What is the Goal of Complex Cell Coding in V1?

A long standing question of biological vision research is to identify the computational goal underlying the response properties of sensory neurons in the early visual system. Some response properties of visual neurons such as bandpass filtering and contrast gain control have been shown to exhibit a clear advantage in terms of redundancy reduction. The situation is less clear in the case of complex cells whose defining property is that of phase invariance. While it has been shown that complex cells can be learned based on the redundancy reduction principle by means of subspace ICA [Hyvarinen Hoyer 2000], the resulting gain in redundancy reduction is very small [Sinz, Simoncelli, Bethge 2010]. Slow feature analysis (SFA, [Wiskott Sejnowski 2002]) advocates an alternative objective function which does not seek to fit a density model but constitutes a special case of oriented PCA by maximizing the signal to noise ratio when treating temporal changes as noise. Here we set out to evaluate SFA with respect to two important empirical properties of complex cells RFs: (1) locality (i.e. finite RF size) and (2) an inverse relationship between RF size and RF spatial frequency. To this end we use an approach similar to that employed by [Field 1987] for sparse coding. Instead of single Gabor functions, however, we use the energy model of complex cells which is built with a (quadrature) pair of even and odd symmetric Gabor filters. We evaluate the objective function of SFA on the energy model responses to motion sequences of natural images for different spatial frequencies and envelope sizes, with patch sizes ranging from 6464 to 512512. We find that the objective function of SFA grows without bound for increasing envelope size (see Figure, blue line). Consequently, SFA by itself cannot explain spatially localized RFs but would need to evoke other mechanisms such as anatomical wiring constraints to limit the size of the RF. It is unlikely, however, that such anatomical constraints are able to reproduce the inverse relationship between RF size and spatial frequency. 64x6 4 2 56x256 512x512 0 1 2 3 4 5 6 Patch size in pixels optimal envelop width/wavelength ICA SFA Range of physiological data [Ringach 2002] In contrast to SFA, the objective function of subspace ICA yields a clear optimum for finite envelope sizes, regardless of assumed patch size (see Figure, red line). In particular, the optimum envelope size is inversely proportional to spatial frequency — just as observed for physiologically measured RFs in primary visual cortex of cat [Field Tolhust 1986] and monkey ([Ringach 2002], histogram see Figure). We conclude that SFA fails to reproduce important features of complex cells. In contrast, the envelope size predicted by subspace ICA lies well within the range of physiologically measured receptive field sizes. As a consequence, if we interpret complex cell coding as a step towards building an invariant representation, the underlying algorithm is more likely to resemble a sparse coding strategy as employed by subspace ICA than the covariance based learning rule employed by SFA