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We are interested in characterization of synchronization transitions ofbursting neurons in the frequency domain. Instantaneous population firing rate(IPFR) $R(t)$, which is directly obtained from the raster plot of neuralspikes, is often used as a realistic collective quantity describing populationactivities in both the computational and the experimental neuroscience. For thecase of spiking neurons, a realistic time-domain order parameter, based on$R(t)$, was introduced in our recent work to characterize the spikesynchronization transition. Unlike the case of spiking neurons, the IPFR $R(t)$of bursting neurons exhibits population behaviors with both the slow burstingand the fast spiking timescales. For our aim, we decompose the IPFR $R(t)$ intothe instantaneous population bursting rate $R_b(t)$ (describing the burstingbehavior) and the instantaneous population spike rate $R_s(t)$ (describing thespiking behavior) via frequency filtering, and extend the realistic orderparameter to the case of bursting neurons. Thus, we develop thefrequency-domain bursting and spiking order parameters which are just thebursting and spiking "coherence factors" $\beta_b$ and $\beta_s$ of thebursting 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 relativewidth). Through calculation of $\beta_b$ and $\beta_s$, we obtain the burstingand spiking thresholds beyond which the burst and spike synchronizations breakup, respectively. Consequently, it is shown in explicit examples that thefrequency-domain bursting and spiking order parameters may be usefully used forcharacterization of the bursting and the spiking transitions, respectively.

Frequency-Domain Order Parameters for the Burst And Spike Synchronization Transitions of Bursting Neurons