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Recent work on Long Term Potentiation in brain slices shows that Hebb's ruleis not completely synapse-specific, probably due to intersynapse diffusion ofcalcium or other factors. We extend the classical Oja unsupervised model oflearning by a single linear neuron to include Hebbian inspecificity, byintroducing an error matrix E, which expresses possible crosstalk betweenupdating at different connections. We show the modified algorithm converges tothe leading eigenvector of the matrix EC, where C is the input covariancematrix. When there is no inspecificity, this gives the classical result ofconvergence to the first principal component of the input distribution (PC1).We then study the outcome of learning using different versions of E. In themost biologically plausible case, arising when there are no intrinsicallyprivileged 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 ofinputs n. We analyze this error-onto-all case in detail, for both uncorrelatedand correlated inputs. We study the dependence of the angle theta between PC1and the leading eigenvector of EC on b, n and the amount of input activity orcorrelation. (We do this analytically and using Matlab calculations.) We findthat theta increases (learning becomes gradually less useful) with increases inb, particularly for intermediate (i.e. biologically-realistic) correlationstrength, although some useful learning always occurs up to the trivial limit Q= 1/n. We discuss the relation of our results to Hebbian unsupervised learningin the brain.

Hebbian Inspecificity in the Oja Model