Analysis of Linsker's application of Hebbian rules to linear networks

Linsker has reported the development of structured receptive fields in simulations of a Hebb-type synaptic plasticity rule in a feedforward linear network. The synapses develop under dynamics determined by a matrix that is closely related to the covariance matrix of input cell activities. We analyse the dynamics of the learning rule in terms of the eigenvectors of this matrix. These eigenvectors represent independently evolving weight structures. Some general theorems are presented regarding the properties of these eigenvectors and their eigenvalues. For a general covariance matrix four principal parameter regimes are predicted. We concentrate on the Gaussian covariances at layer B -> C of Linsker's network. Analytic and numerical solutions for the eigenvectors at this layer are presented. Three eigenvectors clonunate the dynamics: a DC eigenvector, in which all synapses have the same sign; a bi-lobed, oriented eigenvector; and a circularly symmetric, centre-surround eigenvector. Analysis of the circumstances in which each of these vectors dominates yields an explanation of the emergence of centre-surround structures and symmetry-breaking bi-lobed structures. Criteria are developed estimating the boundary of the parameter regime in which centre-surround structures emerge. The application of our analysis to Linsker's higher layers, at which the covariance functions were oscillatory, is briefly discussed.