The rise of mobile multi-agent robotic platforms is outpacing control paradigms for tasks that require operating in complex, realistic environments. To leverage inertial, energetic, and cost benefits of small-scale robots, critical future applications may depend on coordinating large numbers of agents with minimal onboard sensing and communication resources. In this article, we present the perspective that adaptive and resilient autonomous control of swarms of minimal agents might follow from a direct analogy with the neural circuits of spatial cognition in rodents. We focus on spatial neurons such as place cells found in the hippocampus. Two major emergent hippocampal phenomena, self-stabilizing attractor maps and temporal organization by shared oscillations, reveal theoretical solutions for decentralized self-organization and distributed communication in the brain. We consider that autonomous swarms of minimal agents with low-bandwidth communication are analogous to brain circuits of oscillatory neurons with spike-based propagation of information. The resulting notion of 'neural swarm control' has the potential to be scalable, adaptive to dynamic environments, and resilient to communication failures and agent attrition. We illustrate a path toward extending this analogy into multi-agent systems applications and discuss implications for advances in decentralized swarm control.