The recurrence analysis of dynamic systems has been studied since Poincaré’s seminal work. Since then, several approaches have been developed to study recurrence properties in nonlinear dynamical systems. In this work, we study the recently developed entropy of recurrence microstates. We propose a new quantifier, the maximum entropy (${\mathrm{S}}_{\text{max}}$ ). The new concept uses the diversity of microstates of the recurrence plot and is able to set automatically the optimum recurrence neighborhood ($ϵ$ —vicinity), turning the analysis free of the vicinity parameter. In addition, $ϵ$ turns out to be a novel quantifier of dynamical properties itself. We apply ${\mathrm{S}}_{\text{max}}$ and the optimum $ϵ$ to deterministic and stochastic systems. The ${\mathrm{S}}_{\text{max}}$ quantifier has a higher correlation with the Lyapunov exponent and, since it is a parameter-free measure, a more useful recurrence quantifier of time series.

Anomalous phase synchronization describes a synchronization phenomenon occurring even for the weakly coupled network and characterized by a non-monotonous dependence of the synchronization strength on the coupling strength. Its existence may support a theoretical framework to some neurological diseases, such as Parkinson’s and some episodes of seizure behavior generated by epilepsy. Despite the success of controlling or suppressing the anomalous phase synchronization in neural networks applying external perturbations or inducing ambient changes, the origin of the anomalous phase synchronization as well as the mechanisms behind the suppression is not completely known. Here, we consider networks composed of $N=2000$ coupled neurons in a small-world topology for two well known neuron models, namely, the Hodgkin-Huxley-like and the Hindmarsh-Rose models, both displaying the anomalous phase synchronization regime. We show that the anomalous phase synchronization may be related to the individual behavior of the coupled neurons; particularly, we identify a strong correlation between the behavior of the inter-bursting-intervals of the neurons, what we call neuron variability, to the ability of the network to depict anomalous phase synchronization. We corroborate the ideas showing that external perturbations or ambient parameter changes that eliminate anomalous phase synchronization and at the same time promote small changes in the individual dynamics of the neurons, such that an increasing individual variability of neurons implies a decrease of anomalous phase synchronization. Finally, we demonstrate that this effect can be quantified using a well known recurrence quantifier, the “determinism.” Moreover, the results obtained by the determinism are based on only the mean field potential of the network, turning these measures more suitable to be used in experimental situations.

Metastability, characterized by a variability of regimes in time, is a ubiquitous type of neural dynamics. It has been formulated in many different ways in the neuroscience literature, however, which may cause some confusion. In this Perspective, we discuss metastability from the point of view of dynamical systems theory. We extract from the literature a very simple but general through the concept of as long-lived but transient epochs of activity with unique dynamical properties. This definition serves as an umbrella term that encompasses formulations from other works, and readily connects to concepts from dynamical systems theory. This allows us to examine general dynamical properties of metastable regimes, propose in a didactic manner several that generate them, and discuss a theoretical tool to characterize them quantitatively. This Perspective leads to insights that help to address issues debated in the literature and also suggests pathways for future research.

Here we investigate the mechanism for explosive synchronization (ES) of a complex neural network composed of nonidentical neurons and coupled by Newman-Watts small-world matrices. We find a range of nonlocal connection probabilities for which the network displays an abrupt transition to phase synchronization, characterizing ES. The mechanism behind the ES is the following: As the coupling parameter is varied in a network of distinct neurons, ES is likely to occur due to a bistable regime, namely a chaotic nonsynchronized and a regular phase-synchronized state in the phase space. In this case, even small coupling changes make possible a transition between them. The onset of ES occurs via a saddle-node bifurcation of a periodic orbit that leads the network dynamics to display a locally stable phase-synchronized state. The presence of this regime is accompanied by a hysteresis loop on the network dynamics as the coupling parameter is adiabatically increased and decreased. The end of the hysteresis loop is marked by a frontier crisis of the chaotic attractor which also determines the end of the coupling strength interval where ES is possible.