Sunday, September 7, 2008

What is Hierarchy? Revisited: The Structure of the Cortex

Extending this discussion, I'm still curious about the best way to describe the structure of the cortex in terms of either graph theory or as a data structure.

From what I know of graph theory, the cortex can best be described as:

1) A directed graph (meaning there is directionality between nodes, in this case the flow of information)
2) Cyclical (the flow of information is not in a uniform direction, but cycles or loops through at least some of the nodes)

In terms of data structure, any given node in the cortex can have:

1) 0, 1, or many peers
2) 0, 1, or many parents
3) 0, 1, or many children

Connections (or edges, using graph theory lingo) can occur between any two nodes in the system. However, there is a strong bias in connectivity: It tends to be local. Which means that while there are long-range connections in the cortex, and connectivity sometimes jumps levels in the hierarchy, the bulk of it is between immediate parents, children, and peers.

There does seem to be a bias toward fan-in as you move up the hierarchy. This means that there tend to be fewer nodes in lower levels of the hierarchy than higher levels (e.g. the primary visual cortex is bigger than the secondary visual cortex). But this again is a bias, not true in all instances.

So is there a word or phrase that best captures this particular kind of structure? Network seems too general and anarchic. You could just say it's a hierarchical network with a bias toward local connectivity. My advisor has used the term lattice or lattice hierarchy. I think those are interesting terms, though I'm not entirely sold on them.

Any thoughts?


Kenny Wyland said...

Hm, so what would you say the real difference is between a network and a lattice? Just the local preference for connectivity?

Derek James said...

Actually, I thought a network was more general than a lattice, that a network didn't say anything in particular about connectivity or ranking of elements. According to wikipedia, a lattice is a partially-ordered set, so it's usefulness comes from describing the fact that some elements are ranked either above or below others...I think.

Kenny Wyland said...

Ah, my bad, I didn't even go read the specific definition of a lattice. I was just thinking about it from my generic mental image of a lattice.

After reading a bit on wikipedia, I'd say that a network is more general than a lattice. A lattice appears to have an undirected repeating structure while a network implies a directed graph which can connect in any fashion.

How regular or repetitive are the neural structures? From what I learned about graphs in school and what I read on wikipedia about lattices just now, it seems that a network is the best description unless there is a high incidence of repeating structures.

Derek James said...

I honestly don't know much about the internal structure of most brain areas...I'm mostly interested in the hippocampus and neocortex, both of which exhibit a high degree of regularity (though that degree is still debated among neuroscientists).

Your neocortex is about 2-3mm thick, covering your entire brain. It is composed of 6 layers. Within these layers are small units called minicolumns, which span the entire height of the cortex, but are very small in diameter. Minicolumns grouped together form larger clusters.

Connectivity follows certain distinct patterns. Incoming connections from lower areas always terminate on cells in layer 4 (and to a lesser extent layer 6). Cells in layer 4 connect forward to cells in layers 2 and 3, which send their projections forward to regions in the same and opposite hemisphere. Feedback connections terminate mostly in the shallow layers, like layer 1.

There are differences between cortical areas, but to a first approximation, the structure is the same in all areas, and between mammalian species. Vernon Mountcastle has suggested this means all neocortex is computing essentially the same algorithm.

Right now we just don't know, but as for the neocortex, it is not an undifferentiated mass of random connectivity. It is highly structured.

Kenny Wyland said...

Well, in that case, lattice is probably a pretty good term.