Transition to complete synchronization and global intermittent synchronization in an array of time-delay systems

Complex networks exhibit intermittent synchronization

The path toward the synchronization of an ensemble of dynamical units goes through a series of transitions determined by the dynamics and the structure of the connections network. In some systems on the verge of complete synchronization, intermittent synchronization, a time-dependent state where full synchronization alternates with non-synchronized periods, has been observed. This phenomenon has been recently considered to have functional relevance in neuronal ensembles and other networked biological systems close to criticality. We characterize the intermittent state as a function of the network topology to show that the different structures can encourage or inhibit the appearance of early signs of intermittency. In particular, we study the local intermittency and show how the nodes incorporate to intermittency in hierarchical order, which can provide information about the node topological role even when the structure is unknown.

Time-Delayed Models of Gene Regulatory Networks

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternative modelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems.

Characterizing an ensemble of interacting oscillators: the mean-field variability index

We introduce a way of characterizing an ensemble of interacting oscillators in terms of their mean-field variability index κ, a dimensionless parameter defined as the variance of the oscillators' mean field r divided by the mean square of r. Based on the assumption that the overall mean field is the sum of a very large number of oscillators, each giving a small contribution to the total signal, we show that κ depends on the mutual interactions between the oscillators, independently of their number or spectral properties. For purely random phasors, or a noninteracting ensemble of oscillators, κ converges on 0.215. Interactions push κ in different directions: lower where there is interoscillator phase coherence, tending to zero for complete phase synchronization, or higher for amplitude synchronization or intermittent synchronization. We calculate κ for several different cases to illustrate its utility, using both numerically simulated data and electroencephalograph signals from the brains of human subjects while awake, while anesthetized, and while undergoing an epileptic fit.