Monday, July 27, 2009

Neurons and functions

I've been reading Bickel's Philosophy and Neuroscience, A Ruthelessly Reductive Account, and today I've come across a holy grail.

Individually, neurons are performing thresholded temporal summation. This I've known for a while (and I learned the explicit mechanisms for it in Bickel's class last quarter). As compelling as the evidence for these mechanisms is, I hadn't yet heard any good explanation of how this temporal thresholded summation acts in concert with other cells doing the same thing to result in complex memories and actions (though Jeff Hawkins has some insight, from a different approach).

What I learned from Bickel's book today is this: in groups, neurons are performing vector subtraction.

Pretty neat theory, but what's amazing is that Bickel demonstrates this empirically with saccade-sequencing (in two dimensions). That is to say: the instructions for moving the eyes from one point in two dimensional space (say <3,10>) to another point (say <12,8>) are found by summing the negative of the first point and the second point (thus, the vector <9,-2> is found, which are the instructions for moving from the first point to the second point in a straight line). Where does the negative portion come in? Separate sets of neurons fire before and after a saccade, their outputs are always opposed. Thus, after the first the move to <3,10> is executed, a representation of its opposite fires: <-3,-10>. By summing this opposite coordinate with the next coordinate <12,8>, the needed path is obtained (again, <9,-2>).

Bickel posits that this may be a general model for all neural processes involving sequencing. This would include body movement (in three dimensions) and speach (in how many dimensions?).

My question: how are things (like speech) that don't have obvious dimensional properties dimensionalized? What is the dimensionality of verbal space? (I think that can be re-phrased as "how many variables does speach involve?" ) Is it consistent across human individuals? Is it something fixed and absolute (like three dimensional space) that humans have to discover through experience and experimentation? Has the experimentation been done over evolutionary time, so that the structure of language-space is encoded in our DNA? Or is it non-absolute and variable between people with different levels of education and methods of thought?

I think the former is Chompsky's position (as well as Baum's), whereas the latter seems to be more intuitively plausible given the understanding that cortical material is uniform (ala Hawkins). Godel's essential incompleteness also seems to suggest that there is no ultimate, perfect space of language, since reflecting on the existing space creates a new dimension as many times as one cares to do the reflection, infinitely.

Maybe a simpler question: how many dimensions are neural systems capable of representing? Do three-dimensional movements get collapsed into two? Or are there arbitrarily many superflous, greater dimensions involved?

[Update: Churchland speculates that one dimension per neuron can be represented, plus gradients for synaptic weights and firing rates. So the representative capacity is somewhere above 10^100 (a googol) distinct concepts... greater than the number of elementary particles in the universe. Mind-bender, eh? That's not to say that we actually have that many distinctly stored concepts, just that that's the theoretical capacity. The truth is that the fine-grained differences are probably not important]