The cortex can directly represent ``two-and-one-half dimensional'' axonal fields. By ``two-and-one-half dimensional'' we mean a discrete stack of two-dimensional continua; for example, we might have six continua corresponding to six layers in the cortex. (Although synaptic and dendritic fields are embedded in three-dimensional space, the complex structure of the dendritic tree gives them a more complex non-Euclidean topology, therefore the notion of dimension is not directly applicable to them.) Some fields are naturally two dimensional, for example, a light intensity field over the retina or a pressure field over the skin.
There are many cases where the cortex must represent fields
defined over more than two dimensions.
For example, since cells in VI are selective for orientation
as well as retinal position 
, the
activity fields are naturally three-dimensional,
.
Furthermore, there is substantial evidence
(surveyed, for example, in MacLennan 1991)
that they are sensitive to spatial frequency f as
well, so we actually have four-dimensional fields
.
In these cases, representation in the cortex requires that the field be reduced to two dimensions in a way that does as little violence to the proximity relations as possible. The simplest way to do this is to ``slice'' the field, as we might slice a pepperoni, and arrange the pieces in a plane. More generally, the field must be cut into ``nearly two-dimensional'' parts that can then be arranged systematically in a plane. This is one reason for the striate and columnar structure found in many brain areas.
Non-Euclidean fields are found in neuropil (the dense nets comprising the tangled dendritic trees of many neurons) and other places where the pattern of connections alters the effective distance between points of activity. Such fields may be defined over spaces with unusual (e.g. nonmetric) topologies since, for example, the distance a signal must travel in going from A to B may be different from the distance from B to A.