Unbalanced clustering and solitary states in coupled excitable systems

Spiking activities in small neural networks induced by external forcing

Neurons in an excitable mode do not show spiking activity and, therefore, do not contribute to information transfer transmission and its processing. However, some external influences, coupling, or time delay can lead to the appearance of oscillations in individual systems or networks. The main goal of this paper is to uncover the connection parameters and parameters of external influences that lead to the arising of spiking behavior in a small network of locally coupled FitzHugh-Nagumo oscillators. In this study, we analyze the dynamics of a small network in the absence and presence of several types of external influences. First, we consider the impact of periodic-pulse exposure generated as a periodic sequence of Gaussian pulses. Second, we show what behavior can be induced by far less regular pulsed influence (Lévy noise) and its special case called white Gaussian noise. For all types of influences, we have identified the appropriate parameters (local coupling strength, intensity, and frequency) that induce spiking activity in the small network.

Controlling spatiotemporal dynamics of neural networks by Lévy noise

We explore numerically how additive Lévy noise influences the spatiotemporal dynamics of a neural network of nonlocally coupled FitzHugh-Nagumo oscillators. Without noise, the network can exhibit various partial or cluster synchronization patterns, such as chimera and solitary states, which can also coexist in the network for certain values of the control parameters. Our studies show that these structures demonstrate different responses to additive Lévy noise and, thus, the dynamics of the neural network can be effectively controlled by varying the scale parameter and the stability index of Lévy noise. Specifically, introducing Lévy noise in the multistability mode can increase the probability of observing chimera states while suppressing solitary states. Nonetheless, decreasing the stability parameter enables one to diminish the noise effect on chimera states and amplify it on solitary states.

Adaptation rules inducing synchronization of heterogeneous Kuramoto oscillator network with triadic couplings

A class of adaptation functions is found for which a synchronous mode with different number of phase clusters exists in a network of phase oscillators with triadic couplings. This mode is implemented in a fairly wide range of initial conditions and the maximum number of phase clusters is four. The joint influence of coupling strength and adaptation parameters on synchronization in the network has been studied. The desynchronization transition under variation of the adaptation parameter occurs abruptly and begins with the highest-frequency oscillator, spreading hierarchically to all other elements.

Chimera patterns with spatial random swings between periodic attractors in a network of FitzHugh-Nagumo oscillators

For the study of symmetry-breaking phenomena in neuronal networks, simplified versions of the FitzHugh-Nagumo model are widely used. In this paper, these phenomena are investigated in a network of FitzHugh-Nagumo oscillators taken in the form of the original model and it is found that it exhibits diverse partial synchronization patterns that are unobserved in the networks with simplified models. Apart from the classical chimera, we report a new type of chimera pattern whose incoherent clusters are characterized by spatial random swings among a few fixed periodic attractors. Another peculiar hybrid state is found that combines the features of this chimera state and a solitary state such that the main coherent cluster is interspersed with some nodes with identical solitary dynamics. In addition, oscillation death including chimera death emerges in this network. A reduced model of the network is derived to study oscillation death, which helps explaining the transition from spatial chaos to oscillation death via the chimera state with a solitary state. This study deepens our understanding of chimera patterns in neuronal networks.

Solitary routes to chimera states

We show how solitary states in a system of globally coupled FitzHugh-Nagumo oscillators can lead to the emergence of chimera states. By a numerical bifurcation analysis of a suitable reduced system in the thermodynamic limit we demonstrate how solitary states, after emerging from the synchronous state, become chaotic in a period-doubling cascade. Subsequently, states with a single chaotic oscillator give rise to states with an increasing number of incoherent chaotic oscillators. In large systems, these chimera states show extensive chaos. We demonstrate the coexistence of many of such chaotic attractors with different Lyapunov dimensions, due to different numbers of incoherent oscillators.

Mixed-mode chimera states in pendula networks

We report the emergence of peculiar chimera states in networks of identical pendula with global phase-lagged coupling. The states reported include both rotating and quiescent modes, i.e., with non-zero and zero average frequencies. This kind of mixed-mode chimeras may be interpreted as images of bump states known in neuroscience in the context of modeling the working memory. We illustrate this striking phenomenon for a network of N = 100 coupled pendula, followed by a detailed description of the minimal non-trivial case of N = 3. Parameter regions for five characteristic types of the system behavior are identified, which consist of two mixed-mode chimeras with one and two rotating pendula, classical weak chimera with all three pendula rotating, synchronous rotation, and quiescent state. The network dynamics is multistable: up to four of the states can coexist in the system phase state as demonstrated through the basins of attraction. The analysis suggests that the robust mixed-mode chimera states can generically describe the complex dynamics of diverse pendula-like systems widespread in nature.

Patched patterns and emergence of chaotic interfaces in arrays of nonlocally coupled excitable systems

We disclose a new class of patterns, called patched patterns, in arrays of non-locally coupled excitable units with attractive and repulsive interactions. The self-organization process involves the formation of two types of patches, majority and minority ones, characterized by uniform average spiking frequencies. Patched patterns may be temporally periodic, quasiperiodic, or chaotic, whereby chaotic patterns may further develop interfaces comprised of units with average frequencies in between those of majority and minority patches. Using chaos and bifurcation theory, we demonstrate that chaos typically emerges via a torus breakup and identify the secondary bifurcation that gives rise to chaotic interfaces. It is shown that the maximal Lyapunov exponent of chaotic patched patterns does not decay, but rather converges to a finite value with system size. Patched patterns with a smaller wavenumber may exhibit diffusive motion of chaotic interfaces, similar to that of the incoherent part of chimeras.

Response of solitary states to noise-modulated parameters in nonlocally coupled networks of Lozi maps

We study numerically the impact of heterogeneity in parameters on the dynamics of nonlocally coupled discrete-time systems, which exhibit solitary states along the transition from coherence to incoherence. These partial synchronization patterns are described as states when single or several elements demonstrate different dynamics compared with the behavior of other elements in a network. Using as an example a ring network of nonlocally coupled Lozi maps, we explore the robustness of solitary states to heterogeneity in parameters of local dynamics or coupling strength. It is found that if these network parameters are continuously modulated by noise, solitary states are suppressed as the noise intensity increases. However, these states may persist in the case of static randomly distributed system parameters for a wide range of the distribution width. Domains of solitary state existence are constructed in the parameter plane of coupling strength and noise intensity using a cross-correlation coefficient.