  In this review, we discuss critical dynamics of simple nonequilibrium models
on large connectomes, obtained by diffusion MRI, representing the white matter
of the human brain. In the first chapter, we overview graph theoretical and
topological analysis of these networks, pointing out that universality allows
selecting a representative network, the KKI-18, which has been used for
dynamical simulation. The critical and sub-critical behaviour of simple, two-
or three-state threshold models is discussed with special emphasis on
rare-region effects leading to robust Griffiths Phases (GP). Numerical results
of synchronization phenomena, studied by the Kuramoto model, are also shown,
leading to a continuous analog of the GP, termed frustrated synchronization in
Chimera states. The models presented here exhibit avalanche scaling behaviour
with exponents in agreement with brain experimental data if local homeostasis
is provided.
