Due to their significant resemblance to the biological brain, spiking neural networks (SNNs) show promise in handling spatiotemporal information with high time and energy efficiency. Two-terminal memristors have the capability to achieve both synaptic and neuronal functions; however, such memristors face asynchronous programming/reading operation issues. Here, a three-terminal memristor (3TM) based on oxygen ion migration is developed to function as both a synapse and a neuron. We demonstrate short-term plasticity such as pair-pulse facilitation and high-pass dynamic filtering in our devices. Additionally, a 'learning-forgetting-relearning' behavior is successfully mimicked, with lower power required for the relearning process than the first learning. Furthermore, by leveraging the short-term dynamics, the leaky-integrate-and-fire neuronal model is emulated by the 3TM without adopting an external capacitor to obtain the leakage property. The proposed bi-functional 3TM offers more process compatibility for integrating synaptic and neuronal components in the hardware implementation of an SNN.