We study the application of a neural network architecture for identifying charged particle trajectories via unsupervised learning of delays and synaptic weights using a spike-time-dependent plasticity rule. In the considered model, the neurons receive time-encoded information on the position of particle hits in a tracking detector for a particle collider, modeled according to the geometry of the Compact Muon Solenoid Phase II detector. We show how a spiking neural network is capable of successfully identifying in a completely unsupervised way the signal left by charged particles in the presence of conspicuous noise from accidental or combinatorial hits. These results open the way to applications of neuromorphic computing to particle tracking, motivating further studies into its potential for real-time, low-power particle tracking in future high-energy physics experiments.