1. Sparsely Synchronized Brain Rhythm in A Small-World Neural Network
Woochang Lim1 and Sang-Yoon Kim2
1Department of Science Education, Daegu National University of Education, Korea
2Research Division, LABASIS Co., Korea
Introduction
 Sparsely Synchronized Brain Rhythms
Associated with diverse cognitive functions (e.g., sensory perception, feature
integration, selective attention, and memory formation)
Population level: Fast oscillations [e.g., beta rhythm (15~30Hz), gamma rhythm
(30~100Hz), and sharp-wave ripple (100~200Hz)]
Cellular level: Stochastic and Intermittent discharges
 Complex Brain Network
Network Topology: Complex (small-worldness and scale-freeness)
Investigation of Sparsely-Synchronized Brain Rhythms in the Watts-Strogatz
model for small-world networks
Population and Individual Behaviors of
Synchronized States
 Raster Plot and Global Potential
With increasing p, the zigzagness degree in the raster plot becomes reduced.
p>pmax (~0.5): Raster plot composed of stripes without zigzag.
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 VG
 Stochastic phase locking leading to Stochastic Spike Skipping
Small-World Neural Network
 Watts-Strogatz Model for the Small-World Network
on A One-Dimensional Ring
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
 Inhibitory Population of Subthreshold Morris-Lecar Neurons
Connection Weight Matrix W
(determined by the small-world network topology):
Economic Small-World Network
 Synchrony Degree M
Spiking measure Mi of the ith stripe in the raster plot
= Occupation degree Oi (representing the density of the ith stripe)
 Pacing degree Pi (denoting the smearing of the ith stripe)
<Oi> is nearly the same (~0.11), independently of p.
(due to stochastic spike skipping)
With increasing p,<Pi> increases rapidly (due to appearance of long-range
connections), and saturates for p=pmax (Number of long-range shortcuts for
p=pmax is sufficient for the maximal pacing degree).
With increasing p, synchrony degree M is increased until p=pmax 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.
 Dynamical Efficiency Factor
Tradeoff between Synchrony
and Wiring Economy
 Optimally Sparsely Synchronized
Rhythm
Emerges at a minimal wiring cost in
an economic small-world network for
p=p*DE (~0.24).
Real Inhibitory Synapse mediated by GABAA receptors:
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 (p=0)
Raster plot: Zigzag pattern
intermingled with inclined
partial stripes
VG tends to be nearly stationary
as N
 Unsynchronized population state
 Synchronized State for p=0.2
Raster plot: Little zigzagness
VG displays more regular
oscillation as N
 Synchronized population state
Synchrony-Asynchrony Transition
 Thermodynamic Order Parameter
Mean Square Deviation of VG:
As N  then O  non-
zero (zero) limit value for
coherent (incoherent) states.
Occurrence of Population Synchronization for p>pth (~ 0.044)
Summary
 Emergence of Sparsely Synchronized Rhythms in A Small-World Network
of Inhibitory Subthreshold ML Neurons
Occurrence of Sparsely Synchronized Brain Rhythm as The Rewiring Probability
Passes A Threshold pth (~0.044)
Emergence of Optimally Sparsely Synchronized Brain Rhythm at a Minimal
Wiring Cost in An Economic Small-World Network for p=p*DE (~0.24)