Supplementary material

Within channels’ PC analysis

Percentages of the variation in the original set of variables which the principal components account for, equal the normalized eigenvalues of the covariance matrix (Tr is for transpose) where . represented eye, trunk and head-on-trunk angular displacement during the mth trial, and their arithmetic means respectively. By multiplying the array of original variables () with the array of eigenvectors () of we get , where is the array of the PCs. These are organized in columns of an increasing order of importance, such that the 3rd column contains the 1st principal component accounting for the of the variance of the original variables. In general, the jth column of contains the jth principal component accounting for the of the variance of the original variables. Then the contribution of the original variable to the principal component is . Eigenvalues and their corresponding eigenvectors were computed by using the MATLABTM function eig.

For each trial, the eigenvalue and weight are calculated. Further, by averaging across all trials the mean eigenvalue and mean weight have been calculated.

‘Within trials’ PC analysis

Having a matrix of covaried traces , all assigned to the same channel with covariation matrix , corresponding vector of eigenvalues and matrix of eigenvectors , PCA creates a new matrix of orthogonal vectors called PCs. PCs are derived in a decreasing order of importance so that the first, the second and generally the jth PC are the vectors and respectively. The percentage of variance explained by the jth PC is equal to the normalized eigenvalue .

The above method has been applied separately on the traces of gaze, eye-in-orbit, head-in-space, head-on-trunk and trunk-in-space displacements (mean value from its trace was removed) as well as their velocities.

PCs’ statistics

Bartlett’s test of sphericity was used to validate the hypothesis that a principal component was significantly different from all those of lower importance, i.e. . This has to be accepted if the test statistic function is greater than the critical value of X2 distribution with degrees of freedom (Cooley and Lohnes, 1971). As Bartlett’s test is insufficient for testing the significance of the last two principal components, i.e. those of the lowest importance (j = 1, 2), a test for proportion of total variance was applied post hoc. A desired proportion of total variance at 99.0% was selected and excluded all principal components which accounted for less than 1% of the total variance. In PC analysis within channels (3 channels), one sample independent t-test was performed for the arithmetic means of variance proportion of 2nd and 3rd PCs and tested whether they were greater than 1%. All PCs were tested whether they were normally distributed according to a one-sample Kolmogorov-Smirnov non-parametric test.

Multilinear Regression

Average eye-in-orbit (), head-on-trunk () and trunk () displacement trajectories across all N-trials, separately for each condition, were calculated. Any point of trunk displacement trajectory () is represented as a linear combination of,: where are fitting errors. This can be written as a matrix equation by introducing vectors ,, and the matrix : . The estimate of which minimizes the sum square error, as in multiple linear regression, is then given by . The new estimate of trunk displacement trajectory is and, in principle, the 3D-vector lies on a surface-plane. The coefficient is an estimate of the goodness of fit of the surface-plane to the vector of displacement trajectories . Finally, are the angles of orientation (in radians) of the surface-plane with respect to trunk-in-space/head-on-trunk and trunk-in-space/eye-in-orbit planes respectively (Fig. 5).

FIGURES in Supplementary Material

Relationship between mean gaze components displacements in the position-space defined by the contributing angles of eye-in-orbit, head-on-trunk and trunk-in-space rotations, separately for the outbound trials, inbound trials, single-step gaze shifts and trials to visual targets. Data stem from primary gaze shifts of at least 250, 333, 417 and 250 ms duration respectively. Each point represents the mean 3-D vector with coordinates eye-in-orbit, head-on-trunk and trunk-in-space (samples at the rate of 240 / s). They lie approximately on a single plane (goodness of fit > 98%, Table 4 and Fig. 5). Implementation in MATHEMATICA allows rotation of the coordinate system