  For bilayer graphene in a magnetic field at the neutral point, we derive and
solve a full set of gap equations including all Landau levels and taking into
account the dynamically screened Coulomb interaction. There are two types of
the solutions for the filling factor $\nu=0$: (i) a spin-polarized type
solution, which is the ground state at small values of perpendicular electric
field $E_{\perp}$, and (ii) a layer-polarized solution, which is the ground
state at large values of $E_{\perp}$. The critical value of $E_{\perp}$ that
determines the transition point is a linear function of the magnetic field,
i.e., $E_{\perp,{\rm cr}}=E_{\perp}^{\rm off}+a B$, where $E_{\perp}^{\rm off}$
is the offset electric field and $a$ is the slope. The offset electric field
and energy gaps substantially increase with the inclusion of dynamical
screening compared to the case of static screening. The obtained values for the
offset and the energy gaps are comparable with experimental ones. The
interaction with dynamical screening can be strong enough for reordering the
levels in the quasiparticle spectrum (the $n=2$ Landau level sinks below the
$n=0$ and $n=1$ ones).
