Modelling spine locations on dendrite trees using inhomogeneous Cox point processes

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 one of the spine point pattern data sets may be described by an inhomogeneous Poisson process model, 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.