# Towards a theory of striate cortex

Zhaoping Li and J.J. Atick

We explore the hypothesis that linear cortical neurons are concerned with building a particular type of representation of the visual world --- one which not only preserves the information and the efficiency achieved by the retina, but in addition preserves spatial relationships in the input --- both in the plane of vision and in the depth dimension. Focusing on the {\it linear} cortical cells, we classify all transforms having these properties. They are given by representations of the scaling and translation group, and turn out to be labeled by rational numbers $(p+q)/p$' ($p, q$ integers). Any given $(p,q)$ predicts a set of receptive fields which come at different spatial locations and scales (sizes) with a bandwidth of $\log_2[(p+q)/p]$ octaves, and, most interestingly, with a diversity of $q$' cell varieties. The bandwidth affects the trade-off between preservation of planar and depth relations, and, we think, should be selected to match structures in natural scenes. For bandwidths between $1$ and $2$ octaves, which are the ones we feel provide the best matching, we find for each scale a minimum of two distinct cell types that reside next to each other and in phase quadrature, i.e., differ by $90^o$ in the phases of their receptive fields, as are found in the cortex, they resemble the even-symmetric'' and odd-symmetric'' simple cells in special cases. An interesting consequence of the representations presented here is that the pattern of activation in the cells in response to a translation or scaling of an object remains the same but merely shifts its locus from one group of cells to another. This work also provides a new understanding of color coding changes from the retina to the cortex.

Published in Neural Computation 6, 127-146 (1994).