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)