  We describe a numerical method that simulates the interaction of the helium
atom with sequences of femtosecond and attosecond light pulses. The method,
which is based on the close-coupling expansion of the electronic configuration
space in a B-spline bipolar spherical harmonic basis, can accurately reproduce
the excitation and single ionization of the atom, within the electrostatic
approximation. The time dependent Schr\"odinger equation is integrated with a
sequence of second-order split-exponential unitary propagators. The asymptotic
channel-, energy- and angularly-resolved photoelectron distributions are
computed by projecting the wavepacket at the end of the simulation on the
multichannel scattering states of the atom, which are separately computed
within the same close-coupling basis. This method is applied to simulate the
pump-probe ionization of helium in the vicinity of the $2s/2p$ excitation
threshold of the He$^+$ ion. This work confirms the qualitative conclusions of
one of our earliest publications [L Argenti and E Lindroth, Phys. Rev. Lett.
{\bf 105}, 53002 (2010)], in which we demonstrated the control of the $2s/2p$
ionization branching-ratio. Here, we take those calculations to convergence and
show how correlation brings the periodic modulation of the branching ratios in
almost phase opposition. The residual total ionization probability to the
$2s+2p$ channels is dominated by the beating between the $sp_{2,3}^+$ and the
$sp_{2,4}^+$ doubly excited states, which is consistent with the modulation of
the complementary signal in the $1s$ channel, measured in 2010 by Chang and
co-workers~[S Gilbertson~\emph{et al.}, Phys. Rev. Lett. {\bf 105}, 263003
(2010)].
