Anomalous random neural network's guide to Hopfield neural networks

To describe the heterogeneity of the neural network, we propose an anomalous random neural network (ARNN) with arbitrary distributed waiting times, which is a generalization of the random neural network. We investigate the signal flow process in ARNN based on the renewal process and obtain the generalized master equations for the time evolution of the probability of the state vector of neurons. From the obtained master equations, we obtain the generalized rate equations for the time evolution of the average potential of each neuron in both closed and open ARNN systems. It is proved that when the distribution of waiting time is exponential, the generalized rate equations for open ARNN systems can reduce to Hopfield neural networks; when the distribution of waiting time is power-law, the corresponding rate equations become the generalized fractional-order Hopfield neural networks. Particularly, for a single neuron, we derive a power-law firing rate that matches the experiment [Lundstrom et al., Nat. Neurosci. 11, 1335 (2008)].