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Frequency-Domain Order Parameter for Synchronization
Transition of Bursting Neurons
1
Lim
Woochang
and Sang-Yoon
1Department of Science Education, Daegu National University of Education
2Research Division, LABASIS Co.
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
g  ( xi )  1 /[1  e
( x  x*s )
].
Characterization of Intraburst Spiking
Transition Based on Rs
 Investigation of Intraburst Spiking States via Separation of Fast Time
Scale and in the Frequency Domain
I DC  1.3, J  0.3
IPSR Rs 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 Rs decreases.
 Peak in the power spectra
of Rs decreases.
 Frequency-Domain Spiking Order Parameter
Unsynchronized Spiking State:
N, then <s>r0
Synchronized Spiking State:
N, then <s>r Non-zero value
Bursting Transition occurs for D ( ~0.032)
*
s
Inhibitory Population of Bursting Neurons
dxi
3
2
 yi  axi  bxi  zi  I DC  Di  I syn,i ,
dt
dyi
dzi
2
 c  dxi  yi ,
 r[ s( xi  xo )  zi ],
dt
dt
dgi
 g  ( xi )(1  gi )  gi , i  1, ..., N ,
dt
N
J
I syn,i 
g j (t )( xi  X syn ),

N  1 j ( i )
2
Kim
Parameters in the single
Hindmarsh-Rose (HR) neuron
a  1, b  3, c  1, d  5,
r  0.001, s  4, xo  1.6
Parameters for the synaptic
current
X syn  2, xs*  0,   30,
  10 ms -1 ,   0.1 ms -1
Characterization of Bursting Transition Based
on Bursting Onset and Offset Times
I  1.3, J  0.3
DC
 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
(off )
(on )
R
R
times and IPBR b and b
Bursting Activity in the HR Neuron
 Bursting Transition in the Single Neuron
Transition to bursting occurs IDC=I*DC (~1.268)
I DC  1.3
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
(fc=10Hz)  IPBR Rb
As D is increased, bursting
stripes in the raster plots
becomes smeared and
overlap.
(on )
 Amplitude of both Rb
(off )
R
and b decreases.
 Peak in the power
(off )
(on )
R
spectra of Rb and b
of decreases.
 Frequency-Domain Bursting Order Parameter
Unsynchronized Bursting State:
( on )
( off )



and


r  0
N, then
b
r
b
Synchronized Bursting State:
N, then
  b( on )  r and   b( off )  r  Non-zero value
Bursting Transition occurs for D ( ~0.068)
*
b
As D is increased, bursting
bands in the raster plots
becomes smeared and
overlap.  Amplitude of Rb decreases.
 Investigation of Population
Bursting States in the
Frequency Domain
Summary
I DC  1.3, J  0.3
 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
Power spectra of Rb
 b  H pQ; Q  f p / f p
Bursting Coherence Factor
Hp and fp: Height and frequency of bursting peak
fp: width of the bursting peak at the height of e-1/2 Hp.
 Frequency-Domain Bursting Order Parameter
Unsynchronized Bursting State:
N, then <b>r0
Synchronized Bursting State:
N, then <b>r Non-zero value
Bursting Transition occurs for D ( ~0.068)
*
b
IPFR R  IPBR Rb (Describing the Slow Bursting Behavior)
and IPSR Rs (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)