Biological brains are increasingly taken as a guide toward more efficient forms of computing. The latest frontier considers the use of spiking neural-network-based neuromorphic processors for near-sensor data processing, in order to fit the tight power and resource budgets of edge computing devices. However, a prevailing focus on brain-inspired computing and storage primitives in the design of neuromorphic systems is currently bringing a fundamental bottleneck to the forefront: chip-scale communications. While communication architectures (typically, a network-on-chip) are generally inspired by, or even borrowed from, general purpose computing, neuromorphic communications exhibit unique characteristics: they consist of the event-driven routing of small amounts of information to a large number of destinations within tight area and power budgets. This article aims at an inflection point in network-on-chip design for brain-inspired communications, revolving around the combination of cost-effective and robust asynchronous design, architecture specialization for short messaging and lightweight hardware support for tree-based multicast. When validated with functional spiking neural network traffic, the proposed NoC delivers energy savings ranging from 42% to 71% over a state-of-the-art NoC used in a real multi-core neuromorphic processor for edge computing applications.