  We conjecture the existence of two new non-gravitational six-dimensional
string theories, defined as the decoupling limit of NS5-branes in the
background of near-critical electrical two- and three-form RR fields. These
theories are space-time non-commutative Little String Theories with open
branes. The theory with (2,0) supersymmetry has an open membrane in the
spectrum and reduces to OM theory at low energies. The theory with (1,1)
supersymmetry has an open string in the spectrum and reduces to 5+1 dimensional
NCOS theory for weak NCOS coupling and low energies. The theories are shown to
be T-dual with the open membrane being T-dual to the open string. The theories
therefore provide a connection between 5+1 dimensional NCOS theory and OM
theory. We study the supergravity duals of these theories and we consider a
chain of dualities that shows how the T-duality between the two theories is
connected with the S-duality between 4+1 dimensional NCOS theory and OM theory.
