(version 1, will try to expand)
Imagine that you have an alphabet consisting of some letters . Now imagine another symbol; call it . Let us say that I know how to construct words; i.e., know how to construct sequences of letters, using only the alphabet. Now a friend of mine can interrupt me at any point of the sequence by shouting . Suppose now that I only know the letters and each time I try to spell a 5-letter word using these letters a friend of mine interrupts me by shouting the letter . If she is really aggressive, the outcome could be something like . If not I could say something like .
A word containing at least one interruption is called a root.
Now imagine that someone else listens to us. If I start saying the word ‘door’ and my friend interrupts me the listener can guess the following:
- I said
- I said
- I said
- I said
This set is called the combinatorial line of the root .
Speaking about the previous a little more strictly, let be a finite letter alphabet. Let be the new symbol. Words are considered as sequences of letters of the alphabets without containing the letter . Sequences containing at least one character are called roots. If in each such root we replace the with each letter of the alphabet we get a collection of words rooted by the specific root. A combinatorial line is the set of words that stem from simultaneous replacement each time of the characters by one of the alphabet letters.
Excerice: Suppose that there exists an alphabet consisting of 0 and 1. Calculate the number of combinatorial lines for sequences of length .
Now our alphabet is or . What I will do here is play a little bit with combinatorial lines and AAL coordinates. AAL labels are are neuroanatomical labels of the brain (in a brain-coordinate system) commonly used in fMRI. Here I used the 116 predefined labels and coordinates as found in the BrainNet Viewer tool. I will encode the areas based on a specific binary encoding. Obviously they way to assign each area to the aforementioned encoding is completely arbitrary.
For example I can have something like this:
- ; Hippocampus
- ; Posterior Cingulate
- ; …
The pair can be considered as a combinatorial line (with root ) whereas the pair cannot. What we are going to do is, for every entry in the encoding, we are going to search all other entries and see if as a pair they become a combinatorial line.
What I did for this post is to encode the areas based on their Euclidean distance from a reference point. For each area I calculated the Euclidean distance between its coordinates and the point. Then I sorted them in an ascending order and used a 8 bit binary encoding. Therefore the will be the area with smallest distance from the will be the area with somewhat bigger distance and so on.
What I did afterwards was to find all the combinatorial lines in this encoding for each area. Remember that since we are using binary encoding, a combinatorial line line will be a pair of entries, e.g. and could be the combinatorial line with root (there can be other roots for this pair). Therefore for each area I found all the combinatorial pairs that belong to the encoding.
Having all the combinatorial lines for each each area, I tried to plot some!
In the next pictures (you may need to click and zoom for a better resolution) I depict as the big yellow node the area that I considered. The other nodes stand for the areas that the combinatorial lines of the considered area consist of.
I was hoping for a default-mode network but didn’t see a clear picture of it : ). For the precuneus I saw some medial frontal and angular areas as well as some areas of the cerebellum. For the hippocampus I identified the caudate and some angular and orbital areas.
Question: Is there any truly connection between the topology of the brain functional networks and multi-dimensional combinatorial theorems (such as Hales-Jewett)?