Sparsely Synchronized Brain Rhythms

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Sparsely Synchronized Brain Rhythms in
A Small-World Neural Network
W. Lim (DNUE) and S.-Y. KIM (LABASIS)
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Silent Brain Rhythms via Full Synchronization
? Brain Rhythms for the Silent Brain
Sleep Spindle Rhythm M. Steriade, et. Al. J.
Neurophysiol. 57, 260 (1987). Brain rhythm
(714Hz) with large amplitude during deep sleep
without dream
Alpha Rhythm H. Berger, Arch. Psychiatr
Nervenkr.87, 527 (1929) Slow brain rhythm
(312Hz) with large amplitude during the
contemplation with closing eyes
? 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
Connection Weight Matrix W
? Firings in the Single Type-II ML Neuron
? Type-II Excitability of the Single ML Neuron
Regular Firing of the Suprathreshold case for
IDC95
Noise-Induced Firing of the Subthreshold case for
IDC87 and D20
Type-II Excitability (act as a resonator)
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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
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
Small-world-ness measure S(p) forms a bell-shaped
curve. ? Optimal small-world network exists for
ppsw ( 0.037)
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Emergence of Synchronized Population States
Investigation of collective spike synchronization
using the raster plot and population-averaged
membrane potential

• Unsynchronized State in the Regular Lattice
• (p0)

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

• Synchronized State for p0.2

Raster plot Little zigzagness 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
(VG Population-Averaged Membrane Potential)
Incoherent State N??, then O?0 Coherent State
N??, then O? Non-zero value
Occurrence of Population Synchronization for
pgtpth ( 0.044)
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Population and Individual Behaviors of
Synchronized States

• Raster Plot and Global Potential

With increasing p, the zigzagness degree in the
raster plot becomes reduced. pgtpmax (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

• Synchrony Degree M

Corri(0) Normalized cross-correlation
function between VG and vi for the
zero time lag
With increasing p, synchrony degree is increased
because global efficiency of information transfer
becomes better.
? Wiring Length ?
Wiring length increases linearly with respect to
p. ? With increasing p, the wiring cost becomes
expensive.
Optimally sparsely synchronized beta rhythm for
ppDE ( 0.24) Raster plot with a zigzag
pattern due to local clustering of spikes
(C0.31) Regular oscillating global potential
? Dynamical Efficiency Factor
Tradeoff between Synchrony and Wiring Economy
Optimal beta rhythm emerges at a minimal wiring
cost in an economic small-world network for
ppDE (0.24).
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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 ppDE (0.24)

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