dc.description.abstracteng | Synchrony is one of the most common
dynamical states emerging on networks. The speed of convergence
towards synchrony provides a fundamental collective time scale for
synchronizing systems. Here we study the asymptotic synchronization
times for directed networks with topologies ranging from completely
ordered, grid-like, to completely disordered, random topoplogies -
including intermediate, partially disordered, the so-called
small-world, topologies. These small-world topologies
simultaneously exhibit large local clustering as well as short
average path length and have therefore been suggested to support
network synchronization. We investigate the asymptotic speed of
synchronization of coupled oscillators in dependence on the degree
of randomness of their interaction topology in generalized
Watts-Strogatz ensembles. We find that networks with fixed
in-degree synchronize faster the more random they are, with small
worlds just appearing as an intermediate case. For any generic
network ensemble, if synchronization speed is at all extremal at
intermediate randomness, it is slowest in the small-world regime.
This phenomenon occurs for various types of oscillators, intrinsic
dynamics and coupling schemes. Randomly rewiring real-world neural,
social and transport networks confirms this picture. We extend the
approach of master stability functions to quantify synchronization
times. Synchronization dynamics on small-world networks emerge in a
broad range of systems with their spectra characterizing
fundamental asymptotic features. We derive analytic mean-field
predictions for the spectra of small-world models. These
theoretical predictions agree well with the actual spectra
(obtained by numerical diagonalization) for undirected and directed
networks and from fully regular to strongly random topologies.
These results may provide analytical insights to empirically found
features of dynamics on small-world networks from various research
fields, including biology, physics, engineering, and social
science. Another network architecture, the scale-free topology, is
characterized by a heavy-tailed distribution of degree per node
with no characteristic scale. For neural circuits this means,
although most nerve cells (neurons) display local connectivity, a
small number of hub neurons, characterized by long-range
connections linking large numbers of cells, can confer
synchronicity on the network. Thus, the presence of hub neurons,
which act as super-connected nodes, has been postulated as a
substrate for widespread neural synchronization. Intriguingly, by
stimulating single hub neurons one may remove the synchronous
collective oscillations completely. When the stimulation is
switched off, the synchronicity , but if a non-hub unit driven in
the same way, the oscillation stays almost identically as in the
undriven state. Such synchronous oscillations constitute one of the
most dominant collective dynamics of complex networks. They occur
not only in circuits of neurons, but in a large range of systems,
ranging from metabolic and gene regulatory networks within cells to
food webs of crossfeeding species or even to oscillations in the
global climate system. Thus, understanding the functional role of
hubs – not only in neuronal circuits – is a task of paramount
importance and has recently attracted widespread attention.
However, the mechanisms underlying the suppression of global
oscillations in a neurobiological system have not yet been
understood. We set up a detailed theoretical framework and
numerically investigate different neuronal network models to reveal
potential mechanisms that underlie the experimentally discovered
phenomena. | de |