The main question asked in this paper concerns cross-orientation interaction. While the grating orientation
θ 1 in
Figure 1A has a direct effect on key-presses, is it also true that a preceding orientation
θ 2 influences key pressing through its interaction with
θ 1? To answer this question, we constructed two-dimensional probability plots such as the one at the top of
Figure 4. The horizontal axis gives orientation
θ 1 and the other axis gives the orientation
θ 2, which immediately precedes
θ 1 (that is, with an inter-stimulus interval of 33 ms). The gray level at point (
θ 1,
θ 2) indicates the probability that a key-press is preceded by the consecutive orientations
θ 2 and
θ 1. A probability scale for the gray levels is shown at the right.
The two-dimensional plot is dominated by a bright spot at its centre, indicating that a key-press is likely to be preceded by consecutive gratings that both align with the target. This is hardly surprising as the two gratings could independently lead to a key-press. What is more interesting is the possibility that facilitatory interactions between the two gratings lead to a probability higher than that expected of their independent effects. The probability expected if
θ 1 and
θ 2 have independent influences on key pressing was calculated as follows. The influence of
θ 1 acting alone can be calculated by summing the probabilities across all values of
θ 2. The resulting marginal density,
p 1(
θ 1), is shown underneath the two-dimensional plot. Similarly, the marginal density,
p 2(
θ 2), was obtained by summing across all values of
θ 1 and is shown at the right of the two-dimensional plot. The probability that a key-press will result from the independent effects of
θ 1 and
θ 2 is the product of the marginal densities (Papoulis & Pillai,
2002):
The independence model is shown in the middle plot of
Figure 4, and the difference between the observations and the independence model is shown at the bottom. If
θ1 and
θ2 had independent effects on a key-press, the difference plot would be uniformly gray. The presence of light and dark therefore indicates cross-orientation facilitation and suppression, respectively.
Close inspection of the difference plot in
Figure 4 shows that there is a pattern to the light and the dark areas: The light areas run along the main diagonal, and the dark areas along the minor diagonals. Each diagonal represents a constant orientation difference,
θ 1 −
θ 2, indicating that it might be easier to visualize the plot by graphing the probabilities obtained with constant orientation differences. This is done in
Figure 5, which shows the observations and the independence model along two diagonals,
θ 1 −
θ 2 = 0° and
θ 1 −
θ 2 = 90°. The result is quite different for these two cases. In the first case,
θ 1 and
θ 2 are equal, and the observed probabilities exceed the predictions of the independence model. The two gratings are more likely to be aligned than is expected from their independent effects, indicating that alignment leads to facilitation. When the two gratings are misaligned, however, the observed probabilities fall below the independence model, indicating cross-orientation suppression.
The difference between the observations and the independence model can be shown more compactly and completely by summing probabilities along each and all diagonals. The results of this calculation are shown in
Figure 6. Each point in
Figure 6A was obtained by summing the probabilities in the top or the middle plots of
Figure 4 along a single diagonal, where the orientation difference defining the diagonal is given on the horizontal axis. The observations exceed the independence model when
θ 1 and
θ 2 are equal, or nearly so, and fall below the independence model when
θ 1 and
θ 2 differ markedly. The difference between observations and independence model is provided explicitly in
Figure 6B for three targets, the mean across targets is shown in
Figure 6C, and data for all five subjects are shown in
Figure 6D. The magnitude of the cross-orientation interaction differs between subjects, but the results agree in that there is facilitation between the two gratings when their orientations differ by no more than 36° and that there is cross-orientation suppression otherwise.
One interesting feature of the curves in
Figure 6D is that they generally decline monotonically away from the peak, with no upturns at large orientation difference. In particular, the curves do not have a “Mexican hat” profile. This in turn suggests that the suppression is constant across orientation differences, that is, that the suppression is non-oriented. We used this finding to model the facilitation curve, as shown in
Figure 7. Here, facilitation is assumed to be a Gaussian function of orientation difference and to be added to a constant suppression value. The goodness-of-fit of model to observations can be quantified with the root-mean-square error that, for this subject, was 0.0011. Root-mean-square errors for the other four subjects were slightly less than this value, indicating that their fits were no worse than that shown in
Figure 7. The most interesting results from data fitting were the model parameters: The bandwidth at half-height averaged across five subjects was 38°, and the suppression constant differed significantly from zero (one-tailed
t-test,
P = 0.037).