  The recurrent outbursts that characterise low-mass binary systems reflect
thermal state changes in their associated accretion discs. The observed
outbursts are connected to the strong variation in disc opacity as hydrogen
ionises near 5000 K. This physics leads to accretion disc models that exhibit
bistability and thermal limit cycles, whereby the disc jumps between a family
of cool and low accreting states and a family of hot and efficiently accreting
states. Previous models have parametrised the disc turbulence via an alpha (or
`eddy') viscosity. In this paper we treat the turbulence more realistically via
a suite of numerical simulations of the magnetorotational instability (MRI) in
local geometry. Radiative cooling is included via a simple but physically
motivated prescription. We show the existence of bistable equilibria and thus
the prospect of thermal limit cycles, and in so doing demonstrate that
MRI-induced turbulence is compatible with the classical theory. Our simulations
also show that the turbulent stress and pressure perturbations are only weakly
dependent on each other on orbital times; as a consequence, thermal instability
connected to variations in turbulent heating (as opposed to radiative cooling)
is unlikely to operate, in agreement with previous numerical results. Our work
presents a first step towards unifying simulations of full MHD turbulence with
the correct thermal and radiative physics of the outbursting discs associated
with dwarf novae, low-mass X-ray binaries, and possibly young stellar objects.
