The exploration of flexible electronic devices with information processing functions of biological neurons is of great significance for the development of intelligent wearable technologies. Due to lack of inherent mechanical flexibility, conventional threshold-switching memristor based on rigid materials that can implement the computing functions of biological neurons is difficult to fulfill the requirements for potential applications in the future. In this work, an intrinsically stretchable threshold-switching memristor was prepared by using silver nanowire-polyurethane composite as the dielectric layer and liquid metal as the electrodes, respectively. Under application of a sweeping voltage, the device exhibited reliable threshold switching characteristics, which was switched from the high resistance state (HRS) to the low resistance state (LRS) during device programming and spontaneously relaxed to the HRS upon voltage application. Further analysis shows that the underlying mechanism can be attributed to the dynamic formation and rupture of discontinuous silver conductive filaments formed between silver nanowires. In the pulse programming mode, memristor device is able to emulate the integration and firing characteristics of biological neurons, suggesting its great potential as an artificial neuron. Moreover, the pulse amplitude and pulse interval modulated neuronal spiking behaviors are successfully replicated using such devices. Under 20% tensile strain, the threshold-switching memristor shows negligible changes in the operating parameters during device switching and neuronal function implementations, suggesting its excellent mechanical flexibility and stability. This work provides important guidelines for the development of high-performance stretchable artificial neuronal devices and next-generation intelligent wearable systems.