  Many systems can display a very short, rapid changing stage
(quasi-discontinuous region) inside a relatively very long and slowly changing
process. A quantitative definition for the "quasi-discontinuity" in these
systems has been introduced. We have shown by a simplified model that
extra-large Feigenbaum constants can be found inside some period-doubling
cascades due to the quasi-discontinuity. As an example, this phenomenon has
also been observed in Rose-Hindmash model describing neuron activities.
