Computing correlations analytically
Neurons in the brain, especially nearby neurons, are often correlated, in the sense that activity in one set of neurons partially predicts the activity in another. These correlations are of interest because they play a major role in both representation and computations. Importantly, the precise role they play depends in detail on the correlational structure (for a review, see Averbeck et al., 2006). It would, therefore, be highly desirable to know what that structure is. Directly measuring it would seem desirable, but this is hard because the amount of data one needs is exponential in the size of the population (and large populations is the regime of interest). The alternative is to compute the correlational structure as a function of connectivity and single neuron properties. While this has the disadvantage that connectivity in the brain is not known, it has the advantage that it can provide us with general principles about the relationship between connectivity and correlations, and between correlations and computations.
As a first pass at this problem, we consider a very simple model consisting of McCullough-Pitts neurons receiving correlated input. Following Roudi and Latham (2007), the connectivity consists of two components: strong, random background connectivity, and weak structured connectivity. (Here “strong” and “weak” refer to scaling with the number of connections per neuron: strong connectivity scales as 1/sqrt(K) and weak as 1/K, where K is the number of connections per neuron).
For this model, we are able to compute the full correlational structure analytically. Our main finding is that the cross-correlation matrix between pairs of neurons is related, in a nontrivial way, to the eigenvalue spectrum of the full connectivity matrix. This is reminiscent of the results found by Ginzburg and Sompolinsky (1994) for weakly coupled stochastic networks, although at a quantitative level the relationship is, of course, very different. An important corollary to this finding is that the random and structured components of the connectivity matrix make approximately equal contributions to the correlational structure, even though the random background connectivity is sqrt(K) times stronger (30-100 times stronger for cortical networks). This is because the largest eigenvalues of the two connectivity matrices are on the same order (both are order(1)). The fact that both contribute rules out a simple relationship between connectivity and correlational structure.
This analytical calculation paves the way for several more interesting questions: First, given an input that contains information about some quantity in the outside world (e.g., the orientation of a bar), how does the information in the network about that quantity depend on network connectivity? Second, and much harder, what are the tradeoffs in terms of computations? More precisely: are there situations in which networks in the brain would reduce the ability of a network to retain information in order to enhance its computational abilities?
References
1. Averbeck, Latham, Pouget. Nature Reviews Neuroscience, 7:358-366, 2006.
2. Roudi and Latham. PLoS Computational Biology, 3:1679-1700, 2007.
3. Ginzburg and Sompolinsky. Physical Review E, 50:3171-3191, 1994.
Conference:
Computational and systems
neuroscience 2009, Salt Lake City, UT, United States, 26 Feb - 3 Mar, 2009.
Presentation Type:
Poster Presentation
Topic:
Poster Presentations
Citation:
(2009). Computing correlations analytically.
Front. Syst. Neurosci.
Conference Abstract:
Computational and systems
neuroscience 2009.
doi: 10.3389/conf.neuro.06.2009.03.123
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Received:
02 Feb 2009;
Published Online:
02 Feb 2009.