Fig. 1: Capturing the hierarchical organisation using the FDT.

a Hierarchical level is determined by the level of asymmetry of causal interactions between brain regions arising from the breaking of the detailed balance in a system. In equilibrium, the interaction of brain regions is symmetrical, that is, information flows in a reciprocal manner. These symmetrical relationships are in detailed balance, leading to a non-hierarchical organisation. In contrast, in non-equilibrium, the asymmetrical interactions break the detailed balance, introducing hierarchical organisation in the system. b The level of non-equilibrium can be measured by the deviation of FDT, which can then be employed to characterize the hierarchical organisation. In an equilibrium system, spontaneous fluctuations predict the dissipation following the perturbation. Nevertheless, in a non-equilibrium system, the intrinsic fluctuations are not able to forecast the dissipation after a perturbation, leading to a violation of the FDT. c FDT is combined with a whole-brain model fitted to empirical neuroimaging data, incorporating functional and structural connectivity. Each node’s local dynamics of the model is described as the normal form of a supercritical Hopf bifurcation. The optimised model provides the effective connectivity, the FDT deviations as well as the perturbability maps for different brain states.