  Bursting is a periodic transition between a quiescent state and a state of
repetitive spiking. The phenomenon is ubiquitous in a variety of neurophysical
systems. We numerically study the dynamical properties of a normal form of
subcritical Hopf oscillator (at the bifurcation point) subjected to a nonlinear
feedback. This dynamical system shows an infinite-period or a saddle-node on a
limit cycle (SNLC) bifurcation for certain strengths of the nonlinear feedback.
When the feedback is time delayed, the bifurcation scenario changes and the
limit cycle terminates through a homoclinic or a saddle separatrix loop (SSL)
bifurcation. This system when close to the bifurcation point exhibits various
types of bursting phenomenon when subjected to a slow periodic external
stimulus of an appropriate strength. The time delay in the feedback enhances
the spiking rate i.e. reduces the interspike interval in a burst and also
increases the width or the duration of a burst.

Chaotic Dynamics