  We are interested in characterization of synchronization transitions of
bursting neurons in the frequency domain. Instantaneous population firing rate
(IPFR) $R(t)$, which is directly obtained from the raster plot of neural
spikes, is often used as a realistic collective quantity describing population
activities in both the computational and the experimental neuroscience. For the
case of spiking neurons, a realistic time-domain order parameter, based on
$R(t)$, was introduced in our recent work to characterize the spike
synchronization transition. Unlike the case of spiking neurons, the IPFR $R(t)$
of bursting neurons exhibits population behaviors with both the slow bursting
and the fast spiking timescales. For our aim, we decompose the IPFR $R(t)$ into
the instantaneous population bursting rate $R_b(t)$ (describing the bursting
behavior) and the instantaneous population spike rate $R_s(t)$ (describing the
spiking behavior) via frequency filtering, and extend the realistic order
parameter to the case of bursting neurons. Thus, we develop the
frequency-domain bursting and spiking order parameters which are just the
bursting and spiking "coherence factors" $\beta_b$ and $\beta_s$ of the
bursting and spiking peaks in the power spectral densities of $R_b$ and $R_s$
(i.e., "signal to noise" ratio of the spectral peak height and its relative
width). Through calculation of $\beta_b$ and $\beta_s$, we obtain the bursting
and spiking thresholds beyond which the burst and spike synchronizations break
up, respectively. Consequently, it is shown in explicit examples that the
frequency-domain bursting and spiking order parameters may be usefully used for
characterization of the bursting and the spiking transitions, respectively.
