Bayesian estimation of sensory priors in human spatial orientation

Abstract

Perception of spatial orientation relies on integration of sensory information encoding body position with external cues, such as visual inputs from the environment. Inherent variability in sensory inputs plays a critical role in how sensory information is processed. Unlike physical stimuli, neural encoding and transformation of these signals is subject to noise introduced at multiple levels within the sensory pathway. The Bayesian observer model provides a robust computational framework for understanding how the brain accommodates uncertainty to generate earth-vertical perception. Within the Bayesian framework, Gaussian prior distributions serve to constrain noisy sensory signals, allowing the brain to generate reliable estimates even in the presence of ambiguous input. These sensory priors are typically assumed to be centred at zero based on the premise that the brain defaults to upright body orientation when interpreting sensory signals. In this study, we compute the sensory priors associated with the eye and head orientation as two key sensory contributors to spatial orientation. These priors were estimated as free parameters within the Bayesian spatial model by combining measurements of subjective visual vertical (SVV) with perceived head and eye orientations. Our findings reveal that the Bayesian framework can effectively identify the sensory priors for eye and head orientation that contribute to spatial orientation. The results demonstrate the potential of using sensory priors as individualized functional markers for the neural mechanisms underlying spatial orientation.

Graphical Abstract

NEW & NOTEWORTHY

By combining measurements of subjective visual vertical (SVV) with perceived head and eye orientations, we applied the Bayesian observer model to derive the sensory priors for eye and head orientation. Our findings reveal that the Bayesian spatial model can effectively identify the sensory priors of eye and head orientation that contribute to spatial orientation. This approach demonstrates the potential of sensory priors as individualized functional markers for the neural mechanisms underlying spatial orientation.

INTRODUCTION

Perception of spatial orientation relies on integration of sensory information encoding body position with external cues, such as visual inputs from the environment (1). This integration process is critical, as the brain must constantly update its neural representations to accommodate changes in head, eye, and body position, ensuring a stable sense of spatial orientation. The Bayesian observer model offers a robust computational framework for studying this integration process (2–4). This model is based on the optimal combination of inherently noisy sensory signals, taking into account their uncertainty level. In the Bayesian framework, incoming sensory signals encoding eye, head, and body positions are combined with prior knowledge to generate neural estimates that are subsequently integrated for the perception of spatial orientation. Within this framework, sensory priors are key measures of neural mechanisms that compensate for sensory noise, ensuring that spatial orientation remains stable even as sensory inputs fluctuate.

A widely used method to assess spatial orientation is the subjective visual vertical (SVV) task, in which observers indicate the perceived alignment of a visual line with the earth vertical (1,5). In the upright position, where the body orientation is aligned with gravity, SVV errors are typically small, and the perceived body orientation closely matches the actual body orientation. When the body is tilted relative to gravity, the ocular counter-roll (OCR) response changes the orientation of the eye within the head, compensating for the head tilt but with a relatively low gain of 10–25% (6,7). This incomplete compensation results in a misalignment between the head and eye reference frames, causing a change in the orientation of the visual line on the retina due to the altered eye-in-head position. As a result, the brain must integrate information about both the head-in-space and eye-in-head positions to accurately estimate the visual line orientation relative to gravity. The pattern of SVV bias that emerges with head tilt, however, does not simply reflect changes in the eye and head position, but instead indicates more complex sensory processing to maintain orientation constancy. Specifically, at larger tilt angles, SVV errors tend to shift in the direction of the head tilt – known as the A-effect, while at smaller tilt angles they shift in the opposite direction – termed the E-effect (1,8–11).

Studies have shown that SVV biases across different body orientation are well described by Bayesian observer models (3,4,12–14). As illustrated in Figure 1, the sensory contributions within the Bayesian model incorporate the effect of sensory priors in generating the eye and head position estimates. These estimates contribute to the spatial bias measured in the SVV task (8,11). In this framework, the propagation of the sensory noise into spatial perception is reduced by relying on the knowledge about which head and eye positions are most likely on a priori basis. Accordingly, head-in-space (H_S) and eye-in-head (E_H) signals are combined with their respective sensory priors to generate internal estimates of H_S and E_H orientations. Because humans spend most time in the upright position, these priors are typically centered at upright, resulting in underestimation of actual H_S and E_H positions. Once H_S and E_H estimates are computed, their integration with visual inputs can reproduce characteristic SVV errors (3,4,12–14). At large tilt angles, underestimation of H_S explains the A-effect while at small tilt angles, underestimation of E_H accounts for the E-effect.

Figure 1.

Schematic representation of the sensory integration model with sensory measurements including SVV task, Head-in-space visual (Hs-visual) task, and Head-in-space haptic (Hs-haptic) task. ${\text{H}}_{\text{S}}$ and ${\text{E}}_{\text{H}}$ represent the physical orientation of the head in space (blue lines) and the actual torsional orientation of the eyes with respect to the head (red lines). Sensory inputs, denoted by a hat symbol (^), are modeled as a Gaussian with standard deviation reflecting the sensory noise. The optimal estimates of head in space (${\tilde{\text{H}}}_{\text{S}}$) and eye in head (${\tilde{\text{E}}}_{\text{H}}$) result from Bayesian computation that integrates sensory inputs with prior knowledge (green lines). In Hs-visual task, the estimate of head position is determined by integrating the head-in-space (${\widehat{\text{H}}}_{\text{S}}$), eye-in-head (${\widehat{\text{E}}}_{\text{H}}$), and retinal (${\tilde{\text{L}}}_{\text{E}}$) inputs. In the Hs-haptic task, the estimate of head position is determined by the head-in-space input (${\widehat{\text{H}}}_{\text{S}}$) and a bias (Hb) caused by the hand sensory input. ${\tilde{\text{H}}}_{\text{S}}$ and ${\tilde{\text{E}}}_{\text{H}}$ are combined to derive eye orientation in space (${\tilde{\text{E}}}_{\text{S}}$) which is then combined with retinal signal (${\tilde{\text{L}}}_{\text{E}}$) to obtain an estimate of the orientation of the line with respect to gravity (SVV).

As part of the Bayesian framework, Gaussian prior distributions can effectively represent how the brain minimizes uncertainty and maintains perceptual stability for spatial orientation (15). These priors serve to constrain noisy sensory signals, allowing the brain to generate reliable estimates even in the presence of ambiguous inputs. Although these priors could, in principle, serve as functional markers of central neural processing, the Bayesian observer model has not yet been applied to empirically derive them. Instead, they are typically assumed to be centred at zero —reflecting the default assumption that upright is the most likely body orientation. The present study aimed to investigate these priors, with a specific focus on deriving H_S and E_H prior estimates that contribute to spatial orientation. If two individuals are exposed to the same ambiguous sensory input but arrive at different perceptual outcomes, examining their internal priors may help explain this divergence. Identifying these sensory priors holds substantial implications, as they offer insight into the neural mechanisms that transform sensory input into neural estimates that contribute to coherent perception of spatial orientation.

To determine the role of sensory priors, here we used two psychophysical tasks to assess perceived head tilt position through visual and haptic methods (Figure 1). In these tasks, the observer reports their perceived head tilt position using a visual line or a haptic bar stimulus (16–18). Because head tilt perception in the haptic task is not influenced by the eye-in-head position, the eye-in-head signal can be derived as the difference between the visual and haptic measurements (18,19). Additionally, here we directly measured the eye position during head tilt (i.e., ocular counter-roll or OCR) to compare with the estimated E_H value. Using these measurements, we applied the Bayesian observer model to derive the sensory priors for eye and head position in individual participants. Our findings reveal that the Bayesian framework can effectively identify the sensory priors of eye and head orientation that contribute to spatial orientation. This approach demonstrates the potential of sensory priors as functional markers for the neural mechanisms underlying spatial orientation.

MATERIALS AND METHODS

Participants

Nineteen right-handed healthy volunteers (10 male, 9 female), aged between 19 and 37 yrs. (mean ± SD age, 26.1 ± 5.9 years), participated in the experiments. They were all reported to be in good health without visual, vestibular, neurologic, or psychiatric illness following completion of a health screening questionnaire. The experiment protocols were reviewed and approved by the Johns Hopkins University institutional review board, and informed written consent was obtained from all participants.

Experiments

Written and verbal instructions were given at the beginning of each task. All participants were given a few practice trials to get used to the paradigm. We used SVV task to measure orientation of a visual line relative to gravity and visual (Hs-visual) and haptic (Hs-haptic) tasks to measure perceived head tilt position in space (Figure 2A). In all tasks, the participant sat with their body upright and their head was either upright or tilted 20° laterally to the left or right shoulder. Each task at each head position was run on a block of 70 trials (3 tasks × 3 head positions, 9 blocks total). SVV was always presented as the first task whereas the order between Hs-visual and Hs-haptic task was counterbalanced among participants. For each task, blocks of three different head positions always started from the upright position and alternated between left and right tilt randomly across participants. To avoid residual effects from the semicircular canal stimulation during head movement, all tasks with head tilt did not start until 30 s after each time the head was tilted to the left or right side. Between the recording blocks, lights were switched on and participants could move their head and relax. A block of 70 trials for the visual tasks lasted about 3 min, while a block of 70 trials for the haptic tasks lasted about 5 min. The entire experiments took about 90 min to complete, including the time for instructions, practice, and rest breaks between the blocks.

Figure 2.

(A) Subjective visual vertical (SVV) task, head-in-space visual (Hs-visual) task, and head-in-space haptic (Hs-haptic) task with the head tilted 20° to the right as an example. The green line represents the earth vertical axis and blue line represents the head axis. Top: the stimulus is a red radius line in the SVV task, a red diameter line in the Hs-visual task, and a haptic bar in the Hs-haptic task. Bottom: in the SVV task, the orientation of the stimulus was reported with respect to the perceived vertical orientation (dashed green line). In the Hs-visual and Hs-haptic tasks, the orientation of the stimulus was reported with respect to the perceived head orientation (dashed blue line). (B) Sample time course of 70 trials with responses from the paradigm used in three tasks. The y-axis shows the angle of the stimulus presented and each circle represents one trial. The left tilt responses are in blue and the right tilt responses in red. The stimulus angles were presented randomly within a range that started at 90° and then adjusted based on previous responses (gray shade). (C) An example of psychometric fit to the responses in the paradigm using a cumulative Gaussian function. The mean of the Gaussian (μ) provides a measure of point of subjective equality where the probabilities of right and left responses are equal. The width of the curve (σ, angle difference between the 50% and 84% points) was used as a measure for the variability.

Subjective visual tasks (SVV and Hs-visual).

In these tasks, participants clicked a left or right button using a controller bimanually to report whether the visual line was tilted to the left or right of the perceived vertical orientation (SVV task) or perceived head orientation (Hs-visual task). In darkness, the visual stimulus was a red line (width: 0.13° of visual angle) superimposed on a red fixation dot (diameter: 0.33°) at the eye level. The length of the line was 3.8° (9 cm, radius from the fixation dot) for SVV task or 7.6° (18 cm, diameter through the fixation dot) for Hs-visual task. The line orientation was randomly selected within a range that was adjusted in sub-blocks of 10 trials (see details under adaptive psychophysical paradigm). In each trial, the fixation dot appeared first for 1 s, after which the visual line was presented for a minimum of 300 ms and maximum of 1.5 s until the participant responded. If the participant did not respond within 1.5 s, the trial was aborted and the next trial started after a button click. In such cases, the missed angle was presented again later within the same sub-block.

Subjective haptic tasks (Hs-haptic).

In these tasks, participants touched the two edges of a haptic bar with a pinch grip using the thumb and index fingertips of their right hand. With the left hand, they reported whether the stimulus was tilted to the left or right of perceived head orientation by clicking the left or right button of a computer mouse. In darkness, the haptic bar was rotated to specific orientations at each trial and an auditory signal (a double beep at 1000 Hz) indicated that participants can start touching the bar. The bar orientation was randomly selected within a range that was adjusted in sub-blocks of 10 trials. As soon as participants provided a response after retracting their hand from the haptic bar, the next trial began with another auditory signal (a single beep at 1000 Hz) to indicate that the motor was starting to move. There was a window of 4 s to feel the stimulus and respond at each trial. If the participant did not respond within this period, the trial was aborted and the next trial started after a button was clicked. In such cases, the missed angle was presented again later within the same sub-block.

Setup

In all experiments, the participant sat with their body upright in a completely dark room that was designed to perform experiments in the dark with all walls, floor, and ceiling in black color and light sealed. The head was immobilized using a molded bite bar, which was mounted on a rotatory motor so that the head tilt position could be set accurately by remotely controlled rotation. Participants were monitored remotely through an infrared camera to ensure that they remained on the bite bar and were following instructions during the experiment.

In the visual experiments (SVV and Hs-visual), the line stimulus appeared at the center of a CRT monitor (1280 by 1024 pixels) that was covered by black cardboard with a circular opening (diameter, 28 cm). To further eliminate any possible visual cues from the CRT monitor, the brightness and contrast levels of the screen were set to the minimum at a luminance of < 0.5 cd/m². The CRT monitor was positioned 135 cm away in front of the participant and its height was adjusted to align the center of the monitor with the eye level for each participant.

During the SVV task, torsional eye position was recorded simultaneously using RealEyes xDVR system (Micromedical Technologies Inc.) and a custom software. This system uses two cameras (Firefly MV, PointGrey Research Inc., Richmond, BC, Canada) mounted on goggles to capture infrared images of each eye. To measure and track torsional eye position, we used a method based on iris recognition, which operates binocularly in real time at 100 Hz with a noise level less than 0.1° (20).

In the haptic experiments (Hs-haptic), the haptic stimulus was a metal bar that consisted of two pieces placed 30 cm in front of the participant’s right hand. The center of the bar was aligned with the participant’s hand so that they could comfortably extend the forearm to touch the bar while resting the elbow on a flat armrest. This haptic stimulus was described in detail in previous studies (18,19). The haptic bar was 26 cm long, and each piece had an edge of 1.5 mm wide and 1.4 cm thick with a gap of 7 mm between the two pieces. The center of the bar was mounted on a stepper motor with a holding torque of 0.26 Nm and a step size of 1.8° (Stepperonline). An MPU-9250 accelerometer sensor was attached to the haptic bar. Both the stepper motor and accelerometer sensor were connected to an Arduino Uno board and controlled by an Arduino–MATLAB interface (Arduino). The stepper motor allowed the bar to rotate in the roll plane. The accelerometer recorded the angle of the haptic stimulus, relative to gravitational vertical with a ± 0.1° accuracy. The stepper motor moved to adjust the haptic stimulus based on the difference between the accelerometer reading of the angle and the target angle given by the paradigm.

Adaptive Psychophysical Paradigm

The orientation of the stimulus at each trial in all tasks (i.e., visual line or haptic bar) was determined by a custom adaptive two-alternative forced-choice (2AFC) paradigm, written in MATLAB using Psychtoolbox (21). This method is described in detail in previous studies (10,18,19), and also illustrated in Figure 2B. There was a total of 70 trials in each task. At the beginning of the paradigm for the first 10 trials, the angles were selected randomly from the range of 90° (at intervals 0, ±40% and ±80% of the range) and centered around a random angle within ±7.5° of a reference orientation (0° for vertical task and head tilt angle for head orientation tasks). In subsequent trials, the center and range of probing angles were adjusted every 10 trials by calculating a new center derived from the responses in previous trials (i.e., point of subjective equivalence; see data analysis for calculation). The width of the range decreased to 45°, 30°, 22°, 12° and 10° until the paradigm reached the 60 trials, and it remained at 10° for the rest of the trials.

Data Analysis

The clockwise tilt angles of the head and the stimulus were defined positive. For each task and participant, the perceptual measure was quantified as the percentage of responses recorded to the right of the stimulus orientation (Figure 2C). A psychometric curve was then fitted to these responses using a cumulative Gaussian function and a generalized linear regression model (MATLAB fitglm with probit link function). The mean of the Gaussian (μ) provides a measure of point of subjective equality (PSE) that is the orientation at which the visual line is perceived to be vertical (in the SVV task), or the visual line or haptic bar is perceived to be aligned with head orientation (in the Hs-visual and Hs-haptic tasks). The width of the curve (σ, angle difference between the 50% and 84% points) was used as a measure for the participant’s variability in the visual or haptic task and is inversely related to precision. Deviations of the PSE from 0° represent a biased estimate of verticality in the SVV task and deviations of the PSE from the head tilt angle represent a biased estimate of the head orientation in the Hs-visual and Hs-haptic tasks. Here we refer to these deviations as “errors”. Accordingly, positive errors represent clockwise bias relative to the reference orientation and negative errors represent leftward bias.

Torsional orientation of the eyes during the SVV task was measured as the median value of torsional eye position (average of both eyes) during the SVV task. For these torsion values, positive represents a clockwise roll and negative represents a counterclockwise roll.

Sensory Integration Model

The Bayesian observer model is presented using the following conventions: physical variables are denoted by a capital with a subscript denoting the reference frame. For example, ${\text{H}}_{\text{S}}$ represents the orientation of the head in space. Sensory signals are denoted by a hat symbol (^), as in ${\widehat{\text{H}}}_{\text{S}}$, reflecting the orientation of the head in space as measured by the sensors. The outcome of a Bayesian computation is denoted by a tilde symbol (~), as in ${\tilde{\text{H}}}_{\text{S}}$, which represents the optimal estimate of head-in-space orientation according to sensory information and prior knowledge. This model which is based on an earlier model (3) includes the head position prior with mean ${\text{H}}_{\text{SP}}$ and standard deviation ${\sigma }_{{\text{H}}_{\text{SP}}}$, and eye torsion prior with mean ${\text{E}}_{\text{HP}}$ and standard deviation ${\sigma }_{{\text{E}}_{\text{HP}}}$. As a result, the SVV error can be determined from the errors in the head-in-space estimate and eye-in-head estimate.

Head-in-space estimation

To obtain an estimate of the orientation of the head in space, the brain can use the integration of sensory information from tilt sensors and prior knowledge about head-in-space orientation. The sensory head tilt signal (${\widehat{\text{H}}}_{\text{S}}$) is represented by a Gaussian probability distribution with mean value (${\mu }_{{\widehat{\text{H}}}_{\text{S}}}$) and variance (${\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2$). The prior is represented by a Gaussian distribution that is centered at ${\text{H}}_{\text{SP}}$ with a variance of ${{\sigma }^2}_{{\text{H}}_{\text{SP}}}$. According to Bayes’ rule, the optimal estimate of head-in-space orientation (${\tilde{\text{H}}}_{\text{S}}$) is given by the peak of the posterior distribution resulting from the multiplication of the two Gaussian distributions (3,22,23). This results in the following relation:

$${\mu }_{{\tilde{\text{H}}}_{\text{S}}}=\frac{{\sigma }_{{\text{H}}_{\text{SP}}}^2}{{\sigma }_{{\text{H}}_{\text{SP}}}^2+{\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2}\bullet {\mu }_{{\widehat{\text{H}}}_{\text{S}}}+\frac{{\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2}{{\sigma }_{{\text{H}}_{\text{SP}}}^2+{\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2}\bullet {\text{H}}_{\text{SP}}$$ (1)

The variance in the head-in-space estimates can be derived from the variance in head prior and head sensory inputs as the following:

$${\sigma }_{{\tilde{\text{H}}}_{\text{S}}}^2={\left(\frac{{\sigma }_{{\text{H}}_{\text{SP}}}^2}{{\sigma }_{{\text{H}}_{\text{SP}}}^2+{\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2}\right)}^2\bullet {\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2$$ (2)

Eye-in-head estimation

Similar to the computation of the head-in-space estimate, an estimate of the torsional orientation of the eyes with respect to the head (${\tilde{\text{E}}}_{\text{H}}$) is provided by the integration of sensory information about torsional eye-in-head orientation and prior knowledge about eye torsion. We approximate this prior as a Gaussian distribution with a mean ${\text{E}}_{\text{HP}}$ and variance ${\sigma }_{{\text{E}}_{\text{HP}}}^2$. The sensory eye torsion signal (${\widehat{\text{E}}}_{\text{H}}$) based on proprioception and/or efference copy is represented by a Gaussian probability distribution with mean value (${\mu }_{{\widehat{\text{E}}}_{\text{H}}}$) and variance (${\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2$). The posterior distribution resulting from the multiplication of the two Gaussian distributions would then provide the estimate of eye-in-head orientation, which is given by the following:

$${\mu }_{{\tilde{\text{E}}}_{\text{H}}}=\frac{{\sigma }_{{\text{E}}_{\text{HP}}}^2}{{\sigma }_{{\text{E}}_{\text{HP}}}^2+{\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2}\bullet {\mu }_{{\widehat{\text{E}}}_{\text{H}}}+\frac{{\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2}{{\sigma }_{{\text{E}}_{\text{HP}}}^2+{\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2}\bullet {\text{E}}_{\text{HP}}$$ (3)

$${\sigma }_{{\tilde{\text{E}}}_{\text{H}}}^2={\left(\frac{{\sigma }_{{\text{E}}_{\text{HP}}}^2}{{\sigma }_{{\text{E}}_{\text{HP}}}^2+{\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2}\right)}^2\bullet {\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2$$ (4)

SVV accuracy

To compute the SVV, the brain requires estimates of head in space, eye in head and retinal line orientation. For simplicity, the estimated retinal line orientation is assumed to be unbiased as the retinal input is limited to the visual stimulus without other visual cues. This assumption is supported by previous studies showing that systematic errors in perceived earth vertical depend on the tilt angle of the observer, irrespective of the retinal orientation of the stimuli (5,24). Thus, according to the model, the SVV error can be determined as the difference between the actual head and eye positions and the eye and head estimates, as shown in the following relation:

$$\begin{gathered} {\mu }_{\text{SVV}}=\left({\text{H}}_{\text{S}}-{\mu }_{{\tilde{\text{H}}}_{\text{S}}}\right)+\left({\text{E}}_{\text{H}}-{\mu }_{{\tilde{\text{E}}}_{\text{H}}}\right) \\ ={(\text{H}}_{\text{S}}-\frac{{\sigma }_{{\text{H}}_{\text{SP}}}^2}{{\sigma }_{{\text{H}}_{\text{SP}}}^2+{\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2}\bullet {\mu }_{{\widehat{\text{H}}}_{\text{S}}}-\frac{{\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2}{{\sigma }_{{\text{H}}_{\text{SP}}}^2+{\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2}\bullet {\text{H}}_{\text{SP}})+{(\text{E}}_{\text{H}}-\frac{{\sigma }_{{\text{E}}_{\text{HP}}}^2}{{\sigma }_{{\text{E}}_{\text{HP}}}^2+{\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2}\bullet {\mu }_{{\widehat{\text{E}}}_{\text{H}}}-\frac{{\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2}{{\sigma }_{{\text{E}}_{\text{HP}}}^2+{\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2}\bullet {\text{E}}_{\text{HP}}) \end{gathered}$$ (5)

In Equation 5, ${\mu }_{\text{SVV}}$ represents SVV error, ${\text{H}}_{\text{S}}$ is the actual head-in-space position, ${\mu }_{{\tilde{\text{H}}}_{\text{S}}}$ is the estimated head-in-space position, ${\mu }_{{\widehat{\text{H}}}_{\text{S}}}$ is the sensory input encoding the head position, ${\text{E}}_{\text{H}}$ is the actual eye-in-head position or OCR that was directly measured, ${\mu }_{{\tilde{\text{E}}}_{\text{H}}}$ is the estimated eye-in-head position and ${\mu }_{{\widehat{\text{E}}}_{\text{H}}}$ is the sensory input encoding the eye-in-head position.

SVV precision

Based on the model, the standard deviation in SVV responses (${\sigma }_{\text{SVV}}$) can be determined by a combination of noise in the head-in-space (${\sigma }_{{\tilde{\text{H}}}_{\text{S}}}$) and eye-in-head estimates (${\sigma }_{{\tilde{\text{E}}}_{\text{H}}}$).

$${\sigma }_{\text{SVV}}^2\approx {\sigma }_{{\tilde{\text{H}}}_{\text{S}}}^2+{\sigma }_{{\tilde{\text{E}}}_{\text{H}}}^2={\left(\frac{{\sigma }_{{\text{H}}_{\text{SP}}}^2}{{\sigma }_{{\text{H}}_{\text{SP}}}^2+{\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2}\right)}^2\bullet {\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2+{\left(\frac{{\sigma }_{{\text{E}}_{\text{HP}}}^2}{{\sigma }_{{\text{E}}_{\text{HP}}}^2+{\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2}\right)}^2\bullet {\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2$$ (6)

Equation 6 is the summation of equation 2 and 4, in which ${\sigma }_{\text{SVV}}$ represents SVV standard deviation.

Model Fits

The Bayesian model makes predictions of SVV errors and variability as a function of head and eye position. Here we measured the inputs to the model using the Hs-visual and Hs-haptic tasks. In the Hs-visual task, the error in the head orientation (ΔHS_V) can be determined by the errors in the head-in-space and eye-in-head inputs (${\mu }_{{\widehat{\text{H}}}_{\text{S}}}$ & ${\mu }_{{\widehat{\text{E}}}_{\text{H}}}$) (Equation 7). Here the ${\text{H}}_{\text{S}}$ is the actual head tilt position and ${\text{E}}_{\text{H}}$ is the actual eye position directly measured as OCR.

$$\Delta {\text{HS}}_{\text{V}}={(\mu }_{{\widehat{\text{H}}}_{\text{S}}}{-\text{H}}_{\text{S}})+{(\mu }_{{\widehat{\text{E}}}_{\text{H}}}-{\text{E}}_{\text{H}})$$ (7)

As shown in previous studies, Hs-haptic task can be influenced by a fixed bias from the hand sensory input (Hb). Thus, the error in the head orientation (ΔHS_H) can be determined by the error in the head-in-space input (${\mu }_{{\widehat{\text{H}}}_{\text{S}}}$) and Hb (Equation 8) (18,19). Here we accounted for Hb by subtracting Hs-visual from Hs-haptic measurements with the head upright in the same participant (Equation 9). This is because the Hs-visual measurement with the head upright is minimally affected by the eye position (i.e., eye and head orientation are both aligned) while the effect of hand bias is still present in the Hs-haptic measurement with the head upright.

$$\Delta {\text{HS}}_{\text{H}}={(\mu }_{{\widehat{\text{H}}}_{\text{S}}}{-\text{H}}_{\text{S}})+\text{Hb}$$ (8)

$$\text{Hb}=\Delta {\text{HS}}_{\text{H}}\text{in the upright}-\Delta {\text{HS}}_{\text{V}}\text{in the upright}$$ (9)

From equation 8, we can derive the value of ${\widehat{\text{H}}}_{\text{S}}\left({\mu }_{\widehat{{\text{H}}_{\text{S}}}}\right)$ (Equation 10). The value of ${\widehat{\text{E}}}_{\text{H}}\left({\mu }_{{\widehat{\text{E}}}_{\text{H}}}\right)$ can be derived from equations 7 and 8 (Equation 11).

$${\mu }_{{\widehat{\text{H}}}_{\text{S}}}={\Delta \text{HS}}_{\text{H}}+{\text{H}}_{\text{S}}-\text{Hb}$$ (10)

$${\mu }_{{\widehat{\text{E}}}_{\text{H}}}=\Delta {\text{HS}}_{\text{V}}+{\text{E}}_{\text{H}}-\Delta {\text{HS}}_{\text{H}}+\text{Hb}$$ (11)

The variance of eye position signal (${\sigma }_{{\widehat{\text{E}}}_{\text{H}}}^2$) can be determined from the variance in the visual task measurements (${\sigma }_{{\text{HS}}_{\text{V}}}^2$), while the variance of head position signal (${\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2$) can be determined from the variance in the haptic task measurements (${\sigma }_{{\text{HS}}_{\text{H}}}^2$).

All the parameters above were used to fit the model to SVV accuracy (Equation 5) and SVV precision (Equation 6) measured at different head tilt positions (upright and ±20°) in 19 participants. A total of four free parameters were used for fitting the model (${\text{H}}_{\text{SP}},{\sigma }_{{\text{H}}_{\text{SP}}},{\text{E}}_{\text{HP}}$, and ${\sigma }_{{\text{E}}_{\text{HP}}}$). We obtained the best-fit values of these parameters for each participant by minimizing the negative log-likelihood of the data using the Matlab fmincon function (3). The log-likelihood function $\text{L}\left(\theta \right)$ is defined as $\text{L}\left(\theta \right)={\sum }_{i=1}^n\text{log}\left({\text{P}}_{\theta }\right[{\text{N}}_i\left(\text{CW}\right)\left|\theta \right])$, in which ${\text{P}}_{\theta }\left[{\text{N}}_i\right(\text{CW}\left)\right|\theta ]$ represents the chance of obtaining ${\text{N}}_i\left(\text{CW}\right)$, the number of ‘CW’-responses at a particular combination of head tilt and line orientation, for a given parameter set $\theta$. ${\text{P}}_{\theta }\left[{\text{N}}_i\right(\text{CW}\left)\right|\theta ]$ was computed by first calculating ${\mu }_{\text{SVV}}$ and ${\sigma }_{\text{SVV}}$ at each tilt angle for a given parameter set, using Equations 5 and 6. 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. Subsequently, the chance of obtaining ${\text{N}}_i$ ‘CW’-responses (# repetitions at a given line orientation) was specified by the binomial distribution, $\text{B}(\#,\text{P}[\text{CW}\left]\right)$. We used globalsearch solver to make sure that the fitting procedure found a global minimum rather than a local minimum.

RESULTS

Here we used the Bayesian observer model to derive eye and head estimates that contribute to the subjective visual vertical (SVV). We first describe the data and then provide the model fit results.

Psychometric and OCR Measurement Results

The quantitative assessment of spatial orientation was done in the SVV task in which participants judged the orientations of a visual line with respect to gravity. The perception of head tilt position was also measured in all participants using the Hs-visual and Hs-haptic tasks. The results of a representative participant are shown in Fig. 3 for all three tasks at three head positions. The top row shows the OCR measurement during the SVV task, with the values of 6.3°, 0.9° and −3.5° for the 20° left, upright, and 20° right head tilt positions respectively. These values are directed opposite to the head tilt, and partially compensate for the amount of head tilt. The second to bottom rows show the proportion of rightward responses in the SVV, Hs-visual and Hs-haptic tasks. For each head tilt position, a cumulative Gaussian function was fitted to the data from each task to obtain the mean (μ) and SD (σ) values. The μ (red line) indicates the stimulus orientation is perceived equally tilted to the right and left of the reference orientation (green line, 0° in the SVV task and head tilt angle in Hs-visual and Hs-haptic tasks). The σ (width of the psychometric curve) is a measure of variability and is inversely related to precision. In the SVV task (second row), the perceived vertical is near zero (μ = −1.5°) when the head is upright but shifted in the direction opposite to the head tilt (i.e., E effect) when the head is tilted to ±20° (μ = 1.3° for left tilt and μ = −5.5° for right tilt). In the Hs-visual task (third row), the perceived head angle is close to zero (μ = −2.4°) in the upright head position but shifted in the direction of head tilt (μ = −48.5° for left tilt and μ = 45.2° for right tilt). In the Hs-haptic task (bottom row), the perceived head position shows the effect of hand bias across the three head positions (μ = 8.6° for 0°, μ = −12.0° for −20°, and μ = 30.1° for 20°), consistent with previous reports of hand bias in haptic tasks (18,19).

Figure 3.

The results of a representative subject in all tasks for three head tilt positions (left column, head tilted 20° to the left; middle column, head upright; right column, head tilted 20° to the right). Top row, median OCR is plotted against stimulus orientation during the SVV task. The OCR is in the opposite direction of the head tilt (positive torsion represents clockwise direction). Second to bottom row shows the proportion of right responses (gray dots) and cumulative Gaussian fits (black lines, characterized by mean μ and standard deviation σ) plotted against stimulus orientation during SVV, Hs-visual and Hs-haptic tasks. Vertical green lines denote the references for different tasks, that is vertical in the SVV task and head orientation in the Hs-visual and Hs-haptic tasks. Vertical red lines denote μ, that is the orientation at which the stimulus is perceived to be vertical in the SVV task and the orientation at which the head was perceived in Hs-visual and Hs-haptic tasks.

The individual and mean results in the three tasks from 19 participants are shown in Fig. 4 and Table 1. The OCR value for all participants was in the opposite direction of the head tilt (Fig. 4A). The average OCR for the left, upright, and right head tilts were 5.1 ± 0.3° (Mean ± SEM), 0.0 ± 0.2°, and −4.7 ± 0.3°, which corresponds to the gains of ~25 ± 2% for 20° head tilt. There was no significant asymmetry between the OCR for the right and left head tilts (paired t-test p = 0.2). Fig. 4B–D show the errors for the SVV, Hs-visual and Hs-haptic tasks for three head positions. The average SVV errors were 1.5 ± 1.5° with the left head tilt, −0.2 ± 0.4° with the head upright, and −5.1 ± 1.1° with the right head tilt (Fig. 4B). Overall, the head position showed a significant effect on the SVV error (One-way repeated measures ANOVA; F (2,36) = 9.85, p < 0.001, η2ₚ = 0.35).

Figure 4.

OCR in the SVV task (A) along with errors in the SVV (B), Hs-visual (C) and Hs-haptic (D) tasks are shown when the head was upright (UP), tilted to the left (left ear down or LED) and tilted to the right (right ear down or RED). The grey lines represent individual subjects, and the black line represents the group mean. (E) The average error in the Hs-visual and Hs-haptic tasks after removing the hand bias ($\text{Hb}$) across head tilt positions. Because head tilt perception in the haptic task is not influenced by the eye-in-head position, the difference between these Hs-visual and Hs-Haptic can provide Eh signal. Positive OCR value represents clockwise counter-roll and positive errors in all three tasks represent rightward bias relative to the true reference orientation, and vice versa for the negative errors. (F) The average variability in the SVV, Hs-visual and Hs-haptic tasks across head tilt positions. Error bars show the SEM.

Table 1.

Error, variability, and OCR measured during SVV task and fit results of the Bayesian model at three head positions

Variables (Mean ± SEM)	Left tilt	Upright	Right tilt
SVV error	1.5 ± 1.5°	−0.2 ± 0.4°	−5.1 ± 1.1°
SVV variability	2.6 ± 0.4°	0.9 ± 0.2°	3.0 ± 0.5°
OCR	5.1 ± 0.3°	0.0 ± 0.2°	−4.7 ± 0.3°
SVV fit	0.4 ± 1.3°	−0.6 ± 0.4°	−3.2 ± 1.3°
SVV variability fit	4.5 ± 0.5°	3.1 ± 0.3°	4.0 ± 0.4°
${\text{E}}_{\text{H}}$ estimate	−3.5 ± 1.8°	0.0 ± 0.3°	4.9 ± 2.1°
${\text{H}}_{\text{S}}$ estimate	−11.8 ± 1.9°	0.6 ± 0.4°	13.7 ± 1.8°

The average Hs-visual errors were −14.7 ± 2.7° with the left head tilt, −0.4 ± 0.4° with the head upright, and 11.2 ± 3.4° with the right tilt (Fig. 4C and 4E). This is consistent with a bias in the perception of head tilt position in the same direction of head tilt in the visual task. The average Hs-haptic errors were 1.4 ± 2.6° with the left head tilt, 3.8 ± 1.1° with the head upright, and 5.7 ± 2.4° with the right head tilt (Fig. 4D). The hand bias in individual participant was quantified as the difference between Hs-haptic and Hs-visual measurements with the head upright (Equation 9). The average hand bias in the haptic task was 4.1 ± 1.2°. After removing the hand bias, the average Hs-haptic errors were −2.7 ± 2.5° with the left head tilt and 1.6 ± 2.7° with the right head tilt (Fig. 4E).

The average variability in the SVV, Hs-visual and Hs-haptic measurements for the three head positions are shown in Fig. 4F. There was a significant effect of task but no significant effects of head position or their interaction on the precision of perceptual measurements (Two-way repeated ANOVA; factor of task: F (1,18) = 395.71, p < 10⁻¹², η2ₚ = 0.96; factor of head position: F (1,18) = 0.015, p = 0.90, η2ₚ = 0.001; interaction of task and head position: F (1,18) = 0.02, p = 0.88, η2ₚ = 0.001). Multiple comparison showed that the SVV task was significantly more precise than the Hs-visual task (p = 0.008) and Hs-haptic task (p < 10⁻⁷), and Hs-visual task was more precise than Hs-haptic task (p = 0.007).

Model Fit Results

We used the Bayesian observer model to fit the data from individual participants. By fitting the model, we could quantify the accuracy and precision of the priors that contribute to the eye and head estimates for spatial orientation in individual participants. These prior values represent internal brain estimates that are used to modify sensory bias or noise by using prior information about the most likely eye and head positions. Figure 5 illustrates the fit results of the model to the SVV errors (gray line), and R²-values are provided to show how well the model fits for the SVV errors. The model fits the error data quite accurately with R² ≥ 0.94 in eleven participants; R² ranged between 0.56 and 0.77 in six participants; R² was negative in two participants (S10 and S14) due to the unusual error pattern.

Figure 5.

Model fitting and data for SVV errors in 19 individual subjects. The last plot shows the mean from all subjects with SEM as the shaded area. The blue dash line shows the error based on the head estimate (${\text{H}}_{\text{S}}$ estimate), the red dash line shows the error based on the eye estimate (${\text{E}}_{\text{H}}$ estimate), and the gray line shows the Bayesian model fit. Black filled circles show the actual SVV errors measured at each head tilt position. R²-values represent goodness-of-fit of model to SVV errors.

Since the SVV errors in the Bayesian model is quantified by a combination of errors in the head-in-space estimate and in the eye-in-head estimate (see Equation 5), here we accounted for these different sensory contributions (Figure 5; head-in-space (${\text{H}}_{\text{S}}$) estimate error in blue and eye-in-head (${\text{E}}_{\text{H}}$) estimate error in red). The model fit results show that in most participants the ${\text{E}}_{\text{H}}$ estimate or ${\text{H}}_{\text{S}}$ estimate error alone does not predict the SVV error and rather the combination of both is a more accurate estimate. Overall, the ${\text{E}}_{\text{H}}$ estimate errors were 8.6 ± 1.7° and −9.6 ± 2.1° for the left and right tilts, and the ${\text{H}}_{\text{S}}$ estimate errors were −8.2 ± 1.9° and 6.3 ± 1.8° for the left and right tilts. The ${\text{E}}_{\text{H}}$ estimate did not correlate with the actual eye-in-head position measured as the OCR (Pearson correlation analysis, p = 0.11). The SVV variability from the model fit showed a trend of overestimating the actual variability measured from the SVV task (Table 1).

Best-fit average values for the eye and head prior and associated SEM among all participants are shown in Figure 6A. The mean head prior was 2.1 ± 2.9° and eye prior was 1.3 ± 2.0°. The width of the head prior and eye prior distribution was 12.6 ± 3.3° and 4.3 ± 0.9°, respectively. To provide further explanation about the Bayesian computations underlying the SVV, Figure 6B illustrates graphically that the optimal estimate (red lines) of head tilt (first row) and eye-in-head orientation (second row) relies on the product of the sensory input likelihood (blue lines) and prior distribution (green lines), and SVV performance (third row) at three head positions. Here we used average fit values of the head and eye priors and average sensory inputs as calculated in Equations 10 and 11. As Fig. 6B shows, prior knowledge biases estimate toward smaller angles but reduces uncertainty caused by sensory noise.

Figure 6.

(A) Head prior and eye prior average distributions from the SVV data fit among all subjects. The black arrow points to the mean of the gaussian distribution center for head (${\text{H}}_{\text{SP}}$) and eye (${\text{E}}_{\text{HP}}$). The width of green shaded area indicates the SEM of the distribution center. The gray shaded area represents the SEM of gaussian distribution width. Imposed fit bounds, ${\text{H}}_{\text{SP}}$: −20–20°, ${\sigma }_{{\text{H}}_{\text{SP}}}$: 0–50°, ${\text{E}}_{\text{HP}}$: −10–10°, ${\sigma }_{{\text{E}}_{\text{HP}}}$: 0–15°. (B) Bayesian simulation illustrates distribution of sensory inputs (blue lines) and priors (green lines) contributing to the mean neural estimates of eye and head position (i.e., ${\text{H}}_{\text{S}}$ and ${\text{E}}_{\text{H}}$ estimates). The first and second rows show the resulting posterior distributions (red lines) for mean ${\text{H}}_{\text{S}}$ and ${\text{E}}_{\text{H}}$ estimates. The third row shows mean values (black lines) of measured SVV error and precision across all subjects. Vertical gray lines denote true vertical. Left column, head tilted 20° to the left; middle column, head upright; right column, head tilted 20° to the right.

DISCUSSION

Main Findings

While previous studies have shown that systematic errors in spatial orientation conform to the rules of Bayes optimal observer theory, this framework has not been applied to extract prior estimates as a measure of neural mechanisms that modulate sensory signals that contribute to perception of spatial orientation. Here we implemented psychophysical tasks to derive perceptual estimates of head-in-space and eye-in-head positions and determine how these estimates can account for the SVV bias as a key measure of spatial orientation. In this framework, a representation of gravity is used to determine body orientation with respect to the external world.

According to the Bayesian framework, to achieve an optimal estimate of head or eye position, the brain must incorporate a prior knowledge of a more-likely head or eye position. This is represented by a prior distribution, expressed as a Gaussian curve with a standard deviation centered around a prior estimate. In earlier studies, this distribution was centered at zero based on the assumption that the brain defaults to the upright position as the most probable body position. Our approach, however, makes it possible to derive the prior estimates without making such assumption, accounting for individual variability in processing sensory signals that contribute to perception of spatial orientation.

Bayesian Model of Spatial Orientation

In biological systems, inherent variability plays a critical role in how sensory information is processed. Unlike physical stimuli, the encoding neural signals are subject to noise introduced at multiple levels of the sensory pathway. Bayesian inference offers a powerful framework to account for optimal integration of these noisy sensory signals (15,25,26). This optimal integration refers to the process that minimizes perceptual uncertainty by combining multiple sensory signals in proportion to their reliability. The perceptual estimate within the Bayesian observer model is the internal estimation of gravitational vertical (i.e., SVV), which is derived by combining noisy sensory signals (i.e., sensory likelihood) with a priori knowledge (3). This is determined by a weighted sum of sensory likelihood of eye and head positions, using their respective priors. Both the prior and sensory likelihoods are represented as Gaussian distributions, characterized by parameters defining their mean and variance. The optimal combination of these distributions results in posterior distributions that represent neural estimates of the eye and head positions. Unlike the sensory signals from the eye and head, visual inputs encoding the orientation of retinal images are less noisy and therefore do not undergo the same processing (3,5,24). The sum of the neural estimates of the eye and head position ultimately defines the eye-in-space estimate, which serves as the primary source of SVV error. As shown in the results, these combined estimates, rather than individual head or eye estimates, can effectively model the SVV bias across different head tilt angles.

Significance of Prior Estimation

The effect of prior enhances the precision of sensory estimates by consistently modulating sensory inputs, but this comes with the cost of reduced accuracy in generating the head-in-space and eye-in-head estimates. Such trade-off can account for the undercompensation for the head tilt, which contributes to the A-effect at larger tilt angles. Similarly, undercompensation for ocular counter-roll with an effect opposite to that of head tilt or a varying pattern of sensory noise distribution across tilt angles can result in the E-effect at smaller tilt angles (27). Together, these estimates can account for the systematic errors in SVV through two distinct computational stages with the opposing bias effects. In this framework, centering the prior distributions at zero can result in underestimation of both eye and head positions, predicting, for instance, that the estimated eye-in-head position will consistently fall below the actual OCR measurement. However, as demonstrated by our results, perceptual estimates of the eye-in-head position may not align with the actual OCR or head position, suggesting that sensory estimates may deviate from the true physical positions of the eye and head. Assuming that prior distributions necessarily mirror physical stimuli may therefore compromise the accuracy of perceptual measurements. For this reason, here we derived priors for eye or head positions from individual data rather than assuming a default distribution centered at zero. This individualized approach enables the model to more accurately capture discrepancies between perceptual and physical measurements.

Previous studies have accounted for the role of the eye-in-head estimate in SVV bias without directly recording the eye position (3,4,12–14). As previously shown for SVV measurements, our simultaneous measurement of OCR reveals no correlation with the eye-in-head estimate (10). This finding suggests that the neural computation of the eye-in-head estimate is not simply based on an uncompensated ocular counter-roll. Instead, the eye-in-head estimate may be influenced by how the brain establishes a prior estimation of the eye position to accommodate the physical changes in ocular counter-roll.

Impact of Measurement Bias

When visual spatial context is removed, the brain must rely on other sensory signals to determine orientation of external stimuli. In this scenario, psychophysical tasks haptic stimuli depend on different sensory inputs, and any disparity between these visual measurements may reflect modality-specific biases in how the brain processes the eye position for the visual task versus the hand position for the haptic task. In line with such task-specific sensory contributions, previous studies have demonstrated measurable differences between visual and haptic tasks (18,19). In a haptic task, the observed bias is associated with the hand use, referred to as the hand bias. This bias remains relatively constant across measurements at different head tilt positions and can therefore be subtracted from these measurements. Once the hand bias is removed, the haptic and visual measurements show similar spatial bias patterns. These findings suggest that the disparity in visual and haptic measures is primarily due to a task bias that must be accounted for. To derive the hand bias, here we subtracted the Hs-visual and Hs-haptic measurements in the upright head position (Equation 9). This approach can isolate the hand bias as the Hs-visual measurement with the head upright is minimally influenced by the eye position since both eye and head orientation are aligned, whereas the hand bias remains present in the Hs-haptic measurement under the same condition.

As our results indicate, the haptic task measurements generally exhibit higher variability compared to the visual task. This increased variability is also related to the higher noise inherent to the haptic measurements. Since the variability caused by hand bias could not be eliminated in our model, it will be integrated into the overall variance in the head position signal (${\sigma }_{{\widehat{\text{H}}}_{\text{S}}}^2$). This limitation can account for the overall higher variability in the SVV estimates generated in the model compared with the actual variability measured in the SVV task. Such variability introduced by haptic measurements could be addressed by incorporating it as an independent parameter within the model. However, this would necessitate including additional tilt angles beyond the three positions used in this study to reduce reliance on fitting multiple parameters.

Model-Based Interpretation of Neural Mechanisms

Much of the evidence supporting the use of Bayesian inference to optimally integrate noisy sensory signals arises from studies showing that noise levels within individual sensory modalities can reliably predict performance in combined sensory conditions. In this study, we evaluated the performance of healthy individuals in visual and haptic psychophysical tasks to assess the perception of head tilt and spatial orientation.

Our findings highlight that perceptual accuracy and precision in these tasks are directly related to the weighting of sensory signals involved in spatial orientation processing. With these measurements, we were able to isolate the sensory priors associated with the eye and head positions, as key sensory contributions to spatial orientation. These results provide valuable insights into the neural computations underlying sensory integration and their role in maintaining accurate spatial perception.

The sensory priors can serve as the quantifiable markers for neural mechanisms underlying perception of spatial orientation. Through these priors, the Bayesian framework can provide a mechanistic insight into neural processes through dynamics of sensory weighting and integration. Importantly, these priors can be measured across individuals, enabling comparisons between healthy and clinical populations. In this context, the estimated prior distribution can serve as a proxy for internal neural estimates that shape the interpretation of incoming sensory information. Quantifying these internal estimates can be useful for probing the mechanisms of sensory integration for spatial orientation. For instance, in individuals with chronic dizziness—who are often afflicted by daily symptoms related to spatial disorientation—these priors may be altered or maladaptive, contributing to a perceptual dysfunction even when peripheral sensory inputs are intact. In such cases, the Bayesian observer model can help disentangle whether there is abnormal sensory encoding, distorted internal priors, or impaired integration between the two. Isolating these distinct sensory components can help guide diagnosis and target interventions aligned with specific deficits in individual patients.

DATA AVAILABILITY

Data can be made available upon reasonable request by contacting the corresponding author.