Properties of neural networks storing spatially correlated patterns

The author studies the behaviour of a feedforward neural network supplied with spatially organized data. This inner structure is taken into account by a matrix C_ij, whose coefficients equal the average correlation between two pixels i and j of the input patterns. The storage capacity alpha is computed as a function of the required stability and of the eigenvalues of C. The author proposes a geometrical transformation allowing an intuitive interpretation of these results. Numerical simulations using real and binary patterns show a very good agreement with the theory. Finally, the author analyses the synaptic couplings correlations resulting from the training of the network with structured patterns. Focusing on exponentially decreasing correlations one and two dimensions, the author finds that they exhibit a 'Mexican hat' profile, the excitatory centre size of which depends on alpha.