ABSTRACT
The patterns characterizing the spatiotemporal dynamics of excitable media, such as spiral waves as well as spatiotemporal chaos arising from successive breakup and collisions between such waves, are observed in a wide variety of natural systems. They range from chemical systems such as the Belousov-Zhabotinsky reaction [Zaikin and Zhabotinsky 1970] and the heart in the throes of life-threatening arrhythmias [Gray et al. 1998] to neocortical slices from the mammalian brain [Huang et al. 2004]. In the latter system, it has been speculated that the observed spiral waves may provide a spatial framework for persistent cortical oscillations [Huang et al. 2004]. Theoretical work on persistent activity that can occur in excitable media has tended to focus on either the existence of specialized clusters of cells capable of oscillatory self-activation, e.g., the cells in the sinus node of the heart, or the presence of noise in the environment of the system that acts as a source of stochastic stimulation [Garc´ıa-Ojalvo and Schimansky-Geier 1999]. However, one can argue that, in such cases the activity in the medium is not truly self-sustained as the signal from the pacemaker region or the noise is akin to external intervention that is necessary for the initiation and persistence of spiral waves [Hou and Xin 2002; Jung and Mayer-Kress 1995a]. If such patterns are to be seen as spontaneously emerging from arbitrary initial conditions, then the pattern formation should be an outcome of the internal structure of the system alone. Furthermore, small variations in this structure may result in transitions between different dynamical regimes characterized by distinct spatiotemporal patterns.
