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.

Reply to "comment on 'how to obtain extreme multistability in coupled dynamical systems' ".

In this Reply we answer the two major issues raised by the Comment. First, we point out that the idea of constructing extreme multistability in simple dynamical systems is not new and has been demonstrated previously by other authors. Furthermore, we emphasize the importance of the concept of a conserved quantity and its consequences for the dynamics, which applies to all the examples in the Comment. Second, we show that the design of controllers to achieve extreme multistability in coupled systems is as general as described in Phys. Rev. E 85, 035202(R) (2012) by providing two examples which do not lead to a master-slave dynamics.

Generalized synchronization in relay systems with instantaneous coupling.

We demonstrate the existence of generalized synchronization in systems that act as mediators between two dynamical units that, in turn, show complete synchronization with each other. These are the so-called relay systems. Specifically, we analyze the Lyapunov spectrum of the full system to elucidate when complete and generalized synchronization appear. We show that once a critical coupling strength is achieved, complete synchronization emerges between the systems to be synchronized, and at the same point, generalized synchronization with the relay system also arises. Next, we use two nonlinear measures based on the distance between phase-space neighbors to quantify the generalized synchronization in discretized time series. Finally, we experimentally show the robustness of the phenomenon and of the theoretical tools here proposed to characterize it.

How to obtain extreme multistability in coupled dynamical systems.

We present a method for designing an appropriate coupling scheme for two dynamical systems in order to realize extreme multistability. We achieve the coexistence of infinitely many attractors for a given set of parameters by using the concept of partial synchronization based on Lyapunov function stability. We show that the method is very general and allows a great flexibility in choosing the coupling. Furthermore, we demonstrate its applicability in different models, such as the Rössler system and a chemical oscillator. Finally we show that extreme multistability is robust with respect to parameter mismatch and, hence, a very general phenomenon in coupled systems.

Phase multistability and phase synchronization in an array of locally coupled period-doubling oscillators.

We consider phase multistability and phase synchronization phenomena in a chain of period-doubling oscillators. The synchronization in arrays of diffusively coupled self-sustained oscillators manifests itself as rotating wave regimes, which are characterized by equal amplitudes and phases in every site which are shifted by a constant value. The value of the phase shift is preserved while the shape of motion becomes more complex through a period-doubling cascade. The number of coexisting attractors increases drastically after the transition from period-one to period-two oscillations and then after every following period-doubling bifurcation. In the chaotic region, we observe a number of phase-synchronized modes with instantaneous phases locked in different values. The loss of phase synchronization with decreasing coupling is accompanied by intermittency between several synchronous regimes.

Stabilization due to predator interference: comparison of different analysis approaches.

We study the influence of the particular form of the functional response in two-dimensional predator-prey models with respect to the stability of the nontrivial equilibrium. This equilibrium is stable between its appearance at a transcritical bifurcation and its destabilization at a Hopf bifurcation, giving rise to periodic behavior. Based on local bifurcation analysis, we introduce a classification of stabilizing effects. The classical Rosenzweig-MacArthur model can be classified as weakly stabilizing, undergoing the paradox of enrichment, while the well known Beddington-DeAngelis model can be classified as strongly stabilizing. Under certain conditions we obtain a complete stabilization, resulting in an avoidance of limit cycles. Both models, in their conventional formulation, are compared to a generalized, steady-state independent two-dimensional version of these models, based on a previously developed normalization method. We show explicitly how conventional and generalized models are related and how to interpret the results from the rather abstract stability analysis of generalized models.

Preference of attractors in noisy multistable systems.

A model system exhibiting a large number of attractors is investigated under the influence of noise. Several methods for discriminating two qualitatively different regions of the noise intensity are presented, and the phenomenon of noise-induced preference of attractors is reported. Finally, the relevance of our findings for detection of multiple stable states of systems occurring in nature or in the laboratory is pointed out.