  Like many other biological processes, calcium dynamics in neurons containing
an endoplasmic reticulum are governed by diffusion-reaction equations on
interface-separated domains. Interface conditions are typically described by
systems of ordinary differential equations that provide fluxes across the
interfaces. Using the calcium model as an example of this class of ODE-flux
boundary interface problems, we prove the existence, uniqueness and boundedness
of the solution by applying comparison theorem, fundamental solution of the
parabolic operator and a strategy used in Picard's existence theorem. Then we
propose and analyze an efficient implicit-explicit finite element scheme which
is implicit for the parabolic operator and explicit for the nonlinear terms. We
show that the stability does not depend on the spatial mesh size. Also the
optimal convergence rate in $H^1$ norm is obtained. Numerical experiments
illustrate the theoretical results.
