indigorose1
commented
9 years ago Interesting concept, I can shed a little light on the biological accuracy. So far, there is evidence that neurons interconnect themselves massively with all the neurons in a region that secrete one of a set of chemical signals. Then, as information flows through the system, the neurons that are connected correctly survive and their connections grow stronger and the neurons that aren't get their connections weakened and eventually pruned and then the neurons die.
mrjiaruiwang
I am rather inexperienced with graph theory so I don't know where this line of thought leads, but when I was toying around with how to produce a generative model with a null hypothesis kinda-sorta-similar to what Bock et al. were trying to look at, I came up with this brain fart which may or may not be interesting. Basically, the assumption is that excitatory neurons don't "collude" and mutually cause each other to synapse on the same target at the same time; rather, one neuron attaches first, and then "influences" subsequent "similar" neurons to attach. (Is this biologically accurate? No idea!)
(Also, I call the circle as a set "T" for the record; bad math habit.)
So we start with N vertices, and categorize each of them as "E" or "I" via N iid Bernoulli(p_c) variables. We'll call the set of all E vertices E, and the set of all I vertices I, and say we have some indexing on each of these sets. Cool. Next, we will assign to each vertex E_i in E a tuning number A_i uniformly on T (or some discretization of T, w/e; and, iid, of course). Cool.
Now, we proceed to connect the Es to the Is in the following scheme:
For each i in E -- For each j in E : j < i -- -- For each k in I : E_j ~ I_k -- -- -- Connect E_i ~ I_k with probability f( d( A_i, A_j ) ) -- -- End For (k) -- End For (j) -- For each k in I -- -- Connect E_i ~ I_k with probability p0 -- End For (k) End For (i)
Where: d: Distance on T (i.e. smallest angle between points on the circle) f: The "attachment function". This is where the similarity-dependence happens; i.e., f(0) > f(pi/2) = 0. p0: The base attachment probability without preference. (Note that once something is connected, it's connected, even if it gets connected again. This model could probably support multiply-connected vertices, but ... eh.)
The null hypothesis for Bock et al., in this scheme, is H0: f(x) = k \in [0, 1] i.e., that the distance between tuning of neurons doesn't influence their attachment.
(Note that, for k != 0, the events (E_i ~ I_m) and (E_j ~ I_m), i != j are not independent, because there is still a bias (k) toward attaching to things that have already been attached to, regardless of tuning. You could, I guess, restrict this to a smaller null hypothesis: H0: f(x) = 0 This, however, would be making a stronger statement about the data, namely that no preferential attachment occurs.)
Symbolically deriving what this means mathematically would probably be a pain in the behind; but, you definitely can generate graphs under this scheme, which means that, with a good enough test statistic (which I can't think of at the moment) you could, theoretically, use the bootstrap / nonparametric methods to give you some kinda-sorta-meaningful p-value.
Please let me know if this is nonsense.