Dimension reduction, generalized deformable models and the development of ocularity and orientation

It has been proposed that the spatial organization of cells in visual cortex satisfies an abstract optimization criteria based on a smooth mapping criterion. We apply this criteria to the development of ocularity and show that it can be expressed in terms of a generalized deformable model. From this model we can derive two alternative algorithms, one being an elastic net and the other being of Hebbian form. This shows a surprising mathematical relation between the models of Goodhill and Willshaw (1990, Network, 1, 41-59) and Miller et al. (1989, Science, 245, 605-615), even though their motivations are very different. Applying the same techniques to the development of orientation shows mathematically that a variant of the dimension reduction criterion (Durbin & Mitchison, 1990, Nature, 343, 644-647) can be closely approximated by an elastic net.