Grammar acquisition is of significant importance for mastering human language. As the language signal is sequential in its nature, it poses the challenging task to extract its structure during online processing. This modeling study shows how spike-timing dependent plasticity (STDP) successfully enables sequence learning of artificial grammars that include non-adjacent dependencies (NADs) and nested NADs. Spike-based statistical learning leads to synaptic representations that comply with human acquisition performances under various distributional stimulus conditions. STDP, therefore, represents a practicable neural mechanism underlying human statistical grammar learning. These findings highlight that initial stages of the language acquisition process are possibly based on associative learning strategies. Moreover, the applicability of STDP demonstrates that the non-human brain possesses potential precursor abilities that support the acquisition of linguistic structure.