Interaural self-motion linear velocity thresholds are shifted by roll vection

Abstract

The otolith organs respond equivalently to changes in gravitational force due to head tilt and to changes in inertial force due to linear acceleration. It has been shown that the central nervous system (CNS) uses internal models of the laws of physics to distinguish tilt from translation. Models with these internal models predict that illusory tilt, if large enough, will be accompanied by an illusion of linear motion. To investigate this prediction, we measured interaural, self-motion, direction-detection thresholds in darkness and with roll optokinetic stimulation. Each lateral translation consisted of a single cycle of sinusoidal acceleration, after which subjects indicated whether they translated to the left or right. We found that the interaural direction-detection threshold measured during clockwise and counterclockwise optokinetic stimulation shifted in opposite directions relative to thresholds in darkness. Using a generalized linear model, we determined that this finding was statistically significant (P < 0.005) and is consistent with the prediction that illusory tilt should be accompanied by a non-zero neural estimate of linear velocity that, if large enough (supra-threshold), contributes to translation perception.

Appendix: GLM implementation

For the individual normal cumulative distribution (iNCD) fit, we implemented a generalized linear model (McCullagh and Nelder 1983; Collett 1991) using the function glmfit in Matlab 7.3 (The Mathworks). Based on published data using similar translational motion stimuli (Benson et al. 1986), we hypothesized that our perceptual responses follow a probit distribution, which is the inverse cumulative distribution function of a normal distribution; it is a common model used to describe responses that are binomial. It is worth noting that glmfit is implemented using an iterative re-weighted least squares algorithm that converges to the maximum-likelihood estimate for each regression coefficient.

For our optokinetic data, the predictor variable X has 12 columns and 420 rows. The first 10 columns—one per subject—indicate the subject and are referred to as indicator columns. For example, the first ten elements of a row that represents subject #3 would be [0, 0, 1, 0, 0, 0, 0, 0, 0, 0]. Column #11 indicates the interaural velocity (i.e., a number between −25 and 25 cm/s in 2.5 cm/s increments) for each trial. Column #12 indicates the optokinetic stimulation direction (1 for CW and −1 for CCW). Since we have 10 subjects, 21 levels of accelerations and 2 directions of optokinetic stimulation, the indicator matrix X must have 10 × 21 × 2 = 420 rows.

Since our data are binomial, Y is a 420-by-1 vector indicating the number of to-the-left responses. This will be 0, 1, 2, or 3 for 3 repeated trials for each direction of optokinetic stimulation. The percentage of to-the-left responses equals Y/n where n = 3 is the number of repeated optokinetic trials. The multiple regression model can be expressed in matrix notation by:

$$ {\text{norminv}}({Y}) = {X} \times {B} + {E} $$

(A.1)

where B is a 12-by-1 vector of regression coefficients (e.g., the 12th coefficient of regression corresponds to optokinetic stimulation direction), E is a 420-by-1 vector of residuals, and norminv is the inverse of a normal cumulative distribution function. Regression coefficients were obtained using the function glmval in Matlab 7.3.

Because we are using a probit model, the fitted models ${\hat{\text{Y}}}_{i,\varepsilon } /n$ are normal cumulative distribution functions defined for each subject (i) and direction of optokinetic stimulation (ε = 1 for CW and ε = −1 for CCW) by:

$$ {\hat{\text{Y}}}_{i,\varepsilon } \left( x \right)/n = 1/\sqrt {2\pi \sigma^{2} } \int\limits_{ - \infty }^{x} {\exp \left( { - \left( {\xi - \mu_{i,\varepsilon } } \right)^{2} /2\sigma^{2} } \right)} d\xi $$

(A.2)

where variables ξ and x are interaural linear velocity (in cm/s), and parameters σ and μ _i,ɛ are functions of the elements of the regression coefficient vector B = (b _j):

$$ \sigma = \frac{1}{{\sqrt {2\pi } b_{12} }}\quad {\text{and}}\quad \mu_{i,\varepsilon } = \frac{{ - b_{i} - \varepsilon b_{11} }}{{b_{12} }}. $$

(A.3)

We used a similar model for data in the dark except that there is no column for direction of the optokinetic stimulation. Therefore, the predictor variable X is a 147-by-8 matrix. The first 7 columns serve as subjects’ indicators and column #8 indicates the interaural velocity level. Similarly, Y is a 1-by-147 vector indicating the number of to-the-left responses (out of 10 repeated trials). The multiple regression model is similar to the one for data with optokinetic stimulation (Eq. A.2) with ε = 0.