1. Sparsely Synchronized Brain Rhythms in A Small-World Neural Network W. Lim (DNUE) and S.-Y. KIM (LABASIS)

2. 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 1 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 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)  Fully Synchronized Brain Rhythm

3. Behaving Brain Rhythms via Sparse Synchronization 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 2  Cortical Behaving Rhythms for the Awake Brain Gamma rhythm in visual cortex of behaving monkey 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  Sparsely Synchronized Brain Rhythms

4. Beta Rhythm via Sparse Synchronization 3  Sparsely Synchronized Beta Rhythm Population Rhythm ~ 15-30Hz  Beta Oscillation Individual 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)

5. Network of Inhibitory Subthreshold Morris-Lecar Neurons  Coupled Morris-Lecar (ML) Neurons on A One-Dimensional Ring 4  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 I DC =95 Noise-Induced Firing of the Subthreshold case for I DC =87 and D=20 Connection Weight Matrix W:

6. Optimal Small-World Network  Small-World Network of Inhibitory Subthreshold Morris-Lecar Neurons 5  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 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. 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.

7. Emergence of Synchronized Population States 6 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 V G : 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 V G becomes smaller & duration of incoherent parts becomes longer V G tends to be nearly stationary as N   Unsynchronized population state  Synchronized State for p=0.2 Raster plot: Little zigzagness V G displays more regular oscillation as N   Synchronized population state

8. Synchrony-Asynchrony Transition 7  Investigation of Population Synchronization by Increasing the Rewiring Probability Occurrence of Population Synchronization for p>p th (~ 0.044) Incoherent State: N , then O  0 Coherent State: N , then O  Non-zero value Investigation of Population Synchronization Using a Thermodynamic Order Parameter: (V G : Population-Averaged Membrane Potential)

9. Population and Individual Behaviors of Synchronized States 8  Raster Plot and Global Potential  Population Rhythm  Firing Rate of Individual Neurons  Interspike Interval Histograms Power spectra of V G with peaks at population frequencies ~ 18Hz  Beta Rhythm Average spiking frequency ~ 2Hz  Sparse spikings Multiple peaks at multiples of the period of the global potential  Stochastic phase locking leading to Stochastic Spike Skipping With increasing p, the zigzagness degree in the raster plot becomes reduced. p>p max (~0.5): Raster plot composed of stripes without zigzag and nearly same pacing degree. Amplitude of V G increases up to p max, and saturated.

10. Economic Small-World Network 9  Synchrony Degree M  Wiring Length  Dynamical Efficiency Factor With increasing p, synchrony degree is increased because global efficiency of information transfer becomes better. Corr i (0): Normalized cross-correlation function between V G and v i for the zero time lag 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 Tradeoff between Synchrony and Wiring Economy

11. 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 p th (=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=p DE (=0.24) 10