  We investigate the entrainment of a neuron model exhibiting a chaotic
spiking-bursting behavior in response to a weak periodic force. This model
exhibits two types of oscillations with different characteristic time scales,
namely, long and short time scales. Several types of phase synchronization are
observed, such as 1 : 1 phase locking between a single spike and one period of
the force and 1 : l phase locking between the period of slow oscillation
underlying bursts and l periods of the force. Moreover, spiking-bursting
oscillations with chaotic firing patterns can be synchronized with the periodic
force. Such a type of phase synchronization is detected from the position of a
set of points on a unit circle, which is determined by the phase of the
periodic force at each spiking time. We show that this detection method is
effective for a system with multiple time scales. Owing to the existence of
both the short and the long time scales, two characteristic phenomena are found
around the transition point to chaotic phase synchronization. One phenomenon
shows that the average time interval between successive phase slips exhibits a
power-law scaling against the driving force strength and that the scaling
exponent has an unsmooth dependence on the changes in the driving force
strength. The other phenomenon shows that Kuramoto's order parameter before the
transition exhibits stepwise behavior as a function of the driving force
strength, contrary to the smooth transition in a model with a single time
scale.
