Fig. 1

Schematic sketch of the orbit structure in phase space. a Excitable media with activator–inhibitor kinetics; the activator diffuses and the inhibitor is immobilized. These systems are multistable, the quiescent state is the homogeneous steady state (green dot), and at least one traveling wave solution must exit as the excited state (red dot). The basins of attraction between these states are separated by the stable manifold (blue) of a critical nucleation solution (white dot). Such solutions have only one unstable direction; the corresponding unstable manifold consists of two heteroclinic connections to the stable solutions (green and red dots). b Excitable media with one activator and two inhibitors, one of which is immobilized, the other fast diffusing or realized by mean field inhibition. The orbit structure in phase space is similar to a but the traveling wave solution is now localized similar to the critical nucleation solution to which it is connected. c Medium that lost spatial excitability, that is, traveling wave solutions do not exist. Note that the traveling wave solution disappears by a collision of the traveling wave solution with its nucleation solution, that is, in a saddle-node bifurcation. Such systems show ghost behavior, which influences the dynamics in form of local excitability (see text)