We study the spatio-temporal dynamics of a qualitative model of the glycolytic reaction catalyzed by phosphofructokinase. Adding the nonlinear regulation by fructose-2,6-bisphosphate, the model is found to exhibit complex oscillations and two types of chaos. In particular, a region of excitable chaos is found in parameter space. In a spatially extended system, this excitable chaos gives rise to spontaneous irregular spiking as well as self-organized formation of spots of high ATP concentration with "center-surround" characteristics. The deterministic spiking has noise-like temporal proper-ties but shows a pronounced "memory", which we attribute to the transient aparition of a Turing-pattern that exists nearby in parameter space. The model thus provides an example of an excitable medium that prohibits (rather than permits) the propagation of concentration waves. This could be exploited by the living cell to self-organize pronounced local amplification of ATP.