## Abstract

The brain integrates sensory input from the otolith organs, the semicircular canals, and the somatosensory and visual systems to determine self-orientation relative to gravity. Only the otoliths directly sense the gravito-inertial force vector and therefore provide the major input for perceiving static head-roll relative to gravity, as measured by the subjective visual vertical (SVV). Intraindividual SVV variability increases with head roll, which suggests that the effectiveness of the otolith signal is roll-angle dependent. We asked whether SVV variability reflects the spatial distribution of the otolithic sensors and the otolith-derived acceleration estimate. Subjects were placed in different roll orientations (0–360°, 15° steps) and asked to align an arrow with perceived vertical. Variability was minimal in upright, increased with head-roll peaking around 120–135°, and decreased to intermediate values at 180°. Otolith-dependent variability was modeled by taking into consideration the nonuniform distribution of the otolith afferents and their nonlinear firing rate. The otolith-derived estimate was combined with an internal bias shifting the estimated gravity-vector toward the body-longitudinal. Assuming an efficient otolith estimator at all roll angles, peak variability of the model matched our data; however, modeled variability in upside-down and upright positions was very similar, which is at odds with our findings. By decreasing the effectiveness of the otolith estimator with increasing roll, simulated variability matched our experimental findings better. We suggest that modulations of SVV precision in the roll plane are related to the properties of the otolith sensors and to central computational mechanisms that are not optimally tuned for roll-angles distant from upright.

## INTRODUCTION

Self-orientation in space is achieved by integrating multiple sensory inputs that originate from the otolith organs (utriculus and sacculus), the semicircular canals (SCCs), the somatosensory system, and the visual system. From these inputs, the brain computes head and trunk orientation relative to gravity. By studying reflexive motor behavior in response to changing whole body orientation, one can gain indirect insight in the process of graviception. For instance, head tilts in the roll plane modify the torsional position of the eye in the head via the vestibulo-ocular reflex (VOR), referred to as ocular counter-roll (OCR) (Collewijn et al. 1985; Diamond and Markham 1983; Miller and Graybiel 1962; Nagel 1868; Palla et al. 2006). The perception of self-orientation relative to gravity is most frequently measured with the subjective visual vertical (SVV) (Aubert 1861; Bisdorff et al. 1996; Howard 1982, 1986; Mueller 1916; Van Beuzekom and Van Gisbergen 2000). Static SVV adjustments are essentially determined by otolith signals, because darkness excludes visual references and the absence of rotation excludes a contribution by SCC stimulation. The influence of proprioception on the SVV is small, as was shown by comparing the whole body roll-dependent SVV modulation on land with the modulation under water, minimizing proprioception (Graybiel et al. 1968; Jarchow and Mast 1999; Wade 1973).

The SVV exhibits systematic roll-dependent deviations. Aubert (1861) first observed roll undercompensation at angles >60° (A-effect), peaking around 130° (Van Beuzekom and Van Gisbergen 2000), whereas Mueller (1916) reported the opposite phenomenon at roll angles <60°, resulting in roll overcompensation (E-effect). Roll overcompensation was later studied in more detail by others and was found to be either small or even absent (De Vrijer et al. 2008; Howard 1982; Kaptein and Van Gisbergen 2004; Mittelstaedt 1983; Van Beuzekom and Van Gisbergen 2000; Wade and Curthoys 1997). Finally, at roll angles >135–150°, a shift from undercompensation back to overcompensation has been described, and it has been suggested that this shift is a consequence of a switch in the reference frame from the head to the feet (Kaptein and Van Gisbergen 2004, 2005). Whereas E-effects at small roll angles evolve gradually, the shift from A- back to E-effect in the transition zone at large roll angles is abrupt.

Mittelstaedt (1983) formulated a hypothesis on physiological earth-vertical misestimations that can explain both A- and E-effects. Based on anatomical observations made by Rosenhall (1972), Mittelstaedt (1983) postulated an imbalance in the tilt signal caused by an unequal number of hair cells in the utriculus and sacculus. The pattern of observed deviations could be the downside of an optimal strategy for dealing with these imperfections in the tilt signal. By adding a body-fixed constant vector (idiotropic vector) to the otolith signal, perceived vertical was biased toward the body-longitudinal axis, and the experimentally observed deviations were successfully simulated. This proposed strategy reduces roll overcompensation (which is postulated at small roll angles) and is therefore optimized for roll angles close to upright. For larger roll angles, however, roll undercompensation is increased by the addition of the idiotropic vector. Mittelstaedt's hypothesis was later reinterpreted by Eggert (1998) using a Bayesian framework. He proposed that the idiotropic vector in Mittelstaedt's model is equivalent to the role of prior knowledge for the optimal interpretation of a noisy head-roll signal. Bayesian models allow integrating various sources of information to optimize performance in the context of optimal observer theory (Knill and Pouget 2004; Kording and Wolpert 2004; Laurens and Droulez 2007; MacNeilage et al. 2007). De Vrijer et al. (2008) found that both Mittelstaedt's idiotropic vector model and a Bayesian observer model reproduce precisely the observed roll undercompensations in a motion vertical (random-dot pattern) and in a SVV task.

The otolith organs in both vestibular labyrinths are separated into the utriculus and the sacculus, two independent linear acceleration-sensitive organs. They consist of curved membranes, whose planar approximations to the surfaces are oriented approximately perpendicular to each other (Curthoys et al. 1999; De Burlet 1930; Naganuma et al. 2001, 2003; Quix 1925; Sato et al. 1992; Takagi and Sando 1988). Within the otolith organs, changes in the orientation of the head relative to gravity generate shear forces (Jaeger et al. 2002; Kondrachuk 2002; Nam et al. 2005; Schoene 1964; Shotwell et al. 1981) that, in turn, deflect macular hair cells. Hair bundle deflection, and therefore the neural activity transduced by the populations of otolith hair cells, depends on the orientation of the maculae relative to the acceleration vector (Fernandez and Goldberg 1976a; Fernandez et al. 1972). In contrast to the population of semicircular canal afferents, which has three almost orthogonal preferred directions, the population of otolith afferents presents a broad distribution of preferred directions. Importantly, this distribution is not uniform and shows a preponderance of directions around the horizontal plane, mostly along the foreaft axis and around the vertical plane along the head-vertical axis (Fernandez and Goldberg 1976a; Jaeger et al. 2008). A close agreement between the directions of the functional polarization vectors (i.e., the vector describing the direction of an acceleration stimulus leading to maximal neural response in the vestibular fiber) and the morphological (i.e., the vector describing the direction of hair bundle displacement leading to maximal depolarization of the cell) polarization vectors of otolith afferents was reported by Jaeger et al. (2008).

In this study, we asked whether the anatomical and physiological aspects of the otolith organs are reflected in the accuracy (i.e., the degree of veracity) and precision (i.e., the degree of reproducibility) of SVV adjustments in the roll plane. The standard deviation (SD) computed from repetitive adjustments of the SVV with the head in the same position relative to gravity has been used as a measure for the precision of verticality perception in the roll plane. As reported by others, the SD of SVV adjustments within subjects increases with increasing head roll (De Vrijer et al. 2008; Dichgans et al. 1974), having sharp peaks in the range of 120–150° roll (Lechner-Steinleitner 1978; Mittelstaedt 1983; Schoene and Udo de Haes 1968, 1971; Udo de Haes 1970). These observations were explained by a decreasing “effectiveness” of the otolith organs with increasing roll of the head (Lechner-Steinleitner 1978; Schoene and Udo de Haes 1968). However, the concept of “decreasing otolith effectiveness” has not been put into relation with the anatomy and the neurophysiology of the otolith organs and with central processing of the otolith input to fortify this hypothesis. Modeling of sensory estimate variability proposed previously either predicted (Mittelstaedt 1983) or assumed (De Vrijer et al. 2008) monotonically decreasing SVV precision with increasing roll. However, the peaks of SVV variability around 120–150° head-roll observed are not reflected in these simulations. Also, the influence of adaptation on SVV variability over time has not been clarified thus far. As subjects remained in a given roll orientation to collect a series of SVV adjustments, the resulting spread is likely to be affected by drifts in perceived vertical caused by adaptation, as noted by several authors (Lechner-Steinleitner 1978; Schoene and Udo de Haes 1968, 1971). Furthermore, these drifts in perceived vertical are not uniform within the entire roll plane, but depend on the whole body roll angle (Lechner-Steinleitner 1978). SD measurements in SVV experiments that focus on the constancy of roll estimates over time are therefore not suited to study the trial-to-trial SVV variability in the absence of central adaptation effects. By studying the spread of perceived roll angles when each measurement is followed by a change in the subject's roll orientation, one could extend our understanding of the precision of the sensory systems involved and of the reproducibility of roll estimates in an environment with a frequently changing head relative to gravity orientation. By minimizing visual, proprioceptive, and semicircular canal input, the perception of earth-verticality relies mainly on otolith input.

To clarify the possible contribution of adaptation to the accuracy and precision of SVV adjustments, we collected SVV adjustments in the entire roll plane and changed the subject's roll orientation after each trial. To test whether the decreasing SVV precision could be of otolithic origin, as postulated by others, we modeled principle anatomical and neurophysiological aspects of the otolith organs and studied whether the observed pattern of SVV precision could be reproduced by our simulations. We hypothesized that the pattern of variability is related to the spatial distribution of otolith afferents, which do not cover the roll-plane uniformly. To study whether this poses a limitation in the accuracy and precision of the estimate of vertical, we considered an efficient estimation of acceleration based on the neural firing of a population of otolith afferents with nonlinear tuning functions as reported in the literature (Fernandez and Goldberg 1976a; Fernandez et al. 1972).

## METHODS

### Subjects

Nine healthy human subjects (4 females and 5 males; 27–42 yr old) were included in the first paradigm (measurements of the SVV in 4 principal roll positions). Two subjects were familiar with the paradigm, whereas the remaining seven subjects were naïve. Later, five of these subjects were included in the second paradigm (measurements of the SVV over the entire roll plane in steps of 15°), together with two additional naïve subjects (2 females, 5 males; 27–42 yr old). Informed consent of all subjects was obtained after full explanation of the experimental procedure. The protocol was approved by a local ethics committee and was in accordance with the ethical standards laid down in the 1964 Declaration of Helsinki for research involving human subjects.

### Experimental setup

Subjects were seated upright on a turntable with three servo-controlled motor driven axes (Acutronic, Jona, Switzerland). The head was restrained with an individually molded thermoplastic mask (Sinmed, Reeuwijk, The Netherlands). Subjects were positioned so that the roll axis of the turntable intersected the center of the interaural line. Pillows and safety belts minimized movements of the body. Turntable acceleration during changes of both head and trunk (i.e., whole body) roll position (duration: 8 s) was ±10°/s^{2}; peak turntable velocity for roll position shifts of 180° was ±41°/s. Because the otolith organs, which have the largest impact on SVV, are situated in the head, the subjects' orientation in the roll plane will be referred as head roll orientation, although roll movements on the turntable were whole body, i.e., included both head and trunk. An arrow that was projected from a turntable-fixed laser onto the center of a sphere in front of the subject was used to indicate perceived vertical. The inner surface of the sphere was located 1.5 m from the subject's eyes. At this distance, the arrow (length: 500 mm; width: 3 mm) extended over the central 9.5° of the visual field. Subjects with myopia were allowed to wear their glasses. Turntable position and arrow orientation signals were digitized at 200 Hz with 16-bit resolution and stored on a computer hard disk for off-line processing.

### Experimental protocol

Subjects were asked to rapidly (<3 s) adjust the orientation of the arrow using a remote control box to earth-vertical by the smallest angle of rotation and to confirm the completion of the adjustment by pressing a button. In a pilot experiment, subjects were able to comfortably adjust the SVV to vertical within 3 s. Before data collection, subjects were required to practice SVV adjustments until these could be performed within the time limit. This same time limit was used in all roll orientations to avoid possible influences of trial time on the variability of measurements. Whenever the confirm button was not pressed within 3 s, the arrow disappeared, and the missed trial was repeated later in the experiment. The percentage of missed trials was <10% in all subjects. We selected a short interval because previous studies showed that SVV SD changes with head roll position (Dichgans et al. 1974; Lechner-Steinleitner 1978; Mittelstaedt 1983; Schoene and Udo de Haes 1968; Udo de Haes 1970). It is likely that subjects experience more difficulties in setting the visual line to vertical at some (probably larger) head roll angles than at some other (probably smaller) angles. Subjects could potentially compensate the roll-dependent imprecision by spending more time adjusting the line to vertical; consequently, comparisons of SVV SD at different roll angles would be hampered by unequal SVV adjustment times. By setting the time limit short enough that subjects will spend equal time for SVV adjustments in all roll positions, such an effect can be avoided. Because the rotating visual stimulus itself can influence torsional eye position and thereby perceived vertical (Mezey et al. 2004; Wade and Curthoys 1997), the direction of arrow roll to complete the task was pseudo-randomized [clockwise (CW) vs. counterclockwise (CCW)]. By randomizing the direction of arrow rotation and by restricting the time to complete adjustments, we controlled both for possible visual consequences and strategies used by the subjects. The presentation of the arrow started 5 s after the turntable came to a full stop. Rotations with accelerations above the threshold of the SCC were found to modify errors in SVV (Jaggi-Schwarz and Hess 2003; Pavlou et al. 2003). To quantify the contribution of the SCC, we checked for postrotatory torsional ocular drift and nystagmus in two subjects using dual scleral search coils (Skalar, Delft, The Netherlands). Average torsional eye velocity at the time subjects confirmed arrow adjustments was found to be small (0.24°/s). ANOVA (2-way ANOVA) showed no significant differences in torsional eye velocity right before and 5 s after turntable roll [*F*(1,2) = 0.01, *P* = 0.932], which confirms that signals from the torsional ocular-motor velocity storage mechanism were not a major factor at the time of arrow adjustment.

### Paradigm 1: SVV measurements at four principal roll positions

After repositioning the subject in the roll plane, a verbal instruction indicated in a pseudo-randomized fashion whether the arrow should point up or down along the perceived earth-vertical. Also in pseudo-randomized order, the starting roll orientation of the arrow was offset CW or CCW between 28 and 82°. Adjustments of the arrow were tested in four different head roll orientations {0, 90 [i.e., right-ear down (RED)], 180 (i.e., upside-down), and 270° [i.e., left-ear down (LED)]}. All subjects performed two experimental sessions on separate days. During the first session, head roll position was alternating between 0 and 180°, whereas during the second session, roll position was alternating between 90 and 270°. The angle of head roll reorientation between consecutive trials was always 180°, but the direction of turntable rotation was pseudorandom, i.e., subjects either crossed upright or upside-down orientation. By rotating the subjects from different directions (i.e., CW vs. CCW), we controlled for hysteresis. The time to complete the arrow adjustments was restricted to 2 s in this paradigm.

Between trials, subjects remained in darkness to prevent visual hints of verticality. In each head roll position, 4 different trial types were defined according to the direction of turntable rotation and the direction of arrow orientation, resulting in a total of 16 trial types. Twenty trials were run for each trial type, resulting in a total of 320 trials per subject.

### Paradigm 2: SVV measurements in 15° steps

In this paradigm, SVV adjustments were studied in 20 additional head roll positions with steps of 15° (from 15 to 75°, from 105 to 165°, from 195 to 255°, and from 285 to 345°) to obtain a resolution of SVV adjustments of 15° in the entire roll plane together with the data from paradigm 1. As in paradigm 1, these positions were studied in pairs having a shift in head roll position of 180° between trials. Turntable peak velocity and acceleration were identical to those used in paradigm 1. All subjects performed two sessions on separate days, running five pairs of head roll positions in each session. Because it was shown in paradigm 1 that the direction of arrow orientation (up vs. down) did not yield statistically significant differences (see results), subjects were required to adjust the arrow with the arrow head pointing up in all trials. At each roll position, a total of 24 trials were collected, having the direction of turntable reorientation pseudorandomized (CW vs. CCW), resulting in a total of 480 trials for each subject in paradigm 2. As in paradigm 1, the initial offset of the arrow ranged between 28 and 82° in either the CW or CCW direction. Two subjects included in paradigm 2 did not participate in paradigm 1; therefore the roll positions from paradigm 1 (0, 90, 180, and 270°) were collected additionally in the same fashion as the other roll positions in paradigm 2 in these two subjects. In paradigm 2, which was recorded later, the time limit was slightly longer (<3 s), because several subjects included were unable to adjust the arrow within 2 s in the head roll orientations studied in paradigm 2.

### Data analysis

Trials were sorted according to head roll orientation, direction of turntable reorientation, and direction of arrow orientation. Outliers were defined as data points differing >3 SD from the mean. In total, 0.23% of all trials were discarded by this criterion. Average deviations relative to the desired arrow angle and ±SD were calculated for each subject. In the following, we will use the term “intraindividual variability” whenever we report intraindividual SD.

In the range of 135–150° roll, a sudden shift from A-effect back to E-effect has been reported (Kaptein and Van Gisbergen 2004, 2005). As a consequence of this shift, two separate clusters of data points could be observed at certain roll angles in some of the subjects studied. This shift is considered a consequence of central processing (a shift in the reference frame used) rather than a consequence of the otolith sensors (Kaptein and Van Gisbergen 2005; Vingerhoets et al. 2008). Whenever such bistability occurred, we calculated intraindividual variability from single clusters and not from all data points. Thirty-five individual trial conditions with bistability, i.e., with adjustments both toward the A- and E-effect, were identified by histograms of SVV adjustments at a given roll angle with both directions of previous turntable roll. Cluster analysis with an assumed number of clusters of *n* = 2 (clusterdata.m, Matlab 7, The Mathworks) was performed in these trial conditions and, whenever cluster analysis resulted in two clusters with three or more data points each, the intraindividual variability was calculated separately for these clusters and averaged thereafter. However, if less than three data points were assigned to a single cluster, these data points were discarded because no reliable SD could have been calculated. Statistical analysis was done using ANOVA (Minitab, Minitab, State College, PA). Tukey's correction was used to compensate for multiple comparisons. Besides *P* values, degrees of freedom (df) taking into consideration both the number of conditions (df_{a}) and the number of participants for each condition (df_{b}) are provided along the *F-*values.

### Framework of the proposed otolith–SVV model

Previous SVV models either relied on an inaccurate sensory input combined with a tendency to shift the subjective vertical toward the subject's body-longitudinal axis (idiotropic vector) to minimize errors close to upright position (Mittelstaedt 1983) or on a Bayesian framework combining an accurate, but noisy sensory signal with prior knowledge (De Vrijer et al. 2008; Eggert 1998). Here we extend published Bayesian models by adding an otolith afferent input to derive an estimate of the sensory signal indicating vertical direction. This allowed us to study the contribution of the otoliths' firing characteristics to the SVV.

We assumed that noisy otolith afferents induce variability in the verticality estimate. The model has two stages: the first deals with the vertical estimate from otolith afferents (the otolith estimation model) and the second with the combination of the resultant estimate with a central bias (the SVV model).

In stage 1, we apply concepts from information theory to determine the best possible performance of a vertical estimator that relies on realistic otolith afferents properties (Fernandez and Goldberg 1976a; Fernandez et al. 1972; Loe et al. 1973). In particular, we make use of the Cramer-Rao bound (Cox and Hinckley 1974), which provides the minimum achievable variability in the estimate. In the description of Stage 1 below, we give a detailed mathematical derivation of this bound; here we give a qualitative description. At any tilt angle, the population of otolith afferents has a certain activity that varies from trial to trial and the brain has to reconstruct the direction of gravity from the noisy activity. Each afferent provides information about the component of acceleration along its preferred direction. The *top panel* in Fig. 1 shows this point by showing the average tuning functions of eight sample afferents when the earth-vertical forms 45° with the body-longitudinal axis. Because of noise, the size of this component has some uncertainty represented in the *bottom row* as an intensity plot with more likely values of the stimulus represented with lighter shades. The uncertainty is related to the flatness of the tuning curve: the flatter the curve, the bigger the range of stimuli consistent with the observed firing rate and the level of noise. That is, the most informative afferents are the most sensitive to changes in the stimulus, not the ones firing at their peak (Butts and Goldman 2006). Indeed, from the eight afferents shown in Fig. 1, it is the ones with preferred directions perpendicular to the actual vertical direction that constrain the stimulus the most, as indicated by their narrower bands. The combination of all afferents leads to a localized likelihood with a certain spread. The variance of the angle estimate is related to the spread of possible directions of vertical defined by a given change of roll-tilt. The Cramer-Rao bound quantifies this directional spread, which is the minimum achievable given the otolith firing characteristics. For the output of stage 1. we assume an accurate estimate of vertical with the minimum possible variability. This is represented at the output of C in Fig. 1. We choose an unbiased estimator because subjective body tilt experiments suggest that without a visual input the average error in tilt estimate is much smaller than that for SVV (Kaptein and Van Gisbergen 2004; Mittelstaedt 1983; Van Beuzekom et al. 2001; Vingerhoets et al. 2008). We will use this lower bound to explore the limits that the otolith afferents place on the reconstruction of head roll angle and implicitly on the SVV.

In stage 2, the estimate of vertical obtained by stage 1 is combined with central processing by including a bias pointing toward the body-longitudinal axis. This second stage is closely related to the Bayesian model proposed by De Vrijer et al. (2008), but it also accounts for the abrupt switch between the A- and E-effect by allowing two bias directions (Kaptein and Van Gisbergen 2004, 2005), one toward the head and the other toward the feet, whose relative probabilities change with roll angle following two complementary sigmoid functions (Fig. 1, *bottom panel*). The bias direction is always toward the head for small roll-tilt angles and always toward the feet for roll angles close to upside-down. At roll-tilt angles around 120°, a transition occurs, and both outcomes are possible, leading to a bistable range. The output of stage 2 is the model's SVV estimate and is given by the angle at the maximum of the posterior distribution that results from combining the bias toward the body-longitudinal axis and the optimal distribution obtained in stage 1.

### Notation

Vectors will be denoted by an overhead arrow. We define vectors in two different spaces. On the one hand, we consider the two-dimensional space of acceleration vectors in the roll plane. Because we are concerned with the estimation of a vertical vector, these are denoted by *v⃗* with varying subscripts. The two components are defined in an egocentric head-reference frame: the longitudinal (LON) component along the body's vertical *v*_{LON}, and the interaural (IA) component *v*_{IA}, i.e., *v⃗* ≡ (*v*_{IA}, *v*_{LON}). In this frame, the real direction of the upward vertical changes with the body roll-tilt angle *v⃗* = (sinθ_{tilt}, cosθ_{tilt}), where θ is the angle of head roll. We also define vectors in the space spanned by the firing rates of *N* otolith afferents, *r⃗* ≡ (*r*_{1}, *r*_{2}, … *r*_{N}). Integrals over this N-dimensional space will be abbreviated as ∫*d**r⃗* ≡ ∫*dr*_{1} … *dr*_{N}. Finally, vectors of unit length are denoted with a hat *p̂*.

During the task, when the subjects have to report the SVV, the only linear acceleration is gravity *g⃗*, which determines the downward vertical. Because subjects have to report the upward vertical instead, we consider the input to be a vector *v⃗*, which we term vertical vector and which is the opposite of the acting acceleration, that is *v⃗* = −*g⃗*. The angles of the estimates are given in the reference frame of the subject, and they correspond to the estimate of upward earth-vertical. In this frame, zero corresponds to upward toward the head.

### Stage 1: the otolith estimation model

The linear acceleration acting on the head is represented in a distributed manner in the firing of the population of noisy otolith afferents. Here we consider the maximum achievable performance of an estimator of linear acceleration that relies on the information delivered by these afferents. This is given by the Cramer-Rao bound, which provides the minimum variability achievable by an estimator for a given fixed stimulus, which is a given body tilt in our case. Its calculation is based on the Fisher information matrix (Cox and Hinckley 1974) derived from the conditional response distribution *P*(*r⃗*|*v⃗*). The Fisher information matrix (FIM) associated with the input stimulus v⃗ is given by (1) Because the stimulus is two dimensional, the FIM is a 2 × 2 matrix, *k* and *m* denote the indices. The inverse of the FIM itself sets a bound on the covariance of the acceleration vector estimate. We are focusing on the head-roll angle θ derived from this vector estimate. The angle corresponding to a given vector *v⃗* is θ(*v⃗*) ≡ arc tan(*v*_{IA}/*v*_{LON}). The Cramer-Rao bound for any unbiased estimator of θ is (Cox and Hinckley 1974) (2) where θ_{e}(*r⃗*) is the estimator and E(θ_{e}|*v⃗*) = θ(*v⃗*) is the average estimate over many trials, which we assume is not biased *E*(θ_{e}|*v⃗*) = θ(*v⃗*).

To write the response conditional to the stimulus, *P*(*r⃗*|*v⃗*), we assume uncorrelated Gaussian trial-to-trial variability around a mean response that gives the otolith afferent discharge as a function of linear acceleration. The average otolith afferent response, the tuning function, has been well characterized (Fernandez and Goldberg 1976a; Fernandez et al. 1972; Loe et al. 1973). The most salient feature is the existence for each afferent of a different preferred direction with respect to the head: the afferents discharge maximally/minimally when the net gravito-inertial force is parallel/antiparallel (i.e., points into the opposite direction) to their preferred direction. To a good approximation the activity of an afferent can be described as a function of the projection of the linear inertial acceleration along its preferred direction *p̂*, which we will denote by *u* ≡ *v⃗*·*p̂*. For an afferent *a*, we write the firing rate *r*_{a} (3) *f*_{0,a} is a constant background firing rate, μ_{a}, a random variable with zero average and SD σ_{a}. *K*_{a} is the magnitude of the afferent's modulation in spikes per second per g (we use *g* = 9.81 m/s/s as the unit for acceleration) and *h*(·) is a normalized monotonic tuning function based on those reported by Fernandez and colleagues (Fernandez and Goldberg 1976a; Fernandez et al. 1972) who described a nonlinear response. The two nonlinearities implemented here are *1*) a stronger modulation in response to excitation than to inhibition and *2*) saturation (4) The parameters *A* and *B* determine the nonlinearities. *A* is a saturation parameter with units of acceleration and *B* defines the level of asymmetry between excitation and inhibition. Higher values of *A* lead to a more linear response and positive values of *B* to a stronger response to excitation than inhibition. The first expression represents the modulation of the afferent for a linear acceleration of magnitude μ aligned with the afferent's preferred direction; the second represents the modulation in response to a linear acceleration of constant magnitude and varying direction. Therefore the response of the population of otolith afferents for a given roll-tilt of the head represented by a vertical vector *v⃗* can be described by the following multivariate probability distribution *P*(*r⃗*|*v⃗*) (5) The task of the brain is to infer the acceleration given the response of the afferents in single trials *r⃗* = *r*_{1}… *r*_{N}. The FIM for the distribution provided above is (6) Where *h*′(*u*) = 1/[*A*cosh(*u*/*A* − *B*)^{2}] is the derivative with respect to *u*. In the last equality, we changed the sum over afferents to a sum over preferred directions. It is possible to write the matrix more economically in terms of the signal-to-noise ratio per preferred direction Then (7) The product *p̂*_{k}*p̂*_{m} defines a projection matrix onto the preferred direction of the afferent. For linear tuning functions, *h*′(v⃗·*p̂*) = constant and therefore the FIM is the same for all body orientations. This is not the case for more realistic, nonlinear tuning functions.

We can interpret the FIM given by *Eqs. 6* and *7* as a cumulative reduction in the uncertainty about linear acceleration. Each afferent contributes to the reduction of uncertainty about linear acceleration only for the component of acceleration along its preferred direction. The strength of the contribution decreases with intrinsic variability σ_{a} and increases with the afferent's sensitivity to changes of acceleration along its preferred direction.

Finally, we use *Eq. 2* to determine the lower limit of the variability of θ(*v⃗*) ≡ arc tan(*v*_{e,IA}/*v*_{e,LON}). We need the following partial derivatives (8) where stands for the unitary vector perpendicular to *v⃗*. For stimulus vectors corresponding to varying body roll-tilt *v⃗* = (sinθ_{tilt}, cosθ_{tilt}) and Then Δθ is bound by (9) The expression for the Cramer-Rao bound can be understood as follows. The inverse of the FIM gives a bound on the covariance of the vector estimate. The Cramer-Rao bound in *Eq. 11* results from the projection of this covariance along the direction determined by a change in roll angle, which is the perpendicular to the vertical vector in the head. To make the meaning of *Eq. 11* more clear, it is convenient to write it in terms of the eigenvectors ê_{1}(*v⃗*), ê_{2}(v⃗) and eigenvalues 1/σ_{1}(*v⃗*)^{2}, 1/σ_{2}(*v⃗*)^{2} of *FIM*. Then (10) The output of the otolith model is the estimated direction of acceleration θ_{oto} and is shown as a likelihood distribution in the roll plane with a given variance σ_{oto} (see Fig. 1, output from step C). We assume that it is unbiased *E*(θ_{oto}**|**θ) = θ and that, being an efficient estimator, it saturates the CR bound, that is, is given by *Eq. 11* and *Eq. 12*.

### Stage 2: the SVV model

We model the final setting of the visual line in the SVV task as the result of the combination of the estimated direction of acceleration coming from the otoliths and a bias that is either toward the subject's head or toward the subject's feet (Kaptein and Van Gisbergen 2004; Vingerhoets et al. 2008). To implement the switch hypothesis, we postulate that for a given otolith estimate θ_{oto}, there is a certain probability that the bias is up or down. When the otolith roll estimate is small compared with a critical value θ_{switch}, bias up (toward the head) is always chosen. Similarly, when the otolith estimate is greater than θ_{switch} and close to upside down, the bias down (toward the feet) is chosen. This can be described by two complementary sigmoid functions *A*_{up}(θ_{oto}) and *A*_{dn}(θ_{oto}) ≡ 1 − *A*_{up}(θ_{oto}).

For a given trial, one would obtain one of two possible estimates, θ_{SVV} = *f*_{1}(θ_{oto}) or θ_{SVV} = *f*_{2}(θ_{oto}), for the bias toward the head (up) or toward the feet (dn), respectively. That is (11) To determine the estimate corresponding to each choice of bias, we consider the maximum a posteriori estimate (MAP). The posterior probability for the bias up is (12) A similar expression applies for the bias down: *P*_{2}(θ|θ_{oto}). Because θ is an angular variable, we use von Mises distributions, the circular analog of the normal distribution (Evans et al. 2000). For the unbiased otolith estimate, we approximate the likelihood with (13) where is inversely related to the variance of the distribution, (see appendix). Similarly, for the two biases, θ_{up} = 0° (toward the head) and θ_{dn} = 180° (toward the feet) (14) The mechanism adding a bias toward the head leads to a posterior with parameters κ_{1} and θ_{1} defined by the vector sum (15) (where κ gives the length of each vector and θ the angle with the vertical, that is, κ and θ define a vector in polar coordinates; see appendix) and the bias toward the feet to a posterior with parameters κ_{2} and θ_{2} through (16) The corresponding maximum a posteriori estimate (MAP) for the bias up is (17) A similar expression applies for *f*_{2}(θ_{oto}) with the bias down.

We are interested in the average estimates associated with each of the bias directions over different trials with the same body roll-tilt. For a given roll angle, the otolith estimate is centered at θ with a spread given by σ_{oto}(θ). The average estimates are therefore obtained by evaluating them at the true roll angle θ. The variability can be approximated as follows (18) where the vertical bars denote the absolute value and the index *k* refers to the MAP estimates for the bias up (*k* = 1) and bias down (*k* = 2).

The final step is to determine the average SVV and its variability over trials with the same body tilt angle θ, which we denote by θ_{SVV}(θ) and σ_{SVV}(θ). We approximate these averages as the sum of the two possible outcomes weighted by their relative frequencies (19) (20) with (21) (22) where *A*_{1}(θ) is a monotonically decreasing function going from 1 for θ << θ_{switch} to 0 for θ >> θ_{switch}. *A*_{2}(θ) = 1 − *A*_{1}(θ) is the complementary function, and it has the opposite behavior. They represent the probability that, at a given roll angle, the strategy chosen is either up or down. Because proposing a mechanism for the switch is not our priority, we directly parameterize *A*_{1}(θ) and *A*_{2}(θ) as sigmoid functions characterized by two parameters, θ_{switch} and Δθ_{switch} (23)

The expression for σ_{SVV} (*Eq. 21*) is defined in a way that parallels the procedure used to derive this quantity from the data in the presence of separable “clusters.” In the narrow region where both outcomes are observed, two clusters would appear, each with its own intracluster variability σ_{1} and σ_{2}. The only difference is that, during data analysis, the average is calculated from an equally weighted average, and here the values are weighted according to their relative frequencies, which is a reasonable approximation.

### Fitting procedure

The otolith–SVV model was used to fit the averaged data across subjects. We simultaneously fit deviations of the settings with respect to real vertical Δ = θ − θ_{SVV} and their variability σ_{SVV}. We restricted the range of roll-tilts from 0 to 180°, and the average data were symmetrized accordingly so that var(θ) = var(−θ) and deviation(θ) = −deviation(−θ), where θ is the angle of head roll.

To define the otolith population we need to specify *S*_{p}, the signal-to-noise ratio distribution, and the tuning functions. We assumed left-right symmetry and divided the roll-tilt space of preferred directions into 5° bins. Therefore *S*_{p} should be understood as a measure of the signal-to-noise ratio associated with afferents with preferred directions within a bin 5° wide and centered at the preferred direction determined by *p̂*. We parameterize *S*_{p} with von Mises functions centered at θ_{ut,R/L} = ±90° and θ_{sac} = 0° with fixed widths σ_{ut} = 11.5° and σ_{sac} = 4σ_{ut} and two free parameters *C*_{ut} and *C*_{sac}, which control the size of the peaks of the utricular and saccular component, respectively (24)

Initially, we considered a fourth term corresponding to afferents with preferred directions around head-down position. However, the fitting algorithm drove this term to zero, so final fits were done explicitly excluding that term. We assume the same tuning function parameters (*Eq. 3*) for all afferents: *A* = 2*g* and *B* = 0.87 (Fernandez and Goldberg 1976b). For the second stage, we restricted the strength of the biases to satisfy σ_{up} = σ_{dn}. Finally, we defined the switch of the direction of the prior. We fixed the width of the switch Δ_{switch} = 12° (Kapstein and Van Gisbergen 2005) and left θ_{switch} as a free parameter. A standard MATLAB function was used to find the values of the four free parameters, *C*_{ut}, *C*_{sac}, σ_{up}, and θ_{switch}, that give a best simultaneous fit of deviation and variability in the least squares sense. As a measure of the goodness of fit, the variance accounted for (VAF) was calculated for deviation and variability data independently (25)

## RESULTS

### SVV in upright, upside-down, and 90° ear down positions (paradigm 1)

Statistical analysis (3-way ANOVA) of arrow orientations (Fig. 2*A*) showed no main effect for the reorientation movement of the turntable before each trial [CW vs. CCW, *F*(1,16; df_{a}, df_{b}) = 0.17, *P* = 0.684] and the desired arrow orientation [up vs. down, *F*(1,16) = 1.37, *P* = 0.244]. We therefore pooled the data from these conditions. In the upright position, the overall average SVV (−0.1°) was not significantly different from earth-vertical (*t*-test, *P* > 0.05), whereas in both 90 and 270° roll positions, the overall average of arrow adjustments deviated clearly from earth-vertical arrow orientation (90°: 14.9 ± 2.4°; 270°: −14.4 ± 2.5°; average ± SD). There was no significant difference between the absolute values of arrow adjustments in 90 (RED) and 270° (LED) head roll positions (3-way ANOVA: *P* > 0.05). Adjustments in these two positions, however, were significantly (*P* < 0.001) different from adjustments in both upright and upside-down positions using pairwise comparisons including Tukey correction. In the upside-down position, the overall average SVV (−2.1°) was not significantly (*P* > 0.05) different from earth-vertical; however, the interindividual SD was larger than in the upright position (1.4 vs. 0.6°; upside-down vs. upright).

Three-way ANOVA of intraindividual variabilities (Fig. 2*B*) showed no main effect for the direction of the preceding turntable rotation [*F*(1,16) = 0.56, *P* = 0.456] and the desired arrow direction [*F*(1,16) = 0.39, *P* = 0.532]. We therefore pooled these data as well. Pairwise comparisons showed that average intraindividual variability in the upright position [1.5 ± 0.3° (SD)] was significantly smaller (*P* < 0.001) than in the upside-down position (2.6 ± 0.8°). Although average intraindividual variability was not significantly different between 90 (3.9 ± 0.8°) and 270° (3.8 ± 0.7°; *P* > 0.05), variability was significantly larger in both RED and LED than in the upright and upside-down positions (*P* < 0.001).

### SVV in roll positions with 15° resolution (paradigm 2)

In seven subjects, the modulation of intraindividual SVV variability was studied with a finer resolution of 15° between roll positions. Individual average SVV adjustments (including data from paradigm 1 for those subjects who also participated in paradigm 2) are shown in Fig. 3. As the shift from the A-effect back to the E-effect with associated bistability may occur at varying roll angles depending on the direction of the preceding turntable rotation, we present trials with CW and CCW turntable rotations separately. Most subjects showed hysteresis as indicated by the differences in arrow deviations depending on the direction of preceding turntable reorientation at some head roll angles; this could be observed most clearly in *subjects DP* and *RG*. The term “hysteresis” describes a retardation of an effect, when the forces acting on a body are changed (Merriam Webster definition). Shifts from the A-effect back to the E-effect usually occurred between 105 and 135° roll, and the angle of the shift in individual subjects could be different for RED and LED roll angles.

Depending on whether subjects were rotated CW or CWW, average errors in adjusted SVV were consistently different, suggesting hysteresis (Fig. 4). Peak A-effects were found to occur in the range of 90–105° roll, whereas E-effects were largest in the range of 135–150° roll relative to upright. For small roll angles, individual deviations were negligible.

Figure 5 shows the shift from the A-effect to the E-effect in a single subject (*subject AP*) with a bistability leading to two clusters at 120° head roll. Calculating intraindividual variability without considering the two clusters would lead to large variability peaks in the bistability zone, which may confound the variability estimates (Kaptein and Van Gisbergen 2005).

Individual SVV variability is shown in Fig. 6. Because two-way ANOVA yielded no significant differences between intraindividual variabilities for both CW and CCW turntable reorientations [*F*(1,12) = 0.03, *P* = 0.854], results were pooled for further analysis. In all subjects, SVV variability increased with roll angle, but had a local minimum at upside-down. The difference between peak variability and the variability near or in upside-down orientation varied between single subjects. Whereas a clear decrease around upside-down orientation could be observed in some subjects (AT, AP, RB, and DM), the difference was markedly smaller in the other subjects (DS, DP, and RG). SVV variability (RED and LED pooled) peaked either at 120 or 135° head roll relative to upright.

The average intraindividual SVV variability (Fig. 7) shows a distinct pattern of head-roll–dependent modulation resembling an m-shaped curve, whereby the middle foot (upside-down orientation) of the m does not reach the base. Variability was minimal in upright orientation and increased with head-roll angle, peaking in the range of 120–135° roll. With further increasing head-roll angles, intraindividual variability decreased, reaching a relative minimum in upside-down orientation. The minimum average intraindividual variability in upside-down, however, was clearly larger than the variability in upright orientation.

### Predictions from the otolith–SVV model

For linear tuning functions, the *FIM* does not vary with *v⃗*. Because of linearity and left-right symmetry, we can conclude that the eigenvectors of the FIM are aligned with the body axes. From *Eq. 12*, it follows (26) Therefore for linear tuning functions, the modulation of the variance of the otolith estimate predicted by the model falls into one of the following three categories. *1*) If the covariance of the acceleration vector estimate is isotropic is constant and does not modulate with the roll-tilt angle. *2*) If the signal-to-noise ratio is higher for utricles than for saccules, and Variability peaks at 90° roll-tilt and has local minima at 0° roll-tilt (upright) and 180° roll-tilt (head down). *3*) If the signal-to-noise ratio is higher for saccules than for utricles , In this case, variability peaks at 0 and 180° roll-tilt, with local minima at 90° roll-tilt.

For nonlinear tuning functions, the tilt dependence of the Cramer-Rao bound is more complex. However, it is still true that higher effectiveness of one of the two sensors, as given by a higher signal-to-noise ratio, leads to a modulation of variability within the roll plane. When in the upright position, the afferents with the highest signal-to-noise ratio belong to the saccular population. These constrain the component of the acceleration along the body vertical. However, the spread in roll-tilt angle depends on the horizontal component of the vector estimate, which is constrained by afferents with horizontal preferred directions, that is, utricular afferents. Therefore in the upright position, the saccules determine the accuracy of the vertical component of acceleration and the utricles the horizontal component. If there was the same number of utricles and saccules modulating with the same strengths (similar *K*), the vertical component of acceleration would be more precise than the horizontal. We would expect that the thresholds for motion detection would be lower for vertical than horizontal motion. However, in the upright position, horizontal motion is easier to detect than vertical motion (Benson et al. 1986). Furthermore, a larger number of utricular afferents than saccular afferents were proposed (Rosenhall 1972), suggesting that the signal-to-noise ratio for the utricles is higher than for the saccules. In contrast, in a side position, the spread of tilt angle is related to the component of the estimate along the body's vertical axis, which is of saccular rather than utricular origin. Because there is evidence that utricles are more informative than saccules, we expect higher variability with the body in side position than in vertical position, consistent with category 2 as defined above.

### Modeling results

We applied the two-stage model to fit the deviation and variability data simultaneously under the assumption of an efficient otolith estimator (i.e., with the minimal variance as defined by the Cramer-Rao bound) and common nonlinearity parameters for all afferents and obtained the following values for the free parameters: *C*_{ut} = 3.4, *C*_{sac} = 3.0, σ_{up} = 16, and θ_{switch} = 127°. The results of the fit are presented in Fig. 8, *A* (SVV deviations) and *B* (SVV variability); for comparison, the experimentally obtained values are provided as well. The differences between the otolith-derived estimate (stage 1) and the final estimate in Fig. 8, *A* and *B*, indicate the contribution of the second stage to the deviations and the variability of perceived vertical. Whereas the SVV deviations originate from stage 2, SVV variability is mostly a result of the otolith-derived estimate of vertical (stage 1), showing only slightly reduced variability values after applying stage 2. The nonmonotonic increase originating from stage 1 of the otolith–SVV model is preserved. The VAF obtained by the full otolith–SVV model is 0.89 for deviations and 0.69 for the variability. Figure 8*A* shows that the combined otolith–SVV model closely matches both the A-effect and the shift back to the E-effect at large roll angles. However, the otolith estimation model combined with the SVV model was able to reproduce only part of the average pattern of SVV variability. In particular, it was not possible to reproduce the large variability in head-down compared with upright. Larger differences between upright and head-down could be achieved only with afferents with saturation levels four times stronger than our reference value taken from the literature (Fernandez and Goldberg 1976b). The implications of these shortcomings of the otolith model will be further addressed in the discussion.

The otolith signal-to-noise ratio distribution based on the preferred directions for the utricles and saccules as defined in methods is shown in Fig. 8*C*. Here, the signal-to-noise ratio is plotted against the polarization direction relative to the head-longitudinal axis, showing that the narrow peaks of the signal-to-noise ratio in the plane of the utricles are higher than the wider peak in the plane of the saccules (polarization direction in the plane of the head-longitudinal axis). This would be in accordance to reports of a bigger number of utricular than saccular afferents (Rosenhall 1972) and to their higher sensitivity (Fernandez et al. 1972).

## DISCUSSION

### Main observations from our experimental data

Our results confirmed that the intraindividual variability in SVV becomes larger when stationary head-roll positions are increased from upright to ear down; however, this increase is not monotonic. Because the roll angle of the turntable was changed before each SVV setting, we can exclude that the roll-dependent modulations in SVV variability are solely caused by adaptive roll-dependent effects, which were not excluded in previous studies, where subjects remained in a given roll position for several minutes while performing the SVV task (Lechner-Steinleitner 1978; Schoene and Udo de Haes 1971; Udo de Haes 1970). We found an average peak intraindividual variability of 8.8° at head roll angles of ∼126° and a clear decrease in variability for larger roll angles. Whereas nonsignificant roll overcompensation at small angles was observed, considerable roll undercompensation, peaking between 90 and 105° roll, was followed by a sudden shift back to overcompensation at roll angles between 120 and 135°.

### Comparison of experimental findings with previous SVV studies

Similar experimental findings have been reported in previous studies, having peak variabilities around 120 (Lechner-Steinleitner 1978) or 150° (Schoene and Udo de Haes 1971; Udo de Haes 1970) head roll. As head roll increased further in these studies, variability decreased as in our data. Whereas we minimized adaptation over time by changing the subject's roll position after each trial, Udo de Haes (1970), Schoene and Udo de Haes (1971) and Lechner-Steinleitner (1978) collected repetitive SVV adjustments over a period of 8 min in a given roll position. In a publication by Udo de Haes (1970), individual SD values were in a similar range (<5–7°) as our findings for roll angles ≤120°. Considerable increases in SD (≤17° without cluster analysis) were observed for larger head roll angles (between 120 and 165° relative to upright) in four of seven subjects in our study, which are similar to peak SD ≤18° reported by Udo de Haes. Whereas we determined separate clusters for trials with an A- or E-effect within the transition zone described by Kaptein and Van Gisbergen (2004), Udo de Haes (1970) did not. Thereby SVV variability values provided by Udo de Haes might be confounded by the clustering as pointed out by Kaptein and Van Gisbergen (2005), which would explain the larger peak variabilities. However, clustering the data points did not qualitatively change the m-shaped pattern we observed, speaking against the hypothesis that the peaks around 120–150° are solely caused by shifts in the direction of errors within the transition zone.

The m-shaped modulation of SVV variability within the roll plane observed here and in previous studies (Lechner-Steinleitner 1978; Udo de Haes 1970) is in contrast to the approximately linear increase of SVV variability with head roll reported by Kaptein and Van Gisbergen (2005) with peak variability in the upside-down position. To allow a better comparison with Kaptein and Van Gisbergen, we used the same approach to analyze the data as suggested in their study. We calculated SVV variability values from pooled single trials from all subjects after assigning them to one of two clusters (using kmean, Matlab 7.0, The Mathworks). Variability values obtained were somewhat smaller (range: ∼1 to 12–14) than reported by Kaptein and Van Gisbergen (2005) (range: ∼2 to 15–20), and the m-shaped pattern was preserved. The origin of this discrepancy therefore remains unclear; however, we would like to point to the considerable differences in the experimental conditions. Whereas we recorded pairs of whole body roll positions 180° apart and were alternating between these two positions, Kaptein and Van Gisbergen moved subjects to different whole body roll positions always starting from upright position.

### Main observations from the otolith–SVV model

##### IMPLICATIONS OF THE CRAMER-RAO BOUND.

The first task of our model was to derive an estimate of linear acceleration from the firing rates of otolith afferents. Because subjective body tilt (SBT) experiments show smaller errors than the SVV (Van Beuzekom and Van Gisbergen 2000; Van Beuzekom et al. 2001; Vingerhoets et al. 2008), we assumed that the brain has access to an unbiased estimate of the acceleration vector. Under this hypothesis, the roll estimate derived from the otoliths corresponds simply to the angle that the estimated linear acceleration vector forms with the body's longitudinal axis.

We did not propose an explicit estimate; rather, we explored the limit on precision set by the otolith afferents by deriving the Cramer-Rao bound based on Fisher information (Cox and Hinckley 1974). The absolute limit on variance for an unbiased estimator of linear acceleration is given by the Cramer-Rao bound, which depends on the conditional response probability *P*[*r⃗*|*v⃗*(θ_{tilt})] of otolith afferents with responses given by *r⃗* when the tilt is θ_{tilt}. The bound does not depend directly on the roll angle because we assume that there is a necessary intermediate step that estimates the acceleration vector.

Maximum likelihood estimators approach the Cramer-Rao bound asymptotically as the signal-to-noise ratio in the population increases (Xie 2002), and neural networks are capable of performing such computations (Deneve et al. 1999, 2001). The minimum attainable variability of the angle estimate derived from the otoliths modulates with roll-tilt angle caused by a nonuniform distribution of the signal-to-noise ratio in the roll plane. We propose that, at each roll angle, the variance of the angle estimate is dominated to a different degree by the two main sensors: the utricles and saccules. Thus it is mostly determined by the signal-to-noise ratio of the utricles when upright and by the signal-to-noise ratio of saccules for 90° roll-tilts. Because the signal-to-noise ratio seems higher for utricular than for saccular afferents (Fernandez and Goldberg 1976a; Fernandez et al. 1972), we can conclude that, in general, variability has local minima in upright and head-down positions with a maximum at intermediate roll-tilts. For linear tuning functions, left-right symmetry forces variability to peak at 90° roll-tilt and the minima at 0 and 180° roll-tilt to be the same. For nonlinear tuning functions, the location of the peak and the relative size of variability in upright and head-down positions depends on the nonlinearities of the tuning functions and the particular distribution of preferred directions.

Head-down SVV variability was ∼2.4 times larger than upright. In both positions, because of left-right symmetry, the FIM has eigenvectors aligned with the body axes. On the other hand, a change in roll-tilt direction is represented by a horizontal vector, which means that the FIM is dominated by the utricular afferents, and the Cramer-Rao bound can be approximated by (27) where ε is 1 in upright and −1 in head-down and the sum extends over utricular afferents. The two values are different only if there is an up-down asymmetry in the distribution of the signal-to-noise ratio. This would be the case if the average angle of the utricular afferents with the horizontal, which we denote β, is not zero and if each utricle is polarized, that is, it is excited mainly by ipsilaterally directed accelerations.

We calculated the predicted ratios for two saturation levels *A*, where higher values of *A* lead to a lower saturation. For the typical value reported in the squirrel monkey (*A* = 2*g*) (Fernandez and Goldberg 1976b), the mean tilt angle with respect to the horizontal needed to achieve the observed ratio, >20°, seems too high. However, lack of human data precludes a categorical rejection of this possibility. If the saturating nonlinearity was stronger (*A* = 0.5*g*), the required ratio would be reached with mean preferred direction inclinations of <10°. A positive average angle of the utricular afferents with respect to the horizontal β corresponds to a direction of the approximated surface of the utricular macula with the lateral side down. Interestingly, anatomical measurements in human temporal bones by Naganuma et al. (2003) found such a tilt of the utricular macular surface. This contrasts with the functional preferred direction distribution found in squirrel monkeys (approximately −8°) (Fernandez and Goldberg 1976a) and with those from other species (Jaeger et al. 2008).

Based on the otolith afferents and a signal-to-noise ratio higher for the utricular than the saccular afferents, an efficient estimator would show variability with local minima of similar size in upright and upside-down positions. The fact that the observed variability in upside-down is much higher than in upright suggests that it might not be the result of an intrinsic limitation of the information provided by the otolith afferents. For this to be the case, the nonlinearities, in particular the saturating nonlinearity, should be stronger than that reported in the literature. One should keep in mind that we took tuning function parameters from squirrel monkey data (Fernandez and Goldberg 1976b), and it is not clear whether such parameters can be extrapolated to humans.

### Decreasing the effectiveness of the otolith estimator by modulating the Cramer-Rao bound

The conclusions listed in the previous paragraph rely on model simulations assuming an efficient estimator at every roll-tilt angle (results shown in Fig. 8). However, the efficiency of the estimator is likely shaped by behavioral demands and would therefore be most efficient for roll-tilt angles encountered in everyday life. We assume that the system does not have the opportunity to tune itself to maximum efficiency at every roll-tilt angle but only for angles near upright. If we explicitly include a decrease in effectiveness with roll angle by an appropriate modulation of the Cramer-Rao bound—as it is shown in Fig. 9 —such as (28)

an m-shaped pattern that more closely fits the experimental SVV variability will develop (Fig. 9*B*; VAF = 0.86), whereas fitting of the experimental deviations in SVV remains mostly unchanged (Fig. 9*A*; VAF = 0.91) compared with the initial simulations assuming an efficient estimator at every roll angle (Fig. 8). Under these conditions, the following values for the free parameters were obtained: *C*_{ut} = 3.8, *C*_{sac} = 3.5, σ_{up} = 16, and θ_{switch} = 127°.

Precision in SVV after combining the initial otolith estimate with prior knowledge was increased only marginally in the upright position. Based on Bayesian principles, using prior information, one would have expected noise reduction. This, however, was not noticeable in the upright position because the variability for the otolith output was much smaller than that for the prior, which was set to 16°. As a result, the prior did not influence SVV variability in upright significantly. As the uncertainty from the otolith estimate increased at higher roll-tilt angles, the prior had a bigger weight and the reduction in variability became apparent.

### Proposed functional polarization of the saccules to explain the asymmetries in head-up and head-down orientation

Variability could be closely matched when otolith afferents with functional polarization vectors in the average plane of the utricles were assigned larger signal-to-noise ratios as shown in Figs. 8*C* (assuming an optimal estimator at all roll angles) and 9*C* (assuming a decreasing efficiency of the estimator with increasing roll). Factors that can contribute to a higher signal-to-noise ratio include a larger number of utricular hair cells as proposed by Rosenhall (1972) or a larger modulation of the firing rate (Fernandez and Goldberg 1976a; Fernandez et al. 1972). Utricular afferents are distributed on an approximately horizontal plane, and therefore only those with significant projections on the frontal plane modulate with changes in roll tilt to contribute to tilt perception on the roll plane.

A surprising feature of the fitted population is the absence of strong signals from afferents that fire most in the head-down position. This was obtained by assigning the same tuning function parameters for every afferent. This implicitly introduces a correlation between preferred direction and maximum sensitivity; afferents are most sensitive when gravity is parallel to them. Thus afferents with preferred direction close to the head-down position would be maximally sensitive to acceleration around the head-down position. Therefore the correct interpretation of the distribution in Figs. 8*A* and 9*A* is not that there are no afferents with preferred directions close to head-down, but rather that those afferents do not have maximum sensitivity close to head-down. If we postulate that all saccular afferents respond with maximum sensitivity close to the upright position and modify their tuning functions accordingly, in terms of Fisher information, they would be indistinguishable [*h*′_{up}(*v⃗*) = −*h*′_{dn}(*v⃗*)]. Indeed, behavioral demands, mainly maintenance of posture, would tend to make the system more sensitive close to upright position. Furthermore, saccular units with opposite sensitivities fire at the same level in upright position, even if one is receiving excitation and the other is receiving inhibition (Fernandez and Goldberg 1976a). This suggests a functional polarization that makes all the saccular afferents act as a whole, as if they had a uniform preferred direction. Uchino et al. (1997) found that vestibular neurons receive monosynaptic excitation from one population of saccular hair cells and disynaptic inhibition through interneurons from saccular hair cells located on the opposite side of the striola. Behavioral evidence also points to a similar functional polarization of the utricles. Lempert et al. (1998) showed that 1 wk after unilateral vestibular nerve section, subjects showed asymmetric linear VOR responses to lateral acceleration.

### Comparison with previous models

Mittelstaedt (1983) hypothesized that the length of the resultant of the gravity vector and the constant idiotropic vector, which points along perceived vertical, indicate the person's certainty about its location. He found that the SD of SVV adjustments increased with head roll because the length of the resultant vector decreased steadily with increasing roll, being maximal in the upside-down orientation. The model provided by Mittelstaedt, however, does not account for the sudden switching from roll undercompensation to roll overcompensation at roll angles of 135−150° found here and observed by others (Kaptein and Van Gisbergen 2004, 2005). If such a switch was implemented in Mittelstaedt's model by changing the direction of the idiotropic vector, the length of the resultant would increase again with increasing roll, resembling more closely the experimentally observed pattern of SVV variability. However, we found that a constant idiotropic vector that changes direction to reflect the shift from A- to E-effect did not fit the deviation and variability data simultaneously as well as our model.

Mittelstaedt's model can be made mathematically equivalent to adding a bias in an optimal observer model (Eggert 1998). In this interpretation, the angle of the vectors with the vertical corresponds to the estimated angle and bias direction, respectively. The length of the vectors is not related to the magnitude of the estimated acceleration but rather to the uncertainty of the estimate that is inversely related to the variance. When this reinterpretation of the vectors is made, the combination of the estimate and the bias corresponds to vector addition (see methods and the appendix). The resulting estimate is the angle of the resultant vector. In this reinterpretation, our model can be seen as a generalization of Mittelstaedt's model in several respects. First, we consider a second vector corresponding to the bias down; second, we do not assume any deviations in the initial estimate; and third, we consider a roll-tilt–dependent variance of the initial estimate instead of a constant as implicit in Mittelstaedt's model. As a consequence, the angle that maximizes the posterior distribution in general does not align with that predicted by the vector sum described above, which is only valid when the variance of the initial estimate is constant.

The SVV model by De Vrijer et al. (2008) showed that both Mittelstaedt's vector model and a Bayesian observer model successfully reproduced adjusted SVV accuracy at ≤120° roll. Variability, however, was overestimated by a factor of ∼2. Our model can be seen as an extension of the Bayesian model proposed by De Vrijer and colleagues. Both are based on the idea that the settings observed in the SVV task are the result of the combination of the otolith estimate with a prior that shifts the estimated gravity-vector toward the body-longitudinal axis. Our model goes a step further because it adds an explicit process that transforms otolith afferent discharge into an otolith angle estimate through an intermediate step that estimates the linear acceleration vector acting on the head (the gravity vector in roll-tilt paradigms). This allowed us to study the relation between observed SVV variability and features of otolith afferents' discharge as a function of applied shear force. We also studied roll angles ≤180° and dealt with change from underestimation to overestimation at large tilt angles by adding a tilt-dependent shift in the direction of the bias.

De Vrijer et al. assumed that noise increases linearly with roll angle, whereas we derived the variability from a population of otolith afferents. This led to variability with local minima in upright and head-down or to an m-shaped response if we allow for a monotonic decrease in the efficacy of the otolith estimator.

Vingerhoets et al. (2008) had subjects report SVV in a dynamic task where they were continuously rotated, which therefore included the whole range of possible tilts. To describe the deviation pattern, similar to static SVV, they added a constant idiotropic vector along the subject's longitudinal axis, as in Mittelstaedt's model, to the initial estimate of the gravity vector, the equivalent to the otolith estimate for linear acceleration in our model. To account for the sudden switch from A- to E-effect, they changed the idiotropic vector from pointing toward the head to pointing toward the feet and to fit the deviation pattern they had to assume a higher weight for the idiotropic vector toward the feet. As we have mentioned and as the authors themselves argue in their discussion, the same deviations would be obtained if the vector sum is reinterpreted in terms of an optimal observer adding a bias. In this case, the same errors in head down would be obtained if a lower weight to the otolith estimate in upside-down compared with the weight to the otolith estimate in upright were assigned. This is the same as saying that the weight of the bias is the same for all roll-tilt angles and that the strength of its influence on the initial estimate gets bigger for large roll-tilt angles because the initial otolith estimate becomes less reliable. This alternative interpretation allows us to relate the ratio of the weights given to the up and down bias vectors to the ratio of otolith variability in upright and head down: σ_{oto}(head down)/σ_{oto}(upright) = |*w*_{E}|/|*w*_{A}|. The median of this ratio for the values they used to fit the static SVV data is 3.14 compared with the 2.12 that we found for our average data.

### Conclusions

Simulations combining the estimated direction of acceleration coming from a population of noisy otolith afferents with an internal bias reflecting prior knowledge about head roll led us to the following conclusions. *1*) Implementing the general features of otolith afferents described in the literature (larger representation of the head cardinal axes, larger number and stronger modulation of utricular versus saccular afferents, stronger response to excitation than inhibition) results in a m-shaped pattern in the variability of the otolith estimate of the direction of acceleration. *2*) Simulated variability matched our experimental findings better when allowing the effectiveness of the otolith estimator to decrease with increasing roll. We therefore suggest that the precision of SVV is limited by the effectiveness of the otolith sensors and by central computational mechanisms that are not optimally tuned for tilted roll angles.

## APPENDIX: COMBINATION OF VON MISES DISTRIBUTIONS

The SVV model approximates all the distributions on the circle with von Mises distributions (29)

They are characterized by two parameters κ and μ. It is useful to see this pair of parameters as defining a vector in polar coordinates *v⃗*(κ, μ) = κ(sinμ, cosμ). κ is related to the variance of the distribution, and μ defines its center. If the spread in degrees is σ, then κ ∼ (180/π)^{2}/σ^{2}.

For localized distributions on the circle, this is indistinguishable from a Gaussian distribution. However, the interaction of two such localized distributions cannot be well approximated by the interaction of two Gaussian distributions when the distance of the peaks is comparable to 180°, as is the case for the big range of body roll-tilts considered here. The combination of two such distributions is again a von Mises distribution. The parameters defining the product distribution come from the vector sum *v⃗*(κ_{1}, μ_{1}) + *v⃗*(κ_{2}, μ_{2}) ≡ *v⃗*(κ, μ). Explicitly (30) (31)

Each of the priors is therefore determined by a constant vector whose length is the inverse of the uncertainty. Assuming an unbiased otolith estimate, the otolith likelihood is represented by a vector parallel to the true vertical direction but with varying length, reflecting that the variance of the otolith estimate is tilt dependent.

## GRANTS

This work was supported by the Swiss National Science Foundation (3200B0-105434), the Betty and David Koetser Foundation for Brain Research, Zurich, Switzerland, and the Center of Integrative Human Physiology, University of Zurich, Switzerland.

## Acknowledgments

The authors thank A. Züger for technical assistance and J. Laurens for critically reading the manuscript.

- Copyright © 2009 the American Physiological Society

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