Localized activity profiles and storage capacity of rate-based autoassociative networks

We study analytically the effect of metrically structured connectivity on the behavior of autoassociative networks. The steady state equations are derived for a generic input-output function, and then we focus on their solutions in the case of networks composed of three alternative simple rate-based model neurons: threshold-linear, binary or smoothly saturating units. For a connectivity which is short range enough the threshold-linear network shows localized retrieval states. The saturating and binary models also exhibit spatially modulated retrieval states if the highest activity level that they can achieve is above the maximum activity of the units in the stored patterns. We show that this saturation level together with the linear gain of the transfer function are important parameters that determine the possibility of localized retrieval. If the ratio of the number of stored patterns to the number of connections per unit goes to zero, while the latter goes to infinity, it is possible to derive an analytical formula for the critical value of the connectivity width, below which one observes spatially nonuniform retrieval states. The formula is also shown to offer a good first approximation for higher storage loads. We show that even in the case of localized retrieval the storage capacity remains proportional to the number of connections per neurons, with the proportionality constant lower by a factor of 3-4 compared to uniform retrieval. The approach that we present here is generic in the sense that there are no specific assumptions on the single unit input-output function nor on the exact connectivity structure.