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An attractor neural network architecture with an ultra high information capacity: numerical results

Attractor neural network is an important theoretical scenario for modeling memory function in the hippocampus and in the cortex. In these models, memories are stored in the plastic recurrent connections of neural populations in the form of "attractor states". The maximal information capacity for conventional abstract attractor networks with unconstrained connections is 2 bits/synapse. However, an unconstrained synapse has the capacity to store infinite amount of bits in a noiseless theoretical scenario: a capacity that conventional attractor networks cannot achieve. Here, I propose a hierarchical attractor network that can achieve an ultra high information capacity. The network has two layers: a visible layer with $N_v$ neurons, and a hidden layer with $N_h$ neurons. The visible-to-hidden connections are set at random and kept fixed during the training phase, in which the memory patterns are stored as fixed-points of the network dynamics. The hidden-to-visible connections, initially normally distributed, are learned via a local, online learning rule called the Three-Threshold Learning Rule and there is no within-layer connections. The results of simulations suggested that the maximal information capacity grows exponentially with the expansion ratio $N_h/N_v$. As a first order approximation to understand the mechanism providing the high capacity, I simulated a naive mean-field approximation (nMFA) of the network. The exponential increase was captured by the nMFA, revealing that a key underlying factor is the correlation between the hidden and the visible units. Additionally, it was observed that, at maximal capacity, the degree of symmetry of the connectivity between the hidden and the visible neurons increases with the expansion ratio. These results highlight the role of hierarchical architecture in remarkably increasing the performance of information storage in attractor networks.