  Dendritic spines, which are small protrusions on the dendrites of a neuron,
are of interest in neuroscience as they are related to cognitive processes such
as learning and memory. We analyse the distribution of spine locations on six
different dendrite trees from mouse neurons using point process theory for
linear networks. Besides some possible small-scale repulsion, { we find that
two of the spine point pattern data sets may be described by inhomogeneous
Poisson process models}, while the other point pattern data sets exhibit
clustering between spines at a larger scale. To model this we propose an
inhomogeneous Cox process model constructed by thinning a Poisson process on a
linear network with retention probabilities determined by a spatially
correlated random field. For model checking we consider network analogues of
the empirical $F$-, $G$-, and $J$-functions originally introduced for
inhomogeneous point processes on a Euclidean space. The fitted Cox process
models seem to catch the clustering of spine locations between spines, but also
posses a large variance in the number of points for some of the data sets
causing large confidence regions for the empirical $F$- and $G$-functions.
