Thermodynamic and Statistical-Mechanical Measures for Synchronization of Bursting

NeuronsW. 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

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

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

R

Instantaneous Population Spiking Rate (IPSR) Describing the Fast Spiking Behavior

sR

bRThermodynamic 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

Characterization of Burst Synchronization Based on BurstingOnset 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 TransitionStatistical-Mechanical Bursting Measure Characterization of Degree of Burst Sync.

Inhibitory Population of Bursting Hindmarsh-Rose Neurons

Coupled Bursting Hindmarsh-Rose (HR) Neurons

5

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sxxi

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iiii

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isyniDCiiiii

exg

XxtgN

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Niggxgdt

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zxxsrdt

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dt

dy

IDIzbxaxydt

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Bursting Activity of the Single HR Neuron

Resting State for IDC=1.2 Bursting State for IDC=1.3 Hedgehog-like attractor

- Parameters in the single HR neuron

6.1,4,001.0,5,1,3,1 oxsrdcba

- Parameters for the synaptic current-1-1* ms1.0,ms10,30,0,2 ssyn xX

Transition to a bursting state 26.1~*

DCI

6

Characterization of Bursting Transition Based on

Thermodynamic Bursting Order Parameter2))()(( tRtR bbb O

Unsynchronized Bursting State: N, then Ob0Synchronized Bursting State: N, then Ob Non-zero value

Bursting Transition occurs for

)068.0~(*bD

Investigation of Bursting States via Separation of Slow Timescale

3.0,3.1 JIDC

3.0,3.1 JIDC

ms1h

Separation of the slow timescale from IPFR via low-pass filtering (fc=10Hz) IPBR

As D is increased, bursting bands in the raster plots becomes smeared and overlap. Amplitude of decreases.

bRR

bR

bR

Characterization of Bursting Transition Based on and

7

)(onbR )(off

bR

Investigation of Bursting States Based on Bursting Onset and Offset Times

Thermodynamic Bursting Order Parameters Based on and

2)()()( ))()(( tRtR onb

onb

onb O

Bursting Transition occurs for )068.0~(*bD

2)()()( ))()(( tRtR offb

offb

offb O

)(onbR )(off

bR

ms50h

3.0,3.1 JIDC

3.0,3.1 JIDC

Another Raster plots of bursting onset and offset times and smooth IPBR kernel estimates and )(on

bR )(offbR

As D is increased, bursting stripes in the raster plots becomes smeared and overlap. Amplitude of both and decreases.

)(onbR )(off

bR

Statistical-Mechanical Bursting Measure Based on and

8

)(onbR )(off

bR

),(),(),( onbi

onbi

onbi POM

Bursting measure of ith bursting stripe in the raster plot times Occupation degree of bursting onset times

N

NO

bionb

i

)(),(

: No. of bursting HR neurons in the ith bursting stripe

Pacing degree of bursting onset times: contribution of bursting onset times to the macroscopic IPBR

: Global bursting phase based on IPBR

)(

1)(

),( cos1

oniB

kkon

i

onbi B

P

bN

i

onbi

b

onb M

NM

1

),()( 1

33.0~

33.0~)(

)(

offb

onb

O

O

92.0~

94.0~)(

)(

offb

onb

P

P

30.0~

31.0~)(

)(

offb

onb

M

M

2/][ )()( offb

onbb MMM

: Total No. of microscopic bursting onset times in the ith bursting stripe Statistical-Mechanical Bursting

Measure Based on IPBR

Statistical-Mechanical Bursting Measure

0,3.0,3.1 DJIDC

3.0,3.1 JIDC

)(onbR

)(biN

)(onbR

k )(onbR

)(oniB

With increasing D, Ob decreases very slowly, while Pb and Mb decrease rapidly.

Characterization of Intraburst Spiking Transition Based on

9

sR

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)]

sR

Thermodynamic Intraburst Spiking Order Parameter

bN

i

is

bs N 1

)(1OO

Unsynchronized Spiking State: N, then Os0Synchronized Spiking State: N, then Os Non-zero value

Bursting Transition occurs for )032.0~(*sD

3.0,3.1 JIDC

3.0,3.1 JIDC

As D is increased, stripes in the burst band becomes smeared and overlap. Amplitude of decreases.

sR

2)( ))()(( tRtR ssi

s OMean square deviation of in the ith global bursting cycle:sR

Thermodynamic spiking order parameter:

Intraburst Spiking Measure Based on IPSR

10

sR

)(,

)(,

)(,

sji

sji

sji POM Spiking measure of jth global

spiking cycle in the ith bursting cycle Occupation degree of spiking times

N

NO

sjis

ji

)(,)(

, : No. of spiking HR neurons in the jth spiking cycle and the ith bursting cycle

Pacing degree of spiking times: Contribution of spiking times to the macroscopic IPSR

)(

,

1)(

,

)(, cos

1sjiN

kks

ji

sji N

P

sN

j

sji

s

si M

NM

1

)(,

)( 1

bN

i

si

bs M

NM

1

)(1

: Total No. of microscopic spiking times

Statistical-Mechanical Spiking Measure Based on IPSR

Statistical-Mechanical Spiking Measure

sR

)(,sjiN

sR

)(,sjiN

25.0~sO

56.0~sP

14.0~sM

3.0,3.1 JIDC

0 ,3.0,3.1 DJIDC

With increasing D, <Os>r decreases very slowly, while <Ps>r and <Ms>r decrease very rapidly.

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)

R bR sR

bR

sR

sR

)(onbR )(off

bR

)(onbR )(off

bR