Legged robots operating in real-world environments must possess the ability to rapidly adapt to unexpected conditions, such as changing terrains and varying payloads. This paper introduces the Synaptic Motor Adaptation (SMA) algorithm, a novel approach to achieving real-time online adaptation in quadruped robots through the utilization of neuroscience-derived rules of synaptic plasticity with three-factor learning. To facilitate rapid adaptation, we meta-optimize a three-factor learning rule via gradient descent to adapt to uncertainty by approximating an embedding produced by privileged information using only locally accessible onboard sensing data. Our algorithm performs similarly to state-of-the-art motor adaptation algorithms and presents a clear path toward achieving adaptive robotics with neuromorphic hardware.