  Our investigation is specially motivated by the stochastic version of a
common model of potential spread in a dendritic tree. We do not assume the
noise in the junction points to be Markovian. In fact, we allow for long-range
dependence in time of the stochastic perturbation. This leads to an abstract
formulation in terms of a stochastic diffusion with dynamic boundary
conditions, featuring fractional Brownian motion. We prove results on
existence, uniqueness and asymptotics of weak and strong solutions to such a
stochastic differential equation.
