We analyze simple dynamical network models which describe the limited capacity of nodes to process the input information. For a proper range of their parameters, the information flow pattern in these models is characterized by exponential distribution of the incoming information and a fat-tailed distribution of the outgoing information, as a signature of the law of diminishing marginal returns. We apply this analysis to effective connectivity networks from human EEG signals, obtained by Granger Causality, which has recently been given an interpretation in the framework of information theory. From the distributions of the incoming versus the outgoing values of the information flow it is evident that the incoming information is exponentially distributed whilst the outgoing information shows a fat tail. This suggests that overall brain effective connectivity networks may also be considered in the light of the law of diminishing marginal returns. Interestingly, this pattern is reproduced locally but with a clear modulation: a topographic analysis has also been made considering the distribution of incoming and outgoing values at each electrode, suggesting a functional role for this phenomenon.

We analyze the information flow in the Ising model on two real networks, describing the brain at the mesoscale, with Glauber dynamics. We find that the critical state is characterized by the maximal amount of information flow in the system, and that this does not happen when the Ising model is implemented on the two-dimensional regular grid. At criticality the system shows signatures of the law of diminishing marginal returns, some nodes showing disparity between incoming and outgoing information. We also implement the Ising model with conserved dynamics and show that there are regions of the systems exhibiting anticorrelation, in spite of the fact that all couplings are positive; this phenomenon may be connected with some evidences in real brains (the default mode network is characterized by anticorrelated components).

We implement the Ising model on a structural connectivity matrix describing the brain at two different resolutions. Tuning the model temperature to its critical value, i.e. at the susceptibility peak, we find a maximal amount of total information transfer between the spin variables. At this point the amount of information that can be redistributed by some nodes reaches a limit and the net dynamics exhibits signature of the law of diminishing marginal returns, a fundamental principle connected to saturated levels of production. Our results extend the recent analysis of dynamical oscillators models on the connectome structure, taking into account lagged and directional influences, focusing only on the nodes that are more prone to became bottlenecks of information. The ratio between the outgoing and the incoming information at each node is related to the the sum of the weights to that node and to the average time between consecutive time flips of spins. The results for the connectome of 66 nodes and for that of 998 nodes are similar, thus suggesting that these properties are scale-independent. Finally, we also find that the brain dynamics at criticality is organized maximally to a rich-club w.r.t. the network of information flows.

The communication among neuronal populations, reflected by transient synchronous activity, is the mechanism underlying the information processing in the brain. Although it is widely assumed that the interactions among those populations (i.e. functional connectivity) are highly nonlinear, the amount of nonlinear information transmission and its functional roles are not clear. Granger causality constitutes a major tool to reveal effective connectivity, and it is widely used to analyze EEG/MEG data as well as fMRI signals in its linear version. In order to capture nonlinear interactions between even short and noisy time series, a kernel version of Granger causality has been recently proposed. We review kernel Granger causality and show the application of this approach on EEG signals.

The volume describes significant recent advances mainly in strong interaction physics. The new results concern the hadron spectroscopy and the interquark potential, the large baryonic density and high temperature regimes with a discussion of astrophysical consequences and in relativistic heavy ion collisions, the present knowledge of nucleon spin physics. A promising approach to the description of strong interaction regime, AdS/QCD, is debated. Aspects of heavy quark systems are considered, and the role of present and future experiments is analyzed.