Cross-orientation suppression in visual area V2

Object recognition relies on a series of transformations among which only the first cortical stage is relatively well understood. Already at the second stage, the visual area V2, the complexity of the transformation precludes a clear understanding of what specifically this area computes. Previous work has found multiple types of V2 neurons, with neurons of each type selective for multi-edge features. Here we analyse responses of V2 neurons to natural stimuli and find three organizing principles. First, the relevant edges for V2 neurons can be grouped into quadrature pairs, indicating invariance to local translation. Second, the excitatory edges have nearby suppressive edges with orthogonal orientations. Third, the resulting multi-edge patterns are repeated in space to form textures or texture boundaries. The cross-orientation suppression increases the sparseness of responses to natural images based on these complex forms of feature selectivity while allowing for multiple scales of position invariance.


Third-order model Gabors Orthogonal Gabors
Second-order reconstruction 0.6 Supplementary Figure 5: Softplus rectifier nonlinearity yields better predictive power than models with logistic function for second layer nonlinearity. Models using a softplus rectifier have higher correlations with observed responses from held out test set than equivalent models with a logistic function as the second layer nonlinearity. (Wilcoxon signed rank test, two-sided, n = 80, quadratic invariant p < 10 -10 , quadratic non-invariant p < 10 -3 , linear invariant p < 10 -2 , and linear non-invariant p < 10 -2 ).

Test of the algorithm on model neurons
To illustrate and test how the reconstruction algorithm works in practice, we created a model neuron whose response properties mimicked the salient aspects of neural responses in area V2.
The model included graded position invariance as well as selectivity to combinations of Gabor features at each retinotopic location. The Gabor features were included in pairs where all Gabor parameters were identical for the two features forming a pair except for the spatial phase, which differed by 90°. Two Gabors pairs were excitatory and one Gabor pair was suppressive, cf. Supplementary Fig. 1. The two pairs of excitatory features had identical parameters except for a spatial offset and had slightly different orientations such that they combined to form a curved line. The suppressive pair was centered over the intersection of the excitatory pair with an orientation orthogonal to the mean orientation of the excitatory pair. The joint action of these features can be summarized by plotting the resulting quadratic kernel J, shown in panel c. The steps involved in the construction of the quadratic kernel are schematized in panels a and b. The relative contributions of each grid location to the model neuron's firing rate was weighted according to a circularly symmetric Gaussian with a standard deviation of 1.75 pixels.
The reconstruction algorithm successfully characterized the feature selectivity and invariance properties of our model V2 neuron. By diagonalizing J and determining the number and magnitude of eigenvalues that make statistically significant contribution to it (see above subsection on Eigenvector significance for details), the algorithm correctly identified that the model neuron was selective for a 6-D stimulus space. Further, the algorithm correctly identified that four of these features were excitatory and two were suppressive based on the sign of the corresponding contributions (panel d). The subspace formed by the relevant features was also correctly identified, because the orthogonal representations of both the model and reconstructed features closely resembled each other, cf. Supplementary Fig. 1 e, h. The average reconstruction per dimension was 0.9578 ± 0.0019. The correlation in the firing rate between the true and reconstructed model was 0.956 ± 0.005. As a final step, we were also able to successfully recover the actual relevant features underlying the response properties of this model neuron. Because the Gabor features of the model are not orthogonal to each other, their orthogonal representation shown in panel h deviates from the original features shown in panel g. Fitting the reconstructed quadratic kernel J (from panel c) as arising from combinations of Gabor pairs, we were able to match the original Gabors shown in Supplementary Fig. 1 g up to the spatial phase of the Gabors. [The spatial phase of Gabor does not affect the neural firing because of the squaring nonlinearity and thus does not represent a relevant parameter of the model].
In Supplementary Fig. 2 we show how reconstruction quality is expected to change as a function of dataset size. Overall, reconstruction remains robust across two orders of magnitude in dataset size.
To further test the algorithm, we verified that cross-orientation suppression does not inherently follow from the optimization procedure ( Supplementary Fig. 3) We applied the reconstruction algorithm to the responses of a model neuron whose suppressive features did not have orthogonal Gabor orientation to the orientation of its excitatory features. The correct reconstruction of the relevant subspace by the algorithm (subspace projection of 0.957) indicates that cross-orientation suppression observed for V2 responses is not induced by the optimization procedure.