Tag Archives: self-organizing

Entering “chora”, the infinitesimal place

There is a whole discussion around the key phrases “The map is not the territory” and “The map is the territory”. From the wiki entry on the map-territory relation, we learn that Korzybski‘s dictum “the map is not the territory” means that:

A) A map may have a structure similar or dissimilar to the structure of the territory,

B) A map is not the territory.

Bateson, in “Form, Substance and Difference” has a different take on this: he starts by explaining the pattern-substance dichotomy

Let us go back to the original statement for which Korzybski is most famous—the statement that the map is not the territory. This statement came out of a very wide range of philosophic thinking, going back to Greece, and wriggling through the history of European thought over the last 2000 years. In this history, there has been a sort of rough dichotomy and often deep controversy. There has been violent enmity and bloodshed. It all starts, I suppose, with the Pythagoreans versus their predecessors, and the argument took the shape of “Do you ask what it’s made of—earth, fire, water, etc.?” Or do you ask, “What is its pattern?” Pythagoras stood for inquiry into pattern rather than inquiry into substance.1 That controversy has gone through the ages, and the Pythagorean half of it has, until recently, been on the whole the submerged half.

Then he states his point of view:

We say the map is different from the territory. But what is the territory? […] What is on the paper map is a representation of what was in the retinal representation of the man who made the map–and as you push the question back, what you find is an infinite regress, an infinite series of maps. The territory never gets in at all.

Always the process of representation will filter it out so that the mental world is only maps of maps of maps, ad infinitum.

At this point Bateson puts a very interesting footnote:

Or we may spell the matter out and say that at every step, as a difference is transformed and propagated along its pathways, the embodiment of the difference before the step is a “territory” of which the embodiment after the step is a “map.” The map-territory relation obtains at every step.

Inspired by Bateson, I want to explore from the mathematical side the point of view that there is no difference between the map and the territory, but instead the transformation of one into another can be understood by using tangle diagrams.

Let us imagine that the exploration of the territory provides us with an atlas, a collection of maps, mathematically understood as a family of two operations (an “emergent algebra”). We want to organize this spatial information in a graphical form which complies with Bateson’s footnote: map and territory have only local meaning in the graphical representation, being only the left-hand-side (and r-h-s respectively) of the “making map” relation.

Look at the following figure:

In the figure from the left, the “v” which decorates an arc, represents a point in the “territory”, that is the l-h-s of the relation, the “u” represents a “pixel in the map”, that is the r-h-s of a relation. The relation itself is represented by a crossing decorated by an epsilon, the “scale” of the map.

The opposite crossing, see figure from the right, is the inverse relation.

Imagine now a complex diagram, with lots of crossings, decorated by various
scale parameters, and segments decorated with points from a space X which
is seen both as territory (to explore) and map (of it).

In such a diagram the convention map-territory can be only local, around each crossing.

There is though a diagram which could unambiguously serve as a symbol for
“the place (near) the point x, at scale epsilon” :

In this diagram, all crossings which are not decorated have “epsilon” as a decoration, but this decoration can be unambiguously placed near the decoration “x” of the closed arc. Such a diagram will bear the name “infinitesimal place (or chora) x at scale epsilon”.

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).