Sparsely Synchronized Brain Rhythms in A Small-World Neural Network

W. Lim (DNUE) and S.-Y. KIM (LABASIS)

Silent Brain Rhythms via Full Synchronization

Alpha Rhythm[H. Berger, Arch. Psychiatr Nervenkr.87, 527 (1929)]Slow brain rhythm (3~12Hz) with large amplitude during the contemplation with closing eyes

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Sleep Spindle Rhythm[M. Steriade, et. Al. J. Neurophysiol. 57, 260 (1987).]Brain rhythm (7~14Hz) with large amplitude during deep sleep without dream

Brain Rhythms for the Silent Brain

Individual Neurons: Regular Firings like ClocksLarge-Amplitude Slow Population Rhythm viaFull 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)

Fully Synchronized Brain Rhythm

Behaving Brain Rhythms via Sparse Synchronization

Sparsely Synchronized RhythmsDesynchronized EEG: Appearance of fast brain rhythms [Beta Rhythm (15-30Hz), GammaRhythm (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

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Cortical Behaving Rhythms for the Awake Brain

Gamma rhythm in visual cortex of behaving monkey

Individual Neurons: Intermittent and Stochastic Firings like Geiger CountersSmall-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

Sparsely Synchronized Brain Rhythms

Beta Rhythm via Sparse Synchronization

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Sparsely Synchronized Beta Rhythm

Population Rhythm ~ 15-30Hz Beta OscillationIndividual neurons show intermittent and irregular firing patterns like Geiger counters

[V. Murthy and E. Fetz, J. Neurophysiol. 76, 3968 (1996)]

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)

Network of Inhibitory Subthreshold Morris-Lecar Neurons

Coupled Morris-Lecar (ML) Neurons on A One-Dimensional Ring

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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 casefor IDC=95

Noise-Induced Firing of the Subthreshold case for IDC=87 and D=20

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Connection Weight Matrix W:

Optimal Small-World Network

Small-World Network of Inhibitory Subthreshold Morris-Lecar Neurons

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Small-World-ness Measure

Small-world-ness measure S(p) forms a bell-shaped curve. Optimal small-world network exists for p=p*

sw (~ 0.037)

Clustering Coefficient and Average Path Length

Length PathAverage Normalized

tCoefficien Clustering Normalized)( pS

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.

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N

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.

Emergence of Synchronized Population States

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Investigation of collective spike synchronization using the raster plot and population-averaged membrane potential

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Unsynchronized State in the Regular Lattice (p=0)

Raster plot: Zigzag pattern intermingled with inclined partial stripesVG: 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 longerVG 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

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Investigation of Population Synchronization by Increasing the Rewiring Probability

Occurrence of Population Synchronization for p>pth (~ 0.044)

Incoherent State: N, then O0Coherent State: N, then O Non-zero value

Investigation of Population Synchronization Using a Thermodynamic Order Parameter:

(VG: Population-Averaged Membrane Potential)22 ))()(()( tVtVV GGG O

50,20,3,87 kDJIDC

Population and Individual Behaviors of Synchronized States

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Raster Plot and Global Potential

Population Rhythm

Firing Rate of Individual Neurons

Interspike Interval Histograms

50,10,20,3,87 3 kNDJIDC

Power spectra of VG with peaks at population frequencies ~ 18Hz Beta Rhythm

Average spiking frequency ~ 2Hz Sparse spikings

Multiple peaks at multiples of the period of the global potentialStochastic phase locking leading to Stochastic Spike Skipping

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.

Economic Small-World Network

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Synchrony Degree M

Wiring Length

Dynamical Efficiency Factor

With increasing p, synchrony degree is increased because global efficiency of information transfer becomes better.

N

iiCorr

NM

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)0(1 Corri(0): Normalized cross-

correlation function between VG and vi for the zero time lag

LengthWiring Normalized

DegreeSynchrony )( p

Wiring length increases linearly with respect to p. With increasing p, the wiring cost becomes expensive.

Optimal beta rhythm emerges at a minimal wiring cost in an economic small-world network for p=p*

DE (~0.24).

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

50,10,20,3,87 3 kNDJIDC

50,10,20,3,87 3 kNDJIDC 50,10,20,3,87 3 kNDJIDC

Tradeoff between Synchrony and Wiring Economy

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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