Probing the basins of attraction of a recurrent neural network

A recurrent neural network is considered that can retrieve a collection of patterns, as well as slightly perturbed versions of this ‘pure’ set of patterns via fixed points of its dynamics. By replacing the set of dynamical constraints, i.e., the fixed point equations, by an extended collection of fixed-point-like equations, analytical expressions are found for the weights wij(b) of the net, which depend on a certain parameter b. This so-called basin parameter b is such that for b = 0 there are, a priori, no perturbed patterns to be recognized by the net. It is shown by a numerical study, via probing sets, that a net constructed to recognize perturbed patterns, i.e., with values of the connections wij(b) with b 6= 0, possesses larger basins of attraction than a net made with the help of a pure set of patterns, i.e., with connections wij(b = 0). The mathematical results obtained can, in principle, be realized by an actual, biological neural net.