  We study the appearance of a novel phenomenon for linearly coupled identical
bursters: synchronized bursts where there are changes of spike synchrony within
each burst. The examples we study are for normal form elliptic bursters where
there is a periodic slow passage through a Bautin (codimension two degenerate
Andronov-Hopf) bifurcation. This burster has a subcritical Andronov-Hopf
bifurcation at the onset of repetitive spiking while end of burst occurs via a
fold limit cycle bifurcation. We study synchronization behavior of two and
three Bautin-type elliptic bursters for a linear direct coupling scheme. Burst
synchronization is known to be prevalent behavior among such coupled bursters,
while spike synchronization is more dependent on the details of the coupling.
  We note that higher order terms in the normal form that do not affect the
behavior of a single burster can be responsible for changes in synchrony
pattern; more precisely, we find within-burst synchrony changes associated with
a turning point in the spiking frequency.
