  In this paper, by making use of Duan's topological current theory, the
evolution of the vortex filaments in excitable media is discussed in detail.
The vortex filaments are found generating or annihilating at the limit points
and encountering, splitting, or merging at the bifurcation points of a complex
function $Z(\vec{x},t)$. It is also shown that the Hopf invariant of knotted
scroll wave filaments is preserved in the branch processes (splitting, merging,
or encountering) during the evolution of these knotted scroll wave filaments.
Furthermore, it also revealed that the "exclusion principle" in some chemical
media is just the special case of the Hopf invariant constraint, and during the
branch processes the "exclusion principle" is also protected by topology.

Pattern Formation and Solitons