  The geometry of neurons is known to be important for their functions. Hence,
neurons are often classified by their morphology. Two recent methods,
persistent homology and the topological morphology descriptor, assign a
morphology descriptor called a barcode to a neuron equipped with a given
function, such as the Euclidean distance from the root of the neuron. These
barcodes can be converted into matrices called persistence images, which can
then be averaged across groups. We show that when the defining function is the
path length from the root, both the topological morphology descriptor and
persistent homology are equivalent. We further show that persistence images
arising from the path length procedure provide an interpretable summary of
neuronal morphology. We introduce {topological morphology functions}, a class
of functions similar to Sholl functions, that can be recovered from the
associated topological morphology descriptor. To demonstrate this topological
approach, we compare healthy cortical and hippocampal mouse neurons to those
affected by progressive tauopathy. We find a significant difference in the
morphology of healthy neurons and those with a tauopathy at a postsymptomatic
age. We use persistence images to conclude that the diseased group tends to
have neurons with shorter branches as well as fewer branches far from the soma.
