Spatial periodic synchronization of chaos in coupled ring and linear arrays of chaotic systems

Synchronization states and multistability in a ring of periodic oscillators: experimentally variable coupling delays

We experimentally study the complex dynamics of a unidirectionally coupled ring of four identical optoelectronic oscillators. The coupling between these systems is time-delayed in the experiment and can be varied over a wide range of delays. We observe that as the coupling delay is varied, the system may show different synchronization states, including complete isochronal synchrony, cluster synchrony, and two splay-phase states. We analyze the stability of these solutions through a master stability function approach, which we show can be effectively applied to all the different states observed in the experiment. Our analysis supports the experimentally observed multistability in the system.

Effect of duration of synaptic activity on spike rate of a Hodgkin-Huxley neuron with delayed feedback

A recurrent loop consisting of a single Hodgkin-Huxley neuron influenced by a chemical excitatory delayed synaptic feedback is considered. We show that the behavior of the system depends on the duration of the activity of the synapse, which is determined by the activation and deactivation time constants of the synapse. For the fast synapses, those for which the effect of the synaptic activity is small compared to the period of firing, depending on the delay time, spiking with single and multiple interspike intervals is possible and the average firing rate can be smaller or larger than that of the open loop neuron. For slow synapses for which the synaptic time constants are of order of the period of the firing, the self-excitation increases the firing rate for all values of the delay time. We also show that for a chain consisting of few similar oscillators, if the synapses are chosen from different time constants, the system will follow the dynamics imposed by the fastest element, which is the oscillator that receives excitations via a slow synapse. The generalization of the results to other types of relaxation oscillators is discussed and the results are compared to those of the loops with inhibitory synapses as well as with gap junctions.

Variability of spatio-temporal patterns in non-homogeneous rings of spiking neurons

We show that a ring of unidirectionally delay-coupled spiking neurons may possess a multitude of stable spiking patterns and provide a constructive algorithm for generating a desired spiking pattern. More specifically, for a given time-periodic pattern, in which each neuron fires once within the pattern period at a predefined time moment, we provide the coupling delays and/or coupling strengths leading to this particular pattern. The considered homogeneous networks demonstrate a great multistability of various travelling time- and space-periodic waves which can propagate either along the direction of coupling or in opposite direction. Such a multistability significantly enhances the variability of possible spatio-temporal patterns and potentially increases the coding capability of oscillatory neuronal loops. We illustrate our results using FitzHugh-Nagumo neurons interacting via excitatory chemical synapses as well as limit-cycle oscillators.

Periodic patterns in a ring of delay-coupled oscillators

We describe the appearance and stability of spatiotemporal periodic patterns (rotating waves) in unidirectional rings of coupled oscillators with delayed couplings. We show how delays in the coupling lead to the splitting of each rotating wave into several new ones. The appearance of rotating waves is mediated by the Hopf bifurcations of the symmetric equilibrium. We also conclude that the coupling delays can be effectively replaced by increasing the number of oscillators in the chain. The phenomena are shown for the Stuart-Landau oscillators as well as for the coupled FitzHugh-Nagumo systems modeling an ensemble of spiking neurons interacting via excitatory chemical synapses.

Scaling and synchronization in a ring of diffusively coupled nonlinear oscillators

Chaos synchronization in a ring of diffusively coupled nonlinear oscillators driven by an external identical oscillator is studied. Based on numerical simulations we show that by introducing additional couplings at (mN(c)+1)-th oscillators in the ring, where m is an integer and N(c) is the maximum number of synchronized oscillators in the ring with a single coupling, the maximum number of oscillators that can be synchronized can be increased considerably beyond the limit restricted by size instability. We also demonstrate that there exists an exponential relation between the number of oscillators that can support stable synchronization in the ring with the external drive and the critical coupling strength ε(c) with a scaling exponent γ. The critical coupling strength is calculated by numerically estimating the synchronization error and is also confirmed from the conditional Lyapunov exponents of the coupled systems. We find that the same scaling relation exists for m couplings between the drive and the ring. Further, we have examined the robustness of the synchronous states against Gaussian white noise and found that the synchronization error exhibits a power-law decay as a function of the noise intensity indicating the existence of both noise-enhanced and noise-induced synchronizations depending on the value of the coupling strength ε. In addition, we have found that ε(c) shows an exponential decay as a function of the number of additional couplings. These results are demonstrated using the paradigmatic models of Rössler and Lorenz oscillators.

Routes to complex dynamics in a ring of unidirectionally coupled systems

We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures, and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators.

Predicting the synchronization time in coupled-map networks

An analytical expression for the synchronization time in coupled-map networks is given. By means of the expression, the synchronization time for any given network can be predicted accurately. Furthermore, for networks in which the distributions of nontrivial eigenvalues of coupling matrices have some unique characteristics, analytical results for the minimal synchronization time are given.

Dynamics, correlation scaling, and synchronization behavior in rings of delay-coupled oscillators

We study the dynamics of unidirectionally delay-coupled nonlinear oscillators. Cascading them within a ring of fixed total propagation delay, we demonstrate simple scaling behavior of correlation properties. In fact, the correlation properties of a ring with N elements can be deduced from the autocorrelation of the single delayed feedback system. Coupling a ring element to a chain of unidirectionally coupled identical oscillators, we achieve complete synchronization between elements of chain and ring, evidencing generalized synchronization among the other elements, even if uncorrelated.

Layered synchronous propagation of noise-induced chaotic spikes in linear arrays

Stable propagation of noise-induced synchronous spiking in uncoupled linear neuron arrays is studied numerically. The chaotic neurons in the unidirectionally coupled linear array are modeled by Hindmarsh-Rose neurons. Stability analysis shows that the synchronous chaotic spiking can be successfully transmitted to cortical areas through layered synchronization in the neural network under certain conditions of the network structure.

Long-time anticipation of chaotic states in time-delay coupled ring and linear arrays

We study the time-delay and unidirectionally coupled ring and linear arrays of chaotic systems, and find that under certain conditions, the linear array can spatial periodically "copy" the chaotic dynamics of the ring with very long anticipation times. Numerical calculations of the Lyapunov exponents show that the delay times can result in unsynchronized chaotic waves, periodic waves, and stable states in the ring that are replicated in the linear array, but have no effect on the absolute stability of the anticipatory synchronization. Our results show that such configurations can both enhance the absolute stability of the synchronization manifolds and minimize the effects of convective instabilities.