Memory capacity of networks with stochastic binary synapses

In standard attractor neural network models, specific patterns of activity are stored in the synaptic matrix, so that they become fixed point attractors of the network dynamics. The storage capacity of such networks has been quantified in two ways: the maximal number of patterns that can be stored, and the stored information measured in bits per synapse. In this paper, we compute both quantities in fully connected networks of N binary neurons with binary synapses, storing patterns with coding level [Formula: see text], in the large [Formula: see text] and sparse coding limits ([Formula: see text]). We also derive finite-size corrections that accurately reproduce the results of simulations in networks of tens of thousands of neurons. These methods are applied to three different scenarios: (1) the classic Willshaw model, (2) networks with stochastic learning in which patterns are shown only once (one shot learning), (3) networks with stochastic learning in which patterns are shown multiple times. The storage capacities are optimized over network parameters, which allows us to compare the performance of the different models. We show that finite-size effects strongly reduce the capacity, even for networks of realistic sizes. We discuss the implications of these results for memory storage in the hippocampus and cerebral cortex.