Synchrony and Asynchrony for Neuronal Dynamics Defined on Complex Networks

• Published in: Bulletin of mathematical biology Vol. 74; no. 4; pp. 769 - 802
• Main Authors: DeVille, R. E. Lee, Peskin, Charles S.
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01-04-2012; Springer Nature B.V
• Subjects: Cell Biology; Computer Simulation; Humans; Life Sciences; Mathematical and Computational Biology; Mathematical models; Mathematics; Mathematics and Statistics; Models, Neurological; Nerve Net - physiology; Neurons - physiology; Original Article; Stochastic Processes; Neuronal network; Random graphs; Erdős–Rényi; Complex networks; Stochastic integrate-and-fire; Neural network; Scale-free networks; Small world networks; Synchrony; Mean-field analysis
• ISSN: 0092-8240; 1522-9602
• DOI: 10.1007/s11538-011-9674-0

We describe and analyze a model for a stochastic pulse-coupled neuronal network with many sources of randomness: random external input, potential synaptic failure, and random connectivity topologies. We show that different classes of network topologies give rise to qualitatively different types of synchrony: uniform (Erdős–Rényi) and “small-world” networks give rise to synchronization phenomena similar to that in “all-to-all” networks (in which there is a sharp onset of synchrony as coupling is increased); in contrast, in “scale-free” networks the dependence of synchrony on coupling strength is smoother. Moreover, we show that in the uniform and small-world cases, the fine details of the network are not important in determining the synchronization properties; this depends only on the mean connectivity. In contrast, for scale-free networks, the dynamics are significantly affected by the fine details of the network; in particular, they are significantly affected by the local neighborhoods of the “hubs” in the network.