Noise-Induced Burst and Spike Synchronizations in An Inhibitory   Small-World Network of Subthreshold Bursting Neurons
  For modeling complex synaptic connectivity, we consider the Watts-Strogatz small-world network which interpolates between regular lattice and random network via rewiring, and investigate the effect of small-world connectivity on emergence of noise-induced population synchronization in an inhibitory population of subthreshold bursting Hindmarsh-Rose neurons. Thus, noise-induced slow burst synchronization and fasg spike synchronization are found to appear in a synchronized region of the $J-D$ plane. As the rewiring probability $p$ is decreased from 1 (random network) to 0 (regular lattice), the region of spike synchronization shrinks rapidly in the $J-D$ plane, while the region of the burst synchronization decreases slowly. Population synchronization may be well visualized in the raster plot of neural spikes which can be obtained in experiments. Instantaneous population firing rate, $R(t)$, which is directly obtained from the raster plot of spikes, is a realistic population quantity exhibiting collective behaviors with both the slow bursting and the fast spiking timescales. Through frequency filtering, we separate $R(t)$ into $R_b(t)$ (describing the slow bursting behavior) and $R_s(t)$ (describing the fast intraburst spiking behavior). Then, we develop thermodynamic order parameters and statistical-mechanical measures, based on $R_b (t)$ and $R_s (t)$, for characterization of the burst and spike synchronizations of the bursting neurons and show their usefulness in explicit examples. With increase in $p$, both the degrees of the burst and spike synchronizations are found to increase because more long-range connections appear. However, they become saturated for some maximal values of $p$ because long-range short-cuts which appear up to the maximal values of $p$ play sufficient role to get maximal degrees of the burst and spike synchronizations. 
Sang-Yoon Kim and Woochang Lim
q-bio.NC physics.bio-ph