  We construct a new type of spectral sequences for the mixed Hodge structures
on the cohomology of locally symmetric varieties. These spectral sequences
converge to the edge components in the Hodge triangles, and the E1 terms are
expressed by simple geometric invariants associated to the cusps. They already
degenerate at E1 in a certain range, which gives a simple expression of some
Hodge components. An identity of holomorphic Euler numbers is obtained as a
consequence. The theory of cellular (co)sheaves is used in the construction.
  When studying some examples, we also combine this approach with
Hodge-theoretic analysis of the restriction map to the Borel-Serre boundary.
The byproduct contains a simple proof of the surjectivity of the restriction
map in some cases which does not use Eisenstein series.
