Bioelectronic medicine treats chronic diseases by sensing, processing, and modulating the electronic signals produced in the nervous system of the human body, labeled 'neural signals'. While electronic circuits have been used for several years in this domain, the progress in microelectronic technology is now allowing increasingly accurate and targeted solutions for therapeutic benefits. For example, it is now becoming possible to modulate signals in specific nerve fibers, hence targeting specific diseases. However, to fully exploit this approach it is crucial to understand what aspects of the nerve signals are important, what is the effect of the stimulation, and what circuit designs can best achieve the desired result. Neuromorphic electronic circuits represent a promising design style for achieving this goal: their ultra-low power characteristics and biologically plausible time constants make them the ideal candidate for building optimal interfaces to real neural processing systems, enabling real-time closed-loop interactions with the biological tissue. In this paper, we highlight the main features of neuromorphic circuits that are ideally suited for interfacing with the nervous system and show how they can be used to build closed-loop hybrid artificial and biological neural processing systems. We present examples of neural computational primitives that can be implemented for carrying out computation on the signals sensed in these closed-loop systems and discuss the way to use their outputs for neuro-stimulation. We describe examples of applications that follow this approach, highlight open challenges that need to be addressed, and propose actions required to overcome current limitations.