Efficient information processing in the human brain is achieved by dynamics of neurons and synapses, motivating effective implementation of artificial spiking neural networks. Here, the dynamics of spin-orbit torque switching in antiferromagnet/ferromagnet heterostructures is studied to show the capability of the material system to form artificial neurons and synapses for asynchronous spiking neural networks. The magnetization switching, driven by a single current pulse or trains of pulses, is examined as a function of the pulse width (1 s to 1 ns), amplitude, number, and pulse-to-pulse interval. Based on this dynamics and the unique ability of the system to exhibit binary or analog behavior depending on the device size, key functionalities of a synapse (spike-timing-dependent plasticity) and a neuron (leaky integrate-and-fire) are reproduced in the same material and on the basis of the same working principle. These results open a way toward spintronics-based neuromorphic hardware that executes cognitive tasks with the efficiency of the human brain.