1. Thermodynamic and Statistical-Mechanical Measures for Synchronization of Bursting Neurons W. Lim (DNUE) and S.-Y. KIM (LABASIS)  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 1

2. Burst and Spike Synchronizations of Bursting Neurons 2  Synchronization of Bursting Neurons Two Different Synchronization Patterns Due to the Slow and Fast Time Scales of Bursting Activity Synchrony on the Slow Bursting Timescale Temporal coherence between the active phase (bursting) onset or offset times of bursting neurons  Burst Synchronization Synchrony on the Fast Spiking Timescale Temporal coherence between intraburst spikes fired by bursting neurons in their respective active phases  (Intraburst) Spike Synchronization

3. Characterization of Burst and Spike Synchronizations via Separation of the Slow (Bursting) and Fast (Spiking) Time Scales 3 Raster Plot of Neural Spikes: Population synchronization may be well visualized. Obtained in experiments Instantaneous Population Firing Rate (IPFR) Describing the population behaviors Separation of the Slow and Fast Timescale of Bursting Activity via Frequency Filtering Instantaneous Population Bursting Rate (IPBR) Describing the Slow Bursting Behavior Instantaneous Population Spiking Rate (IPSR) Describing the Fast Spiking Behavior Thermodynamic Intraburst Spiking Order Parameter Characterization of Intraburst Spiking Transition Statistical-Mechanical Intraburst Spiking Measure Characterization of Degree of Intraburst Spike Sync. Thermodynamic Bursting Order Parameter Characterization of Bursting Transition

4. 4 Characterization of Burst Synchronization Based on Bursting Onset and Offset Times Raster Plots of Active Phase (Bursting) Onset or Offset Times More direct visualization of bursting behavior Instantaneous Population Bursting Rates (IPBRs) for Active Phase Onset and Offset Times Raster Plot of Neural Spikes Thermodynamic Bursting Order Parameter Characterization of Bursting Transition Statistical-Mechanical Bursting Measure Characterization of Degree of Burst Sync.

5. Inhibitory Population of Bursting Hindmarsh-Rose Neurons  Coupled Bursting Hindmarsh-Rose (HR) Neurons 5  Bursting Activity of the Single HR Neuron Resting State for I DC =1.2 Bursting State for I DC =1.3 Hedgehog-like attractor - Parameters in the single HR neuron - Parameters for the synaptic current Transition to a bursting state

6. 6 Characterization of Bursting Transition Based on  Thermodynamic Bursting Order Parameter O Unsynchronized Bursting State: N , then O b  0 Synchronized Bursting State: N , then O b  Non-zero value Bursting Transition occurs for  Investigation of Bursting States via Separation of Slow Timescale Separation of the slow timescale from IPFR via low-pass filtering (f c =10Hz)  IPBR As D is increased, bursting bands in the raster plots becomes smeared and overlap.  Amplitude of decreases.

7. Characterization of Bursting Transition Based on and 7  Investigation of Bursting States Based on Bursting Onset and Offset Times  Thermodynamic Bursting Order Parameters Based on and O Bursting Transition occurs for O Another Raster plots of bursting onset and offset times and smooth IPBR kernel estimates and As D is increased, bursting stripes in the raster plots becomes smeared and overlap.  Amplitude of both and decreases.

8. Statistical-Mechanical Bursting Measure Based on and 8  Bursting measure of i th bursting stripe in the raster plot times  Occupation degree of bursting onset times : No. of bursting HR neurons in the i th bursting stripe  Pacing degree of bursting onset times: contribution of bursting onset times to the macroscopic IPBR : Global bursting phase based on IPBR : Total No. of microscopic bursting onset times in the i th bursting stripe  Statistical-Mechanical Bursting Measure Based on IPBR  Statistical-Mechanical Bursting Measure With increasing D, O b decreases very slowly, while P b and M b decrease rapidly.

9. Characterization of Intraburst Spiking Transition Based on 9  Investigation of Intraburst Spiking States via Separation of Fast Time Scale  IPSR via band-pass filtering [lower and higher cut-off frequencies of 30Hz (high-pass filter) and 90 Hz (low-pass filter)]  Thermodynamic Intraburst Spiking Order Parameter Unsynchronized Spiking State: N , then O s  0 Synchronized Spiking State: N , then O s  Non-zero value Bursting Transition occurs for As D is increased, stripes in the burst band becomes smeared and overlap.  Amplitude of decreases. Mean square deviation of in the i th global bursting cycle: Thermodynamic spiking order parameter:

10. Intraburst Spiking Measure Based on IPSR 10  Spiking measure of jth global spiking cycle in the ith bursting cycle  Occupation degree of spiking times : No. of spiking HR neurons in the jth spiking cycle and the i th bursting cycle  Pacing degree of spiking times: Contribution of spiking times to the macroscopic IPSR : Total No. of microscopic spiking times  Statistical-Mechanical Spiking Measure Based on IPSR  Statistical-Mechanical Spiking Measure With increasing D, r decreases very slowly, while r and r decrease very rapidly.

11. Summary 11  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  IPBR and IPSR  Thermodynamic Bursting Order Parameter Based on  Characterization of Bursting Transition  Thermodynamic (Intraburst) Spiking Order Parameter Based on  Characterization of (Intraburst) Spiking Transition  Statistical-Mechanical (Intraburst) Spiking Measure Based on  Degree of (Intraburst) Spike Synchronization  Characterization of Burst Synchronization Based on Bursting Onset and Offset Times (Direct Investigation of Bursting Behavior without Separation of Timescale)  Thermodynamics Bursting Order Parameter Based on and  Characterization of Bursting Transition  Statistical-Mechanical Bursting Measure Based on and  Degree of Bursting Synchronization  Thermodynamic and Statistical-Mechanical Measures: Realistic Measures Applicable in the Real Experiment (Raster Plots and IPFR: Directly Obtained in the Experiments)