A major point of interest for
INI-ots one and all is the local arrangement and connectivity of neurons in the
neocortex. But I'm working right now at
Kinderspital Zurich, so I won't get into any of that nonsense. What I'm going to talk about is the
long-range connectivity of neurons. And
then I'll get a bit meta, but you don't have to read that part.
Duncan Watts and Steven Strogatz
(whose columns for the New York Times are a much better read than this)
published their original paper on so-called “Small World Networks” in
1998. Before I explain what they are,
I'll explain where they are.
Everywhere. For those who remain
unconvinced, more compelling examples are below. A small-world network is an attempt to mesh
two common network types, those that are completely random, and those that are
entirely ordered. Many real-life
networks tend to fall somewhere in between.
The brain is a good example of this: though most connectivity is local,
long-range projections are what make quick computations possible. Computations in a brain comprised of purely
local connections would take so long as to make the incoming information
irrelevant in the first place. Lions as
clumsy as cows would stumble after humans lumbering at a snail's pace, in a
scene more reminiscent of Absinthe than Attenborough.
Watts and Strogatz's contribution
was the idea that a certain amount of randomness in the organization of their
network could decrease the distance between nodes without adding a noticeable
detriment to the local clustering of the network. The sense behind this is apparent: removal of
a few short connections rarely adds more than one new link between neighboring
nodes, while it shortens the number of links between distant clusters
logarithmically, comparable to the advent of the freeway/motorway. Whereas one originally had to take whichever
local roads were available, being able to drive to a high-speed thoroughfare,
even if its access point is in the wrong direction from the start, greatly
decreases the time to get across town.
Even if it cuts off a few local streets (apologies to Arthur Dent).
To prove that this concept is not
limited to the famed “Six Degrees of Separation” theory, they applied this to
collaborations between film actors (Six Degrees of Kevin Bacon seems to be an
overestimate; 3(.65) degrees would suffice on average), connectivity between
generators, transformers, and substations on a power grid, based on the power
lines connecting them, and, more notably, synapses and gap junctions in the
flatworm C. elegans. All showed
the same principles, namely, a reasonably short average path length, with high
amounts of clustering between neighbors.
Principles characteristic of small-world networks. While we have yet to map every connection in
the human brain (a bit more difficult with more than the 302 neurons of C.
elegans), it is nevertheless extremely slick (sorry) that a real-life neural network can be described
by such a simple model.
Figure
1: The left-most graph features only local connectivity. By exchanging one close connection for one
random connection, the average path length decreases dramatically, but the
local clustering remains high. (Creative
Commons Licensed from
http://en.wikipedia.org/wiki/File:Watts-Strogatz-rewire.png)
The gamma rhythm, which many, though
not all, link to our ability to consider detached visual percepts (like your
ability to recognize my feet as being associated with my face, even if the
stuff in the middle is obscured) as being part of the same object, is too fast
to be possible without such long-range connections. Furthermore, small-world networks have been
shown to be bistable, displaying both stable loops across their long-distance
connections while also being capable of quiet behavior. Memory consolidation and retrieval is often
hypothesized to function using such networks; high network activity levels
first imprint a memory in the synaptic strengths of a loop, and the network
then lies quiescent until activity causes the memory to be recalled. So this small-world stuff might be pretty
big.
I promised that this would get a bit
meta, so I'll finish with one last example.
As a tongue-in-cheek tribute to the mathematician Paul Erdős, the Erdős
number commemorates Erdős's penchant for collaboration by giving an Erdős
number of 1 to all of his direct collaborators, a 2 to their collaborators, and
so on. While most scientists haven't
worked on graph theory (Erdős and one of his collaborators, Alfréd Rényi, came
up with a more random precursor to Watts and Strogatz's model), a very large
and even more eclectic spread of scientists have an Erdős number, and it's
average (per the Erdős number project website) is a mere 4.65, matching the
short path length of a random graph, and the high clustering (after all, not
many neuroscientists with Erdős numbers collaborate with the Erdős-endowed
political scientists) of a tightly ordered graph. Just 4.65 links between most scientists, and
one of the forefathers of this concept itself.
A very small world indeed.
Citations:
The
original paper:
D.J.
Watts and S.H. Strogatz. Collective
dynamics of 'small-world' networks. Nature 393, 440-442 (1998).
An
introduction to the Temporal Binding hypothesis:
Wolf
Singer (2007) Binding by synchrony. Scholarpedia, 2(12):1657.
And
its detractors:
Shadlen,
MN and Movshon, JA (1999). "Synchrony unbound: a critical evaluation of
the temporal binding hypothesis." Neuron, 24: 67-77.
A
quick introduction to bistability in small-world networks:
Cohen,
Philip. Small world networks key to memory. New Scientist. 26 May 2004.
Steven
Strogatz's Really Quite Well-Written Series, On the Elements of Math:
http://topics.nytimes.com/top/opinion/series/steven_strogatz_on_the_elements_of_math/index.html
The
Erdős number project:
http://www.oakland.edu/enp
