## Abstract

The vestibular organs in the base of the skull provide important information about head orientation and motion in space. Previous studies have suggested that both angular velocity information from the semicircular canals and information about head orientation and translation from the otolith organs are centrally processed in an internal model of head motion, using the principles of optimal estimation. This concept has been successfully applied to model behavioral responses to classical vestibular motion paradigms. This study measured the dynamic of the vestibuloocular reflex during postrotatory tilt, tilt during the optokinetic afternystagmus, and off-vertical axis rotation. The influence of otolith signal on the VOR was systematically varied by using a series of tilt angles. We found that the time constants of responses varied almost identically as a function of gravity in these paradigms. We show that Bayesian modeling could predict the experimental results in an accurate and consistent manner. In contrast to other approaches, the Bayesian model also provides a plausible explanation of why these vestibulooculo motor responses occur as a consequence of an internal process of optimal motion estimation.

## INTRODUCTION

Natural head movements typically consist of rotations combined with some translations, both of which can be detected by the vestibular organs in the inner ear. The vestibular sensory organs comprise the three ampullae in the semi-circular canals, which contain hair cells that are activated by head angular accelerations and the otolith organs in the utricle and saccule, which sense head orientation relative to gravity and head linear accelerations. In general it is a challenging task to reliably compute head motion in space from vestibular afferent signals. Indeed not only is afferent sensory information generally noisy, but on top of that semicircular canal afferents respond only poorly to middle and low frequency head rotations, whereas otolith afferent information alone does not allow the brain to discriminate between effects of gravity and accelerations due to translations of the head. Many previous studies have tried to describe vestibular information processing in terms of more or less sophisticated cascades of linear and nonlinear filters. Although this kind of approach can perfectly describe a large class of experimental phenomena (Kushiro et al. 2002; Raphan and Cohen 2002; Raphan and Sturm 1991), it cannot account for the considerable challenge that the brain is faced with by the noisy character of afferent information (Roddey et al. 2000; Sadeghi et al. 2007). The insight into the nature of vestibular signal processing that one can hope to reach by this kind of modeling is therefore rather limited. A quite different approach starts from the fundamental assumption that there exist internal representations of head and body motion that are constantly updated by the central processes evaluating the continuous vestibular, visual and somatosensory afferent inflow as well as re-afferent information. This assumption is commonly referred to as the internal model hypothesis and has inspired a large body of theoretical work (Droulez and Darlot 1989; Merfeld 1995a,b; Merfeld et al. 1993; Mergner and Glasauer 1999; Oman 1982; for more recent reviews, see Angelaki and Cullen 2008; Green and Angelaki 2010; MacNeilage et al. 2008; Oman 1998). According to the usual rules of kinematics, head orientation relative to gravity can be computed by integration of head angular velocity and position over time (Guedry 1974; Mayne 1974; Young 1974). This, in turn, allows separating the contribution of gravity and linear acceleration to otolith activation. However, it is well known that continuous integration of imperfect velocity information over time leads to an accumulation of position errors (Holly et al. 2008). Therefore the mentioned limited bandwidth of vestibular velocity information and inherent noise require that estimated head orientation must be continuously refined by integrating other available orientation cues, among which otolith afferent inflow. The internal model hypothesis assumes that appropriate adjustments of the current estimates of head orientation occur through such a mechanism. More recently, Laurens and Droulez (2007, 2008) have built a model of vestibular information processing, which computes optimal estimate of head motion by using Bayesian inference. The basic hypothesis underlying the Bayesian model is that all available information about head motion is extracted from the sensory signals. When sensory noise leads to uncertainty about the estimate of motion, a priori information is used to drive it toward those motion states that are more frequent in every day's life. In the present study, we tested experimentally the predictions of the Bayesian model on two completely different vestibular motion paradigms.

One group of experiments consisted of protocols in which there was a mismatch between vestibular or optokinetic self-motion cues and graviceptive inputs. The other group of experiments consisted in the classical off-vertical axis rotation (OVAR) paradigm (Fig. 1). We studied the influence of otolith signals on the central processing of angular motion information during these protocols by measuring the rotatory vestibuloocular reflex (VOR), which assesses the brain's estimate of head rotation. In the postvestibular rotation and tilt protocols (PVRT, Fig. 1, *A* and *C*), a rotation signal of vestibular origin is induced by suddenly stopping subjects after they had been rotated at constant velocity in complete darkness for an extended period of time (Fig. 1*A*) (Angelaki and Hess 1994, 1995, Dai et al. 1991; Harris 1987; Hess et al. 2005; Jaggi-Schwarz et al. 2000; Merfeld and Young 1993; Waespe et al. 1985). In this situation, the semicircular canal response seems to reflect the onset of a rotation in the opposite direction while the animal is in fact immobile. If the subjects are tilted during this after response, part of the vestibular signal will conflict with the incoming otolith information. This is so because the semicircular canal signals are head-fixed and therefore invariably indicate a head rotation about the same head axis as before the tilt, which is now off vertical. This should cause head orientation relative to gravity to change, e.g., from a nose-up to an ear-down position. The otoliths, however, indicate that the head is in fact stable relative to gravity. A similar mismatch occurs due to a visual after response if subjects suddenly plunge into darkness and tilt after having been exposed to prolonged period of constant optokinetic stimulation (postoptokinetic rotation and tilt, i.e., PORT protocol, Fig. 1, *B* and *C*) (Dai et al. 1991). In both these protocols, the sensory mismatch results in a rapid decrease of the vestibular or visual afterresponse, a phenomenon that has also been called dumping, characterized by a sharp shortening of the postrotatory time constant. A completely different type of protocol consists in rotating a subject at constant velocity about a tilted axis (Angelaki and Hess 1996b; Angelaki et al. 2000, Harris 1988; Kushiro et al. 2002) (Fig. 1*D*). In this situation, the semicircular canals do not detect the on-going rotation while the otoliths are activated by the gravity vector that rotates around the subject. Here a mismatch is created between the absence of an angular velocity signal and the revolving signals from the otoliths. It leads to a slow build-up of a VOR, which ultimately compensates at least in part for the head rotation. The dynamics of this build-up has so far not been thoroughly investigated nor has it been put on a firm modeling basis. One of the purposes of this study was to measure it under a variety of conditions and to show that the canal-otolith interactions underlying these different paradigms are very similar. Indeed during PVRT and PORT, the axis of postrotatory angular velocity mismatches with static graviceptive head orientation signals. During OVAR, the semicircular canals indicate no rotation while the head in fact continues to rotate relative to gravity. The internal model hypothesis predicts that these different mismatches can be resolved by a single mechanism, implying quantitatively similar dynamics in these different paradigms. Specifically, the dumping time constants during PVRT and PORT and the VOR build-up time constant during OVAR should be identical. To make a direct comparison possible, we studied these protocols in the same animals using the same motion parameters. We combine the quantitative analysis of these results with a theoretical analysis, demonstrating that the internal model hypothesis can accurately predict our results. Furthermore to demonstrate the power of optimal information processing, we simulated the responses with the Bayesian model, using a parsimonious version with a very limited number of only four free parameters.

## METHODS

Four rhesus monkeys (Macaca mulatta) were chronically prepared with skull bolts for head restraint and dual search coils implanted under the conjunctiva (Hess 1990) for three-dimensional eye movement recording. All procedures conformed to the National Institutes of Health Guide for the Care and Use of Laboratory Animals and were approved by the Veterinary Office of the Canton of Zurich, Switzerland.

Three-dimensional (3D) eye positions were measured using magnetic search coils (Robinson 1963) with an eye position meter 3000 (Skalar, Delft, The Netherlands). Eye position was calibrated as described in Hess et al. (1992); digitized at a sampling rate of 833.33 Hz (Cambridge Electronic Device, Model No. 1401plus) and stored on a computer for off-line data analysis. Eye angular velocity vectors (**Ω**) were computed as described by Hepp (1990). The three components Ω_{x}, Ω_{y}, Ω_{z} of **Ω** represented the torsional, vertical, and horizontal eye velocity. Slow phase eye velocity (SPV) was computed by detecting and removing quick phase movements using an algorithm similar to the one published by Holden et al. (1992). In the following, we mean with “eye velocity” always “slow phase angular eye velocity.”

### Experimental setup and stimulation protocols

Monkeys were seated in a primate chair with the head restrained in upright position so that the lateral semicircular canals were approximately earth-horizontal (stereotactic plane tilted ∼15° nose-down). The primate chair was mounted on a computer controlled motorized four-axis turntable, which was surrounded by a lightproof sphere of 1.6 m diam.

Monkeys were subjected in random order to three different motion protocols: In the *off-vertical axis rotation (OVAR) protocols,* the animals were initially rotated in upright at constant velocity of 30°/s about the body-vertical ( = yaw) axis. After 90 s, during which time the yaw VOR had decayed to zero, the animal was tilted in the roll (i.e., toward a left-ear down or right-ear down orientation) or in the pitch plane (i.e., toward nose-up or nose-down orientation) through angles between α = ±10° and α = ±90° using triangular tilt velocity profiles, i.e., a constant acceleration of 180°/s^{2}, followed by constant deceleration of −180°/s^{2}. This tilt maneuver elicited a short transient VOR. The subsequent off-vertical axis rotation phase lasted for 48 s, after which the animal was tilted back to upright with the same acceleration-deceleration profile. The yaw rotation still continued for another 60 s to measure the decay of the off-vertical axis rotation-induced VOR. To test the effect of tilt in head-down positions, the animal was tilted after the initial rotation into upside-down position while the yaw rotation was maintained. The light inside the optokinetic sphere, which rotated at the same speed as the animal, was turned on for ∼6 s to suppress uncontrolled drifts of the eyes. Then the light was turned off and the animal was tilted through ±30 or ±60° to a final position of ±120 or ±150° relative to upright. After another 48 s the animal was tilted back to upright, where the decay of the VOR was measured.

To elicit a vestibular tilt-rotation mismatch in the postrotatory phase (*PVRT conflict paradigm*), the animal was first rotated in upright at constant velocity of 30°/s for 72 s. In the interval extending from 15 to 2 s before stop of rotation, the light inside the optokinetic sphere that rotated at the same speed as the animal was turned on to eliminate any remaining vestibular eye velocity drift by fixation suppression. After another 2 s in darkness, the rotation was stopped with a deceleration of 60°/s^{2}. Two seconds after the onset of deceleration, the animal was tilted either in the roll plane toward left or right ear-down (LED, RED) or in the pitch plane toward nose-up or nose-down (NU, ND) through a tilt angle α. The tilt profile consisted of an acceleration pulse followed by a deceleration pulse, each equal to α°/s^{2}, so that the tilt through the angle α lasted 1.4 s in all conditions. The animal was maintained in the tilted position for 60 s after which it was tilted back to upright.

To generate an optokinetic tilt-rotation mismatch in the afterresponse phase (*PORT conflict paradigm*), the optokinetic sphere was rotated about the stationary animal in upright position at constant velocity (30°/s) for 45 s, after which the light was turned off. After a delay of 2 s, the animal was tilted from upright either in the roll or in the pitch plane similar as in the PVRT conflict paradigms. For testing tilt angles exceeding 90°, the animal was first tilted into upside-down position and then subjected to an optokinetic stimulation for 45 s, which generated stable horizontal eye movements. Then the light was turned off and, after a further delay of 2 s, the animal was tilted through an angle α toward upright as described in the PVRT paradigms.

The three stimulation protocols were tested in four rhesus monkeys. We used tilt angles increasing by steps of 30° and added the angles 10, 20, and 45° to obtain an additional resolution for small angles, which correspond to a smaller conflict. Tilt angles of 0 and 180° were used as reference in the PVRT and PORT experiments. For each animal, each protocol and each tilt angle, we performed eight experimental runs, by combining four initial orientations (NU, ND, LED, RED) and two directions of rotation (CW and CCW). The total number of trial was therefore 224 for each animal. The order of all trials was randomized, and each animal performed the trials in a series of experimental sessions.

### Data analysis

The eye velocity of each animal was averaged for each tilt angle and each direction of rotation. For the OVAR conditions at a given tilt angle, we thus obtained eight average traces, one for each animal and each direction of rotation. In this way, we obtained eight independent samples for each experimental condition (assuming that responses to CW and CCW rotations were independent), each of which was the average of four experimental runs. Influences of head orientation relative to gravity (i.e., NU, ND, etc) on eye movements were thereby minimized.

Average slow phase eye velocity curves were fitted with an exponential function of the form *f*(*t*) = *A*·exp(−*t*/*B*) + *C,* using the “lsqcurvefit” routine in Matlab (The MathWorks). The parameters *B* and *C* represent the time constant and the amplitude of steady-state eye velocity. The eye velocity analysis of the PORT and PVRT responses was separated in two phases. In the first phase, immediately after the tilt, horizontal eye velocity decayed quickly, whereas in the second phase decay was slower as the eye velocity vector tended to align with gravity. We defined the first phase as the period during which horizontal eye velocity exceeded 15°/s. Horizontal eye velocity curves during this phase were fitted with an exponential as described in the preceding text.

### Three-dimensional angular eye velocity analysis

To analyze the spatial reorientation of the after response during the second phase of PORT and PVRT experiment, we described the angular eye velocity of the afterresponse as a two-dimensional vector, Ω_{post} = (Ω_{hor} Ω_{orth}), with the cardinal component Ω_{hor} and an orthogonal component Ω_{orth} along the corresponding orthogonal axis in the tilt plane. For instance, Ω_{orth} was identical with Ω_{x} along the *x* axis during backward pitch tilt and with Ω_{y} along the *y* axis during right ear-down tilt. The orthogonal component, Ω_{orth}, was fitted with a double exponential, *f*(*t*) = *A*·exp(−*t*/*B*) + *C*·exp(−*t*/*D*) + *E* (Angelaki and Hess 1994) and its maximum value was computed. Then we analyzed the realignment starting from the point where *f*(*t*) was <80% of this maximum. For each curve, we computed the angle from which the eye velocity vector has rotated as the median of the arc tangent of Ω_{orth}/Ω_{hor}. The time constant of the horizontal eye velocity decay in this phase was evaluated by fitting both Ω_{hor} and Ω_{orth} with two exponentials having the same time constant, i.e., Ω_{hor} (*t*) = *A*_{1}·exp (−*t*/*B*) and Ω_{orth}(*t*) = *A*_{2}·exp(−*t*/*B*).

These analyses were performed for all tilt angles tested in the PORT and PVRT conflict paradigms. For each condition, we averaged the trials obtained from all animals, all directions and all head orientations.

### Bayesian model

The 3D Bayesian model (Laurens 2006; Laurens and Droulez 2007, 2008) is based on the hypothesis that there exists a central representation of head and body motion in space (Angelaki et al. 2004; Bos and Bles 2002; Cullen and Roy 2004; Droulez and Darlot 1989; Glasauer 1992; Merfeld 1995a,b, Merfeld and Glasauer 1997; Merfeld et al. 1993; Mergner and Glasauer 1999; Oman 1982; Reymond et al. 2002; Zupan et al. 2002). Head and body motion in space is estimated optimally on the basis of noisy sensory signals and by using a priori knowledge about the likelihood of naturally occurring motion states. The model is based on the following three assumptions: First, the noise in afferent signals fundamentally limits the accuracy that can be reached in the estimation of head angular velocity, in particular at middle and low frequencies. The uncertainty on estimated head angular velocity leads in turn to a certain scatter of the estimation of head orientations relative to gravity and translational head accelerations. Second, low velocity rotations and low translational accelerations are a priori more likely. Therefore head motions corresponding to lower angular velocities or lower linear accelerations are more probable. Altogether this favors estimates of head motions at orientations that closely correspond to the otolith input. Third, it is assumed that the visual optokinetic pathways indicate head velocity relative to the environment when the light is on with a certain amount of noise (Laurens and Droulez 2008). Note that so far the model does not take the noisy nature of otolith afferent signals into account. We have earlier found that adding otolith noise to the model has mathematically the same effect as lowering the influence of a priori information on linear head accelerations by a certain amount. Although this noise certainly exists, it turns out to have little influence on our model and is therefore omitted for simplicity.

Based on these considerations, we used the Bayesian approach as follows: we simulated noisy semicircular canal as well as otolith afferent signals during the OVAR, PORT and PVRT protocols. Given these signals, the model then computed the probability distributions of head motion variables (angular velocity, position etc) over time (see Laurens and Droulez 2007 for details). An average probability distribution of head angular motion was computed, its sign inverted (to account for the fact that a leftward rotation causes a rightward eye velocity) and used as a simulated eye velocity to be compared with the experimental measurements. We used the same parameters as in Laurens (2006 and Laurens and Droulez (2008) except for one (see following text). In these studies, these parameters were adjusted by hand to reproduce various experimental results on VOR in macaques available in the literature (see discussion for more details on this procedure). The noise in the semicircular canal and visual afferent signals was simulated by a Gaussian noise distribution with SDs of σ_{V} = 10°/s and σ_{o} = 7°/s, respectively. The a priori distributions of angular velocity and linear accelerations were also Gaussian with zero mean and SDs of σ_{Ω} = 40°/s and σ_{A} = 3 m/s^{2}, respectively. In the present study, this last parameter was adjusted to σ_{A} = 1 m/s^{2} to improve the correspondence with our results. Simulations performed with both the original and the adjusted model are presented.

In this formulation of the model, the noise on the sensory as well as the statistical distribution of the head velocity and acceleration were assumed to be Gaussian and time independent. Although such distributions are only an approximation of the real distribution of head motion, they allow building a simpler model. The fact that this model can accurately reproduce a wide range of responses to head motion (Laurens 2006; Laurens and Droulez 2007, 2008) in addition to the present results demonstrates the validity of this approximation.

## RESULTS

### OVAR experiments

In this protocol, the animal was rotated about the body-vertical axis. At the time at which the tilt was completed (taken as *t* = 0), the horizontal semicircular canals had ceased to detect the rotation that was initiated 90 s earlier. Examples of average eye velocity traces during and after OVAR at tilt angles of 20, 45, and 90° are illustrated in Fig. 2. During the OVAR phase, head orientation relative to gravity was constantly changing due to the rotation of the animal. This modulation of gravity was detected by the otoliths and generated a horizontal VOR through a central integration process. Immediately after tilt, the horizontal SPV rose exponentially to reach a steady-state of 17 ± 5°/s on average. When the animal was tilted back to upright, the eye velocity decayed to zero in an exponential manner. As apparent in Fig. 2, the increase of eye velocity at the beginning of OVAR depended on the tilt angle, whereas its decrease at the end of OVAR when the animal was back to upright showed similar rates of decay for all tilt angles.

More specifically, the time constant of eye velocity build-up decreased as tilt angles increased toward 90° (Fig. 3*A*, see slopes of eye velocity increase in Fig. 2). When the animals were tilted beyond 90° toward 120 or 150°, the time constants increased again. This dependence of the time constants on the tilt angle was highly significant [ANOVA, *F*(1,7) = 5.86, *P* = 4*10^{−5}]. In contrast, the time constants of eye velocity decay after OVAR (Fig. 3*B*) were independent of the previous tilt of the animal [ANOVA, *F*(1,7) = 0.46, *P* = 0.86]. The average time constant of the decay was 15.5 ± 2.6 s.

The steady-state velocity during OVAR (Fig. 3*C*) was independent of the tilt angle in the range of 20–90°, as shown previously by Angelaki et al. (2000) and Kushiro at al. (2002) [ANOVA, *F*(1,4) = 1.39, *P* = 0.26]. The average velocity in this tilt range was 17 ± 5°/s. This value will be used as reference in the following analyses. The bias velocity at 10° tilt was significantly lower [*F*(1,5) = 5.48, *P* = 6*10^{−4}]. As a result of the longer time constant of the build-up of eye velocity in this condition, the plateau of eye velocity could not be reached during OVAR runs of 48 s. This casted some uncertainty in our estimation of the bias with this tilt angle, therefore we confirmed this value by repeating the OVAR experiment at 10° tilt with a duration of 100 s. Similarly, the steady-state velocities at 120 and 150° tilt were significantly higher and lower, respectively, compared with the reference velocity [*F*(1,5) = 3.67, *P* = 0.008 for 120° tilt, *F*(1,5) = 2.68, *P* = 0.03 for 150° tilt].

In summary, the results obtained for OVAR at tilt angles ≤90° did show two significant trends: the steady-state eye velocity was similar for all tilt angles between 20 and 90° but lower at 10°, and the time constant of eye velocity build-up decreased as tilt angles increased.

### Dumping protocols

In addition to this dynamic spatial orientation paradigm, we also studied the classical tilt-rotation conflict paradigms during vestibular or optokinetic after responses (Fig. 1). During PVRT, the semicircular canals are activated by the deceleration that precedes the tilt of the head. During PORT, the visual stimulation induces a rotation signal which persists after the tilt. In both paradigms, these signals indicate an off-vertical axis rotation while the otoliths indicate that the head is stationary relative to gravity. Tilting the animals during postrotatory VOR (PVRT) or during the optokinetic after response (PORT) resulted in a faster decay of horizontal VOR (Fig. 4). Eye velocity decayed faster for tilts ≤90°, and then the decay became slower as tilt angles progressively increased toward 180° (Fig. 4, *E* and *F*). Although in upright the optokinetic after response lasted somewhat longer than postrotatory VOR (Fig. 4*A*), this difference vanished for tilt angles between 30 and 90°. In particular, responses obtained for tilts toward 90° were virtually indistinguishable. Notice also that the inter-trial variability became larger when the tilt angles exceeded 90° as indicated by the increased jitter. Furthermore, eye velocity did typically not decay to zero during PVRT and PORT trials with a tilt angle of 180° (Fig. 4*F*).

Closer examination of the eye velocity responses revealed that it frequently decayed quickly during the first seconds after which the decay rate became slower. This pattern is especially apparent when the animals were tilted toward 60° (Fig. 4*C*). During the second phase, we measured vertical and torsional eye velocity components that reflected the tendency of eye velocity to align with the earth vertical. In the present paragraph, we focus on the initial decay time constant. The analysis of the second phase will be presented later.

### Comparison between the experimental paradigms

The average time constants measured during OVAR, PVRT, and PORT are summarized in Fig. 5. We found a striking similarity between the time constants of responses to these three different paradigms. On average, the time constant of horizontal response decreased from 30 ± 12 s in upright to 3.8 ± 1 s at 90° tilt. A two-way ANOVA with tilt angles between 10 and 150° as first factor and type of protocol (OVAR, PVRT, or PORT) as second factor yielded no significant influence of the second factor [*F*(2,1) = 0.05, *P* = 0.37].

### Bayesian modeling

We simulated the eye velocity during OVAR, PVRT, and PORT with the Bayesian model and compared the resulting time constants to the experimentally determined time constants. By adjusting the SD of the a priori on linear acceleration to 1 m/s^{2}, the model reproduced accurately the influence of tilt angle on the time constant of eye movements (Fig. 6, black lines). The original version of the model reproduced this influence only qualitatively and it predicted a less intense dumping effect (Fig. 6, gray lines). Furthermore, the model predicted that steady-state eye velocity should increase between α = 0 and 30° then reach a plateau at around 27°/s while tilt angles increase further ≤90° (both versions of the model predicted identical steady-state velocities). Finally the model predicted that eye velocity should decay in a mirror- symmetric fashion while tilt increases toward 180°. Experimental results corresponded to this trend, although the steady-state eye velocity reached a lower level of 17°/s for tilt angles of 30–90°. Altogether the dependence of eye velocity on the tilt angle in the three protocols was well predicted.

### 3D response to PVRT and PORT

Previous studies (Angelaki and Hess 1994; Dai et al. 1991) have shown that the eye rotation axis tends to align with the earth vertical during PORT and PVRT. For instance, during PORT after a tilt through 90° to an ear-down position, the initially horizontal after response evolves into a vertical after response (as seen from an egocentric frame of reference). Similarly after a tilt through 60°, the eye velocity reorients by appropriately reducing the horizontal and acquiring a vertical response component. We pooled eye velocity recordings that were obtained at different head orientations and found that realignment consistently occurred. Results following a tilt through 60° from upright are shown in Fig. 7*A* in the case of PORT. Because the angular eye velocity vector was initially aligned with the longitudinal body axis, it initially tilted together with the head through 60° in space. Then it started to realign with the spatial vertical (Fig. 7*C*). The angular motion estimate produced by the Bayesian model realigned with gravity in a similar fashion (Fig. 7, *B* and *C*). The angles through which the eye velocity vector rotated during realignment were on average close to the head tilt angles for head tilts ≤90°, indicating a close match between the realigned angular velocity and earth vertical in this range (Fig. 7*D*). For larger tilt angles, the rotation of the eye velocity vector was under compensatory. During PORT and PVRT at 120 or 150° tilt, the eye velocity vector rotated by >90° and reversed sign. In simulations with the Bayesian model, the estimated angular velocity vector always realigned exactly with earth vertical. However, when tilt angle exceeded 90°, the simulated velocity vector rotated by −60 and −30° rather than 120 and 150°. Therefore the model predicted that the eye velocity should always realign with gravity through the smallest possible angle. Experimentally, the eye velocity realigned through the larger of the two possible angles when monkeys were initially placed in close to head-down positions. Because the Bayesian model used here does not distinguish between head-up and -down positions, it cannot explain this phenomenon. Recent modeling work (Tarnutzer et al. 2009) suggests that the sensitivity of the otoliths varies as a function of head orientation relative to gravity. Because we aimed at using the simplest possible model, this or other refinements are beyond the scope of this work. Despite this discrepancy, the model shows realignment also in head down positions, thereby solving the mismatch with otoliths. Once the eye velocity had aligned with the earth-vertical axis, it decayed exponentially toward zero. We observed that the time constant of this decrease was independent of the tilt angle (Fig. 7*E*), in agreement with the simulations and with previous observations (Angelaki and Hess 1994; Merfeld and Young 1993).

### Comparison of the eye velocity time constants during PVRT and PORT

To compare the internal model hypothesis with the previous model of Raphan and Cohen (Raphan and Cohen 2002; Raphan and Sturm 1991; Raphan et al. 1977), we have used both models to simulate the eye velocity during PORT and PVRT with a tilt angle of 90°. As shown earlier, the eye velocity recorded during these protocols were strikingly similar (see Fig. 8*A*), a result which was predicted by the Bayesian model (*B*). The Raphan and Cohen's model assumes that one eye velocity signal *V* is directly proportional to the semicircular canal afferent signal. A central processing element called velocity storage provides an additional velocity command *X*. The velocity storage is a low-pass filter that receives vestibular input during PVRT and visual input during PORT. During PORT, it is charged to an initial activity of 26°/s by the optokinetic stimulation (this value takes into account the 2 s delay between the extinction of the optokinetic stimulus and the tilt of the animal). After the stimulation is stopped and the animal is tilted, it discharges with a time constant τ, which depends of the tilt angle, i.e., the eye velocity can be computed as 26.e^{(−t/τ)}. The constant τ can be directly deduced from the result of the PORT experiment: it is equal to 26.5 s when the animals are upright and 3.5 s when the animals are tilted by 90°. During PVRT, the velocity storage is charged by the postrotatory vestibular signal follows the equation *V* = 30.e^{(−t/ τc),} in which τ_{c}, the time constant of the canals, is equal to 4 s. Furthermore the velocity storage discharges simultaneously, according to d*X*/d*t* = −1/τ·*X* + *k*·*V*, in which k is a gain factor. We deduced this factor from the response to PVRT at a tilt angle of 0° by curve fitting: we found *k* = 0.2, in perfect agreement with (Raphan et al. 1977). According to Raphan and Cohen's model, the constant *k* doesn't depend from the tilt angle. Therefore one can simulate the response during PVRT with 90° tilt by using *k* = 0.2 and τ = 3.5 s, while the response during PORT is computed with τ = 3.5 s (Fig. 8*C*). The simulated response during PORT matched closely the experimental results, which is logic because it is in fact an exponential fit to these results. In contrast, we found that the simulated response during PVRT lasted longer than both the experimentally measured eye velocity.

As a general manner, the Raphan and Cohen's model fails to reproduce the similarity of the time constant recorded during PVRT and PORT. Indeed this model predicts that these constants are similar when the time constant of the velocity storage (τ) is largely superior to the time constant of the canals (τ_{c}). This is the case during rotation around a vertical axis (i.e., τ = 26.6 s and τ_{c} = 4 s). However, as the tilt angle increases, the time constant τ that governs the response during PORT decreases. This leads to a discrepancy between the simulated response during PVRT and PORT. This effect is apparent on Fig. 8*D* in which these time constants are plotted one against another (gray line). In contrast, the values measured experimentally and simulated with the Bayesian model align perfectly with the unity line in Fig. 8*D*.

In the Bayesian framework, motion information provided by the semicircular canals or by the velocity storage are fundamentally equivalent. Indeed the factor that governs the influence of otolith information on angular velocity estimation is the amount of uncertainty on the estimate of velocity. During PORT, the optokinetic signal indicates that the head rotates at a given velocity. When the light is turned off, the absence of stimulation of the canals indicates that the head doesn't endure any angular acceleration, and therefore the velocity estimate remains stable. If the amount of uncertainty on the canals was equal to zero, then the estimate of velocity would remain stationary indefinitely. During PVRT, the canals are activated by the acceleration, then their signal decays. This pattern of activation corresponds to a constant-velocity rotation. In the absence of uncertainty, this signal would be interpreted as a constant-velocity rotation of indefinite duration. During both PVRT and PORT, the estimate of rotation velocity can only decrease when a certain amount of uncertainty on head velocity builds up, allowing for the otolith to influence of estimate. This uncertainty originates from the canals being noisy, and is equivalent in both cases. This explains why the dynamic of the eye velocity during PORT and PVRT is identical despite the origin of the VOR being different.

### Bayesian processing of conflicting afferent motion information

The Bayesian model can qualitatively describe the principles that underlie the responses to conflicting motion signals. In the following paragraph, we illustrate the inference process that might take place during the vestibular and optokinetic postrotatory tilt and the OVAR experiments.

In the vestibular tilt experiments, the animal is at rest after the rotation had been stopped and the subsequent tilt was completed (Fig. 9*A*, real motion). In contrast, the postrotatory activity of the semicircular canal afferents indicates a rotation in yaw at 30°/s (black circle in Fig. 9*A*, initial distribution). While this information is coupled with a certain amount of uncertainty (intensity plot in Fig. 9*A*), it is compared with other indirect information about head-in-space orientation that comes from the otoliths. Because the head is in fact stationary, any nonzero rotation estimate about an off-vertical axis would have to modify head orientation and therefore result in a mismatch between the internal estimate and the incoming otolith signals. As a result there would be a nonzero translational acceleration estimate, as demonstrated experimentally by Merfeld et al. (1999). The a priori knowledge on linear acceleration thus tends to favor rotation estimates which are close to the earth-vertical (Fig. 9*A*, a priori on acceleration). Additionally, the a priori on rotation favors low angular velocities about any axes in space. Altogether, these factors drive the estimated head orientation toward the earth-vertical line and thereby increase its rate of decay toward zero (Fig. 9*A*, final distribution, and gray arrow). This mechanism causes the rapid decrease and realignment of the rotation estimate toward gravity. Once the rotation estimate has realigned with gravity, the mismatch described in the preceding text disappears. The rotation estimate decreases then slowly under the effect of the prior. Because the rotation estimate is no more affected by gravity, rate of decrease is independent of head orientation relative to gravity, as observed experimentally (Fig. 7*E*). During the postrotatory optokinetic tilt experiment, the model behaves in an essentially similar fashion. In this case, the initial velocity signal is the consequence of the after response to an optokinetic stimulation, which can also be explained by Bayesian inference (Laurens and Droulez 2008; for details, see Laurens 2006).

Although the VOR generated during OVAR is geometrically more complex due to the fact that the head now rotates relative to gravity, it can be inferred by the same principles. As illustrated in Fig. 9*B*, the initial head velocity estimates are centered at zero at the beginning of OVAR. To minimize the estimation of translational acceleration, the internal rotation estimate needs to correspond to the displacement of the head relative to gravity. A rotation estimate of 30°/s about the head-vertical axis is one obvious solution to this problem because it is identical to the real motion of the head (square in Fig. 9*B*). Furthermore, adding another rotation vector aligned with gravity would result in an equally satisfying solution because rotations around an earth-vertical axis do not affect the head orientation relative to gravity. In other words, the rotation estimates that minimize the translational acceleration estimation are placed around an earth-vertical (in the Ω_{y}, Ω_{z} plane) during both OVAR (Fig. 9*B*, a priori on acceleration) and postrotatory tilt (*A*). This line necessarily passes through the point that represents the real motion of the head, i.e., the point (Ω_{y} = 0, Ω_{z} = 0) during PVRT and the point (Ω_{y} = 0, Ω_{z} = 30) during OVAR. During OVAR, this causes the internal estimate of rotation to shift upward and rightward (Fig. 9*B*, final distribution). The rightward shift implies that some vertical response should be generated during OVAR. However, this response component is quickly modified because the head moves further relative to gravity to reach a left-ear-down position. This causes the earth-vertical line in Fig. 9*C* (a priori on acceleration) to be inverted compared with Fig. 9*B*. As a consequence, the internal estimate of rotation is shifted up- and leftward. By this process, the estimate of rotation about the head-vertical axis continuously increases in magnitude, while the estimated angular velocity about the interaural axis remains close to zero. A similar angular velocity estimate about the naso-occipital axis appears when the head is in nose-up or -down position. Finally, the a priori knowledge on rotation, although weak, is continuously pulling the estimate downward. As a consequence, the motion estimate around the head-vertical axis never reaches 30°/s but stabilizes at a lower steady-state value.

## DISCUSSION

We have studied the influence of otolith afferent information on the central estimation of head angular motion as reflected in the VOR. We found that the build-up time constant of the VOR during off-vertical axis rotation depends in much the same way on the angle of head tilt as do the decay time constants of the VOR during postrotatory responses. Although effects of gravity have previously been extensively studied during off-vertical axis rotation (Angelaki and Hess 1996a,b, Angelaki et al. 2000; Benson and Bodin 1966; Cohen 1983; Harris 1988; Hess et al. 2005; Jaggi-Schwarz et al. 2000; Kushiro et al. 2002) as well as in postrotatory tilt paradigms (Angelaki and Hess 1994, 1995; Merfeld and Young 1993; Waespe et al. 1985), we present here a unified view of the spatiotemporal dynamics obtained by these paradigms in the same subjects using a large range of tilt angles. We show that the gravity-dependent VOR dynamics can be understood as a consequence of a Bayesian estimation process that accurately reproduces the outcome of the two tested widely different vestibular motion paradigms, in agreement with the internal model hypothesis (Droulez and Darlot 1989; Merfeld et al. 1993; Oman 1982).

### Bayesian inference

The Bayesian modeling approach combines the idea of an internal model with an optimal estimation process. The internal model mathematically links head rotation and orientation relative to gravity by combining the processes of estimation of angular motion with the central evaluation of otolith afferent information. The optimal motion estimation process takes both the noise in the afferent signals (Fig, 1*A*) as well as a priori knowledge on motion into account (Fig. 9, *B* and *C*). To implement this approach, we had to define only a minimal set of three parameters: the width of a time-independent Gaussian distribution describing the noisy afferent semicircular signals (σ_{v}) and a priori information about the distribution of angular velocities and accelerations in the head motion repertoire (see methods). An additional (4th) parameter describing the visual afferent signals is used to model the OKN. This latter parameter plays only a minimal role in the present study because it influences the dynamics and the steady-state velocity of OKN but not the time constant of OKAN, which was our main focus.

To adapt the model to our paradigms, we adjusted the parameters by hand, i.e., by a trial and error process, guided by our experience about the effects of these parameters. These effects can be briefly described as follows: the a priori on head angular velocity tends to drive the estimate of angular velocity toward zero in the presence of uncertainty. Thus decreasing the width of the a priori head angular velocity distribution (σ_{Ω}) decreases the time constant of the VOR as the strength of this prior increases. It also decreases the steady-state estimate of angular velocity during OVAR. The a priori on linear acceleration (σ_{A}) tends to align the estimate of gravity orientation with the gravito-inertial signal from the otoliths. This ultimately affects the angular velocity estimation, which is used to compute head orientation relative to gravity. Therefore increasing the strength of this prior by decreasing the distribution width σ_{A} increases the influence of gravity on the angular velocity estimation. This effect in turn reduces the time constant of the VOR during canal-otolith interaction and increases the steady-state velocity of OVAR. Note also that the influence of the priors increases as the uncertainty on the motion estimate increases. The main source of uncertainty is the presence of noise in the semicircular canal afferent signals. Thus increasing this noise (i.e., σ_{V}) decreases the time constant of the VOR and also increases the dumping effect of gravity on this time constant. Simultaneously, it leads to a decrease of the steady-state estimate of angular velocity during OVAR, which as earlier shown (Laurens 2006; Laurens et al. 2008) is an indirect and complex effect. Notice also that these effects can approximately cancel each other if the three parameters are proportionally decreased or increased. In other words, one parameter can be kept fixed while the other twos are adjusted. Our approach of adjusting the various versions of the models was to keep the amount of noise on the semicircular canals fixed to a value of σ_{v} = 10°/s. We have earlier shown (Laurens and Droulez 2007) that this value is consistent with studies of the threshold of rotation perception in humans (reviewed by Guedry (1974). For simplicity, we used the same value for the study in monkeys. Next we adjusted σ_{Ω} to obtain a realistic time constant of the VOR. Finally, we adjusted σ_{A} to match the effects of the otolith signals on the VOR.

The present Bayesian model has originally been developed to reproduce certain perceptual phenomena in humans, in particular during OVAR (Laurens and Droulez 2007). Because the steady-state perception of velocity in humans is weak (almost 0 at small tilt angles, i.e., 30°, but larger at an angle of 90°), σ_{Ω} was originally set to 30°/s and σ_{A} to 5 m/s^{2}. In previous works (Laurens 2006; Laurens and Droulez 2008), we adapted the model to monkey studies by focusing on reproducing the steady-state velocity of the VOR reported by Angelaki at al. (2000) and Kushiro and al. (2002). This VOR has a gain of ∼75% of stimulus velocity at a tilt angle of 30°. Similarly, the time constant of the VOR and OKAN are generally longer in monkeys than in humans. Accordingly, we increased the a prioris on angular velocity (σ_{Ω} = 40°/s) and on linear acceleration (σ_{A} = 3 m/s^{2}) with the effect of increasing the influence of the otoliths. In this way, the model could also reproduce the dumping and realignment effects during postrotatory tilt (Laurens 2006). The model predicted a reduction of the time constant of the VOR to 6.5 s with a tilt angle of 90°; this was consistent with previously reported values (between 4 and 10 s). In the present experiments, we found the time constants of the VOR as low as 3.5 s. Because such small values have also been reported in previous studies, we adjusted our model to provide a better correspondence with the present results by decreasing σ_{A} to 1 m/s^{2}.

### Previous studies on the internal model hypothesis

Oman (1982) originally proposed that motion sickness arises when the brain fails to match an internal representation of motion with the sensory afferent inflow. He suggested that the brain acts as an ideal observer, which computes optimal estimates of motions on the basis of an internal model of motion. This hypothesis is directly implemented by the Bayesian model but also inspired several previous works, such as the models of Droulez (Droulez and Darlot 1989; Reymond et al. 2002), which implement the same principle as the Bayesian model by using an optimization algorithm rather than Bayesian inference. Holly (2008) has used the internal model hypothesis as a basis for a variety of models designed to explain motion perception in humans during various motion paradigms. Merfeld (Merfeld 1995a, b; Merfeld et al. 1993), Glasauer (Glausauer and Merfeld 1997) and Zupan (Zupan et al. 2002) have proposed computationally efficient architectures in which ad-hoc loops compute the anticipated sensory inputs corresponding to the estimated motion. These anticipated inputs are compared with the afferent inflow, and mismatches between both are used to correct the estimate of motion. The feedback loops are also designed in a way that minimizes the estimate of linear acceleration, and these models can also reproduce the dynamics of VOR and OKAN in the absence of sensory conflict. In the paradigms described here, feedback loops as described in the preceding text will behave in a way that is essentially equivalent to the process described in Fig. 9. Therefore it can be expected that these models would reproduce the experimental results presented here. Our choice to use the Bayesian model to analyze our results was based on the limited number of free parameters that it requires. Furthermore, this model allows a detailed analysis of the estimation process, as presented in Fig. 9, by focusing directly on the information provided by various sensory signals.

Some studies have also provided direct demonstrations that the brain fuses semicircular canal and otolith information through an internal model of head motion. These studies have focused on the gravito-inertial ambiguity: because the otolith organs are sensitive to both gravity and linear acceleration, knowledge of head orientation relative to gravity is required to extract the linear acceleration of the head. Angelaki and colleagues (Angelaki et al. 1999, 2004) have demonstrated that cerebellar and brain stem neurons integrate sensory afferent information from the semicircular canals and the otolith to perform this operation. During PVRT and PORT, incorrect rotation information should create a mismatch between the internal estimate of gravity and the head orientation detected by the otoliths. This in turn should lead to an internal estimate of translation. Merfeld and colleagues (Merfeld et al. 1999) have shown evidence that such mismatch indeed results in a linear VOR during PVRT in humans.

In addition to the internal model hypothesis, the notion that the brain processes self-motion information in an optimal way has been addressed in a variety of experimental and modeling works. In particular, the finding that humans tilted in darkness systematically underestimate their body tilt has been explained by a priori knowledge that favors small body tilt angles (De Vrijer et al. 2008; Eggert 1998). Optimal estimation has also been advocated to explain the fusion of visual and vestibular information in human perception of tilt and acceleration (MacNeilage et al. 2007; Vingerhoets et al. 2009), and in a variety of studies on other sensory systems (Ernst and Banks 2002; Fetsch et al. 2009; Weiss et al. 2002). As a conclusion, the Bayesian model implements the internal model hypothesis as it was originally expressed and keeps in line with other modeling works in the vestibular field. It also confirms the efficiency of the optimal estimation theory as a way to model sensory functions in the brain.

### Limitations of previous models

Previous feedforward models were able to reproduce part of our experimental results (Hain 1986; Kushiro et al. 2002; Raphan and Cohen 2002; Raphan and Schnabolk 1988; Raphan and Sturm 1991; Schnabolk and Raphan 1992). However, these models fail to explain the similarity of the results obtained from the postrotatory vestibular and optokinetic experiments (see results, Fig. 8). Another more fundamental limitation of these models is that although they incorporate the hypothesis that otolith signals influence the central processing of angular motion, they cannot explain why this interaction exists. Indeed Raphan and Cohen's model is based on the notion that the velocity storage is influenced by gravity (Kushiro et al. 2002; Raphan and Cohen 2002; Raphan and Sturm 1991). The gravity dependency was modeled by a 3D set of differential equations the parameters of which were fitted to the eye velocity profiles of the postrotatory response by optimization methods (Raphan and Sturm 1991). The same model can be used to fit the results recorded during OVAR by postulating the existence of an additional mechanism that filters the signal from the otoliths and generates an eye velocity command through the velocity storage (Kushiro et al. 2002). A major weakness of this model is that it does not provide any insight into how the realignment of eye velocity with earth vertical or the dumping effect and its dependence on the tilt angle come about as the fitting procedure merely fits the eye velocity traces with one exponential curve along the head's vertical axis which decays faster than during a normal rotation and another double exponential that is aligned with vertical. In fact, the model is unable to predict that the dumping effect gets stronger as the tilt angle increase; it simply fits the results with the aim to provide an accurate description of the dynamics as a function of head orientation relative to gravity. It shares this approach with a number of other models (Angelaki and Hess 1995; Hess et al. 2005; Jaggi-Schwarz et al. 2000).

In contrast, Bayesian inference, embedded within the framework of the internal model hypothesis, is an approach that actually allows explaining the interactions between angular velocity and the otolith signals within an ecological context.

### Internal representation of motion, eye movements and perception

One fundamental hypothesis of our modeling approach is that one central estimate of motion drives the VOR. The analysis of our results supports our hypothesis. Interestingly, studies of motion perception in humans (Merfeld et al. 2005a,b) provided evidence that motion perception and VOR dynamics are not necessarily congruent, suggesting that VOR in humans may involve other mechanisms than an internal estimator of motion. Previous modeling studies have shown that the internal model hypothesis can account for human VOR and perception in a variety of conditions, including OVAR (Laurens and Droulez 2007) and postrotatory tilt protocols (Laurens 2006). In particular, Merfeld's model as well as the Bayesian model can be tuned to reproduce VOR in humans by a simple adjustment of their parameters (Merfeld 1995b; Laurens and Droulez 2007). However, the dumping and realignment of the VOR during postrotatory tilt and the steady-state VOR are limited in humans, indicating that the central integration of vestibular signals is very weak. Therefore other mechanisms that affect the VOR may have prevailed in the experiments of Merfeld et al. (2005a,b). In contrast, central integration mechanisms dominate the VOR in monkeys in conformance with the principles of optimal estimation.

### Central integration of angular motion cues and velocity storage

The Bayesian model can also help in interpreting one of the most apparent manifestations of central processing of vestibular information, which is the prolongation of the time constant of the VOR compared with the semicircular canal afferent signals. This phenomenon has been called velocity storage and represents the central element of the Raphan and Cohen model (Raphan et al. 1979; Raphan and Sturm 1991; Kushiro et al. 2002; Raphan and Cohen 2002). Despite the efforts of more than three decades of research, the ecological relevance of such a “storage” mechanism has not been convincingly demonstrated and remains to be less than clear for a variety of reasons. First of all, prolonged rotations are rather infrequent in everyday life. However, the Bayesian model predicts the effects of velocity storage on the basis of a Gaussian a priori probability about head angular motion according to which long duration rotations are infrequent. Therefore the fact that low-frequency VOR is enhanced by central processing does not conflict with the assumption that low frequency motion is infrequent in natural life. In most other circumstances, the visual system also provides motion information. In these cases, the Bayesian model predicts that there is in fact a strong reduction of the velocity storage (Laurens 2006; Laurens and Droulez 2008), confirming experimental findings (Raphan et al. 1979). This reduced processing still optimizes head rotation estimation, allowing the brain in turn to accurately extract the translational acceleration components from the otolith afferent signals. Another important function of the low-pass filtering performed by the velocity storage mechanisms is to maintain the motion estimate during brief interruptions of visual information, for instance during short fixations of the thumb while moving it in phase with the head. Similarly, it can strengthen the brain's estimation that the head is not rotating. Furthermore, the Bayesian approach also suggests that the central vestibular system processes angular velocity information derived from the otolith afferent signals. Finally, a number studies have suggested that it also integrates proprioceptive information during active or passive locomotion (Brandt et al. 1977; Solomon and Cohen 1992). In the light of the here suggested fundamental role of Bayesian inference, it appears thus that the so-called velocity storage is a misnomer for a central vestibular mechanism the task of which is in the first place to optimally integrate multisensory motion information, whereas its ability to store head angular velocity is just one aspect of this function (Angelaki and Hess 1994, 1995, Green and Angelaki 2004; Green et al. 2005).

All together these considerations suggest that multisensory self-motion information is integrated by neural populations that implement an optimal self-motion detection system based on an internal model of motion. We have shown experimentally that the dynamic of the VOR during canal-otolith interaction conforms to this principle. Our study is a direct demonstration that the brain has the ability to implement an optimal estimator system for processing dynamic and multisensory information.

## GRANTS

This study was supported by the Zurich Center for Integrated Human Physiology, the Koetser Foundation, and the Swiss National Science Foundation.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## ACKNOWLEDGMENTS

We thank C. Bettoni, J. Thomassen, and U. Scheifele for assistance and D. Angelaki for valuable comments to an earlier version of the manuscript.

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