A method and apparatus are disclosed for implementing a neural network having a sleep mode during which capacitively stored synaptic connectivity weights are refreshed. Each neuron outputs an analog activity level, represented in a preferred embodiment by the frequency of digital pulses. Feed-forward synaptic connection circuits couple the activity level outputs of first level neurons to inputs of second level neurons, and feed-back synaptic connection circuits couple outputs of second level neurons to inputs of first level neurons, the coupling being weighted according to connectivity weights stored on respective storage capacitors in each synaptic connection circuit. The network learns according to a learning algorithm under which the connections in both directions between a particular first level neuron and a particular second level neuron are strengthened to the extent of concurrence of high activity levels in both the first and second level neurons, and weakened to the extent of concurrence of a high activity level in the second level neuron and a low activity level in the first level neuron. The network is put to sleep by disconnecting all environmental inputs and providing a non-specific low activity level signal to each of the first level neurons. This causes the network to randomly traverse its state space with low intensity resonant firings, each state being visited with a probability responsive to the initial connectivity weights of the connections which abut the second level neuron representing such state. Refresh is accomplished since the learning algorithm remains active during sleep. Thus, the sleep refresh mechanism enhances the contrast in the connectivity terrain and strengthens connections that would otherwise wash out due to lack of visitation while the system is awake. A deep sleep mechanism is also provided for preventing runaway strengthening of favored states, and also to encourage Weber Law compliance.