  Let ${\mathbb M}_{g, 1}$, $g \geq 1$, be the moduli space of triples $(C,
P_0, v)$ of genus $g$, where $C$ is a compact Riemann surface of genus $g$,
$P_0 \in C$, and $v \in T_{P_0}C\setminus\{0\}$. Using Chen's iterated
integrals we introduce a higher analogue of the period matrix for a triple $(C,
P_0, v)$, {\it the harmonic Magnus expansion}. It induces a flat connection on
a vector bundle over the space ${\mathbb M}_{g, 1}$, whose holonomy gives all
the higher Johnson homomorphisms of the mapping class group. The connection
form, which is computed as an explicit quadratic differential, induces
"canonical" differential forms representing (twisted) Morita-Mumford classes
and their higher relators on ${\mathbb M}_{g, 1}$. In particular, we construct
a family of twisted differential forms on ${\mathbb M}_{g, 1}$ representing the
$(0, p+2)$-twisted Morita-Mumford class $m_{0, p+2}$ combinatorially
parametrized by the Stasheff associahedron $K_{p+1}$.
