Shape dynamics and permeability of a membrane neck connecting a vesicle and plasma membrane are considered. The neck is modeled by a lipid membrane tubule extended between two parallel axisymmetric rings. Within a range of lengths, defined by system geometry and mechanical properties of the membrane, the tubule has two stable shapes: catenoidal microtubule and cylindrical nanotubule. The permeabilities of these two shapes, measured as ionic conductivity of the tubule interior, differ by up to four orders of magnitude. Near the critical length the transitions between the shapes occur within less than a millisecond. Theoretical estimates show that the shape switching is controlled by a single parameter, the tubule length. Thus the tubule connection can operate as a conductivity microswitch, toggling the release of vesicle content in such cellular processes as "kiss-and-run" exocytosis. In support of this notion, bistable behavior of membrane connections between vesicles and the cell plasma membrane in macrophages is demonstrated.