Bistability in the synchronization of identical neurons.

We investigate the role of bistability in the synchronization of a network of identical bursting neurons coupled through an generic electrical mean-field scheme. These neurons can exhibit distinct multistable states and, in particular, bistable behavior is observed when their sodium conductance is varied. With this, we consider three different initialization compositions: (i) the whole network is in the same periodic state; (ii) half of the network periodic, half chaotic; (iii) half periodic, and half in a different periodic state. We show that (i) and (ii) reach phase synchronization (PS) for all coupling strengths, while for (iii) small coupling regimes do not induce PS, and instead, there is a coexistence of different frequencies. For stronger coupling, case (iii) synchronizes, but after (i) and (ii). Since PS requires all neurons being in the same state (same frequencies), these different behaviors are governed by transitions between the states. We find that, during these transitions, (ii) and (iii) have transient chimera states and that (iii) has breathing chimeras. By studying the stability of each state, we explain the observed transitions. Therefore, bistability of neurons can play a major role in the synchronization of generic networks, with the simple initialization of the system being capable of drastically changing its asymptotic space.

Phase-locking intermittency induced by dynamical heterogeneity in networks of thermosensitive neurons.

In this work, we study the phase synchronization of a neural network and explore how the heterogeneity in the neurons' dynamics can lead their phases to intermittently phase-lock and unlock. The neurons are connected through chemical excitatory connections in a sparse random topology, feel no noise or external inputs, and have identical parameters except for different in-degrees. They follow a modification of the Hodgkin-Huxley model, which adds details like temperature dependence, and can burst either periodically or chaotically when uncoupled. Coupling makes them chaotic in all cases but each individual mode leads to different transitions to phase synchronization in the networks due to increasing synaptic strength. In almost all cases, neurons' inter-burst intervals differ among themselves, which indicates their dynamical heterogeneity and leads to their intermittent phase-locking. We argue then that this behavior occurs here because of their chaotic dynamics and their differing initial conditions. We also investigate how this intermittency affects the formation of clusters of neurons in the network and show that the clusters' compositions change at a rate following the degree of intermittency. Finally, we discuss how these results relate to studies in the neuroscience literature, especially regarding metastability.

Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks.

Extracting relevant properties of empirical signals generated by nonlinear, stochastic, and high-dimensional systems is a challenge of complex systems research. Open questions are how to differentiate chaotic signals from stochastic ones, and how to quantify nonlinear and/or high-order temporal correlations. Here we propose a new technique to reliably address both problems. Our approach follows two steps: first, we train an artificial neural network (ANN) with flicker (colored) noise to predict the value of the parameter, [Formula: see text], that determines the strength of the correlation of the noise. To predict [Formula: see text] the ANN input features are a set of probabilities that are extracted from the time series by using symbolic ordinal analysis. Then, we input to the trained ANN the probabilities extracted from the time series of interest, and analyze the ANN output. We find that the [Formula: see text] value returned by the ANN is informative of the temporal correlations present in the time series. To distinguish between stochastic and chaotic signals, we exploit the fact that the difference between the permutation entropy (PE) of a given time series and the PE of flicker noise with the same [Formula: see text] parameter is small when the time series is stochastic, but it is large when the time series is chaotic. We validate our technique by analysing synthetic and empirical time series whose nature is well established. We also demonstrate the robustness of our approach with respect to the length of the time series and to the level of noise. We expect that our algorithm, which is freely available, will be very useful to the community.

Maximum entropy principle in recurrence plot analysis on stochastic and chaotic systems.

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 (Smax). 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 Smax and the optimum ϵ to deterministic and stochastic systems. The Smax 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.

Mechanism for explosive synchronization of neural networks.

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.

Erratum: Phase synchronization and intermittent behavior in healthy and Alzheimer-affected human-brain-based neural network [Phys. Rev. E 99, 022402 (2019)].

This corrects the article DOI: 10.1103/PhysRevE.99.022402.

Phase synchronization and intermittent behavior in healthy and Alzheimer-affected human-brain-based neural network.

We study the dynamical proprieties of phase synchronization and intermittent behavior of neural systems using a network of networks structure based on an experimentally obtained human connectome for healthy and Alzheimer-affected brains. We consider a network composed of 78 neural subareas (subnetworks) coupled with a mean-field potential scheme. Each subnetwork is characterized by a small-world topology, composed of 250 bursting neurons simulated through a Rulkov model. Using the Kuramoto order parameter we demonstrate that healthy and Alzheimer-affected brains display distinct phase synchronization and intermittence properties as a function of internal and external coupling strengths. In general, for the healthy case, each subnetwork develops a substantial level of internal synchronization before a global stable phase-synchronization state has been established. For the unhealthy case, despite the similar internal subnetwork synchronization levels, we identify higher levels of global phase synchronization occurring even for relatively small internal and external coupling. Using recurrence quantification analysis, namely the determinism of the mean-field potential, we identify regions where the healthy and unhealthy networks depict nonstationary behavior, but the results denounce the presence of a larger region or intermittent dynamics for the case of Alzheimer-affected networks. A possible theoretical explanation based on two locally stable but globally unstable states is discussed.

Synchronous patterns and intermittency in a network induced by the rewiring of connections and coupling.

The connection architecture plays an important role in the synchronization of networks, where the presence of local and nonlocal connection structures are found in many systems, such as the neural ones. Here, we consider a network composed of chaotic bursting oscillators coupled through a Watts-Strogatz-small-world topology. The influence of coupling strength and rewiring of connections is studied when the network topology is varied from regular to small-world to random. In this scenario, we show two distinct nonstationary transitions to phase synchronization: one induced by the increase in coupling strength and another resulting from the change from local connections to nonlocal ones. Besides this, there are regions in the parameter space where the network depicts a coexistence of different bursting frequencies where nonstationary zig-zag fronts are observed. Regarding the analyses, we consider two distinct methodological approaches: one based on the phase association to the bursting activity where the Kuramoto order parameter is used and another based on recurrence quantification analysis where just a time series of the network mean field is required.

Neuron dynamics variability and anomalous phase synchronization of neural networks.

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.

Detection of nonstationary transition to synchronized states of a neural network using recurrence analyses.

We study the stability of asymptotic states displayed by a complex neural network. We focus on the loss of stability of a stationary state of networks using recurrence quantifiers as tools to diagnose local and global stabilities as well as the multistability of a coupled neural network. Numerical simulations of a neural network composed of 1024 neurons in a small-world connection scheme are performed using the model of Braun et al. [Int. J. Bifurcation Chaos 08, 881 (1998)IJBEE40218-127410.1142/S0218127498000681], which is a modified model from the Hodgkin-Huxley model [J. Phys. 117, 500 (1952)]. To validate the analyses, the results are compared with those produced by Kuramoto's order parameter [Chemical Oscillations, Waves, and Turbulence (Springer-Verlag, Berlin Heidelberg, 1984)]. We show that recurrence tools making use of just integrated signals provided by the networks, such as local field potential (LFP) (LFP signals) or mean field values bring new results on the understanding of neural behavior occurring before the synchronization states. In particular we show the occurrence of different stationary and nonstationarity asymptotic states.