Collective dynamics of networks of active units with pulse coupling: Review

Aim of the paper is to review of collective dynamics study of networks of active units with pulse couplings. Many network oscillatory systems are characterized by interactions between the nodes in the form of an exchange of short signals, or pulses. The most important class of such network systems is biological neural networks, that is, populations of nerve cells. Main approaches to the study of networks with pulse coupling are described and the results obtained to date are systematized.
The works considered in the review usually use fairly simple models to describe the local dynamics of network elements such as integrate-and-fire and its generalizations. The simplicity of these models allows in many cases to study them analytically, and the main ideas of this analysis are described in the review. As for the structure of the networks, they are quite diverse and include fully-connected networks, networks with sparce coupling, multi-population and modular (cluster) networks.
The review is structured according to the type of collective dynamics observed in networks with pulse coupling. First, works on synchronous dynamics were described, the study of which in networks with impulse communications was historically the first. Next, we turn to asynchronous dynamics, characterized by the absence of a correlation between the moments of pulse generation by various network units. An important special case of such dynamics is irregular asynchronous dynamics, which is discussed in the next section. Finally, partially synchronous regimes are considered, characterized by pronounced fluctuations in the mean field. At the end of the review, modern approaches to reducing network dynamics are systematized, aimed at obtaining low-dimensional dynamic systems that describe network dynamics in terms of averaged variables.