In this concept article, we review recent works on the use of biologically-derived materials, including biomolecules, hybrid biomolecular composites, and natural tissues for achieving stimuli-responsiveness, adaptability, and memory storage for potential neuromorphic computing applications. Unlike solid-state materials, bioderived platforms are often intrinsically modular, including possessing the ability to leverage the diversity of naturally multifunctional biomolecules, cornerstones in biological transduction, signaling, and learning. The primary neuromorphic functionality of bioderived systems considered here is memory resistance, as obtained through combinations of resistive switching, activity-dependent plasticity, and memory storage. In some cases, these capabilities are achieved in bio-derived systems by closely emulating the structure, function, or signaling mechanisms found in living neurons. Our review considers bioderived systems that span multiple length scales and levels of hierarchical complexity: from single-molecule enabled devices to intact tissues, interrogated either in vitro or in vivo on living organisms. Through this exercise, we offer our perspective on research opportunities in the development and use of synapse- and neuron-inspired bioderived systems, including the prospect of biocompatible, and even tissue-like, neuromorphic materials to smartly monitor and treat injury and disease, deliver therapeutics, and interpret neural activity. These capabilities could unlock a new generation of smart, adaptable bioderived composites that autonomously report, learn (i.e., classify, forecast), compute, and actuate in the direct vicinity of living cells and tissues, i.e. at the edge of biology.