Self-trapping in an attractor neural network with nearest neighbor synapses mimics full connectivity

A means of providing the feedback necessary for an associative memory is suggested by self-trapping, the development of localization phenomena and order in coupled physical systems. Following the lead of Hopfield (1982, 1984) who exploited the formal analogy of a fully-connected ANN to an infinite ranged interaction Ising model, we have carried through a similar development to demonstrate that self-trapping networks (STNs) with only near-neighbor synapses develop attractor states through localization of a self-trapping input. The attractor states of the STN are the stored memories of this system, and are analogous to the magnetization developed in a self-trapping 1D Ising system. Post-synaptic potentials for each stored memory become trapped at non-zero valves and a sparsely-connected network evolves to the corresponding state. Both analytic and computational studies of the STN show that this model mimics a fully-connected ANN.