  It has long been argued that neural networks have to establish and maintain a
certain intermediate level of activity in order to keep away from the regimes
of chaos and silence. Strong evidence for criticality has been observed in
terms of spatio-temporal activity avalanches first in cultures of rat cortex by
Beggs and Plenz (2003) and subsequently in many more experimental setups. These
findings sparked intense research on theoretical models for criticality and
avalanche dynamics in neural networks, where usually some dynamical order
parameter is fed back onto the network topology by adapting the synaptic
couplings. We here give an overview of existing theoretical models of dynamical
networks. While most models emphasize biological and neurophysiological detail,
our path here is different: we pick up the thread of an early self-organized
critical neural network model by Bornholdt and Roehl (2001) and test its
applicability in the light of experimental data. Keeping the simplicity of
early models, and at the same time lifting the drawback of a spin formulation
with respect to the biological system, we here study an improved model
(Rybarsch and Bornholdt, 2012b) and show that it adapts to criticality
exhibiting avalanche statistics that compare well with experimental data
without the need for parameter tuning.
