The emergence of artificial intelligence has represented great potential in solving a wide range of complex problems. However, traditional general-purpose chips based on von Neumann architectures face the "memory wall" problem when applied in artificial intelligence applications. Based on the efficiency of the human brain, many intelligent neuromorphic chips have been proposed to emulate its working mechanism and neuron-synapse structure. With the emergence of spiking-based neuromorphic chips, the computation and energy efficiency of such devices could be enhanced by integrating a variety of features inspired by the biological brain. Aligning with the rapid development of neuromorphic chips, it is of great importance to quickly initiate the investigation of the electromagnetic interference and signal integrity issues related to neuromorphic chips for both CMOS-based and memristor-based artificial intelligence integrated circuits. Here, this paper provides a review of neuromorphic circuit design and algorithms in terms of electromagnetic issues and opportunities with a focus on signal integrity issues, modeling, and optimization. Moreover, the heterogeneous structures of neuromorphic circuits and other circuits, such as memory arrays and sensors using different integration technologies, are also reviewed, and locations where signal integrity might be compromised are discussed. Finally, we provide future trends in electromagnetic interference and signal integrity and outline prospects for upcoming neuromorphic devices.