Modelling on the very large-scale connectome

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 certain universality, which allows to select a representative one, the KKI-18, which has been used for dynamical simulation subsequently. The critical and sub-critical behavior of simple, two or three state threshold models is discussed with special emphasis of 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 GP, termed frustrated synchronization in Chimera states. The models presented here exhibit avalanche scaling behavior with exponents in agreement with brain experimental data if local homeostasis is provided.