  In this paper we consider point processes specified on directed linear
networks, i.e. linear networks with associated directions. We adapt the
so-called conditional intensity function used for specifying point processes on
the time line to the setting of directed linear networks. For models specified
by such a conditional intensity function, we derive an explicit expression for
the likelihood function, specify two simulation algorithms (the inverse method
and Ogata's modified thinning algorithm), and consider methods for model
checking through the use of residuals. We also extend the results and methods
to the case of a marked point process on a directed linear network.
Furthermore, we consider specific classes of point process models on directed
linear networks (Poisson processes, Hawkes processes, non-linear Hawkes
processes, self-correcting processes, and marked Hawkes processes), all adapted
from well-known models in the temporal setting. Finally, we apply the results
and methods to analyse simulated and neurological data.
