Let $\mathcal{A}$ an abelian category. Take $M$ an object of $\mathcal{A}$, and $K_*$ a bounded complex in $\mathcal{A}$ equipped with a bounded increasing filtration $F$. By using homological and functorial properties of the $Ext$ functor, one can construct, for all $k \in \mathbb{Z}$, a bigraded complex

$(Ext^{k-\alpha}(M,\frac{F_{\alpha}K_{\beta}}{F_{\alpha-1}K_{\beta}}))_{(\alpha,\beta) \in \mathbb{Z} \times \mathbb{Z}}$

with decreasing indices $\alpha$ and $\beta$.

This bigraded complex (like any bigraded complex) induces two spectral sequences and, since it is bounded (the complex $K_*$ and its filtration $F$ are bounded), they both converge to the homology of the associated total complex, but a priori this last one has not really a nice expression, and remains a kind of "theoritical" abutment.

My question is then the following : is there a "simple" or "nice" expression for the abutment of the spectral sequences induced by the bigraded complex $(Ext^{k-\alpha}(M,\frac{F_{\alpha}K_{\beta}}{F_{\alpha-1}K_{\beta}}))_{(\alpha,\beta) \in \mathbb{Z} \times \mathbb{Z}}$ ?

By advance, thank you very much.