DOMAINS OF ATTRACTION AND THE DENSITY OF STATIC METASTABLE STATES IN SINGLE-PATTERN ITERATED NEURAL NETWORKS

I calculate, for a single-pattern iterated neural network, the density of static metastable states, partial N/partial m, as a function of the field distribution and the symmetry of the synaptic matrix. The features of this function strongly suggest that the boundary upon which this density vanishes gives the critical overlap for the memory state, i.e. gives a measure of the size of its domain of attraction. This heuristic interpretation is shown to agree with the exact result for vanishing symmetry, the only case where a direct calculation can be performed. I explicitly calculate critical overlaps as a function of symmetry and mean field strength when the field distribution is a delta-function and when it is a unit-width Gaussian.