  The Hopfield model is a pioneering neural network model with associative
memory retrieval. The analytical solution of the model in mean field limit
revealed that memories can be retrieved without any error up to a finite
storage capacity of $O(N)$, where $N$ is the system size. Beyond the threshold,
they are completely lost. Since the introduction of the Hopfield model, the
theory of neural networks has been further developed toward realistic neural
networks using analog neurons, spiking neurons, etc. Nevertheless, those
advances are based on fully connected networks, which are inconsistent with
recent experimental discovery that the number of connections of each neuron
seems to be heterogeneous, following a heavy-tailed distribution. Motivated by
this observation, we consider the Hopfield model on scale-free networks and
obtain a different pattern of associative memory retrieval from that obtained
on the fully connected network: the storage capacity becomes tremendously
enhanced but with some error in the memory retrieval, which appears as the
heterogeneity of the connections is increased. Moreover, the error rates are
also obtained on several real neural networks and are indeed similar to that on
scale-free model networks.
