Sparsely Synchronized Brain Rhythms in A Small-World Neural Network

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  Brain Rhythms for the Silent Brain Sleep Spindle Rhythm [M. Steriade, et. Al. J. Neurophysiol. 57, 260 (1987).] Brain rhythm (7~14Hz) with large amplitude during deep sleep without dream Alpha Rhythm [H. Berger, Arch. Psychiatr Nervenkr.87, 527 (1929)] Slow brain rhythm (3~12Hz) 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) 1

3. BehavingBrain 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 2

4. 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) 3

5. 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 IDC=95 Noise-Induced Firing of the Subthreshold case for IDC=87 and D=20 Type-II Excitability (act as a resonator) 4

6. 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 p=p*sw (~ 0.037) 5

7. Emergence of Synchronized Population States Investigation of collectivespike 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: 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 p=0.2 Raster plot: Little zigzagness VG displays more regular oscillation as N  Synchronized population state 6

8. 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 p>pth (~ 0.044) 7

9. 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 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 8

10. 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 p=p*DE (~ 0.24) Raster plot with a zigzag pattern due to local clustering of spikes (C=0.31) 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 p=p*DE(~0.24). 9

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 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) 10