Perception of angular displacement without landmarks: evidence for Bayesian fusion of vestibular, optokinetic, podokinesthetic, and cognitive information

Abstract

The perception of angular displacement during self turning is generally based on a combination of redundant signals from different sources. For example, during active turning in a visually structured environment devoid of landmarks, podokinesthetic, vestibular, and optokinetic velocity signals are fused and integrated over time to yield a unitary percept of the ongoing change in angular position (‘podokinesthetic’ refers to proprioceptive and corollary signals related to leg and foot movement). Previously we have shown that the fusion of two of these afferents improves perceptual accuracy and reliability in comparison to when only one is available. For example, with only a single modality available, slow rotations are perceived to be significantly larger than fast ones, whereas the combination of two modalities greatly reduces this difference. These observations spurred the hypothesis that displacement perception results from a weighted average of bottom-up (sensory) signals and top-down signals (a priori knowledge or expectation), with the weight of the latter decreasing the more sensory information is available. We now ask (1) whether the accuracy of angular displacement estimation can be further improved if it can draw on all three sensory modalities instead of only two, and (2) whether bottom-up sensory and top-down a priori information is combined for displacement estimation in a statistically optimal way. To this end 12 healthy subjects (Ss) standing on a turning platform surrounded by a rotatable optokinetic pattern were exposed to 6 different sensory conditions: pure podokinesthetic (P), vestibular (V), or optokinetic (O) stimulation, and combined podokinesthetic-vestibular (PV), vestibular-optokinetic (VO), or podokinesthetic-vestibular-optokinetic (PVO) stimulation. Stimuli had constant angular velocities of either 15, 30, or 60°/s. Subjects were to press a signal button when they felt that angular displacement had reached a previously instructed magnitude (150–900°). In agreement with earlier observations, the combination of two sensory signals improved the accuracy of displacement perception by reducing both the variance of subjects’ displacement estimates and their dependence on turning velocity. Adding a third sensory signal (condition PVO) led to a further reduction of variance and almost eliminated the effect of velocity. We show that these experimental results are compatible with a probabilistic fusion mechanism based on Bayes’ law. This mechanism would operate on logarithmic representations of turning velocity and proceed in two stages. A first stage fuses all available bottom-up information to create a unitary representation of the velocity signalled by the different sensory modalities. A second stage then fuses this sensory information with top-down a priori information; the latter creates a bias in favour of a ‘default velocity’ that grows as the uncertainty of the sensory information increases. Our experimental data agree with the relation between (1) the variance of displacement estimates and (2) their modulation by velocity predicted by this scheme.

Appendix

We here give supplementary details regarding the assumptions and calculations underlying the Bayesian interpretation paragraph in the Discussion section.

Symbols and assumptions

Referring to Fig. 4, we will use the following symbols:

General

Overlined symbols (e.g., $\bar v_{\rm p})$ denote the mean (experimental data) or central value (probability distributions) of variables. Assumptions are being referred by square brackets [].

Signals and parameters in linear space

v ₀ :

Default velocity or ‘prior’

v _i(i=1–3):

Afferent signals from sensors P, V, O, respectively

v _st :

Stimulus velocity

v _s :

Compound sensory signal corresponding to ψ_s

v _p :

‘Perceived’ velocity (internal estimation of stimulus velocity)

C :

Calibration factor

D _p :

Perceived angular displacement

D _A, D _D :

Achieved and desired displacements as defined in Methods

G _T :

Targeting gain as defined in Methods

GMI :

Gain modulation index as defined in Results

t :

Time from start of trial

Signals and parameters in logarithmic space

ψ _i (i=0–3):

Log-transforms of v _i

ψ_s :

Compound sensory signal; identical to ψ_i when only a single channel i is active (monomodal case); result of fusing several signals in case of multimodal stimulation

ψ_f :

Result of fusing prior and compound sensory signal

ψ_p :

Log-transform of v _p

σ ²₀ , σ ²_s :

Variances of ψ₀ and ψ_s

σ ²_f , σ ²_p :

Variances of ψ_f and ψ_p

σ ²_r :

Residual variance

w ₀, w _s :

Relative weights of, respectively, prior and compound sensory signal during fusion

All numeric values to be reported for these variables are based on the natural logarithm.

Assumptions

• [A1] v _i (i = 1, 2, 3) are statistically independent, noisy signals with approximately normal (Gaussian) distribution all having central values equal to v _st (‘concordant’ information) throughout a run and ‘sufficiently’ (see below) small variance.

• [A2] ψ₀ has normal distribution; its central value (the log equivalent of v ₀) remains constant throughout a run and is unrelated to the actual magnitude of v _st and, by the same token, to that of v _i (i = 1, 2, 3).

• [A3] The residual noise σ ²_r has normal distribution.

• [A4] ψ_p has ‘sufficiently’ small variance.

We call the variances σ²(v) or σ²(ψ) of, respectively, normally distributed variables v or ψ ‘sufficiently’ small, if their transforms log{v} or log⁻¹{ψ} also have an approximately normal distribution. The following approximations then hold; $\overline {{1 \mathord{\left/ {\vphantom {1 v}} \right. \kern-\nulldelimiterspace} v}} \approx {1 \mathord{\left/ {\vphantom {1 {\bar v}}} \right. \kern-\nulldelimiterspace} {\bar v}}, \sigma (v/\bar v) \approx \sigma (\bar v/v), \sigma (\log \{ v\}) \approx {{\sigma (v)} \mathord{\left/ {\vphantom {{\sigma (v)} {\bar v}}} \right. \kern-\nulldelimiterspace} {\bar v}},$ and likewise for ψ. Most variables to be considered below have standard deviations σ of less than 25% of their mean. Although this creates large deviations at the low and high ends of the distribution of their inverse, the standard deviation of the inverse differs by less than 5% from that of the original variable. We therefore will treat all of the above variables (except obviously v _st) as Gaussians.

Basic relations

Subjects were supposed to deliver their ‘on-target’-signal when D _p = C·v _p·t reached the desired value D _D; given [A1] and because D _A = v _st·t the following relation between targeting gain and perceived velocity results:

$$v_{\rm p}= \frac{{v_{\rm st} }}{{C \cdot G_{\rm T}}}$$

(5)

hence

$$\hbox{log}(v_{\rm p})= -\hbox{log}(G_{\rm T})+ \hbox{log} (v_{\rm st}/C).$$

Noting that log(v _st/C) = constant and log(v _p) ≡ ψ_p, we therefore obtain

$$\sigma_{\rm p}^{2}=\sigma^{2}(\hbox{log}\{G_{\rm T}\})$$

(6)

that is, the variance of the perceived velocity’s logarithmic representation can be estimated from the experimentally observed values of the targeting gain.

Bayesian interaction is unlikely to take place in linear space

An interaction between prior and sensory signals on a linear rather than a log-scale cannot explain the experimentally observed characteristics of GMI. As noted in Results, the standard deviation of v _p rises in proportion to v _st from whence it can be inferred that also that of v _s would be a rising function of v _st. Therefore, if signals v ₀ and v _s rather than their log representations ψ₀ and ψ_s were combined according to Eqs. 2 and 3, prior weight w ₀ would be much larger with v _st = 60°/s than with 15°/s and, consequently, GMI(60/30) would be larger than GMI(30/15). Using reasonable first order approximations for v ₀ (30°/s) and w ₀ (0.2), this difference can be estimated to be in the order of 25%. However, the experimentally observed GMI(60/30) did not significantly differ from GMI(30/15) being either slightly smaller (4 conditions) or equal (2 conditions), but definitely not larger (Fig. 2). Hence, a Bayesian fusion in linear space does not yield an appropriate description of our data.

Baysian interaction predicts a lawful relation between GMI and the variance of G _T

According to Fig. 4

$$\psi_{\rm p} = \psi_{\rm f} + \sigma_{\rm r}^{2}.$$

Because the addition of residual noise σ ²_r does not change the central value, Eq. 2 (cf. Discussion) implies

$$\bar \psi _p = w_0 \cdot \bar \psi _0 + w_{\rm s} \cdot \bar \psi _s $$

(7)

or, expressed in terms of the linearly coding signals v _p, v ₀ and v _s, given [A1]:

$$\bar v_{\rm p} = \bar v_0 ^{w_0 } \cdot \bar v_{\rm s}^{w_{\rm s}}.$$

(8)

Let $\bar v_{\rm s1}$ and $\bar v_{\rm s2}$ be the mean sensory velocity signals evoked by v _st1 and v _st2 = 2·v _st1 (e.g. 15 and 30°/s), and $\bar G_{\rm T1}$ and $\bar G_{\rm T2}$ the corresponding mean targeting gains. Combining Eqs. 5 and 8 and noting that $v_{\rm st}=\bar v_{\rm s}[\text{A1}]w$ we obtain

$$GMI = \frac{{\bar G_{\rm T2} }}{{\bar G_{\rm T1} }} = \frac{{\bar v_{\rm s1} ^{w_{\rm s}} \cdot \bar v_{\rm s2} }}{{\bar v_{\rm s2} ^{w_{\rm s} } \cdot \bar v_{\rm s1}}}.$$

Because w _s=1−w ₀, GMI can also be expressed in terms of w ₀:

$$GMI = ({{\bar v_{\rm s2} } \mathord{\left/ {\vphantom {{\bar v_{\rm s2} } {\bar v_{\rm s1} }}} \right. \kern-\nulldelimiterspace} {\bar v_{\rm s1} }})^{w_{0}} $$

or, because [A1] implies $\bar v_{\rm s2}/\bar v_{\rm s1} = 2,$

$$GMI = 2^{w_0}. $$

(9)

Combining Eqs. 3 and 4 and taking into account the residual variance by noting σ ²_f  = σ ²_p − σ ²_r (Fig. 4) w ₀ can be written as

$$w_0 = \frac{{\sigma _{\rm p}^2 - \sigma _{\rm r}^2 }}{{\sigma _0^2 }}$$

(10)

hence Eq. 9 becomes

$$GMI = 2^{\frac{{\sigma _{\rm p}^2 - \sigma _{\rm r}^2 }}{{\sigma _0^2 }}} $$

and, finally, solving for σ ²_p we obtain

$$ \sigma ^{2}_{{\text{p}}} = \sigma ^{2}_{0} \cdot \frac{{\log \{ GMI\} }} {{\log (2)}} + \sigma ^{2}_{{\text{r}}} $$

(11)

Equation 11 predicts a linear relationship between variance σ ²_p and the log transform of GMI. Because both σ ²_p and log{GMI} can be obtained from observed data, we can use Eq. 11 to check the compatibility of our data with the Bayesian hypothesis and estimate the unknown parameters σ ²₀ and σ ²_r by linear regression (Fig. 5). Because the estimate for the standard deviation of ψ₀ obtained in this way (σ₀ = 0.4) is not sufficiently small in the sense of [A1], its standard deviation in linear space was approximated by fitting the log⁻¹-transform of a Gaussian with σ = 0.4 by a new Gaussian (this does not touch the validity of the above equations as none requires ψ₀ to have small variance).

Based on the estimated values of σ ²₀ and σ ²_r and on Eq. 4 the variance (σ ²_s ) of the compound sensory signal can be obtained from

$$\frac{1}{{\sigma _{\rm s}^2 }} = \frac{1}{{\sigma_{\rm p}^2 - \sigma _{\rm r}^2 }} - \frac{1}{{\sigma _0^2 }}.$$

(12)

An expression for the mean default velocity $\bar v_0$ is obtained from Eq. 8:

$$\bar v_0 = \bar v_{\rm p}^{\frac{1}{{w_0}}} \cdot v_{\rm st} ^{ - \frac{{w_{\rm s} }}{{w_0 }}} $$

which, using Eq. 5 becomes

$$\bar v_0 = \frac{{v_{\rm st} }}{{C \cdot (\bar G_{\rm T})^{{1 \mathord{\left/ {\vphantom {1 {w_0 }}} \right. \kern-\nulldelimiterspace} {w_0 }}} }}.$$

(13)