  Let ($\Omega^{\ast}(M), d$) be the de Rham cochain complex for a smooth
compact closed manifolds $M$ of dimension $n$. For an odd-degree closed form
$H$, there are a twisted de Rham cochain complex $(\Omega^{\ast}(M),
d+H_\wedge)$ and its associated twisted de Rham cohomology $H^*(M,H)$. We show
that there exists a spectral sequence $\{E^{p, q}_r, d_r\}$ derived from the
filtration $F_p(\Omega^{\ast}(M))=\bigoplus_{i\geq p}\Omega^i(M)$ of
$\Omega^{\ast}(M)$, which converges to the twisted de Rham cohomology
$H^*(M,H)$. We also show that the differentials in the spectral sequence can be
given in terms of cup products and specific elements of Massey products as
well, which generalizes a result of Atiyah and Segal. Some results about the
indeterminacy of differentials are also given in this paper.
