Projection of topographic maps from curved brain surfaces into a plane

The environment is represented in the brain by topographic maps. These maps and their interrelationship form a dominant feature in the brain, but unfortunately, they are often projected onto surfaces which are variously curved. It would be useful to be able to compare the topography on these surfaces quantitatively with the origin of the map. This comparison would be possible if the curved brain surfaces could be represented in planes. Therefore, it was necessary to develop a procedure for unfolding maps on convex structures like the retina and the optic tectum of the chick. For this purpose a unit sphere was constructed whose origin was close to the centre of gravity of the curved brain structure. The surface of this structure was then projected into the sphere. From its representation on the spherical surface it was unfolded into the plane using a polar azimuthal radially equidistant projection. Our method can safely be applied to convex structures only. However, it is not limited to the visual system. The mathematical procedure is formulated in such a way as to be easily implemented. It is exemplified by experiments on the optic tectum and the developing retina of the chick.