Maps in the brain: fact and explanations

From wikipedia

Retinotopy describes the spatial organization of the neuronal responses to visual stimuli. In many locations within the brain, adjacent neurons have receptive fields that include slightly different, but overlapping portions of the visual field. The position of the center of these receptive fields forms an orderly sampling mosaic that covers a portion of the visual field. Because of this orderly arrangement, which emerges from the spatial specificity of connections between neurons in different parts of the visual system, cells in each structure can be seen as forming a map of the visual field (also called a retinotopic map, or a visuotopic map).

See also tonotopy for sounds and the auditory system.

The existence of retinotopic maps is a fact, the problem is to explain how they appear and how they function without falling into the homunculus fallacy, see my previous post.

One of the explanations of the appearance of these maps is given by Teuvo Kohonen.

Browse this paper (for precise statements) The Self-Organizing map , or get a blurry impression from this wiki page. The last paragraph from section B. Brain Maps reads:

It thus seems as if the internal representations of information in the brain are generally organized spatially.

Here are some quotes from the same section, which should rise the attention of a mathematician to the sky:

Especially in higher animals, the various cortices in the cell mass seem to contain many kinds of “map” […] The field of vision is mapped “quasiconformally” onto the primary visual cortex. […] in the visual areas, there are line orientation and color maps. […] in the auditory cortex there are the so-called tonotopic maps, which represent pitches of tones in terms of the cortical distance […] at the higher levels the maps are usually unordered, or at most the order is a kind of ultrametric topological order that is not easy interpretable.

Typical for self-organizing maps is that they use (see wiki page) “a neighborhood function to preserve the topological properties of the input space”.

From the connectionist viewpoint, this neighbourhood function is implemented by lateral connections between neurons.

For more details see for example Maps in the Brain: What Can We Learn from Them? by Dmitri B. Chklovskii and Alexei A. Koulakov. Annual Review of Neuroscience 27: 369-392 (2004).

Also browse Sperry versus Hebb: Topographic mapping in Isl2/EphA3 mutant mice by Dmitri Tsigankov and Alexei A. Koulakov .

Two comments:

1. The use of a neighbourhood function is much more than just preserving topological information. I tentatively propose that such neighbourhood functions appear out of the need of organizing spatial information, like explained in the pedagogical paper from the post Maps of metric spaces.

2. Just to reason on discretizations (like hexagonal or other) of the plane is plain wrong, but this is a problem encountered in many many places elsewhere. It is wrong because it introduces the (euclidean) space on the back door (well, this and using happily an L^2 space).

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