Translational motion perception and vestiboocular responses in the absence of non-inertial cues

Abstract

Path integration studies in humans show that we have the ability to accurately reproduce our path in the absence of visual information. It has been suggested that this ability is supported by acceleration signals, as transduced by the otolith organs, which may then be integrated twice to produce path excursion. Vestibuloocular responses to linear translations (LVOR), however, show considerable frequency dependence, with substantial attenuation in response to low frequency translational motion. If otolith information were processed similarly by path integration mechanisms, the resulting signal would not be sufficient to account for robust path integration for stimuli typically used in such studies. We hypothesized that such behavior relies upon cognitive skill and transient otolith cues, typically combined with non-directional cues of motion, such as vibration and noise produced by the mechanics apparatus used to produce linear motion. Continuous motion estimation tasks were used to assess translation perception, while eye movement recordings revealed LVOR responses, in 12 normal and 2 vestibulopathic human subjects while riding on a sled designed to specifically minimize non-directional motion cues. In the near absence of such cues, perceptual responses, like the LVOR, showed high-pass characteristics. This implies that otolith signals are not sufficient to support previously observed path integration behaviors, which must be supplemented by non-directional motion cues.

Appendix

Post-hoc comparison of phase:

Statistical treatment of vector data, which contains both gain and phase information, is most conveniently carried out by converting to a rectangular coordinate scheme, and then employing multivariate analysis techniques (Calkins 1998). As described the text, gain (G) and phase (Θ) data are converted to the complex form X + iY (Fig. 8), where X = G·cosΘ, Y = G·sinΘ, and i is the imaginary operator (the square root of −1), and compared using MANOVA on the XY plane.

Fig. 8

Comparison of two vector populations (crosses and circles) using zero-intercept linear fits. The vectors shown (dark and light arrows) reflect the vector means of the respective populations. Dashed lines show the lines of best fit, and the light grey lines, the 95% confidence interval for the fits. Inset: Representation of vector data on complex plane. The magnitude is expressed as the length of the vector, and the phase as the rotation from the real axis

If a difference is shown, post hoc analysis can be used to test gain and phase. Gain comparison can be carried out using simple ANOVA techniques. Phase comparison is complicated by the substantial interactions between gain and phase: as the magnitude of a vector becomes small, the phase becomes more difficult to determine, and as the magnitude approaches zero, phase estimates can become meaningless. Thus, small vectors for which phases are determined with low precision can cause artificially large variance in phase, hampering statistical comparison.

The resulting data points (still represented in Cartesian coordinates) are then submitted to a zero-intercept linear regression, and a resultant slope that is found to statistically differ from zero (equivalent to a non-zero Y component) is then used as an indication of a significant change in phase angle.

To test for differences in phase, we took advantage of the relationship between the angle of the vector population and the slope of the line fit to the equation Y = aX; a linear regression fitting the real component of the vector population to the imaginary component, and forced to pass through the origin (i.e., a zero-intercept linear regression). Two (or more) vector populations can be fit to this equation, and the resulting slopes are then statistically compared, with significant difference indicating a difference in phase (Seidman 2005).

The equation yielding the estimate of slope $(\hat{a})$ in a simple zero intercept regression is:

$$ {\mathop a\limits^ \wedge } = \frac{{{\sum\nolimits_{i = 1}^n {y_{i} x_{i} } }}} {{{\sum\nolimits_{i = 1}^n {x^{2}_{i} } }}}, $$

and the confidence interval on the slope is given by

$$ {\mathop a\limits^ \wedge } - t_{{\alpha /2,n - 1}} {\sqrt {\frac{{{\text{MS}}_{{\text{E}}} }} {{{\sum {x^{2}_{i} } }}}} } \le a \le {\mathop a\limits^ \wedge } + t_{{\alpha /2,n - 1}} {\sqrt {\frac{{{\text{MS}}_{{\text{E}}} }} {{{\sum {x^{2}_{i} } }}}} }, $$

where 1-α is the confidence interval and MS_E is mean-square error.

Figure 8 shows the post-hoc comparison of two simulated data sets (black vs. grey) that are multivariate normally distributed in the complex plane. The mean vectors are shown as arrows extending from the origin. The zero-intercept line of best fit, shown as dashed lines, approach the angle of the vector mean. The 95% confidence intervals on slope are shown as dotted lines. For these particular simulated data sets, phase shows a significant difference, as indicated by clearly different slopes of the lines of best fit (P < 0.0001). Note that vectors that are small with respect to the remainder of the data set will not alter the slope of the lines of best fit by a substantial amount, regardless of their phase, as the residual for relatively small data points will remain small by virtue of proximity to the origin. In contrast, large-magnitude vectors of phase that is different from the remainder of the data set (i.e., phase outliers) can have dramatic effect on the estimated slope, but will also increase the size of the confidence interval because of increased mean-square error.

Testing on simulated data sets with multivariate normal distributions shows that results are quite similar to those of a direct statistical comparison of phase angles. However, as expected, the zero-intercept slope method is quite robust in the presence of phase outliers that are of small magnitude. Of course, a zero-intercept line extends in two directions from the origin, so care must be taken when comparing vector populations that approach 180° phase difference. As this method assesses differences in the slopes of vectors, and a vector has the same slope as a vector that differs in phase by 180°, this method will show no significant difference in this situation.

A complete regimen for statistical testing of vectorial data is thus:

• Representation of data in complex plane.

• MANOVA to determine difference between data sets.

• ANOVA or t test on magnitude.

• Test to see if either data set includes origin in complex plane, which would make phase meaningless.

• Test for difference in slope of zero-intercept fit.

• Check non-significant differences for 180° phase shift, which this method will not detect.