Using the subjective visual vertical task (SVV), previous investigations on the maintenance of visual orientation constancy during lateral tilt have found two opposite bias effects in different tilt ranges. The SVV typically shows accurate performance near upright but severe undercompensation at tilts beyond 60 deg (A-effect), frequently with slight overcompensation responses (E-effect) in between. Here we investigate whether a Bayesian spatial-perception model can account for this error pattern. The model interprets A- and E-effects as the drawback of a computational strategy, geared at maintaining visual stability with optimal precision at small tilt angles. In this study, we test whether these systematic errors can be seen as the consequence of a precision-accuracy trade-off when combining a veridical but noisy signal about eye orientation in space with the visual signal.

To do so, we used a psychometric approach to assess both precision and accuracy of the SVV in eight subjects laterally tilted at 9 different tilt angles (−120° to 120°). Results show that SVV accuracy and precision worsened with tilt angle, according to a pattern that could be fitted quite adequately by the Bayesian model. We conclude that spatial vision essentially follows the rules of Bayes' optimal observer theory.

^{1}of the SVV are compatible with optimal observer theory.

*a priori*basis.

*) be compensated by a neural signal (*

_{E}*) that equals the actual eye orientation in space (*

_{S}*E*). Thus, to obtain a proper compensatory signal, the observer must take account of both the orientation of the head in space (

_{S}*H*) and the orientation of the eye within the head (

_{S}*E*). If the corresponding central estimates (

_{H}*and*

_{H}*) were veridical and precise, the observer would obtain an unbiased and stable percept of line orientation in space (*

_{S}*). However, if*

_{S}*underestimates*

_{S}*H*, this would result in underestimation of eye-in-space angle

_{S}*E*, thus causing an A-effect. By contrast, underestimation of ocular torsion would give rise to overestimating

_{S}*E*, which would cause an E-effect. We now proceed to explain how such biases in

_{S}*and*

_{S}*may be the downside of a noise-coping strategy in handling the raw neural signals from which they are derived (*

_{H}*and*

_{S}*), even though the latter are assumed to be accurate, on average.*

_{S}*a priori*knowledge about head tilt and ocular torsion, expressed in the prior probability distributions, which represent the fact that head tilt and eye torsion are mostly small. To combine the likelihood function and prior distribution optimally, the observer relies on their product, called the posterior distribution. When the subject is tilted, the posterior peaks in-between the peaks of the prior and the likelihood, thus giving rise to systematic errors: both head tilt and ocular torsion are systematically underestimated. However, the posterior distributions are less affected by sensory noise than the likelihood functions, thus yielding a precision that exceeds the precision of the sensory signals (see width of orange sectors). Hence, using prior knowledge affects the head-in-space and eye-in-head tilt estimates in two ways: it biases estimates toward smaller angles (reduced accuracy) but brings down uncertainty caused by sensory noise (increased precision). This strategy, an accuracy-precision trade-off, is particularly useful for small tilt angles, which are most common in daily life. For a full mathematical treatment of the scheme in Figure 2, see Methods section Modeling.

*SD*: 31 ± 14 yrs.), provided written informed consent to participate in the experiments. Participants were free of known vestibular or other neurological disorders and had normal or corrected-to-normal visual acuity.

*H*in total darkness, with right-ear-down angles coded as positive. Rotation was performed at a constant angular velocity of 30°/s, which was reached within 1s using a peak acceleration of 50°/s

_{S}^{2}. After a 30-s waiting period that allowed canal effects to subside, subjects viewed the polarized luminous line with the appearance of an inverted exclamation mark (see Figure 1) for a brief period of 20 ms and indicated whether its orientation in space was clockwise (CW) or counterclockwise (CCW) from their perceived direction of gravity, using a toggle switch. Subsequently, a new trial followed with a different line orientation, picked randomly from a set of 11 line orientations (details follow below). This sequence was repeated until all line orientations had been tested, after which subjects were rotated back to upright and lights were turned on, during a 30s resting period. Positive and negative body tilt angles were alternated regularly. For the 0°-tilt condition, we added an equal number of catch trials, in which subjects were tilted to an angle that was picked randomly from the range of ±5°, using a sub-threshold rotation speed of 2°/s, so that they could not perform the task in body coordinates.

*H*= 0°), test orientations were presented at 0, ±3, ±6, ±9, ±12, and ±15° relative to this value. For upright, where performance was typically more precise, we used a narrower test range at 0, ±2, ±4, ±6, ±8, ±10°. Each set of line orientations was presented in 12 experimental runs in random order, yielding a total of 132 responses for each psychometric curve. For each subject, data were collected in a total of 5 sessions of approximately 45 min. each. Catch trial responses were excluded from further analysis.

_{S}*x*represents line orientation. The mean of the Gaussian,

*μ,*represents the subjective vertical in the SVV task. The width of the curve,

*σ,*serves as a measure for the subject's uncertainty in the SVV and is inversely related to precision. Parameter

*λ,*representing the lapse rate, accounts for stimulus-independent errors caused by subject lapses or mistakes, and was restricted to small values (

*λ*< 0.06). Fits were performed using Matlab 7.0 software (The MathWorks) with the routine “psignifit” (Wichmann & Hill, 2001b).

*E*represents the (physical) roll orientation of the eye (

_{H}*E*) with respect to the head (

*H*). Sensory signals are denoted by a

*hat*symbol (^), as in

*, reflecting the orientation of the eye in the head as measured by the sensors. The outcome of a Bayesian computation is denoted by a*

_{H}*tilde*symbol (∼), as in

*, which represents the optimal estimate of eye-in-head orientation according to sensory information and prior knowledge.*

_{H}*), measured by a variety of tilt sensors, is a noisy but unbiased representation of the physical head tilt angle (*

_{S}*H*). Thus, the sensory tilt signal varies in repeated trials at the same physical tilt angle but the expected value of

_{S}*is a veridical representation of the actual head tilt. Conversely, this means that the brain cannot be sure about the physical angle, based on the sensory signal, and needs a statistical approach to determine the best estimate of head tilt angle. The Bayesian model assumes that the brain is adapted to the noise properties of its sensors, which allows it to deduce the probability of each tilt angle based on the sensory evidence, known as the likelihood function*

_{S}*P*(

*∣*

_{S}*H*). When sensory noise increases, the likelihood function, which is modeled by a Gaussian centered on

_{S}*and with standard deviation*

_{S}*a*

_{0}reflects the noise at

*H*= 0° and

_{S}*a*

_{1}represents the proportional increase of noise with tilt angle (see square bottom panel in Figure 2). Note, however, that by setting the lower limit of parameter

*a*

_{1}to zero, the model did not force tilt-sensor precision to be dependent on head tilt.

*a priori*basis. In the model, this is expressed by the prior distribution

*P*(

*H*), which is modeled by a Gaussian with standard deviation

_{S}*σ*

_{ H S}, centered on zero head tilt (

*H*= 0°), reflecting the knowledge that small head tilts are more likely than large tilts. Multiplication of the likelihood and prior distributions yields the posterior probability distribution

_{S}*P*(

*H*∣

_{S}*) according to Bayes' rule:*

_{S}*P*(

*H*∣

_{S}*) =*

_{S}*k*·

*P*(

*∣*

_{S}*H*) ·

_{S}*P*(

*H*), in which

_{S}*k*serves a normalization purpose. The peak of the posterior probability function is in-between the peaks of the likelihood and prior distributions, depending on their relative widths (Carandini, 2006; Knill & Pouget, 2004; MacNeilage et al., 2007). In the model, the peak of the posterior (

_{S}) is used as the optimal estimate of head tilt angle.

*) of the actual torsional orientation of the eyes with respect to the head,*

_{H}*E*(see Figure 2). Following Palla et al. ( 2006) we approximated eye-torsion by:

_{H}*E*= −

_{H}*A*· sin(

*), in which*

_{S}*A*represents the maximum torsion amplitude and

_{ S}reflects the sensory head-tilt signal. The negative sign reflects the fact that the eyes counterrotate relative to the head. Information about torsional eye-in-head orientation (

*), whether based on an efference copy signal, or on proprioception, or both, is treated as a sensory signal, assumed to be accurate but contaminated by noise (*

_{H}*), by taking into account which orientations are most likely on an*

_{H}*a priori*basis. Here, prior knowledge entails that the eyes are mostly closely aligned with the head (i.e.

*E*∼ 0°). Sensory information about torsional eye position is represented by the likelihood function

_{H}*P*(

*∣*

_{H}*E*) and prior knowledge is represented by a Gaussian centered on 0° with standard deviation

_{H}*σ*

*. The peak of the posterior distribution is used as the optimal estimate of eye-in-head angle (*

_{E H}*).*

_{H}*) and head-in-space (*

_{H}*) are combined to obtain an optimal estimate of the orientation of the eye in space (*

_{S}*), which is then used as the compensating signal in the SVV task. The expected value of*

_{S}*in many repeated trials (*

_{S}*and*

_{S}*. As shown in 1, this results in the following relation:*

_{H}*in repeated trials (*

_{S}^{2}) is given by:

*), required in the SVV task, the central estimate of eye position in space*

_{S}*is added to the estimated retinal line orientation*

_{S}*, which is assumed to be unbiased (*

_{E}*=*

_{S}*+*

_{E}*). Thus, according to the model, the systematic errors in the SVV (*

_{S}*μ*

_{SVV}) are caused exclusively by bias in the eye-in-space estimate, as shown in the following relation:

*a*

_{0}and

*a*

_{1}(see Equation 2) and the head prior,

*σ*

*. Complete fitting of the second term would also involve 3 parameters (*

_{H S}*A*,

*σ*and

_{E H}*E*), representing the uncompensated magnitude of eye torsion based on the following consideration: If we assume that both the noise in the central estimate of eye torsion (

_{H}*σ*

*) are constant (i.e. independent of*

_{E H}*H*), the second term on the right-hand side of Equation 2 reduces to a scaled version of the actual eye torsion

_{S}*E*. In other words, uncompensated torsion is a scaled version of the actual torsion, with the same sinusoidal tilt relation:

_{H}*E*represents the uncompensated part of the eye-in-head amplitude (

_{H}*A*) and

*r*

^{2}reflects the ratio of the variances of the eye-torsion prior and sensory eye-in-head signal (

*σ*

_{ E H}

^{2}/

^{2}). Thus, the narrower the prior relative to the torsion noise distribution, the larger the uncompensated torsion Δ

*E*

_{ H}.

*σ*

_{SVV}) is determined by a combination of head-in-space noise (

*a*

_{0},

*a*

_{1}and

*σ*

*), using:*

_{H S}*σ*

*were somewhat overestimated, but, as we discuss later (see section Effect of simplifying assumptions in Discussion) the effect is probably minor and does not affect our overall conclusion. Note that the impact of eye-in-head noise on the accuracy of the eye-in-space estimate ( Equation 3) was taken into account.*

_{H S}*a*

_{0},

*a*

_{1},

*σ*

*, and Δ*

_{Hs}*E*) that determine the accuracy and the precision of the SVV at each head tilt angle (see Table 1).

_{H}Parameter | Definition | Equation |
---|---|---|

a _{0} [°] | Noise in sensory head-tilt signal ( σ H ^ s) at 0° head tilt | 2 |

a _{1} [°/°] | Increase of noise in sensory head-tilt signal ( σ H ^ s) with tilt angle | 2 |

σ [°]_{H s} | Width of head-tilt prior distribution | 3 and 4 |

Δ E [°]_{H} | Maximum amplitude of uncompensated ocular counterroll | 6 |

*μ*

_{SVV}) and SVV variability (

*σ*

_{SVV}) as a function of head tilt angle

*H*. We used a maximum-likelihood estimation (MLE) procedure to fit the model to the psychophysical responses. We obtained the best-fit values of the four parameters for each subject by minimizing the negative log-likelihood using the fmincon routine (Matlab 7.0; The MathWorks). The log-likelihood function

_{S}*L*(

*θ*) is defined as

*L*(

*θ*) =

*P*

_{ θ}[

*N*

_{ i}(

*CW*)∣

*θ*]), in which

*P*

_{ θ}[

*N*(

_{i}*CW*)∣

*θ*] represents the chance of obtaining

*N*(

_{i}*CW*), the number of ‘

*CW*’-responses at a particular combination of head tilt and line orientation, for a given parameter set

*θ*.

*P*

_{ θ}[

*N*(

_{i}*CW*)∣

*θ*] was computed by first calculating

*μ*

_{SVV}and

*σ*

_{SVV}at each tilt angle for a given parameter set, using Equations 5 and 7. The chance of obtaining a ‘

*CW*’-response (

*P*[

*CW*]) at a certain combination of tilt angle and line orientation was calculated using the normal cumulative distribution function. Moreover, since subjects may have made stimulus-independent lapses, we included a lapse rate (

*λ*) into the distribution function. For simplicity, the lapse rate in these model fits was set at a fixed value of 0.06 (Wichmann & Hill, 2001a). Subsequently, the chance of obtaining

*N*‘

_{i}*CW*’-responses (given 12 repetitions) was specified by the binomial distribution, B(12,

*P*[

*CW*]).

a _{0} [°] | a _{1} [°/°] | σ _{HS} | Δ E [°]_{H} | |
---|---|---|---|---|

JG | 3.6 ± 1.1 | 0.03 ± 0.01 | 8.5 ± 1.0 | 0 ± n/a* |

MV | 3.1 ± 0.4 | 0.03 ± 0.01 | 10.5 ± 1.0 | 0 ± n/a* |

SR | 1.5 ± 0.4 | 0.06 ± 0.02 | 10.7 ± 2.4 | 8.9 ± 2.7 |

PM | 3.3 ± 1.2 | 0.08 ± 0.02 | 15.1 ± 1.9 | 10.6 ± 2.7 |

DB | 2.2 ± 0.9 | 0.15 ± 0.03 | 50.0 ± 1.5 | 5.8 ± 2.8 |

MD | 3.0 ± 0.8 | 0.07 ± 0.02 | 12.3 ± 2.6 | 16.2 ± 3.4 |

FW | 1.5 ± 0.8 | 0.10 ± 0.03 | 21.5 ± 4.7 | 8.9 ± 2.1 |

RV | 4.0 ± 0.7 | 0.03 ± 0.01 | 11.7 ± 1.6 | 0 ± n/a* |

*H*) affected the SVV of a typical subject (RV). Each panel shows how the proportion of ‘

_{S}*CW*’ responses,

*P*(

*CW*), changed as line orientation in space was varied around perceived vertical. At each tilt angle, response rates range from 0 to 1, indicating that the stimulus sets were positioned correctly. In an ideal observer, all psychometric functions would resemble a step centered at zero. In fact, as body tilt increases, psychometric curves shift away from zero and become less steep as a sign that there is decay in both accuracy and precision. For example, an earth-vertical line (0°) is always perceived as “

*CW*from earth vertical” at −60° head tilt, whereas it is always perceived as “

*CCW*from vertical” at 60° head tilt. For each tilt angle, we fitted the data with a cumulative Gaussian function (see Equation 1), which is characterized by three parameters: mean (

*μ*),

*SD*(

*σ*) and lapse rate (

*λ*). We took

*μ*as a measure for accuracy and used 1/

*σ*

^{2}as a measure for the precision of the verticality percept. When precision improves,

*σ*becomes smaller and hence the psychometric curve becomes steeper.

*H*= 0°), the percept of visual verticality is virtually unbiased and relatively precise compared to the other tested tilt angles. In the top panel (

_{S}*H*= −120°), the mean of the psychometric curve is at

_{S}*μ*= −38.2°, which means that the line must be tilted away from true vertical by this angle to be perceived as vertical in space, an expression of the A-effect (Aubert, 1861). In the bottom panel (

*H*= +120°), the curve is centered at

_{S}*μ*= +28.7°, which again reflects an A-effect. To appreciate the deterioration in precision, notice that the curve is steepest at 0° roll tilt (

*σ*= 2.3°) and that

*σ*increases at larger tilt angles, reaching maximum values of 5.8° and 5.1° at

*H*= +90° and

_{S}*H*= −90°, respectively.

_{S}*μ*-values from all subjects to illustrate how the accuracy of the verticality percept changes as a function of tilt angle. Model fits through the data will be discussed below (see section Model fit results). With one notable exception (DB), all subjects show variations of the response pattern known from the literature (De Vrijer et al., 2008; Mittelstaedt, 1983; Udo de Haes, 1970; Van Beuzekom et al., 2001; Van Beuzekom & Van Gisbergen, 2000), with less consistent systematic errors at small tilts and gradually decreasing accuracy, in the form of increasing A-effects, at larger tilt angles. Furthermore, several subjects show E-effects at the intermediate tilt angles, ranging up to −13.2° at 60° roll tilt for subject FW.

*σ*of the fitted psychometric curves as a function of tilt angle for all subjects. Again, model fits will be discussed in the section Model fit results. Invariably, precision is best at 0° tilt and deteriorates with tilt angle (one-way ANOVA;

*F*(8,63) = 5.3,

*P*< 0.001). Values for

*σ*range from ∼2° at zero tilt to a maximum of about 7° (PM) at the largest roll tilt angles. These findings are consistent with anecdotal reports from several subjects that judging the visual vertical was more difficult at the largest tilt angles.

*σ*steeply increases between 0° and ±30° tilt which is then followed by more gradual increments, resulting in the highest

*σ*values at ±120° roll tilt.

*σ*levels, which are coupled in the Bayesian model, were fitted simultaneously. Figure 4 illustrates the fit results of the Bayesian model (dashed lines) in terms of the systematic errors in the SVV. For most subjects, the model fits the systematic error data quite accurately, with

*R*

^{2}-values ≥0.80. Due to the fact that DB has a very unusual error pattern, with only small negative errors at even the largest tilt angles, this fit is considerably worse (

*R*

^{2}< 0)

^{2}. Note that

*R*

^{2}-values are provided merely to show how well the model accounts for the systematic errors, but do not reflect the overall goodness-of-fit, since

*σ*-levels are equally important. Since the Bayesian model attributes systematic SVV errors to a combination of errors in the head-in-space estimate (A-effects) and in the eye-in-head estimate (E-effects), see Equation 5, we also depicted these opposite contributions separately (red and blue line, respectively). In the three subjects without E-effects (JG, MV, and RV), eye-in-head errors are absent (i.e. Δ

*E*= 0°), as illustrated by the blue lines through the abscissa (0°). For the other subjects, the fits indicate the degree of undercompensation for ocular counterroll, reflected by the sinusoidal function. Additional fits of a reduced model that lacked uncompensated ocular counterroll, showed that model fits of JG, MV, and RV did not change. The fits of the five subjects with E-effects worsened significantly (likelihood ratio test,

_{H}*P*≪ 0.01) and parameter a

_{0}became unrealistically small (0°). Precision fits, shown in Figure 5, are equally relevant for a complete evaluation of the model. In most subjects (except DB and FW), model fits and actual data show the same trends. Fits show an increase of

*σ*

_{SVV}with tilt angle, which is similar to the actual increase observed in the data. Responses from subject DB were rather atypical, also in repeated testing, and therefore difficult to interpret. The overestimation of

*σ*

_{SVV}in subject FW appears related to the fact that the systematic error pattern shows increased accuracy at the most negative tilt angle (

*H*= −120°, see Figure 4). The model has no solution to account for this observation other than by increasing the value of

_{S}*σ*

_{SVV}. We confirmed this by performing separate fits at positive and negative tilts for subject FW. This resulted in minor differences with regard to the accuracy fits, but strongly affected precision levels: at negative tilts,

*σ*

_{SVV}levels were still overestimated, but at positive tilts, the fit improved greatly. This example illustrates how overestimation of

*σ*

_{SVV}may be directly related to small discrepancies in the systematic errors of model and data. Moreover, small asymmetries that are present in each observer (see e.g. the

*CW*-shift of subject FW in Figure 3) may also affect the fits, because the present model cannot account for such asymmetry. A possible solution would be to allow a shift of the prior on head-in-space, which could be interpreted as a shift in the internal reference frame of the observer.

*SD*levels are listed in Table 2. The best-fit values of parameter

*a*

_{1}are positive in all subjects (0.03–0.15°/°), which means that noise in the tilt sensors must increase with head-tilt angle if the model is to account for the SVV data. Values of a

_{0}(mean ±

*SD*= 2.8 ± 0.9°), reflecting the sensory head-tilt noise in the upright subject, range from 1.5° for subjects SR and FW, to 4.0° for RV. The width of the head-in-space prior distribution (

*σ*

*) ranges from 8.5° for subject JG to 21.5° for subject FW (the fit of subject DB reached the arbitrarily chosen limit value). This result indicates that prior knowledge about head tilt has a stronger influence in subject JG than in subject FW. The effect of the width of the head-in-space prior (*

_{Hs}*σ*

*) is best illustrated by comparing subjects JG and MV, where this is the only strikingly different parameter. The prior is narrower in JG than in MV (8.5° vs. 10.5°), which explains why his A-effects are larger. The amplitude of uncompensated eye counterroll (Δ*

_{Hs}*E*) is significantly larger than zero for the five subjects in which we observed E-effects (SR, PM, DB, MD, and FW). This parameter, which accounts for systematic errors of tilt overcompensation (E-effects), ranges from 8.9° for subject SR to 16.2° for subject MD. In the other three subjects, Δ

_{H}*E*was zero due to the absence of any E-effects, as shown by Figure 4. A further evaluation of parameter variations among all subjects would be contentious, since parameters

_{H}*a*

_{0},

*a*

_{1}, and

*σ*

*have a combined effect on the systematic errors and uncertainty levels and thus cannot be compared in isolation.*

_{Hs}*a priori*basis, thereby providing a unified explanation of both A- and E-effects.

*a priori*assumption that the eyes are generally nearly aligned with gravity—would only explain A-effects.

- OCR is not taken into account in spatial perception or
- the brain perfectly compensates for the effects of OCR.

*r*> 0.85), with slopes varying between 0.57 and 1.51. A further experiment, rotating upright subjects in yaw (Goonetilleke, Mezey, Burgess, & Curthoys, 2008), which induces ocular counterroll but no tilt perception, also revealed a clear correlation between ocular torsion and visual verticality perception (

*r*between 0.4 and 0.8). However, the slope was not unitary, indicating that there was some level of compensation by the visual system. Pavlou, Wijnberg, Faldon, and Bronstein ( 2003) performing a similar experiment, found clear effects on the SVV, suggesting that approximately 76% of the torsional eye position change was uncompensated and thus affected the SVV. A similar observation was made by de Graaf et al. ( 1992) in roll-tilted subjects, but only in subjects with persistent E-effects. However, conclusions by Mast ( 2000) point in a different direction. In this study, SVV results were found to dissociate from ocular torsional changes induced by centrifugation or barbecue rotation. The Bayesian model provides a rational explanation for the variable results of these previous studies, by suggesting the possibility that OCR may only be

*partially*taken into account during visual verticality perception.

*E*= 5°, for example, had an OCR amplitude of 10° and 50% compensation, or an amplitude of 5° and 0% compensation. All we can do is to regard Δ

_{H}*E*as the

_{H}*minimum*OCR amplitude. This implies that, according to the model, the eyes of subject MD counterrolled by at least 16.5°, whereas JG, MV, and RV may not have had any OCR at all (which is rather improbable). Clearly, direct measurements of ocular counterroll in our study would have helped in clarifying this issue, but these were beyond the scope of the study. Another possibility is that the subjects without E-effects had quite normal OCR amplitudes, but compensated perfectly. In the literature, various peak amplitudes of ocular counterroll during static and very slow (quasi-static) tilts have been reported. Population averages roughly vary between 6 and 10° in normal subjects (Diamond & Markham, 1983; Diamond, Markham, Simpson, & Curthoys, 1979; Kingma, Stegeman, & Vogels, 1997; Palla et al., 2006). However, most studies also reported large differences among subjects and Diamond and Markham even found an amplitude range of 2 to 20 degrees during slow (3°/s) dynamic tilts (Diamond and Markham, personal communication, June 11, 2008). Whether the high inter-subject variability in OCR explains the equally variable E-effect, can only be assessed by simultaneous measurement of both variables.

*a*

_{0},

*a*

_{1},

*σ*

*) also partially reflected the contributions of these additional noise sources (see Methods). We performed several simulations with the Bayesian model to test how large these effects may have been in a worst-case scenario. To do so, we created data through forward simulations of the complete Bayesian model (without simplifications) using the best-fit parameter values of a single subject (PM, see Table 2) combined with a set of values for visual noise (*

_{Hs}*σ*

*) and eye-in-head prior width (*

_{E H}_{0}, reflecting the offset of tilt noise, was affected most (changing from 3.3 to 6.5°), whereas the other parameters showed only minor changes. We conclude that these simplifying assumptions (see Methods section SVV precision) were warranted and that conclusions remain unchanged.

Signal | 0° tilt | 90° tilt | Evidence | References | |
---|---|---|---|---|---|

Visual, measured | ( σ L ^ E) | 1° | 1° | data | Vandenbussche et al., 1986 |

SVV, measured | ( σ _{SVV}) | 2.0 ± 0.6° | 5.0 ± 1.5° | data | present study |

SBT, measured | ( σ _{SBT}) | 4.5 ± 1.0° | 10.5 ± 3.4° | data | 0° tilt: unpublished own data 90° tilt: Mast and Jarchow, 1996 |

SBT, predicted | ( σ H ^ S) | 2.8 ± 0.9° | 7.7 ± 1.9° | fit result | present study |

*SD*values of 2.0° at upright and 5.0° at 90° tilt. Can this tilt dependency and the overall decrease of precision be ascribed to the precision characteristics of the compensatory head-tilt and eye-torsion signals? A measure for the precision of the head-tilt signal comes from a study by Mast and Jarchow ( 1996), who tested subjective body tilt (SBT) in human subjects and found that the average

*SD*of body tilt settings was 10.5 ± 3.4° at 90° body tilt. Unpublished psychometric SBT data from our laboratory show a somewhat lower average

*SD*level of ∼8° at 90° body tilt and an

*SD*of ∼4.5° near upright. It is interesting to compare these experimental data with the head-tilt noise fit results derived from the present experiments. As can be seen by comparing rows 3 and 4 in Table 3, the model prediction based on the population averaged parameters a

_{0}and a

_{1}, amounting to an increase from 2.8° at upright to 7.7° at 90° tilt, shows the same trend as the experimentally obtained values in perceived body-tilt experiments. Taken together with the scatter fits in Figure 5, these findings strongly support the model assumption ( Equation 2) that noise in the head tilt signal increases with tilt angle. Other studies provide indirect support for this notion. For example, the perturbing effect of roll-optokinetic stimulation on the SVV (Dichgans, Diener, & Brandt, 1974) and on body tilt estimates (Young, Oman, & Dichgans, 1975) becomes more pronounced at larger tilt angles. Similarly, after prolonged roll rotations, the SVV is more strongly affected by residual semicircular canal signals at larger tilt angles (Lorincz & Hess, 2008). Diamond and Markham ( 1983) showed that variability in OCR, which is thought to be mediated by the utricles (Suzuki, Tokumasu, & Goto, 1969), increases with tilt angle during dynamic tilting. Likewise, Tarnutzer, Bockisch, and Straumann ( 2007) observed that both SVV and OCR variability increased with tilt angle.

*), for given sensory signal*

_{S}*and prior information, is obtained by applying Bayes' rule, and is defined by:*

_{S}*μ*) of

*in many repeated trials is then specified by:*

_{S}*determines the variance of*

_{S}*according to:*

_{S}*):*

_{H}*is equal to the real head-in-space angle (*

_{S}*H*). For the variance in the eye-in-head estimate we deduce,

_{S}^{1}The term

*accuracy*refers to constant errors (bias) in the response. Precision is linked to variable errors, which reflect noise in the system (Howard, 1982).