal pathway. To provide for otolith supplementation of canal inputs, la is first differentiated to form jerk (the derivative of acceleration).The result is then transformed into canal coordinates through multiplication by matrix OI and then enters the velocity storage mechanism.Finally, provision for modulating velocitystorage feedback gain vector, k, by the linear acceleration vector, la, is made throughmatrix KI.

2 Static and Dynamic Ocular Responses to Linear Acceleration

2.1 Gaze Deviations
Static linear accelerations do not in general induce a sustained eye velocity but cause gaze deviation.We will develop a three dimensional model for gaze deviation in order to simulate the modulation component of OVAN.In man, torsional counterroll of the eye to static lateral tilt has been well described (Miller, 1962; Diamond et al., 1979).Such gaze deviations need not always be torsional -- in rabbits and frogs vertical eye movements are induced by linear accelerations along the interaural axis ( Baarsma and Collewijn, 1975, Hess et al., 1984,).During counterroll the eye does not immediately assume the new position and the dynamics of the movement can be measured by use of sinusoidal linear accelerations. In such experiments the gain of the response decreases for frequencies above about .2 -.5 hz.
Small horizontal gaze deviations also occur to static positioning in rabbits (Lorente de No, 1932), monkeys (Krejcova, Cohen and Highstein, 1971), and humans (Benson, 1974).No quantitative data is available in humans, probably because of variability in the horizontal and vertical gaze position that derives from voluntary eye movements. In humans undergoing barbecue rotation, horizontal gaze deviations, about 25 deg. in amplitude, occur directed away from the gravitational vector, that is, towardsthe earth (Benson and Bodin, 1966 ; Wall and Black, 1984).Gaze deviations can be also elicited by canal inputs (Mishkin et al., 1966)
We will assume that the dynamics of all gaze deviations induced by linear acceleration are the same as those found for counterroll.Hansen and coworkers (1966) described counterroll by relating it to lateral head tilt in the human.Their formula has the transfer function of the form
R(s) = Ggain*e-as / (sTg + 1) (4)
where Ggain is the gain of the reflex, delay a is between 0 and 400 ms and Tg is .32 sec.Although for small lateral deviations from upright, the angle from vertical is proportional to the laterally directed component of acceleration, experiments with centrifuges have shown that in humans, counterroll is more closely related to lateral acceleration than the angle from vertical (Woellner and Graybiel, 1969).In the model, gaze deviation, vector gdev, is produced by multiplying la by a gaze matrix, GI in (5) below.The dynamics are provided by applying (4) allowing Ggain to be -5 and the delay, a, to be 250 msec.The model simulates static and dynamic torsional counterroll as well as compensatory horizontal gaze deviation as reported during OVAN. In section 4 we will show that this small gaze deviation could account for much of the modulation component observed during OVAN.It should be emphasized that the transfer function for horizontal gaze deviations is extrapolated from counterroll data and further studies of horizontal gaze deviations in human subjects are needed. GI (5)
| gdev_x | | 0 0 0 | | la_x | -5e-.25s
| gdev_y | = | -1 0 0 | | la_y | * ---
| gdev_z | | 1 0 0 | | la_z | .32s + 1

2.2 L-nystagmus
Humans also exhibit an eye movement response to time varying linear accelerations exerted along the X and Z axes called 'L-nystagmus' (Jongkees & Phillipzoon, 1962; McCabe, 1964; Niven et al., 1966; Buizza et al.., 1980).For sinusoidal lateral accelerationsthere is approximately 16 deg./sec. of horizontal eye velocity generated per 'g' of linear acceleration with the head upright between .2 and .8 hz (see Fig. 5).L-nystagmus lags head acceleration by about 210 deg. for the same frequency range (Niven et al., 1966; Buizza et al., 1980).This can be compared to OVAN modulation which lags linear acceleration by about 140 deg. at 0.13 hz (Wall and Black, 1984) and about 180 deg. at 0.5 hz (Correia and Guedry, 1966).
Records of L-nystagmus show nystagmus with resetting quick phases superimposed on gaze deviation in the direction of the slow phase (Niven et al., 1966). Therefore, although some part of L-nystagmus could result from the gaze deviation, most must originate from some other source.It is generally thought that Lnystagmus is generated from the derivative of linear acceleration, which is called "jerk", (Niven et al., 1966; Young and Meiry, 1967).The jerk signal could be constructed centrally by differentiating the robust head linear acceleration signal provided by regular otolith afferents (Raphan et al., 1981) or might reflect the output of the irregular otolith afferents since partialderivatives of acceleration are found in their firing patterns (Fernandez and Goldberg, 1976).
It has been suggested that the purpose of L-nystagmus is to compensate for head translations (Buizza et al., 1981).However,to match eye position to head position, eye velocity would need to respond mainly to the integral of linear acceleration, that is, translational head velocity. The amplitude of L-nystagmus, at the frequencies reported, rather is proportional to head acceleration.We propose instead that L-nystagmus, per se, has no purpose. Rather it is a side effect of an otolith mechanism that supplements the semi-circular canals when the orientation of the head is changing with respect to gravity.

2.3 A Mechanism to Determine Angular Velocity from Linear Acceleration Signals
It is known that the otoliths enhance the low frequency characteristics of the vertical VOR of the rabbit (Favilla et al., 1980; Barmack, 1981) and cat (Peterson et al., 1986). Furthermore, the bias component of OVAN is an enhancement of the lowfrequency response that exists for both the horizontal and vertical VOR in humans and monkeys. The mechanism of this enhancement is not well understood. A simple addition of otolith and canal activity is not a feasible method of obtaining this response. For example, the Z axis component of la is decreased both for pitch forward and backward from upright. However, to supplement the VOR the otoliths shouldcontribute an upward slow phase in the former case and downward in the latter.
A solution to this problem can be found by using information from several axes simultaneously. For example, one solution would be the rule that if the Y axis signal is positive, a decrease in Z should induce a downward slow phase but if the Y signal is negative, an upward slow phase should be induced. Further insight can be obtained by examining the spatial relationship in three dimensions between the vector of head angular velocity, w, and the rate of change of linear acceleration, jerk.
When rotation occurs without translation, information about the rate of rotation is found in the jerk vector. In particular, when the rotation has no component about earth vertical, the magnitude of the jerk vector is proportional to head angular velocity.Consider the example of barbecue rotation :if the head is rotated counter-clockwise in about its Z axis from the supine position, the angular velocity vector (relative to the head), w, is {0, 0, w}.The la vector ( which is also defined relative to the head), will rotate with the head and is {-sin(wt), cos(wt), 0}.jerk is then the derivative of la, {-wcos(wt), -wsin(wt), 0}, and its magnitude is w.The direction of jerk is determined by the vector cross product of la and w (Goldstein, 1950) jerk is perpendicular to both and changes continuously during the rotation.
How can this rotating jerk vector be converted into an eye movement compensatory to w ? The problem has two aspects -- calculation of the magnitude of jerk and calculation of thedirection of the resultant so that it is compensatory for w.The problem of direction can be simplified by considering it in the coordinates of the neural signals.We assume that in order to supplement canal inputs, otolith signals are transformed into canal coordinates.The directional mapping between Cartesian linear acceleration and activity in canal coordinates can be determined from observations of L-nystagmus.Linear accelerations along the X axis produce horizontal nystagmus andlinear accelerations along the Z axis produce vertical eye movements.Accelerations along the Y axis do not produce horizontal or vertical nystagmus in humans (Niven et al., 1966).These relationships are expressed in the OI matrix of (6) which was constructed from a CI matrix (3) by making all elements of the 2nd column zero. There is also rearrangement to reproduce the spatial effects noted above and addition of a unit conversion factor, Ogain (100 deg./sec/g), justified in section 2.5.
jerk OI la (6)
| jerk_lrh | | .927 0 0 | | la_x |
| jerk_rpla | = | .156 0 -.673 | s * Ogain | la_y |
| jerk_ralp | | .156 0 .673 | | la_z |

Returning to the example of barbecue rotation, for a bias to be compensatory to w it should be generated mainly by activity in the plane of the lateral canal.The external jerk vector rotates continuously in the plane perpendicular to w (the transverse plane of the head) producing a sinusoidal neural signal mainly in the plane of the lateral canals, with amplitude proportional to w. However, according to (6) there is no neural activity generated by the component of jerk directed about the Y axis. Similarly, for headover heels rotation, there is a sinusoidal Z component of jerk shared between the vertical canal planes.Thus the sinusoidal otolith jerk neural signal is collinear to the compensatory response and the direction problem is partially solved. To determine w, the brain must extract the signed amplitude of this sinusoidal signal.
As noted above, use of otolith information from other axes can be helpful. In particular, we propose that the amplitude of the jerk vector is calculated by correlation of jerk with the Y axis otolith position signal. Correlation, in the mathematical sense, is the process of multiplying one signal by a second and integration of the product. For the example of barbecue rotation, the Y axis otolith signal is cos(wt). The jerk signal in the plane of the lateral canals is .927cos(wt). The integral over one cycle of the product of these two signals is -.463w which is proportional to w and solves the magnitude calculation problem.
The evidence that correlation could occur in the vestibular system will be presented in following sections. In section 2.4 we will review arguments that jerk is connected to the vestibular system via the velocity storage integrator. In section 3 we will show that the actions of the otoliths on velocity storage are explained by a multiplication of the la vector with velocity storage signals.In section 4 we show that using these neural equivalents of multiplication and integration, OVAN can be explained.

2.4 Otolith Signals are connected to Velocity Storage

There is much experimental data to support the hypothesis that otolith signals are connected to the vestibular system indirectly through velocity storage.Raphan and associates suggested such a link to explain their observations that responses due to otolith activity (such as OVAN) build up slowly while the otolith afferent signals have a rapid response.Furthermore, OVAN is abolished when the lateral canal nerve is cut, a procedure that also abolishes velocity storage (Raphan et al., 1981).The known contributions of the otoliths to the VOR as reviewed in section 2.3 are effective only at low frequencies as predicted by this hypothesis.
The main alternative possibility, namely that a form of jerk and a signal derived from the canals is simply added together, is unreasonableas then VOR gains greater than unity could occur. For example, if the head were suddenly accelerated the response could reflect the sum of the canals, which might easily have an initial gain of unity, and an otolith contribution.This problem does not arise if jerk is connected to the vestibular system indirectly through the velocity storage mechanism.Then jerk adds to the VOR only the components that the canals don't provide, extending the low frequency response.

2.5 A Model of L-nystagmus
By using (6) to construct jerk and sending the output to the velocity storage mechanism, a model of Lnystagmus can now be formed. The expression for L-nystagmus is complex because the model is three-dimensional.To understand the general features of theresponse, (7) shows an approximate one-dimensional form.The two terms reflect activity transferred through the velocity storage mechanism and gaze deviation, respectively.
e. k -100s -5se-.25s (7)
-- = --- -------------- + -----------
la 1-k sTc(1/1-k) + 1 .312s + 1

In section 3 we will develop a general expression for k that forms part of the correlation mechanism.However, for the simplified one dimensional form above, k in the upright position for the lateral canal plane can be calculated to be .657 by using (1).This is the value required by a one-dimensional model to generate a time constant of 17.5 sec. given the presumed human cupula time constant, Tc, of 6 sec. The factor of 100 in the numerator corresponds to Ogain in (6) and was chosen to fit the amplitude of simulated Lnystagmus to experimental data.Fig. 5 shows that the full model of Fig. 4 simulates most aspects of the data of Niven (Niven et al., 1966).
--------------------- figure 5 ---------------------

3 Interactions between Linear Acceleration and Velocity Storage

In order to determine the time constant of the VOR alone, without a superimposed OVAN, the axis of head rotation must coincide with the earth vertical axis, or more generally, the resultant of the gravity vector and any externally-applied linear accelerations.Thus under normal circumstances a particular pattern of canal activation is uniquely associated with one orientation of the head with respect to gravity.For example, in order to activate the vertical canalswithout stimulating the otoliths or lateral canals, the axis of rotation and earth vertical axis must coincide and be in the plane of the lateral canals.In order to circumvent this obligatory association of the axes of head rotation and linear acceleration,two strategies have been used. First, external linear acceleration can be applied or removed by eccentric centrifuge rotation or parabolic flight.Second, the post-rotatory response can be measured following a tilt.

Benson and Whiteside (1961) observed that a 3.1 g linear acceleration directed anteriorly, developed by a centrifuge, shortened significantly the decay of horizontal nystagmus induced by rotation in yaw.If the influence of gravity is removed, such as in parabolic flight, the duration of post-rotatory nystagmus is reduced (Yuganov, 1966).These reports demonstrate that the time constant of the horizontal VOR is modulated by static linear acceleration directed along the Y or Z axes.When a linear acceleration is directed along the X axis the effects on the response to rotation depend on the direction of rotation.The duration of per-rotatory nystagmus is increased when the direction of acceleration is opposite to the direction of the horizontal slow phase and decreases for the opposite condition (Lansberg et al., 1965 ; Crampton, 1966).
Considerably more data is available on the effects of tilt on post-rotatory vestibular responses butthis response is harder to interpret than those described above for several reasons. First, the change of head position itself might influence the time constant of an ensuing response independent of the new value of the linearacceleration vector. Second, as the response is often obtained through active tilts of the head, some of the effect may be related to the neck receptors. Finally, during the head tilt there is a Coriolis effect that may alter the subsequent response.Nevertheless, it is useful to determine how much of this data can be explained by our model.
When human subjects are rotated with their head upright, the time constant of their post-rotatory response is approximately 17 seconds.However, if the head is tilted prone (face down) at the onset of the post-rotatory response, the time constant is reduced to about 7 sec. (Benson and Bodin, 1966b; Hain et al., 1985).Tilts of the head supine or into the lateral (roll 90 degrees) positions also lower the time constant (Benson and Bodin, 1966b; Schraeder et al., 1985ab).If the per-rotatory response is elicited by barbecue rotation, the time constant is 8.5 whether the head is stopped supine, prone or in either lateral position (Benson and Bodin, 1966c) (see Table I).For the vertical VOR, tilts from lateral to upright reduce the time constant from 7.5 to 5.9 sec. (Benson and Bodin, 1966b).

n upright prone supine lateral Source
11 17.8 8.5 8.5 8.5 Benson and Bodin, 1966c
8 17.5 7.0 9.7 8.5 Benson and Bodin, 1966b
7 17.2 7.1     Hain et al., 1985
4 16.1 5-6.9     Schraeder et al., 1985a
4 13.4 5.4 9.0 7.2-7.5 Schraeder et al., 1985b

TABLE 1. Time constants of the VOR after repositioning. All responses were obtained by active head tilts during the postrotatory phase of the vestibular response except for the studies of Benson and Bodin. In all studies except for the first the perrotatory response was obtained by rotation about earth vertical. In the former study, rotation was about earth horizontal.

3.1 A Model of Otolith Modulation of the VOR Time Constant
Benson, developing a suggestion of Guedry (1965), proposed that the time constant of the post-rotational response is reduced by disparity between canal and gravireceptor information (Benson, 1974).In vector notation, for simple rotations about earth verticalthe directions of the canal and gravireceptor input vectors, h. and la are identical.Tilts during the post-rotatory response rotate la but not h. which increases the angle (disparity) between the the canal and otolith vectors.Thus tilt reduces the time constant of the response.

This hypothesis can be modeled by allowing the velocity storage feedback gain vector, k, which controls the time constant of the VOR in the model, to be a scaled linear acceleration vector, or in other words, be proportional to la.The time constant for each canal plane is Tv = Tc/(1-k) by (1). As the component of k for each canal plane is proportional to the projection of la on the conjugate canal axis, the time constant will be modulated by la.The component of la along the axis of a canal plane can vary from +g where velocity storage is maximized to -g where it is minimized.

A coordinate transformation must be provided to let Cartesian vector la influence central signals that are in canal coordinates.The appropriate dependence is obtained by providing a coordinate transformation matrix, KI, identical to CI.Conversion from units of 'g' to the dimensionless k is provided by a gain, kgain.
k KI la (8)
| k_lrh | | 0 -.374 .927 | | la_x |
| k_rpla | = kgain * | -.673 .723 .156 | | la_y |
| k_ralp | | .673 .723 .156 | | la_z |

kgain can be calculated from the time constant for a particular canal plane in a given orientation.The exact value of the cupula time constant in humans, Tc, has never been measured.It isgenerally considered to be close to that of the squirrel monkey, 5.7 seconds (Fernandez and Goldberg, 1976).Here and subsequently, we will assume Tc = 6 seconds in humans and the time constant of the horizontal VOR to be 17.5 sec. Neglecting the small amount of horizontal eye movement that derives from the vertical canals, k_lrh must be .657 from (1).By solving (8) for kgain when the head is upright (la is {0, 0, 1}) we can determine it to be .709.

Two modifications of (8) are necessary to simulate observed behavior.First, we must consider the effect of extreme values of la.For linear accelerations of magnitude greater than 1 g, k could become 1 or more which would correspond to an infinite Tv.Thus, k in any canal plane has been limited to a maximum of 0.8 corresponding to a maximum time constant of 30 seconds.

Second, letting k be purely a scaled la implies that in certain head positions, for example the right dependent position, no velocity storage should be present in the plane of the lateral canals.OKAN is thought to originate from the same neuronal pool as vestibular velocity storage andin monkeys a robust horizontal OKAN response in the lateral position has been reported (Raphan and Cohen, 1983).OKAN in this position can be obtained by allowing k to be the sum of an initial value, k0, and a scaled la as previously.Eqn (9) incorporates k0 which is set to {.1, .1, .1}.After adjustment for k0 to maintain a time constant of 17.5 in the upright position, kgain (.601) has been merged into the coefficients of the KI matrix. (9)
k k0 KI la
| k_lrh | | .1 | | .000 -.225 .557 | | la_x |
| k_rpla | = | .1 | + | -.404 .434 .094 | | la_y |
| k_ralp | | .1 | | .404 .434 .094 | | la_z |

Equation (9) expresses the final form of the model's method of causing the otoliths to control the dynamics of the VOR.The modulation of the time constant of the VOR is obtained by multiplication of two signals (denoted in Fig. 4 by a a circle with an 'X').Multiplication of two signals is a nonlinear operation.This differs from the many coordinate transformations of signals used which are obtained through multiplication of signals by constants and thus are linear operations.There is no direct physiological evidence that demonstrates that otolith position signal influence velocity storage in this multiplicative way but there are many possible neural mechanisms that could provide for a neural multiplier.Some neurons exhibit multiplicative input/output relationships (Bullock et al., 1977).Another method might be through the use of neural networks whose sensitivity to one signal is determined by another signal. For example, an ensemble of parallel neural channels (neurons) sharing a common vestibular input and output might exist.These neural channels might be driven into cutoff in a graded fashion by otolith input which would allow the network as a whole to perform multiplication.

3.2 Predicted Effects of Linear Acceleration on the VOR
The effects of externally-applied accelerations on the VOR as described at the beginning of the preceding section are explained by (9). In the experiment of Benson and Whiteside (1961), a positive external acceleration was applied along the Y axis. In vector notation, la was {0, 3.1, 1.0}.This is predicted by (9) to decrease the lateral canal plane time constant to 5.7 sec. which is consistent with the significant reduction in the time constant observed.Removing the effects of gravity is predicted to decrease the lateral canal plane time constant to 6.7 sec.This is consistent with Yuganov's report (Yuganov, 1966).
Accelerations directed along the X axis are predicted to leave the time constant of the lateral canal plane unchanged.Additionally however, X axis accelerations produce L-nystagmus.In the experiments of Lansberg and Crampton described above, a period of acceleration in a centrifuge was followed by a constant velocity period during which the vestibular response was recorded.The direction dependent modulation of the VOR reported is consistent with superposition of L-nystagmus and the VOR (Young, 1967).
For tilt suppression, the model predictions are largely consistent with experimental results (compare table I and table II).The main difference is that the VOR time constant with the head prone is greater than that supine while the opposite pattern has usually been found experimentally.As mentioned above, tilt suppression responses may be influenced by sources of input other than the otoliths which are not included in themodel.Alternatively, there may be a different function relating la and the time constant of the VOR than that used in the model.
---- TABLE II about here -----
The per-rotatory time constant for the lateral canal component of the VOR elicited with the head pitched forwardso as to make the lateral canal horizontal is predicted to be 20.6 sec., slightly greater than the time constant of 17.5 sec. elicited with the head upright.The increase in time constant associated with the head in this position is consistent with findings in humans (Fetter, Hain and Zee, submitted).
Because CI, from which KI was obtained, is a normalized matrix, the maximum time constant for each vertical velocity storage mechanism is also 20.6 sec.In humans the time constant of the vertical VOR with the head in the lateral position is about 9 sec. (Baloh et al., 1983) which is close to the predicted time constant of 10.0 sec.The time constant of the vertical VOR with the head upright is predicted to be 7.5 sec. The reduction of the time constant with the head upright relative to the head lateral position is consistent with the tilt suppression experiment of Benson and Bodin (1966b).

4 Off-Vertical Axis Nystagmus (OVAN)

OVAN is produced by stimuli that have acceleration components that could produce L-nystagmus in addition to components along the head Y axis.As discussed in section 2, a compensatory signal proportional to angular velocity could determined from linear accelerationsignals by correlation of jerk with the component of linear acceleration about the Y axis.Correlation, in the mathematical sense, is obtained by multiplication of two signals followed by integration of their product.It is a signal processing technique that can be used to determine the degree to which one signal that is related to another.We have suggested in section 3 that otolith position signals directly modulate the feedback gain element k by which velocity storage signals are multiplied.Furthermore, in section 2.4 we have reviewed evidence that otolith jerk signals are relayed to the vestibular system through the velocity storage integrator.Thus there is evidence that a multiplication followed by integration, that is, a correlation process, occurs. Because of these characteristics the model simulates the sustained bias of OVAN (compare Fig. 6 to Fig. 1).
--------------------- figure 6 ---------------------
The model also duplicates most other reported behaviors of OVAN.The duration of the post-rotatory response is reduced as reported (Raphan, 1981) for two reasons. First, the bias partially cancels the post-rotatory response.Second, the lateral canal plane VOR time constant has been decreased because of tilt away from its axis.The model also predicts small vertical components in the post-rotatory response that depend on the head orientation on stopping as reported by Harris and Barnes (1985).Finally, the continuous vertical nystagmus induced by head-over-heels rotation (Bodin, 1968) is produced (see Fig. 6C). Although the model output cannot be precisely determined through analytic methods because it is nonlinear, understanding of its behavior can be obtained through consideration of the linear portions.For clockwise barbecue rotation with an angular frequency w in radians, la_x is g*sin(wt) and the scaled derivative of the component in the plane of the lateral canals, jerk_lrh is .927*Ogain*w*cos(wt) by (6).In this plane during barbecue rotation, k_lrh is k0_lrh + KI[1,2]*cos(wt) by (9).Substituting in the actual values, the input to the velocity storage lag for that plane, j'_lrh is jerk_lrh * k_lrh or

j'_lrh = 92.7w*cos(wt)*(.1 - .215cos(wt)) (10)
Rearranging and using the identity cos(w)2 = .5 (1+cos(2w)) gives

j'_lrh = 9.27w*cos(wt) - 10.1w*cos(2wt) + 10.1w (11)

The first two terms of (11) define the sinusoidal input to velocity storage.The amplitude of this signal is greatly reduced by the low-pass filtering characteristics of the velocity storage lag and for a 60 deg./sec. barbecue rotation about 1.5 deg./sec. of modulation originates from this input. For rotations at velocities greater than than 30 deg./sec, the larger portion of modulation is caused by linear acceleration induced gaze deviation as developed in section 2.1.This explains why the frequency dependence of modulation is dissimilar to that of L-nystagmus and why section of the lateral canal nerve, which abolishes velocity storage in thehorizontal plane as well as the bias component of OVAN, does not completely remove modulation (Cohen et al., 1983).Figure 5 shows the frequency behavior of modulation resembles closely that predicted by the model.As the data of Benson and Bodin (1966), which provides the points on Fig. 5 up to 1.3 rad./sec.,was analyzed by a process that removed some of the gaze deviation component, these points were not included in the phase plot.
The second portion of (11) represents the bias.Figure 6D shows that for 10-60 deg./sec. rotational velocities, the predicted bias obtained from the simulation is about 1/2 to 2/3 of that reported by Benson and Bodin (1966a).There are several possible explanations.In the model the gain of the otoliths, Ogain, was adjusted to match the amount of L-nystagmus reported in the study of Niven et al. (1966) which might not match that of the subjects studied by Benson and Bodin.A larger KI[1,2] in (9) or a different (nonlinear) dependence of k on la could also provide more bias.The abolition of bias for the 180 deg./sec. rotational speed reported by Correia and Guedry (1966) is not predicted by the model.

5 Otoliths and Optokinetic Afternystagmus (OKAN)

OKAN is thought to be generated by the same velocity storage mechanism that is used by the VOR (Raphan et al., 1979).Thus modulation of velocity storage by the otoliths should also affect OKAN.The model's transfer function for OKAN, in a one-dimensional approximate form as used for L-nystagmus is: (12)
e. k Rgain
----- = --- --------------
rslip 1-k sTo(1/(1-k)) + 1

which differs from the transfer function for OKAN as derived by Robinson only in an additional factor of k in the numerator (Robinson, 1982).Rgain, the open loop OKN gain, is 3.The time constant of OKAN is To/(1-k) which, in the upright position with To = Tc predicts a time constant of 17.5 sec.
The model predictions are that the time constant of horizontal OKAN should behave like that of the horizontal VOR as determined through tilt-suppression experiments (see table 2).The actual behavior of the time constant of horizontal OKAN when subjects are not in the upright position is unknown.
An interesting reported behavior that is predicted by the model is that OKAN elicited in the head lateral position by horizontal drum rotation (relative to the head) has, in addition to the horizontal response, vertical components.In monkeys, the vertical component direction is upward when the drum rotates toward the dependent side and downward when the drum rotates toward the upmost side (Raphan and Cohen, 1983).In humans a similar pattern is found in the post-rotatory nystagmus induced by off-vertical axis rotation when rotation is stopped with the head lateral (Harris and Barnes, 1985).
--------------------- figure 7 ---------------------
This cross-coupling of horizontal stimulation into vertical OKAN is implicit in the KI matrix as developed above and derivesfrom elements KI[2,1] and KI[3,1].When the head is tilted into the right side dependent position, the la vector becomes {1, 0, 0}.This makes the time constant of velocity storage depend largely on the first column of the KI matrix.By calculating k from (9), it can be determined that the time constant of OKAN in the plane of the lateral canals decreases to 6.7.Similarly, the time constant of the ralp plane is increased to 12.1 sec. while the rpla plane time constant is decreased to 4.6 sec.
These changes in velocity storage cause a vertical component to appear for horizontal stimulation.Retinal slip generated by drum rotation in yaw is connected both to the vertical as well as horizontal velocity storage mechanisms.This occurs because the retinal input matrix, RI, is numerically identical to the the canal input matrix CI as it must be to use velocity storage mechanisms organized in canal planes.Normally, input to both vertical velocity storage elements is equal and the vertical components of their output cancel.When the vertical canal plane inputs become unequal, a vertical nystagmus must result.Furthermore, the increased time constant of the ralp plane perseverates nystagmus in that plane, that is, obliquely right-up or left-down.Drum movement that excites ralp with a rightward movement must therefore produce upward slow phases.Fig. 7 shows simulated horizontal OKAN and demonstrates the vertical components.Presumably this behavior is obscured in reality by the presence of a pursuit system which would null out the inappropriate vertical components as long as the lights remained on but allow them to appear during OKAN. Studies of vertical OKAN in monkeys have shown that the time constant of downward directed OKAN (slow-phase direction) is short with the head upright (about 4 sec) and does not change with head-lateral position.The time constant of upward directed OKAN is also short with the head upright but it becomes considerably greater (about 20 sec) in the head lateral position (Matsuo et al., 1979; Matsuo et al., 1984).OKAN in humans behaves similarly to that of monkeys (Baloh et al., 1983).The model agrees qualitatively with these observations and predicts a relative increase in the time constant of vertical OKAN in the head lateral position (see table 2).

6 Conclusion

This model can simulate a wide variety of otolith-vestibular interactions.The main assumption is that the otoliths modulate the time constant of the vestibular-ocular reflex by altering the gain of a positive feedback loop in the velocity storage mechanism.Off-vertical axis nystagmus is generated by a central correlation mechanism that also provides for otolith supplementation of canal signals.The effects of tilt and external linear accelerations on the VOR time constant are simulated as well as the effects of linear accelerations alone: L-nystagmus and gaze deviation.The model correctly predicts generation of a vertical component to the nystagmus evoked by horizontal optokinetic stimuli in certain head positions.
I appreciate the comments of David Zee and David A.Robinson.This work was supported by NIH grants RO1-EY05505 and K01-NS00914.



Figure Legends
Fig. 1 A-B.Slow-phase velocity profile of off-vertical axis nystagmus.A) Results of barbecue rotation at 60 deg./sec.(from Benson, 1966 with permission).The initial response, mediated by the semicircular canals, is followed by a sustained bias with a superimposed modulation.When the rotation stops, the post-rotatory response is shorter and of lower amplitude than the per-rotatory response.B) Polar plot of eye velocity versus head position for a 60 deg./sec. clockwise barbecue rotation (from Correia and Guedry, 1966 with permission).The absolute value of slow-phase velocity is plotted against head position.At 0 deg. the subject was supine, at 180 degrees prone, and at 90 and 270 degrees, the right and left ear were dependent, respectively.

Fig.2.Model of the optokinetic system and VOR after Robinson (1977).The semicircular canals, SCC, transduce head velocity, h., with the transfer function given.Tc is the time constant of the cupula, 6 seconds, and s is the Laplace complex frequency.The canal signal is added to a stored (lagged) copy to form signal vn which is then multiplied by -1 to produce eye velocity, e..The storage element has a time constant To which is close or identical to To.The degree of supplementation of the canal signal by the lagged velocity storage signal, that is the time constant of the VOR, is determined by a gain, k.Retinal slip, rslip, is connected to the vestibular nucleus indirectly through the velocity storage lag.
Fig. 3 A-B.Coordinate system and relative orientation of the otoliths and semicircular canals. A)Cartesian coordinate system with axes X (left positive), Y (anterior positive) and Z (superior positive).Rotations about these axes are denoted h._x, h._y, and h._z and their sense is determined by the right hand rule.Linear accelerations are positive when they might be evoked by a movement along the positive sense of an axis.Also shown are the geometric axes of the coordinate system defined by the conjugate canal planes made by combining the left-right lateral (lrh), right posterior-left anterior (rpla), and right anterior-left posterior (ralp) pairs.Rotation of the head about any one of these axes induces a response in the orthogonal plane.B)The utricle lies approximately in the earth horizontal plane and the saccule approximately in the parasaggital plane.The utricle is thus oriented to respond to shearing forces developed by linear accelerations directed in the horizontal plane.The saccule responds mainly to forces exerted along the superior-inferior Z axis.

Fig. 4. A three-dimensional model of otolith-vestibular interaction.For vestibular inputs, head velocity, h., is projected upon the canals by the CI matrix to form a canal vector, c.The canal input is summed with activity from the velocity storage system to form vector vn.The vn vector causes the eyes to move with velocity, e., according to the muscle and brainstem matrices, MO and BO.Additionally vn is fed back through a gain, k, to the velocity storage lag (storage element).Velocity storage is multidimensionalhaving one storage element for each canal plane.Retinal slip is converted into canal coordinates by a matrix RI and then relayed to velocity storage.
The left side of the model implements otolith interactions.In the upper pathway la is multiplied by gaze input matrix, GI, to produces static and dynamic gaze deviation.The resulting gaze deviation vector, gdev is sent through a delay and lag element that reproduce the reported dynamics of counterroll and the result is differentiated and added to eye velocity, e..In the middle pathway, la is differentiated and transformed into canal coordinates by matrix OI to produce jerk.The jerk vector is fed into the velocity storage mechanism.In the lower otolith pathway, k, which determines the time constant of velocity storage in each canal plane, is constructed from the sum of a vector constant, k0 and the scaled projection of la on each conjugate canal axis calculated by matrix KI.Signals in the velocity storage pathway and k are multiplied together -- this is denoted by a circle which encloses an 'X'.Not shown in the figure are three scalar gains referred to in the text.Rgain scales retinal slip and is 3.Jerk is scaled by Ogain which is 100.Gaze deviation, gdev, is scaled by Ggain which is 5.

Fig. 5.Frequency response of L-nystagmus and modulation component of OVAN.The circles depict L-nystagmus data reported by Niven et al. (1966). The triangles depict reported modulation components of off-vertical nystagmus (Benson and Bodin, 1966; Correia and Guedry ,1966). The slow phase velocity of L-nystagmus is proportional tohead acceleration over the range .2 - .8 hz.There is no L-nystagmus generated by a DC input.The model predictions of L-nystagmus and modulation are shown as solid lines.

Fig. 6 A-D. A) Simulation of barbecue nystagmus(horizontal OVAN) B) Plot of the steady state OVAN in polar form for clockwise rotation at 10, 20, 40, and 60 deg./sec. (compare with Fig. 1B). The model values are as given in the text.C) Simulated vertical OVAN induced by head-over-heels rotation.D) Bias of OVAN vs.stretcher velocity.The data of Benson and Bodin (1966a) is compared to that predicted by the model.

Fig. 7 A-B.Simulated horizontal optokinetic nystagmus (OKN) and optokinetic afternystagmus (OKAN) with the head upright and in the lateral positions (roll of 90 deg. to either side).The stimulus is clockwise drum rotation about the head-foot axis, 60 deg./sec.The lights are extinguished after 30 seconds of stimulation.A) Horizontal component of OKAN.With the head upright (solid line) the time constant ia 17.5 sec.If the head is positioned in either lateral position (dashed line), the amplitude and time constant of horizontal OKAN is decreased.B).Vertical component of OKAN.With the head upright (solid line), no vertical component occurs.With the left ear down, an upward vertical component appears (upper dashed line).With the right ear down, a downward vertical component appears (lower dashed line).The directions of the vertical components reverse for counterclockwise stimulation (not shown).

TABLE 2. Predicted time constants of the VOR after repositioning.
Time constants are generated by fitting 30 seconds of simulation output to a single exponential decay through least squares. Because there are three velocity storage mechanisms and some activity is coupled to all 6 canals by rotations for most positions, behavior that does not follow a single-exponential decay pattern can occur.The non-single exponential aspects account for the small deviations from the time constant calculated from the formula Tv = Tc/(1-k) and differences between the time constant of the VOR and OKAN.

Stimulus Upright Prone Supine Lateral
Horizontal VOR 17.0 8.4 7.0 6.4
Horizontal OKAN 17.6 8.5 7.8 6.8
Vertical VOR 7.3 6.3 12.4 10.0
Vertical OKAN 7.5 6.3 12.6 10.6





This page is an edited version of a paper published in