Sparsely Synchronized Brain Rhythms
in A Small-World Neural Network
W. Lim (DNUE) and S.-Y. KIM (LABASIS)
Silent Brain Rhythms via Full Synchronization
 Brain Rhythms for the Silent Brain
Alpha Rhythm
[H. Berger, Arch. Psychiatr Nervenkr.87, 527 (1929)]
Slow brain rhythm (3~12Hz) with large amplitude
during the contemplation with closing eyes
Sleep Spindle Rhythm
[M. Steriade, et. Al. J. Neurophysiol. 57, 260 (1987).]
Brain rhythm (7~14Hz) with large amplitude
during deep sleep without dream
 Fully Synchronized Brain Rhythm
Individual Neurons: Regular Firings like Clocks
Large-Amplitude Slow Population Rhythm via
Full Synchronization of Individual Regular Firings
Investigation of this Huygens mode of full
synchronization using coupled oscillators model
 Coupled Suprathreshold Neurons (without
noise or with small noise)
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Behaving Brain Rhythms via Sparse Synchronization
 Cortical Behaving Rhythms for the Awake Brain
Sparsely Synchronized Rhythms
Desynchronized EEG: Appearance of fast brain
rhythms [Beta Rhythm (15-30Hz), Gamma
Rhythm (30-100Hz), Ultrafast Rhythm (100-200Hz)]
With Small Amplitude in the EEG of the Waking Brain.
In contrast for the slow brain rhythm with large
amplitude for silent brain
Gamma rhythm in visual cortex of behaving monkey
 Sparsely Synchronized Brain Rhythms
Individual Neurons: Intermittent and Stochastic Firings
like Geiger Counters
Small-Amplitude Fast Population Rhythm via Sparse
Synchronization of Individual Complex Firings
Coupled oscillators model: Inappropriate for
investigation of the sparsely synchronized rhythms
 Coupled Subthreshold and/or Suprathreshold
Neurons in the Presence of Strong Noise
They exhibit noise-induced complex firing patterns
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Beta Rhythm via Sparse Synchronization
 Sparsely Synchronized Beta Rhythm
[V. Murthy and E. Fetz, J. Neurophysiol. 76, 3968 (1996)]
Population Rhythm ~ 15-30Hz  Beta Oscillation
Individual neurons show intermittent and irregular firing patterns like Geiger counters
Beta Rhythm: Associated with (1) Preparation and Inhibitory Control in the motor
system, (2) Long-Distance Top-Down Signaling along feedback
pathways in reciprocally-connected loop between cortical areas
with laminar structures (inter-areal synchronization associated
with selective attention, working memory, guided search,
object recognition, perception, sensorimotor integration)
Impaired Beta Rhyrhm: Neural Diseases Associated with Cognitive Dysfunction
(schizophrenia, autism spectrum disorder)
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Network of Inhibitory Subthreshold Morris-Lecar Neurons
 Coupled Morris-Lecar (ML) Neurons on A One-Dimensional Ring
C
dvi
  I ion,i  I DC ,i  Di  I syn ,i
dt
dwi
( w ( v )  wi )
  i
, i  1, ..., N .
dt
 R (vi )
I syn ,i 
J
diin
w
j ( i )
ij
wij  1 : Neuron j is presynaptic to neuroni
wij  0 : Otherwise
In - degree : d 
SynapticCoupling:
N
Connection Weight Matrix W:
in
i
s j (t )(vi  Vsyn );
dsi
   s  (vi )(1  si )    si
dt
N
w ,
j ( i )
ij
InhibitorySynapse(mediatedby GABA A receptors):
  10ms-1 ,   0.1ms-1 , Vsyn  80mV
 Type-II Excitability of the
Single ML Neuron
Type-II Excitability (act as a resonator)
 Firings in the Single Type-II ML Neuron
Regular Firing of the
Suprathreshold case
for IDC=95
Noise-Induced Firing of
the Subthreshold case
for IDC=87 and D=20
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Optimal Small-World Network
 Small-World Network of Inhibitory
Subthreshold Morris-Lecar Neurons
Start with directed regular ring lattice with N
neurons where each neuron is coupled to its
first k neighbors.
Rewire each outward connection at random
with probability p such that self-connections
and duplicate connections are excluded.
 Clustering Coefficient and Average Path Length
N  3 103 , k  50
Average path length decreases dramatically with increasing p.
During the drop in the average path length, clustering
coefficient remains almost constant.
 For small p, small-world network with high clustering and
short path lengths appear.
 Small-World-ness Measure
S ( p) 
NormalizedClusteringCoefficient
NormalizedAverage PathLength
Small-world-ness measure S(p) forms a bell-shaped curve.
 Optimal small-world network exists for p=p*sw (~ 0.037)
N  3 103
k  50
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Emergence of Synchronized Population States
Investigation of collective spike synchronization using
the raster plot and population-averaged membrane
potential
1 N
VG (t )   vi (t )
N i 1
I DC  87, J  3, D  20, k  50, p  0
N  103
 Unsynchronized State in the Regular Lattice
(p=0)
Raster plot: Zigzag pattern intermingled with inclined
partial stripes
VG: Coherent parts with regular large-amplitude
oscillations and incoherent parts with irregular
small-amplitude fluctuations.
With increasing N,
Partial stripes become more inclined.
Spikes become more difficult to keep pace
Amplitude of VG becomes smaller & duration of
incoherent parts becomes longer
VG tends to be nearly stationary as N
 Unsynchronized population state
N  104
I DC  87, J  3, D  20, k  50, p  0.2
N  103
 Synchronized State for p=0.2
Raster plot: Little zigzagness
N  104
VG displays more regular oscillation as N
 Synchronized population state
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Synchrony-Asynchrony Transition
 Investigation of Population Synchronization by Increasing the Rewiring
Probability
Investigation of Population Synchronization Using a Thermodynamic Order Parameter:
O  (VG ) 2  (VG (t )  VG (t )) 2
(VG: Population-Averaged Membrane Potential)
Incoherent State: N, then O0
Coherent State: N, then O Non-zero value
Occurrence of Population Synchronization for p>pth (~ 0.044)
I DC  87, J  3, D  20, k  50
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Population and Individual Behaviors of Synchronized States
 Raster Plot and Global Potential
I DC  87, J  3, D  20, N  103 , k  50
With increasing p, the zigzagness
degree in the raster plot becomes
reduced.
p>pmax (~0.5): Raster plot composed
of stripes without zigzag and nearly
same pacing degree.
Amplitude of VG increases up to pmax,
and saturated.
 Population Rhythm
Power spectra of VG with peaks at
population frequencies ~ 18Hz
 Beta Rhythm
 Firing Rate of Individual Neurons
Average spiking frequency ~ 2Hz
 Sparse spikings
 Interspike Interval Histograms
Multiple peaks at multiples of the
period of the global potential
 Stochastic phase locking leading
to Stochastic Spike Skipping
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Economic Small-World Network
I DC  87, J  3, D  20, N  103 , k  50
 Synchrony Degree M
Corri(0): Normalized cross-correlation
function between VG and vi
i 1
for the zero time lag
With increasing p, synchrony degree is increased because
global efficiency of information transfer becomes better.
1
M
N
N
 Corri (0)
 Wiring Length 
Wiring length increases linearly with respect to p.
 With increasing p, the wiring cost becomes expensive.
 Dynamical Efficiency Factor
Tradeoff between Synchrony and Wiring Economy
SynchronyDegree
 ( p) 
NormalizedWiringLength
Optimally sparsely synchronized
beta rhythm for p=p*DE (~ 0.24)
Raster plot with a zigzag pattern due to
local clustering of spikes (C=0.31)
Regular oscillating global potential
Optimal beta rhythm
emerges at a minimal
wiring cost in an economic
small-world network for
p=p*DE (~0.24).
I DC  87, J  3, D  20, N  103 , k  50
I DC  87, J  3, D  20, N  103 , k  50
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Summary
 Emergence of Sparsely Synchronized Beta Rhythm in A
Small-World Network of Inhibitory Subthreshold ML Neurons
Regular Lattice of Inhibitory Subthreshold ML Neurons
 Unsynchronized Population State
Occurrence of Sparsely Synchronized Beta Rhythm as The Rewiring Probability
Passes A Threshold pth (=0.044):
 Population Rhythm ~ 18 Hz (small-amplitude fast sinusoidal oscillation)
 Beta Oscillation
Intermittent and Irregular Discharge of Individual Neurons at 2 Hz
(Geiger Counters)
Emergence of Optimally Sparsely Synchronized Beta Rhythm at A Minimal
Wiring Cost in An Economic Small-World Network for p=pDE (=0.24)
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