The self-trapping attractor neural network. I. Analysis of a simple 1-D model

Attractor neural networks (ANNs) based on the Ising model are naturally fully connected and are homogeneous in structure. These features permit a deep understanding of the underlying mechanism, but limit the applicability of these models to the brain. A more biologically realistic model can be derived from an equally simple physical model by utilizing recurrent self-trapping inputs to supplement very sparse intranetwork interactions. This paper reports the analysis of a one-dimensional (1-D) ANN coupled to a second system that computes overlaps with a single stored memory. Results show that: 1) the 1-D self-trapping model is equivalent to an isolated ANN with both full connectivity of one strength and nearest neighbor synapses of an independent strength; 2) the dynamics of ANN and self-trapping updates are independent; 3) there is a critical synaptic noise level below which memory retrieval occurs; 4) the 1-D self-trapping model converges to a fully connected Hopfield model for zero strength nearest neighbor synapses, and has a greater magnitude memory overlap for nonzero strength nearest neighbor synapses; and (5) the mechanism of self-trapping is an iterative map on the mean overlap as a function of the reentrant input.