  Recent work on Long Term Potentiation in brain slices shows that Hebb's rule
is not completely synapse-specific, probably due to intersynapse diffusion of
calcium or other factors. We extend the classical Oja unsupervised model of
learning by a single linear neuron to include Hebbian inspecificity, by
introducing an error matrix E, which expresses possible crosstalk between
updating at different connections. We show the modified algorithm converges to
the leading eigenvector of the matrix EC, where C is the input covariance
matrix. When there is no inspecificity, this gives the classical result of
convergence to the first principal component of the input distribution (PC1).
We then study the outcome of learning using different versions of E. In the
most biologically plausible case, arising when there are no intrinsically
privileged connections, E has diagonal elements Q and off- diagonal elements
(1-Q)/(n-1), where Q, the quality, is expected to decrease with the number of
inputs n. We analyze this error-onto-all case in detail, for both uncorrelated
and correlated inputs. We study the dependence of the angle theta between PC1
and the leading eigenvector of EC on b, n and the amount of input activity or
correlation. (We do this analytically and using Matlab calculations.) We find
that theta increases (learning becomes gradually less useful) with increases in
b, particularly for intermediate (i.e. biologically-realistic) correlation
strength, although some useful learning always occurs up to the trivial limit Q
= 1/n. We discuss the relation of our results to Hebbian unsupervised learning
in the brain.
