On the Number of Limit Cycles in Diluted Neural Networks

We consider the storage properties of temporal patterns, i.e. cycles of finite lengths, in neural networks represented by (generally asymmetric) spin glasses defined on random graphs. Inspired by the observation that dynamics on sparse systems has more basins of attractions than the dynamics of densely connected ones, we consider the attractors of a greedy dynamics in sparse topologies, considered as proxy for the stored memories. We enumerate them using numerical simulations and extend the analysis to large systems sizes using belief propagation. We find that the logarithm of the number of such cycles is a non monotonic function of the mean connectivity and we discuss the similarities with biological neural networks describing the memory capacity of the hippocampus.