Stability in contractive nonlinear neural networks

Models of the form mu x=-x+p+WF(x), where x=x(t) is a vector whose entries represent the electrical activities in the units of a neural network are considered. W is a matrix of synaptic weights, F is a nonlinear function, and p is a vector (constant or slowly varying over time) of inputs to the units. If the map WF(x) is a contraction, then the system has a unique equilibrium which is globally asymptotically stable; consequently, the network acts as a stable encoder in that its steady-state response to an input is independent of the initial state of the network. Considered are some relatively mild restrictions on W and F(x), involving the eigenvalues of W and the derivative of F, that are sufficient to ensure that WF(x) is a contraction. It is shown that, in the linear case with spatially homogeneous synaptic weights, the eigenvalues of W are simply related to the Fourier transform of the connection pattern. This relation makes it possible, given cortical activity patterns as measured by autoradiographic labeling, to construct a pattern of synaptic weights which produces steady-state patterns showing similar frequency characteristics.