seek some applications of a double complex of special type

Question

I am considering an upper half plane double complex $C^{\bullet}_{\bullet}$ satisfying the following condition: consider the horizontal homology of each position in every row, $H^{i}(C^{\bullet}_{*})=0,\forall i \neq 0,1$(i.e. in each row every position is exact except the 0-th and 1-st position). Then I think there are some relations among $H^{0}(C^{\bullet}_{*}), Tot(C^{\bullet}_{\bullet})$ and $H^{1}(C^{\bullet}_{*})$. And also I am wondering is there any application of this kind of double complex in algebraic topology (like computing the homology of some topological space) or some other fields?