  We present an \textit{ab initio} molecular dynamics study of the
temperature-induced phases of methylammonium lead bromide (MAPbBr$_3$). We
confirm that the low-temperature phase is not ferroelectric, and rule out the
existence of any overall polarization arising from the motion of the individual
sub-lattices. Our simulations at room temperature resulted in a cubic
\textit{Pm-3m} phase with no discernible local orthorhombic distortions. At low
temperatures, such distortions are shown to originate from octahedral
scissoring modes, but they vanish at room temperature. The predicted timescales
of MA motion agree very well with experimental estimates, establishing dynamic
disordering of the molecular dipoles over several orientational minima at room
temperature. We also identify the key modes of the inorganic and organic
sub-lattices that are coupled at all temperatures mainly through the
N-H$\cdots$Br hydrogen-bonds. Estimated lifetimes of the H-bonds correlate well
with MA dynamics indicating a strong connection between these two aspects of
organic inorganic hybrid perovskites. We also confirm that, in addition to
disordering of MA orientations, the transition to the cubic phase is also
associated with displacive characteristics arising from both MA as well as Br
ions in the lattice.
