  In this letter, we first derive the analytical channel impulse response for a
cylindrical synaptic channel surrounded by glial cells and validate it with
particle-based simulations. Afterwards, we provide an accurate analytical
approximation for the long-time decay rate of the channel impulse response by
employing Taylor expansion to the characteristic equations that determine the
decay rates of the system. We validate our approximation by comparing it with
the numerical decay rate obtained from the characteristic equation. Overall, we
provide a fully analytical description for the long-time behavior of synaptic
diffusion, e.g., the clean-up processes inside the channel after communication
has long concluded.
