Existing machines are functionally specific tools that were made for easy prediction and control. Tomorrow's machines may be closer to biological systems in their mutability, resilience, and autonomy. But first they must be capable of learning, and retaining, new information without repeated exposure to it. Past efforts to engineer such systems have sought to build or regulate artificial neural networks using task-specific modules with constrained circumstances of application. This has not yet enabled continual learning over long sequences of previously unseen data without corrupting existing knowledge: a problem known as catastrophic forgetting. In this paper, we introduce a system that can learn sequentially over previously unseen datasets (ImageNet, CIFAR-100) with little forgetting over time. This is accomplished by regulating the activity of weights in a convolutional neural network on the basis of inputs using top-down modulation generated by a second feed-forward neural network. We find that our method learns continually under domain transfer with sparse bursts of activity in weights that are recycled across tasks, rather than by maintaining task-specific modules. Sparse synaptic bursting is found to balance enhanced and diminished activity in a way that facilitates adaptation to new inputs without corrupting previously acquired functions. This behavior emerges during a prior meta-learning phase in which regulated synapses are selectively disinhibited, or grown, from an initial state of uniform suppression.