We consider the problem of neural association for a network of non-binary neurons. Here, the task is to first memorize a set of patterns using a network of neurons whose states assume values from a finite number of integer levels. Later, the same network should be able to recall previously memorized patterns from their noisy versions. Prior work in this area consider storing a finite number of purely random patterns, and have shown that the pattern retrieval capacities (maximum number of patterns that can be memorized) scale only linearly with the number of neurons in the network. In our formulation of the problem, we concentrate on exploiting redundancy and internal structure of the patterns in order to improve the pattern retrieval capacity. Our first result shows that if the given patterns have a suitable linear-algebraic structure, i.e. comprise a sub-space of the set of all possible patterns, then the pattern retrieval capacity is in fact exponential in terms of the number of neurons. The second result extends the previous finding to cases where the patterns have weak minor components, i.e. the smallest eigenvalues of the correlation matrix tend toward zero. We will use these minor components (or the basis vectors of the pattern null space) to both increase the pattern retrieval capacity and error correction capabilities. An iterative algorithm is proposed for the learning phase, and two simple neural update algorithms are presented for the recall phase. Using analytical results and simulations, we show that the proposed methods can tolerate a fair amount of errors in the input while being able to memorize an exponentially large number of patterns.
Quantitative expressions of long-term memory storage capacities of complex neural network are derived. The networks are made of neurons connected by synapses of any order, of the axono-axonal type considered by for example. The effect of link deletion possibly related to aging, is also considered. The central result of this study is that, within the framework of Hebb's laws, the number of stored bits is proportional to the number of synapses. The proportionality factor however, decreases when the order of involved synaptic contact increases. This tends to favor neural architectures with low-order synaptic connectivities. It is finally shown that the memory storage capacities can be optimized by a partition of the network into neuron clusters with size comparable with that observed for cortical microcolumns.