1. Inhibitory Population of Bursting Neurons Frequency-Domain Order Parameter for Synchronization Transition of Bursting Neurons Woochang Lim 1 and Sang-Yoon Kim 2 1 Department of Science Education, Daegu National University of Education 2 Research Division, LABASIS Co. Summary   Synchronization of Bursting Neurons: Burst and Spike Synchronizations Due to the Slow and Fast Timescales of Bursting Activity   Characterization of Burst and Spike Synchronizations via Separation of the Slow (Bursting) and Fast (Spiking) Time Scales by Frequency Filtering IPFR R  IPBR R b (Describing the Slow Bursting Behavior) and IPSR R s (Describing the Fast Spiking Behavior)   Characterization of Burst Synchronization Based on Bursting Onset and Offset Times Direct Investigation of Bursting Behavior without Separation of Timescales   Characterization of Burst and Spike Synchronization in the Frequency Domain Power Spectrum: Practical Analysis Tools in the Experiments  Frequency-Domain Order Parameter: Realistic Order Parameter Applicable in the Real Experiment (Raster Plots, IPFR, and Power Spectrum: Directly Obtained in the Experiments) Characterization of Bursting Transition Based on Bursting Onset and Offset Times  Investigation of Population Bursting States Based on Bursting Onset and Offset Times and in the Frequency Domain Another Raster plots of bursting onset and offset times and IPBR and As D is increased, bursting stripes in the raster plots becomes smeared and overlap.  Amplitude of both and decreases.  Peak in the power spectra of and of decreases.  Frequency-Domain Bursting Order Parameter Unsynchronized Bursting State: N , then Synchronized Bursting State: N , then  Non-zero value Bursting Transition occurs for Characterization of Bursting Transition in Terms of Bursting Order Parameter  Investigation of Population Bursting States via Separation of Slow Timescale Separation of the slow timescale from IPFR R via low-pass filtering (f c =10Hz)  IPBR R b As D is increased, bursting bands in the raster plots becomes smeared and overlap.  Amplitude of R b decreases.  Investigation of Population Bursting States in the Frequency Domain Power spectra of R b Bursting Coherence Factor H p and f p : Height and frequency of bursting peak  f p : width of the bursting peak at the height of e -1/2 H p.  Frequency-Domain Bursting Order Parameter Unsynchronized Bursting State: N , then r  0 Synchronized Bursting State: N , then r  Non-zero value Bursting Transition occurs for Characterization of Intraburst Spiking Transition Based on R s  Investigation of Intraburst Spiking States via Separation of Fast Time Scale and in the Frequency Domain IPSR R s via band-pass filtering [lower and higher cut-off frequencies of 30Hz (high-ass filter) and 90 Hz (low-pass filter)] As D is increased, stripes in the burst band becomes smeared and overlap.  Amplitude of R s decreases.  Peak in the power spectra of R s decreases.  Frequency-Domain Spiking Order Parameter Unsynchronized Spiking State: N , then r  0 Synchronized Spiking State: N , then r  Non-zero value Bursting Transition occurs for Introduction  Burstings with the Slow and Fast Time Scales Bursting: Neuronal activity alternates, on a slow timescale, between a silent phase and an active (bursting) phase of fast repetitive spikings Representative examples of bursting neurons: chattering neurons in the cortex, thalamocortical relay neurons, thalamic reticular neurons, hippocampal pyramidal neurons, Purkinje cells in the cerebellum, pancreatic  -cells, and respiratory neurons in pre-Botzinger complex  Synchronization of Bursting Neurons Two Different Synchronization Patterns Due to the Slow and Fast Time Scales of Bursting Activity  Burst Synchronization: Synchrony on the Slow Bursting Timescale Temporal coherence between the active phase (bursting) onset or offset times of bursting neurons  Spike Synchronization: Synchrony on the Fast Spiking Timescale Temporal coherence between intraburst spikes fired by bursting neurons in their respective active phases Bursting Activity in the HR Neuron  Bursting Transition in the Single Neuron Transition to bursting occurs I DC =I* DC (~1.268) Parameters in the single Hindmarsh-Rose (HR) neuron Parameters for the synaptic current