

Preprint 46/2009
Synchronized chaos in networks of simple units
Frank Bauer, Fatihcan M. Atay, and Jürgen Jost
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Submission date: 04. Aug. 2009 (revised version: January 2010)
Pages: 9
published in: epl, 89 (2010) 2, art-no. 20002
DOI number (of the published article): 10.1209/0295-5075/89/20002
Bibtex
PACS-Numbers: 05.45.-a, 05.45.Ra, 05.45.Xt
Keywords and phrases: synchronization, direct coupling, neuronal networks
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Abstract:
We study synchronization of non-diffusively coupled map
networks with arbitrary network topologies, where the connections
between different units are, in general, not symmetric and can carry
both positive and negative weights. We show that, in contrast to
diffusively coupled networks, the synchronous behavior of a
non-diffusively coupled network can be dramatically different from
the behavior of its constituent units. In particular, we show that
chaos can emerge as synchronized behavior although the dynamics of
individual units are very simple. Conversely, individually chaotic
units can display simple behavior when the network synchronizes. We
give a synchronization criterion that depends on the spectrum of the
generalized graph Laplacian, as well as the dynamical properties of
the individual units and the interaction function. This general
result will be applied to coupled systems of tent and logistic maps
and to two models of neuronal dynamics. Our approach yields an
analytical understanding of how simple model neurons can produce
complex collective behavior through the coordination of their
actions.