  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.
