Disparity-driven heterogeneous nucleation in finite-size adaptive networks.

Phase transitions are crucial in shaping the collective dynamics of a broad spectrum of natural systems across disciplines. Here, we report two distinct heterogeneous nucleation facilitating single step and multistep phase transitions to global synchronization in a finite-size adaptive network due to the trade off between time scale adaptation and coupling strength disparities. Specifically, small intracluster nucleations coalesce either at the population interface or within the populations resulting in the two distinct phase transitions depending on the degree of the disparities.

Modeling brain network flexibility in networks of coupled oscillators: a feasibility study.

Modeling the functionality of the human brain is a major goal in neuroscience for which many powerful methodologies have been developed over the last decade. The impact of working memory and the associated brain regions on the brain dynamics is of particular interest due to their connection with many functions and malfunctions in the brain. In this context, the concept of brain flexibility has been developed for the characterization of brain functionality.

Perspectives on adaptive dynamical systems.

Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems, such as the power grid, social, and neural networks, and they form the backbone of closed-loop control strategies and machine learning algorithms. In this article, we provide an interdisciplinary perspective on adaptive systems.

Synchronization transitions in Kuramoto networks with higher-mode interaction.

Synchronization is an omnipresent collective phenomenon in nature and technology, whose understanding is still elusive for real-world systems in particular. We study the synchronization transition in a phase oscillator system with two nonvanishing Fourier-modes in the interaction function, hence going beyond the Kuramoto paradigm. We show that the transition scenarios crucially depend on the interplay of the two coupling modes.

Collective Activity Bursting in a Population of Excitable Units Adaptively Coupled to a Pool of Resources.

We study the collective dynamics in a population of excitable units (neurons) adaptively interacting with a pool of resources. The resource pool is influenced by the average activity of the population, whereas the feedback from the resources to the population is comprised of components acting homogeneously or inhomogeneously on individual units of the population. Moreover, the resource pool dynamics is assumed to be slow and has an oscillatory degree of freedom.

Critical Parameters in Dynamic Network Modeling of Sepsis.

In this work, we propose a dynamical systems perspective on the modeling of sepsis and its organ-damaging consequences. We develop a functional two-layer network model for sepsis based upon the interaction of parenchymal cells and immune cells cytokines, and the coevolutionary dynamics of parenchymal, immune cells, and cytokines. By means of the simple paradigmatic model of phase oscillators in a two-layer system, we analyze the emergence of organ threatening interactions between the dysregulated immune system and the parenchyma.

Asymmetric adaptivity induces recurrent synchronization in complex networks.

Rhythmic activities that alternate between coherent and incoherent phases are ubiquitous in chemical, ecological, climate, or neural systems. Despite their importance, general mechanisms for their emergence are little understood. In order to fill this gap, we present a framework for describing the emergence of recurrent synchronization in complex networks with adaptive interactions.

Heterogeneous Nucleation in Finite-Size Adaptive Dynamical Networks.

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, coevolving with the nodes' dynamical state.

Exotic states induced by coevolving connection weights and phases in complex networks.

We consider an adaptive network, whose connection weights coevolve in congruence with the dynamical states of the local nodes that are under the influence of an external stimulus. The adaptive dynamical system mimics the adaptive synaptic connections common in neuronal networks. The adaptive network under external forcing displays exotic dynamical states such as itinerant chimeras whose population density of coherent and incoherent domains coevolves with the synaptic connection, bump states, and bump frequency cluster states, which do not exist in adaptive networks without forcing.

Modeling Tumor Disease and Sepsis by Networks of Adaptively Coupled Phase Oscillators.

In this study, we provide a dynamical systems perspective to the modelling of pathological states induced by tumors or infection. A unified disease model is established using the innate immune system as the reference point. We propose a two-layer network model for carcinogenesis and sepsis based upon the interaction of parenchymal cells and immune cells cytokines, and the co-evolutionary dynamics of parenchymal, immune cells, and cytokines.

Generalized splay states in phase oscillator networks.

Networks of coupled phase oscillators play an important role in the analysis of emergent collective phenomena. In this article, we introduce generalized m-splay states constituting a special subclass of phase-locked states with vanishing mth order parameter. Such states typically manifest incoherent dynamics, and they often create high-dimensional families of solutions (splay manifolds).

What adaptive neuronal networks teach us about power grids.

Power grid networks, as well as neuronal networks with synaptic plasticity, describe real-world systems of tremendous importance for our daily life. The investigation of these seemingly unrelated types of dynamical networks has attracted increasing attention over the past decade. In this paper, we provide insight into the fundamental relation between these two types of networks.

Desynchronization Transitions in Adaptive Networks.

Adaptive networks change their connectivity with time, depending on their dynamical state. While synchronization in structurally static networks has been studied extensively, this problem is much more challenging for adaptive networks. In this Letter, we develop the master stability approach for a large class of adaptive networks.

FitzHugh-Nagumo oscillators on complex networks mimic epileptic-seizure-related synchronization phenomena.

We study patterns of partial synchronization in a network of FitzHugh-Nagumo oscillators with empirical structural connectivity measured in human subjects. We report the spontaneous occurrence of synchronization phenomena that closely resemble the ones seen during epileptic seizures in humans. In order to obtain deeper insights into the interplay between dynamics and network topology, we perform long-term simulations of oscillatory dynamics on different paradigmatic network structures: random networks, regular nonlocally coupled ring networks, ring networks with fractal connectivities, and small-world networks with various rewiring probability.

Effect of topology upon relay synchronization in triplex neuronal networks.

Relay synchronization in complex networks is characterized by the synchronization of remote parts of the network due to their interaction via a relay. In multilayer networks, distant layers that are not connected directly can synchronize due to signal propagation via relay layers. In this work, we investigate relay synchronization of partial synchronization patterns like chimera states in three-layer networks of interacting FitzHugh-Nagumo oscillators.

Birth and Stabilization of Phase Clusters by Multiplexing of Adaptive Networks.

We propose a concept to generate and stabilize diverse partial synchronization patterns (phase clusters) in adaptive networks which are widespread in neuroscience and social sciences, as well as biology, engineering, and other disciplines. We show by theoretical analysis and computer simulations that multiplexing in a multilayer network with symmetry can induce various stable phase cluster states in a situation where they are not stable or do not even exist in the single layer. Further, we develop a method for the analysis of Laplacian matrices of multiplex networks which allows for insight into the spectral structure of these networks enabling a reduction to the stability problem of single layers.

Frequency cluster formation and slow oscillations in neural populations with plasticity.

We report the phenomenon of frequency clustering in a network of Hodgkin-Huxley neurons with spike timing-dependent plasticity. The clustering leads to a splitting of a neural population into a few groups synchronized at different frequencies. In this regime, the amplitude of the mean field undergoes low-frequency modulations, which may contribute to the mechanism of the emergence of slow oscillations of neural activity observed in spectral power of local field potentials or electroencephalographic signals at high frequencies.

Hierarchical frequency clusters in adaptive networks of phase oscillators.

Adaptive dynamical networks appear in various real-word systems. One of the simplest phenomenological models for investigating basic properties of adaptive networks is the system of coupled phase oscillators with adaptive couplings. In this paper, we investigate the dynamics of this system.