We introduce a neuromorphic architecture that is suitable for computing the next-hop for bundles in a disruption or delay-tolerant network. We define gateways that use spiking neural networks (SNN) that learn how to make autonomous routing decisions by observing single-hop bundle transmission performance and without requiring knowledge of the global network state. To formulate learning rewards, the gateways predict the end-to-end response time using local information. The rewards are then absorbed into the synapse strengths of the SNNs which estimate the relative Q-values of the next-hop alternatives for the next bundles. The use of the SNNs leads to improved routing performance with respect to performance goals of interest. Unlike comparable routing techniques, the proposed method achieves traffic balancing when multiple outbound links are available to send bundles to their next hop. An experimental evaluation of a proof-of-concept confirms that the proposed architecture is able to achieve high performance despite the presence of different kinds of communication impairments, such as frequent and lengthy link disruptions, long propagation delays and erratic transmission performance.