All maps, whether they are used in the brain or not, represent a mathematical function (v = f(u)) transforming one points in space (the domain, u) to another (the codomain, v). For a map to be accurate, it must be continuous, without any breaks. Also, every pair of nearby points must correspond to two nearby points in the codomain. For example, take the cities Sheffield and Leeds. They represent two geographical points. For a map to be accurate, Sheffield and Leeds on a map must correspond proportionally in terms of distance, direction, etc. with Sheffield and Leeds in real life. The map of South Yorkshire would be the codomain and the real cities would be the domain. In the visual system, these points could be two retinal points (domain) accurately reflected onto the striatal cortex (codomain). In addition, a map of the brain must specify direction. Direction here is not used in the common sense, direction refers to continuity between domain and codomain. For example, mapping from retina to the striate is actually discontinuous; however, mapping from each half of the retina to the striate cortex is continuous, giving rise to the retinotopic map. If you map against the specified direction, it is called inverse mapping. An example would be mapping from the cortex to the retina, which is discontinuous. This is because most nearby points in the striate cortex correspond to nearby points in the retina; however, if two striate points are located on different ocularity stripes, inverse mapping is discontinuous (Blasdel, 1992).
Scientists argue that the reason the striate cortex maintains the retinotopic map is because it economises the length of nerve fibres. Nerve fibres are necessary for inter-hypercolumn communication, and if neighbouring points were spread out, wiring would become chaotic. Of course, mapping does not occur only in the visual system. The brain has a myriad of maps including ones for auditory, touch and motor output and many copies of these maps exist. All of these maps, like the retinotopic map must be continuous, suggesting spatial organisation is key to a healthy, functioning brain. Unfortunately, because there are so many features of the visual system that need to be represented in the map, singularities arise.
Singularities are jumps in continuity (Frisby and Stone, 2010), and they are the result of the packing problem. To put it simply, the brain wants to pack all features of the visual system into the brain; to maximise efficiency, the brain wants all similar variations of a feature in one place. In other words, the brain wants continuity. However, as was discussed with the binding problem, there are far too many variations and features of our visual system to account for them all in every possible detail. Hence, some continuity must be compromised (Hubel, 1981). The brain’s map of the visual system is continuous with respect to retinal position, but the map is discontinuous with respect to orientation. However, the brain still attempts to keep similar preferred orientations close together to maximise efficiency to the limited extent it can (Frisby and Stone, 2010).
Another way of thinking of singularities and the packing problem is in terms of parameters. Two points, as discussed above, define each retinal position otherwise known as position parameters (x, y). Correspondingly, each retinal point must correspond to a point on the cortex (x’, y’). Even though our cortex exists in three dimensions, there is still a finite amount of space to store information and parameters limit us to representing information in 2D. As orientation brings its own parameter (theta), the brain has to represent x, y and theta in 2D. This cannot be done without introducing discontinuities in at least one of the parameters because we are limited to two parameters. As the cortex must maintain a smooth map of the retinal map, discontinuities must be introduced into the representation of orientation. Based on this information the packing problem can be redefined in terms of parameters. The packing problem arises from attempting to pack all three dimensions of a 3D parameter space into a 2D one. As a solution to the packing problem, the cortex treats two of the parameters with varying priority, in this case the domain and codomain. Low priority is given to orientation, hence singularities.
As singularities exist for orientation, the topological index was introduced to describe the number of singularities. Specifically, the index tells us how orientation varies as we move around the centre of a singularity (Frisby and Stone, 2010). This can be done by drawing a circle around a singularity and moving clockwise. If the underlying representation orientation changes clockwise, the singularity is positive; however, if the orientation changes anticlockwise, the singularity is negative. Singularities in the striate cortex rotate no more than 180 degrees, so the singularity variable is always between + – ½. Hypothetically, a pinwheel is a full rotation, with an index of +1. To date, now pinwheels have been found in our visual cortex. Tal and Schwartz (1997) found that for any neighbouring singularities, you could usually draw a smooth curve between them. The remaining cells form columns along the curve with the same orientation preference (iso-orientation). In addition, Tal and Schwartz confirmed that nearby singularities have the same topological indices with opposite signs.
In addition to orientation, ocularity is another parameter that has to be represented in the striate cortex. Ocularity refers to the extent in which cells respond to our eyes. The brain as ocularity stripes meaning columns in the striate alternate between monocular and binocular cells. The stripes suggest that the brain wants to ensure that pairs of L and R stripes process every part of the visual field. Unfortunately, adding another feature parameter only furthers the packing problem. Researchers have found that the brain maximised economy by ensuring that each iso-orientation domain in the orientation maps tends to cover a pair of L-R ocularity columns; in other words, each orientation is represented for each eye (Hubel and Wiesel, 1971). Furthermore, the brain needs to perceive lines of different widths. The brain has solved this problem by having cells tuned to the same orientation, sensitive to various widths of spatial frequencies. As with orientation, representation of spatial frequency is continuous except for some singularities. Lastly, directionality is packed together with orientation. Except for sudden changes in direction (180 degrees), the direction map is continuous. It overlays the orientation map; however, it does not effect the continuity of orientation as orientations defines two possible directions (Frisby and Stone, 2010).
Fortunately, colour does not add to the packing problem! This is because colour is represented exclusively at the centre of orientation singularities. At the centre of singularities, cells have no preferred orientation, so colour does not add any parameter. To fully understand how orientation, ocularity and colour come together, the polymap was constructed to show an overlay of all the parameters. Based on observations from a polymap, in addition to the specific wiring of the brain, some scientists argue that the cortex is not really trying to solve the packing problem. All the maps try to do is to minimise the amount of wiring the brain needs to employ (Frisby and Stone, 2010). In fact, Swindale et al. 2000 found that the cortex does attempt to maximise coverage, and if any small changes were made to the current mapping system, wiring efficiency would be reduced.