We investigate the extent to which the behavior of neural network language models reflects incremental representations of syntactic state. To do so, we employ experimental methodologies which were originally developed in the field of psycholinguistics to study syntactic representation in the human mind. We examine neural network model behavior on sets of artificial sentences containing a variety of syntactically complex structures. These sentences not only test whether the networks have a representation of syntactic state, they also reveal the specific lexical cues that networks use to update these states. We test four models: Two publicly available LSTM sequence models of English (Jozefowicz et al., 2016; Gulordava et al., 2018) trained on large datasets; an RNN Grammar (Dyer et al., 2016) trained on a small, parsed dataset; and an LSTM trained on the same small corpus as the RNNG. We find evidence for basic syntactic state representations in all models, but only the models trained on large datasets are sensitive to subtle lexical cues signalling changes in syntactic state.