Human participants steered toward a briefly cued target location on a virtual ground plane, with varying sensory conditions and control dynamics interleaved across trials. Participants sat on a motion platform in front of a screen displaying a virtual environment (Figure 1A). Stereoscopic depth cues were provided using polarizing goggles. On each trial, a circular target appeared briefly at a random location (drawn from a uniform distribution within the field of view; Figure 1B and C) and participants had to navigate to the remembered target location in the virtual world using a joystick to control linear and angular self-motion. The virtual ground plane was defined visually by a texture of many small triangles which independently appeared only transiently; they could therefore only provide optic-flow information and could not be used as landmarks. The self-motion process evolved according to Markov dynamics, such that the movement velocity at the next time step depended only on the current joystick input and the current velocity (Materials and methods – Equation 1a).

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A time constant for the control filter (control timescale) governed the control dynamics: in trials with a small time constant and a fast filter, joystick position essentially controlled velocity, providing participants with responsive control over their self-motion, resembling regular road-driving dynamics. However, when the time constant was large and the control filter was slow, joystick position mainly controlled acceleration, mimicking high inertia under viscous damping, as one would experience on an icy road where steering is sluggish (Figure 1D right and 1E – top vs. bottom). For these experiments, as the control timescale changed, the maximum velocity was adjusted so that the participant could reach the typical target in about the same amount of time on average. This design ensured that the effect of changing control dynamics would not be confused with the effect of integrating sensory signals over a longer or shorter time.

Concurrently, we manipulated the modality of sensory observations to generate three conditions: (1) a vestibular condition in which participants navigated in darkness, and sensed only the platform’s motion (note that this condition also engages somatosensory cues, see Materials and methods), (2) a visual condition in which the motion platform was stationary and velocity was signaled by optic flow, and (3) a combined condition in which both cues were available (Figure 1E – left to right). Across trials, sensory conditions were randomly interleaved while manipulation of the time constant followed a bounded random walk (Materials and methods – Equation 2). Participants did not receive any performance-related feedback.

We first compared the participants’ stopping locations on each trial to the corresponding target locations, separately for each sensory condition. We calculated the radial distance ${\displaystyle \tilde{r}}$ and angular eccentricity ${\displaystyle \tilde{\theta }}$ of the participants’ final position relative to the initial position (Figure 2A), and compared them to the initial target distance $r$ and angle $\theta$, as shown for all trials (all time constants together) of a typical participant in Figure 2B and C. This revealed biased performance with notable undershooting (participants stopped short of the true target location), in both distance and angle, which was well described by a linear model without intercept (radial distance $R^2$ ± standard deviation – vestibular: 0.39 ± 0.06, visual: 0.67 ± 0.1, combined: 0.64 ± 0.11; angular eccentricity $R^2$ ± standard deviation – vestibular: 0.85 ± 0.06, visual: 0.95 ± 0.05, combined: 0.96 ± 0.04. Adding a non-zero intercept term offered negligible improvement; radial distance ${\Delta R}^2$ – vestibular: 0.02 ± 0.02, visual: 0.03 ± 0.03, combined: 0.03 ± 0.02; angular eccentricity ${\Delta R}^2$ – vestibular: 0.02 ± 0.03, visual: 0.01 ± 0.01, combined: 0.01 ± 0.01). We refer to the slope of the linear regression as ‘response gain’: a response gain of unity indicates no bias, while gains larger (smaller) than unity indicate overshooting (undershooting). As shown with the example participant in Figure 2B and C, there was substantial undershooting in the vestibular condition, whereas performance was relatively unbiased under the combined and visual conditions (see also Figure 2—figure supplement 1A). These results were consistent across participants (Figure 2D, mean radial gain± standard deviation – vestibular: 0.76 ± 0.25, visual: 0.88 ± 0.23, combined: 0.85 ± 0.22, mean angular gain± standard deviation – vestibular: 0.79 ± 0.22, visual: 0.98 ± 0.14, combined: 0.95 ± 0.12), and no significant sex differences were observed (see Figure 2—figure supplement 1B). The difference in response gain between modalities could be traced back to the control exerted by the subjects on the joystick. Both linear and angular components of control input had shorter duration in the vestibular condition (mean ± SEM of total area of joystick input across participants (a.u.): radial – vestibular: 5.62 ± 0.27, visual: 7.31 ± 0.33, combined: 7.07 ± 0.34; angular – vestibular: 2.39 ± 0.30, visual: 3.29 ± 0.42, combined: 3.79 ± 0.46), and produced smaller displacements, as summarized by the response gains (Figure 2, Figure 2—figure supplement 2).

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Radial and angular responses.

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Response gains.

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To examine whether control dynamics affected the response gain, we performed three complementary analyses. First, we recomputed response gains by stratifying the trials into three groups of equal size based on the time constants. We found that smaller time constants (velocity control) were associated with smaller response gains (Figure 3A; Appendix 2—table 1). This relationship was most pronounced in the vestibular condition, where larger time constants (acceleration control) resulted in better (closer to ideal) performance (Figure 3, green; see Discussion). Control dynamics had a smaller but considerable effect on steering responses in the visual and combined conditions, with participants exhibiting modest overshooting (undershooting) when the time constant was large (small) (Figure 3A, cyan/purple).

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Residual errors.

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Regression coefficients between time constant and residual errors.

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Second, we performed a fine-grained version of the above analysis by computing residual errors on each trial, that is, the deviation of the response from the mean response predicted from target location alone (Materials and methods – Equation 3). Since participants try to stop at their believed target location, ideally their mean responses should depend only on target location, and not on control dynamics. In other words, if participants adapted their control appropriately to the varying control dynamics, their responses should cluster around their mean response, and as a result, their residual errors should be centered around zero without any mean dependence on dynamics. However, we found a significant correlation between residual errors and the time constant across trials (Figure 3B, C; Figure 3—figure supplement 1, Appendix 2—table 2, see Materials and methods; no significant sex differences were observed, and therefore are not investigated in subsequent analyses, see also Figure 2—figure supplement 1C). This correlation and the corresponding regression slopes were substantially higher in the vestibular condition (mean Pearson’s r ± SEM: radial component – vestibular: 0.52 ± 0.02, visual: 0.36 ± 0.03, combined: 0.37 ± 0.03; angular component – vestibular: 0.23 ± 0.02, visual: 0.23 ± 0.03, combined: 0.26 ± 0.03; see also Appendix 2—Tables 2 and 3). Thus, for a given target distance, participants tended to travel further when the time constant was larger (acceleration control), indicating they did not fully adapt their steering control to the underlying dynamics.

Third, to quantify the contribution of the time constant in the participants’ responses, we expanded the linear model to accommodate a dependence of response (final stopping position) on target location, time constant, and their interaction. A partial correlation analyses revealed that the time constant contributed substantially to participants’ response gain, albeit only by modulating the radial and angular distance dependence (Appendix 2—table 4; Figure 3—figure supplement 2; see Materials and methods – Equation 4). Again, the contribution of the time constant-dependent term was much greater for the vestibular condition (Figure 3D), especially for the radial distance (p-values of difference in coefficient values across modalities obtained by a paired t-test – radial: vestibular vs. visual: p < 10^–4 , vestibular vs. combined: p < 10^–4; angular: vestibular vs. visual: p = 0.016, vestibular vs. combined: p = 0.013). While perfect adaptation should lead to response gain that is independent of control dynamics, all three independent analyses revealed that control dynamics did substantially influence the steering response gain, exposing participants’ failure to adapt their steering to the underlying dynamics. Adaptation was lowest for the vestibular condition; in contrast, for the visual and combined conditions, the response gain was less affected indicating greater compensation when visual information was available.

We quantified the extent to which participants failed to adapt to the control dynamics, by simulating a null case for no adaptation. Specifically, we generated null-case trajectories by using the steering input from actual trials and re-integrating it with time constants from other trials. In this set of null-case trajectories, the steering control corresponds to different time constants; in other words, steering is not adapted to the underlying dynamics (see Materials and methods). We then grouped these trajectories based on the simulation time constant (as in Figure 3A) and computed the corresponding response gains. We found that the true response gains in the vestibular condition were much closer to the no-adaptation null case, compared to visual/combined conditions (Figure 3E). Interestingly, this finding was more prominent in the radial component of the response gain (Figure 3E insets), consistent with our earlier observations of a stronger influence of the dynamics on the radial component of the responses.

We have shown how various measures of the participants’ final responses (stopping positions,response gain, residual errors) are influenced by the time constant of the dynamics. This large dependence of the final responses on the time constant exposes participants’ failure to fully adapt their steering to the underlying dynamics. In other words, the influence of the dynamics on steering control was relatively weak, especially in the vestibular condition.

For best performance, however, control dynamics should influence the time course of steering behavior. We directly quantified the influence of the control dynamics on steering by comparing participants’ braking (negative control input) across time constants: when the time constant is large, we ideally expect to see more braking as a countermeasure for the sluggish control (Figure 1D) to minimize travel duration (see Materials and methods). Indeed, participants do tend to brake more for higher time constants, but this effect is weaker in the vestibular condition (Figure 4 and inset). Nevertheless, correlations between the time constant and cumulative braking (total area below zero linear control input) were significant in all sensory conditions (mean Pearson’s r ± SEM – vestibular: 0.20 ± 0.03, visual: 0.62 ± 0.04, combined: 0.57 ± 0.04; p-values of Pearson’s r difference from zero – vestibular: p = 10⁻⁵, visual: p < 10⁻⁷, combined: p < 10⁻⁷). Overall, it appears that behavior in the vestibular condition is minimally influenced by the dynamics (i.e. smaller modulation of control input by the time constant, as shown by the cumulative braking). When optic flow is available, however, participants are more flexible in adjusting their control.

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We have shown previously that accumulating sensory noise over an extended time (~10 s) would lead to a large uncertainty in the participant’s beliefs about their position, causing them to undershoot (Lakshminarasimhan et al., 2018). The exact amount of undershooting depends both on the reliability of self-motion cues, which determines the instantaneous uncertainty in the self-motion estimate, and on travel duration, which governs how much uncertainty is accumulated while navigating to the target. With recent findings ascribing uncertainty accumulation to noise in the velocity input (Stangl et al., 2020), the observed differences in navigation performance across sensory modalities can be readily attributed to greater measurement noise (lower reliability) in vestibular signals. On the other hand, we observed performance differences across control dynamics within each sensory modality, so those differences cannot be attributed to differences in the reliability of self-motion cues (instantaneous uncertainty). However, it might seem that this effect of control dynamics must be due to either differences in travel duration or velocity profiles, which would both affect the accumulated uncertainty. We adjusted stimulus parameters to ensure that the average travel time and average velocity were similar across different control dynamics (Materials and methods – Equation 1.2-1.10), however, we found that travel duration and average velocity depend weakly on the time constant in some participants. Simulations suggest that both dependencies are a consequence of maladaptation to the dynamics rather than a cause of the observed effect of the dynamics on the responses. Interestingly, the dependence is stronger in the vestibular condition where there is less adaptation to the dynamics, agreeing with our simulations (Figure 5—figure supplement 1A, B). Difference in velocity profiles is also an unlikely explanation since their expected effect on the participants’ responses (undershoot) is the opposite of the observed effect of the control dynamics (overshooting tendency; Figure 5—figure supplement 1C). Consequently, unlike the effect of sensory modality on response gain, neither instantaneous nor accumulated differences in the uncertainty can fully account for the influence of control dynamics, that is, the time constant. Instead, we will now show that the data are well explained by strong prior expectations about motion dynamics that cause a bias in estimating the time constant.

From a normative standpoint, to optimally infer movement velocity, one must combine sensory observations with the knowledge of the time constant. Misestimating the time constant would produce errors in velocity estimates, which would then propagate to position estimates, leading control dynamics to influence response gain (Figure 5A, middle-right). This is akin to misestimating the slipperiness of an ice patch on the road causing an inappropriate steering response, which would culminate in a displacement that differs from the intended one (Figure 5—figure supplement 2). However, in the absence of performance-related feedback at the end of the trial, participants would be unaware of this discrepancy, wrongly believing that the actual trajectory was indeed the intended one. In other words, participants’ imperfect adaptation to changes in control dynamics could be a consequence of control dynamics misestimation.

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correlations between time constant and model-implied residual errors.

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We tested the hypothesis that participants misestimated the time constant using a two-step model that reconstructs the participants’ believed trajectories according to their point estimate of the time constant τ, as follows. First, a Bayesian observer model infers the participant’s belief about τ on individual trials, that is, the subjective posterior distribution over the time constant (τ inference step; Figure 5A, left). Second, we used the median of that belief to reconstruct the believed trajectory by integrating the actual joystick input according to the estimated time constant on that trial (integration step), resulting in a believed stopping location (Figure 5A, middle-right). In the absence of bias (response gain of one), the believed stopping locations should land on or near the target. However, various unmeasurable fluctuations in that belief across trials should lead to variability clustered around the target location. When the behavior is biased (response gain different from one, as was the case here – Figure 2D), this cluster should instead be centered around the participants’ mean belief for that target location (determined from their biased responses and henceforth referred to as mean stopping location). Since the participants’ goal is to stop as close to their perceived target location as possible, the deviation of believed stopping locations from the mean stopping location for a given target should be small. We call this deviation the subjective residual error. Therefore, we inferred the parameters of the Bayesian model separately for each participant by minimizing the subjective residual errors induced by the control dynamics using the principle of least squares (see Materials and methods for further details). We next describe the parameters of the Bayesian model and then describe the results of fitting the model to our data.

Because the time constant τ is always positive, we model both the prior distribution and the likelihood function over the variable $\phi =\log \tau$ as Gaussians in log-space. We parameterized both the prior and the likelihood with a mean ($\mu$) and standard deviation ($\sigma$). The mean of the prior ($\mu$) was allowed to freely vary across sensory conditions but assumed to remain fixed across trials. On each trial, the likelihood was assumed to be centered on the actual value of the log time constant ${\tau }^{*}$ on that trial according to $\mu ={\phi }^{*}=\log {\tau }^{*}$ and was therefore not a free parameter. Finally, we set the ratio $\lambda$ of prior over likelihood $\sigma$, to freely vary across sensory conditions. Thus, for each sensory condition, we fit two parameters: the $\mu$ of the prior, and the ratio (${\displaystyle \lambda }$) of prior $\sigma$ to likelihood $\sigma$. As mentioned above, we fit the model to minimize the difference between their believed stopping locations and their experimentally measured mean stopping location (subjective residual errors), using a least-squares approach (Materials and methods) and obtained one set of parameters for each condition. Finally, the participant’s estimated time constant $\widehat{\tau }$ on each trial was taken to be the median of the best-fit model, which equals the median of the distribution over $\phi$ (Figure 5A, left). By integrating the subject’s joystick inputs on each trial using $\widehat{\tau }$ rather than the actual time constant $\tau$, we computed the believed stopping location and the subjective residual errors implied by the best-fit model.

We then compared the correlations between the time constant and the residual errors for real responses (from data in Figure 3B and C) or subjective responses (from model), separately for radial and angular components. Because participants try to stop at their believed target location, the believed stopping position should depend only on target location and not on the control dynamics. Any departure would suggest that participants knowingly failed to account for the effect of the control dynamics, which would manifest as a dependence of the subjective residual errors on the time constant τ. In other words, a good model of the participants’ beliefs would predict that the subjective residual errors should be uncorrelated with the time constant τ (Figure 5A inset – red) even if the real residual errors are correlated with the time constant (Figure 5A inset – black). In all cases, we observed that the correlation between residual error and time constant was indeed significantly smaller when these errors were computed using the subjective (believed) rather than real stopping location (Figure 5B). In fact, subjective residual errors were completely uncorrelated with the time constant suggesting that the Bayesian model is a good model of participants’ beliefs, and that the apparent influence of control dynamics on behavioral performance was entirely because participants misestimated the time constant of the underlying dynamics.

We next examined the model posterior estimates to assess how subjects’ internal estimate of the control dynamics departed from the true dynamics. The relationship between real and model-estimated time constants for all participants can be seen in Figure 6A. In the vestibular condition, all participants consistently misestimated τ, exhibiting a substantial regression toward the mean (Figure 6A, green). This effect was much weaker in the visual condition. Only a few participants showed relatively flat estimates, with the majority showing smaller departures from ideal estimates (dashed line). The data for the combined condition followed a similar trend, with properties between those in the visual and vestibular conditions (Figure 6A, purple). These results suggest that the better control adaptation in the visual and combined conditions shown in Figure 3 is due to participants’ improved estimates of the time constant when optic flow was available.

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Real versus estimated time constants.

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Model parameters.

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The source of inaccuracies in the estimated time constant can be understood by examining the model parameters (Figure 6B).The ratio $\lambda$ of prior over likelihood standard deviations was significantly lower in the vestibular condition than other conditions, suggesting stronger relative weighting of the prior over the likelihood (Figure 6B left, green bar; mean ratio $\lambda$± standard SEM – vestibular: 0.30 ± 0.09, visual: 1.02 ± 0.17, combined: 0.80 ± 0.10; p-value of ratio $\lambda$ paired differences obtained by bootstrapping – vestibular vs. visual: p = 0.0007, vestibular vs. combined: p = 0.0087; visual vs. combined: p = 0.016). Notably, the ratio was close to one for the visual and combined conditions, suggesting equal weighting of prior and likelihood. Thus, participants’ estimate of the control dynamics in the vestibular condition was plagued by a combination of strong prior and weak likelihood, which explains the much stronger regression toward the mean in Figure 6A.

To test whether our assumption of a static prior distribution over time constants was reasonable, we fit an alternative Bayesian model in which the prior distribution was updated iteratively on every trial, as a weighted average of the prior on the previous trial and the current likelihood over $\phi$ (Dynamic prior model; see Materials and methods). For this version, the initial prior $\mu$ was taken to be the time constant on the first trial, and we once again modeled the likelihood and prior as normal distributions over the log-transformed variable, $\phi$, where the likelihood was centered on the actual $\phi$ and was therefore not a free parameter. Thus, we fit one parameter: the ratio $\lambda$ of prior $\sigma$ over likelihood $\sigma$. On each trial, the relative weighting of prior and likelihood responsible for the update of the prior depended solely on $\lambda$; that is, the relationship between their corresponding $\sigma$ (i.e. relative widths). The performance of the static and dynamic prior models was comparable in all conditions, for both distance and angle, suggesting that a static prior is adequate in explaining the participants’ behavior on this task (Figure 7; light vs. dark bars). In line with our expectations, when updating the prior in the dynamic model, the weighting of the previous-trial prior received significantly more weight in the vestibular condition (in the range of [0,1]; mean prior weights ± SEM– vestibular: 0.93 ± 0.03, visual: 0.48 ± 0.10, combined: 0.61 ± 0.09; p-value of paired weight differences obtained by bootstrapping – vestibular vs. visual: p = 10^–5 , vestibular vs. combined: p = 4 · 10^–4; visual vs. combined: p = 0.08). The comparable goodness of models with static and dynamic priors suggest that sensory observations were not too reliable to cause rapid changes in prior expectations during the course of the experiment.

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At the other extreme, to test whether participants used sensory observations at all to estimate control dynamics, we compared the static prior Bayesian model to a parsimonious model that assumed a fixed time constant across all trials (i.e. completely ignoring changes in control dynamics). This latter model can be understood as a Bayesian model instantiated with a very strong static prior. In line with our expectations (see Figure 6A), this latter model performed comparably in the vestibular condition, but substantially worse in the visual and combined conditions (Figure 7).

Due to the correlated nature of the random walk process dictating the evolution of time constants, an alternative by which participants could get away without estimating the time constant in the vestibular condition would be to carry over their estimate from the previous combined/visual trial to the current vestibular trial. To test this, we considered two models: the time constant estimate in the current vestibular trial was taken to be either the real time constant or the posterior estimate of the time constant from the previous visual/combined trial. Neither model, however, could account for the participants’ behavior, as they could not fully explain away the correlation between the residual errors and the time constant (Figure 7—figure supplement 1). Intuitively, although choosing actions with respect to the previous trial’s time constant should result in estimates that regress toward the mean, the predicted effect is weaker than that observed in the data.

Finally, we tested a variation of previously suggested sensory feedback control models (Glasauer et al., 2007; Grasso et al., 1999) where a controller relies solely on sensory inputs to adjust their control without explicitly estimating the latent variables governing the control dynamics. Specifically, the model assumes that participants implement a type of bang-bang control that switches at a certain distance from the target (or more accurately, the mean response). However, this model predicts a much stronger dependence of the responses on the dynamics compared to our data, and characteristics of the predicted control input differ significantly from the actual control (Figure 7—figure supplement 2). Overall, our results suggest that optic flow, but not vestibular signals, primarily contributes to inferring the latent velocity dynamics.

In the paradigm used here, participants actively controlled linear and angular motion, allowing us to study multisensory path integration in two dimensions with few constraints. This paradigm was made possible by the development of an MC algorithm to render visual and vestibular cues either synchronously or separately. In contrast, previous studies on human path integration used restricted paradigms in which motion was either 1D or passively rendered, and participants’ decisions were typically reduced to end-of-trial binary evaluations of relative displacement (Campos et al., 2012; Chrastil et al., 2016; Chrastil et al., 2019; Jürgens and Becker, 2006; Koppen et al., 2019; ter Horst et al., 2015; Tramper and Medendorp, 2015). As a result, findings from past studies that evaluate the contributions of different sensory modalities to self-motion perception (Chrastil et al., 2019; Israël et al., 1996; Koppen et al., 2019; Seemungal et al., 2007; ter Horst et al., 2015) may be more limited in generalizing to real-world navigation.

Our results show that, at least in humans, navigation is driven primarily by visual cues under conditions of near-constant travel velocity (velocity control). This dominance of vision suggests that the reliability of the visual cues is much higher than vestibular cues (as generated by our platform), as corroborated by the data from the combined condition in which performance resembles the visual condition. This makes sense because the vestibular system is mainly sensitive to acceleration, exhibiting higher sensitivity to higher-frequency motion compared to the visual system (Karmali et al., 2014). Consequently, it may only be reliable when motion is dominated by acceleration. This interpretation is further supported by the observation that participants’ vestibular performance was a lot less biased in the regime of acceleration joystick control, where accelerations are prolonged during navigation.

Experimental constraints in past navigation studies have also precluded examining the influence of control dynamics. In fact, the importance of accurately inferring control dynamics, which are critical for predicting the sensory consequences of actions, has largely been studied in the context of limb control and motor adaptation (Burdet et al., 2001; Kording et al., 2007; Krakauer et al., 1999; Lackner and Dizio, 1994; Shadmehr and Mussa-Ivaldi, 1994; Takahashi et al., 2001). Here, we provide evidence for the importance of accurately inferring control dynamics in the context of path integration and spatial navigation. Although participants were not instructed to expect changes in the latent dynamics and received no feedback, we showed that they nevertheless partly adapted to those dynamics while exhibiting a bias toward prior expectations about these dynamics. This biased estimation of control dynamics led to biased path integration performance. This result is analogous to findings about the effect of changing control dynamics in motor control: first, adaptation to the dynamics happens even in the absence of performance-related feedback (Batcho et al., 2016; Lackner and Dizio, 1994) and, second, this adaptation relies on prior experience (Arce et al., 2009) and leads to systematic errors when knowledge of the dynamics is inaccurate (Körding et al., 2004). Thus, participants try to exploit the additional information that the dynamics contain about their self-motion in order to achieve the desired displacement.

A Bayesian estimator with a static prior over the dynamics sufficiently explained participants’ beliefs in our data, while results were comparable with a dynamic prior that was updated at every trial. This could be attributed to the structure of the random walk of the control dynamics across trials, as a static prior is not as computationally demanding and potentially more suitable for fast changes in the time constant. These Bayesian models attempt to explain behavior in an optimal way given the task structure. Meanwhile, alternative suboptimal models (fixed estimate, carry-over estimate, sensory feedback model) failed to explain behavior successfully, especially when optic flow was available. These results strongly favor underlying computations within the context of optimality in the presence of optic flow.

Task performance was substantially worse in the vestibular condition, in a manner suggesting that vestibular inputs from motion cueing lack the reliability to precisely estimate control dynamics on individual trials. Nevertheless, the vestibular system could still facilitate inference by integrating trial history to build expectations about their statistics. Consistent with this, the mean of the prior distribution over the dynamics fit to data was very close to the mean of the true sampled distribution, suggesting that even if within-trial vestibular observations are not sufficient, participants possibly combine information about the dynamics across trials to construct their prior beliefs. This is consistent with the findings of Prsa et al., 2015, where vestibular cues were used to infer an underlying pattern of magnitude of motion across trials. However, the measurement of the dynamics in that study substantially differs from ours: here, motion dynamics are inferred using self-motion cues within each trial whereas in Prsa et al., 2015, the dynamics were inferred by integrating observations about the magnitude of the displacement across trials. If vestibular cues can in fact support inference of dynamics – as recent findings suggest in eye-head gaze shifts (Sağlam et al., 2014) – a common processing mechanism could be shared across sensory modalities. Overall, this finding highlights the importance of incorporating estimates of the control dynamics in models of self-motion perception and path integration.

Note that restrictions of our motion platform limited the range of velocities that could be tested, allowing only for relatively small velocities (see Materials and methods). Consequently, trial durations were long, but the motion platform also restricted total displacement, so we could not test larger target distances. We previously studied visual path integration with larger velocities and our results in the visual and combined conditions are comparable for similar travel times (as trials exceeded durations of 10 s, undershooting became more prevalent; Lakshminarasimhan et al., 2018). However, it is unclear how larger velocities (and accelerations) would affect participants’ performance (especially under the vestibular condition) and whether the present conclusions are also representative of the regime of velocities not tested.

The design of the MC algorithm allowed us to circumvent the issues associated with the physical limitations of the platform to a large degree. This was achieved in part by exploiting the tilt/translation ambiguity and substituting linear translation with tilt (see Materials and methods). However, high-frequency accelerations, as those found at movement onset, generated tilts that briefly exceeded the tilt-detection threshold of the semicircular canals (Figure 1—figure supplement 2). Although the duration of suprathreshold stimulation was very small, we cannot exclude the possibility that the perceived tilt affected the interpretation of vestibular inputs. For example, participants may not attribute tilt to linear translation, hence underestimating their displacement. This, however, would lead to overshooting to compensate for the lack of perceived displacement,which is not what we observed in our experiment. Another potential explanation for the poor vestibular performance could be that participants perceive tilt as a conflicting cue with respect to their expected motion or visual cues. In that case, participants would only use the vestibular inputs to a small extent if at all. Manipulating vestibular inputs (e.g. gain, noise manipulations) in future experiments, either alone or in conjunction with visual cues, would offer valuable insights on two fronts: first, to help clarify the efficiency of our MC algorithm and its implications on the design of driving simulators in the future, and second, to precisely quantify the contribution of vestibular cues to path integration in natural settings.

For the sake of simplicity, we modeled each trial’s control dynamics as a single measurement per trial when, in reality, participants must infer the dynamics over the course of a trial using a dynamic process of evidence accumulation. Specifically, participants must measure their self-motion velocity over time and combine a series of measurements to extract information about the underlying dynamics. Although we were able to explain the experimental findings of the influence of control dynamics on steering responses with our model, this approach could be expanded into a more normative framework using hierarchical Bayesian models (Mathys et al., 2011) to infer subjective position estimates by marginalizing over possible control dynamics.

One interesting question is whether providing feedback would eliminate the inference bias of the control dynamics estimation and future studies should explicitly test this hypothesis. Furthermore, it would be interesting to jointly introduce sensory conflict and manipulate sensory reliability to study dynamic multisensory integration such that sensory contributions during navigation can be better disentangled. Although it has been shown that cue combination takes place during path integration (Tcheang et al., 2011), previous studies have had contradicting results regarding the manner in which body-based and visual cues are combined (Campos et al., 2010; Chrastil et al., 2019; Koppen et al., 2019; Petrini et al., 2016; ter Horst et al., 2015). Since visual and vestibular signals differ in their sensitivity to different types of motion (Karmali et al., 2014), the outcomes of their integration may depend on the self-motion stimuli employed. Combined with hierarchical models of self-motion inference that considers the control dynamics, it is possible to develop an integrated, multi-level model of navigation, while constraining dramatically the hypothesized brain computations and their neurophysiological correlates.

Fifteen participants (9 male, 6 female; all adults in the age group 18–32) participated in the experiments. Apart from two participants, all participants were unaware of the purpose of the study. Experiments were first performed in the above two participants before testing others. All experimental procedures were approved by the Institutional Review Board at Baylor College of Medicine and all participants signed an approved consent form.

Customized MATLAB code was written to analyze data and to fit models. Depending on the quantity estimated, we report statistical dispersions either using 95% confidence interval, standard deviation, or standard error in the mean. The specific dispersion measure is identified in the portion of the text accompanying the estimates. For error bars in figures, we provide this information in the caption of the corresponding figure. We report and describe the outcome as significant if p<0.05.

When inferring the participant’s beliefs about the control dynamics, we computed the posterior distribution on trial i as $\mathrm{p}\left(\phi |m_i\right)\propto \mathrm{p}\left(\phi \right)\mathrm{p}\left(m_i\right|\phi )$ (Figure 5A, left) and then selected the median over $\phi$ (equal to the maximum a posteriori estimate), and back-transformed it from log-space to obtain an estimate of the time constant ${\displaystyle {\hat{\tau }}_i}$ for that trial:

Subsequently, ${\displaystyle \hat{\tau }}$ is used to integrate the actual joystick input and produce the participant’s believed trajectory, according to (Equations 1–10) in the Control dynamics section.

The Bayesian model had two free parameters ${\displaystyle \mathrm{\Theta }\ni \left\{{\mu }_{\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}},\lambda \right\}}$. We fit the model by assuming that participants stop as close to the target as possible given their understanding of the task. Specifically, we minimized the mean squared error (MSE) between the measured mean stopping position (computed using the response gains $g_r$ and $g_{\theta }$ from Equation 3) and our model of the participant’s believed stopping location ${\widehat{\mathbf{x}}}_i$ given the inferred dynamics ${\widehat{\tau }}_i$ . For each sensory condition:

where, for each trial ${\displaystyle i,{\hat{\mathbf{x}}}_i}$ is the believed participant’s position, ${\displaystyle {\hat{\tau }}_i}$ is the estimated time constant, ${\mathbf{u}}_i$ is the time series of the joystick control input, ${\mathbf{x}}_i^{\mathrm{t}\mathrm{a}\mathrm{r}}$ is the actual target position, $\mathbf{G}$ is the response gain matrix determined from $g_r$ and $g_{\theta }$ , and $n$ is the total number of trials.