The analysis of the interdependence between time series has become an important field of research in the last years, mainly as a result of advances in the characterization of dynamical systems from the signals they produce, the introduction of concepts such as generalized and phase synchronization and the application of information theory to time series analysis. In neurophysiology, different analytical tools stemming from these concepts have added to the ‘traditional’ set of linear methods, which includes the cross-correlation and the coherency function in the time and frequency domain, respectively, or more elaborated tools such as Granger Causality. This increase in the number of approaches to tackle the existence of functional (FC) or effective connectivity (EC) between two (or among many) neural networks, along with the mathematical complexity of the corresponding time series analysis tools, makes it desirable to arrange them into a unified-easy-to-use software package. The goal is to allow neuroscientists, neurophysiologists and researchers from related fields to easily access and make use of these analysis methods from a single integrated toolbox. Here we present HERMES (http://hermes.ctb.upm.es), a toolbox for the Matlab® environment (The Mathworks, Inc), which is designed to study functional and effective brain connectivity from neurophysiological data such as multivariate EEG and/or MEG records. It includes also visualization tools and statistical methods to address the problem of multiple comparisons. We believe that this toolbox will be very helpful to all the researchers working in the emerging field of brain connectivity analysis.

In this Letter we identify the general rules that determine the synchronization properties of interconnected networks. We study analytically, numerically, and experimentally how the degree of the nodes through which two networks are connected influences the ability of the whole system to synchronize. We show that connecting the high-degree (low-degree) nodes of each network turns out to be the most (least) effective strategy to achieve synchronization. We find the functional relation between synchronizability and size for a given network of networks, and report the existence of the optimal connector link weights for the different interconnection strategies. Finally, we perform an electronic experiment with two coupled star networks and conclude that the analytical results are indeed valid in the presence of noise and parameter mismatches.

Inter-layer synchronization is a dynamical process occurring in multi-layer networks composed of identical nodes. This process emerges when all layers are synchronized, while nodes in each layer do not necessarily evolve in unison. So far, the study of such inter-layer synchronization has been restricted to the case in which all layers have an identical connectivity structure. When layers are not identical, the inter-layer synchronous state is no longer a stable solution of the system. Nevertheless, when layers differ in just a few links, an approximate treatment is still feasible, and allows one to gather information on whether and how the system may wander around an inter-layer synchronous configuration. We report the details of an approximate analytical treatment for a two-layer multiplex, which results in the introduction of an extra inertial term accounting for structural differences. Numerical validation of the predictions highlights the usefulness of our approach, especially for small or moderate topological differences in the intra-layer coupling. Moreover, we identify a non-trivial relationship connecting the betweenness centrality of the missing links and the intra-layer coupling strength. Finally, by the use of multiplexed layers of electronic circuits, we study the inter-layer synchronization as a function of the removed links.

Inter-layer synchronization is a distinctive process of multiplex networks whereby each node in a given layer evolves synchronously with all its replicas in other layers, irrespective of whether or not it is synchronized with the other units of the same layer. We analytically derive the necessary conditions for the existence and stability of such a state, and verify numerically the analytical predictions in several cases where such a state emerges. We further inspect its robustness against a progressive de-multiplexing of the network, and provide experimental evidence by means of multiplexes of nonlinear electronic circuits affected by intrinsic noise and parameter mismatch.

Mild cognitive impairment (MCI) has been considered an intermediate state between healthy aging and dementia. The early damage in anatomical connectivity and progressive loss of synapses that characterize early Alzheimer's disease suggest that MCI co

We consider a set of interacting phase oscillators, with a coupling between synchronized nodes adaptively reinforced, and the constraint of a limited resource for a node to establish connections with the other units of the network. We show that such a competitive mechanism leads to the emergence of a rich modular structure underlying cluster synchronization, and to a scale-free distribution for the connection strengths of the units.