  Reactivity boundaries that divide the origin and destination of trajectories
are crucial of importance to reveal the mechanism of reactions, which was
recently found to exist robustly even at high energies for index-one saddles
[Phys. Rev. Lett. 105, 048304 (2010)]. Here we revisit the concept of the
reactivity boundary and propose a more general definition that can involve a
single reaction associated with a bottleneck made up of higher index saddles
and/or several saddle points with different indices, where the normal form
theory, based on expansion around a single stationary point, does not work. We
numerically demonstrate the reactivity boundary by using a reduced model system
of the $H^+_5$ cation where the proton exchange reaction takes place through a
bottleneck made up of two index-two saddle points and two index-one saddle
points. The cross section of the reactivity boundary in the reactant region of
the phase space reveals which initial conditions are effective in making the
reaction happen, and thus sheds light on the reaction mechanism.
