  In dynamical systems with distinct time scales the time evolution in phase
space may be influenced strongly by the fixed points of the fast subsystem.
Orbits then typically follow these points, performing in addition rapid
transitions between distinct branches on the time scale of the fast variables.
As the branches guide the dynamics of a system along the manifold of former
fixed points, they are considered transiently attracting states and the
intermittent transitions between branchescorrespond to state switching within
transient-state dynamics. A full characterization of the set of former fixed
points, the critical manifold, tends to be difficult in high-dimensional
dynamical systems such as large neural networks. Here we point out that an
easily computable subset of the critical manifold, the set of target points,
can be used as a reference for the investigation of high-dimensional slow-fast
systems. The set of target points corresponds in this context to the adiabatic
projection of a given orbit to the critical manifold. Applying our framework to
a simple recurrent neural network, we find that the scaling relation of the
Euclidean distance between the trajectory and its target points with the
control parameter of the slow time scale allows to distinguish an adiabatic
regime from a state that is effectively independent from target points.
