Polaritons and excitons: Hamiltonian design for enhanced coherence

The primary questions motivating this report are: Are there ways to increase coherence and delocalization of excitation among many molecules at moderate electronic coupling strength? Coherent delocalization of excitation in disordered molecular systems is studied using numerical calculations. The results are relevant to molecular excitons, polaritons, and make connections to classical phase oscillator synchronization. In particular, it is hypothesized that it is not only the magnitude of electronic coupling relative to the standard deviation of energetic disorder that decides the limits of coherence, but that the structure of the Hamiltonian-connections between sites (or molecules) made by electronic coupling-is a significant design parameter. Inspired by synchronization phenomena in analogous systems of phase oscillators, some properties of graphs that define the structure of different Hamiltonian matrices are explored. The report focuses on eigenvalues and ensemble density matrices of various structured, random matrices. Some reasons for the special delocalization properties and robustness of polaritons in the single-excitation subspace (the star graph) are discussed. The key result of this report is that, for some classes of Hamiltonian matrix structure, coherent delocalization is not easily defeated by energy disorder, even when the electronic coupling is small compared to disorder.

Synchronization and Consensus in Networks of Linear Fractional-Order Multi-Agent Systems via Sampled-Data Control

This article addresses synchronization and consensus problems in networks of linear fractional-order multi-agent systems (LFOMAS) via sampled-data control. First, under very mild assumptions, the necessary and sufficient conditions are obtained for achieving synchronization in networks of LFOMAS. Second, the results of synchronization are applied to solve some consensus problems in networks of LFOMAS. In the obtained results, the coupling matrix does not have to be a Laplacian matrix, its off-diagonal elements do not have to be nonnegative, and its row-sum can be nonzero. Finally, the validity of the theoretical results is verified by three simulation examples.

Real-time Inference and Detection of Disruptive EEG Networks for Epileptic Seizures

Recent studies in brain science and neurological medicine paid a particular attention to develop machine learning-based techniques for the detection and prediction of epileptic seizures with electroencephalogram (EEG). As a noninvasive monitoring method to record brain electrical activities, EEG has been widely used for capturing the underlying dynamics of disruptive neuronal responses across the brain in real-time to provide clinical guidance in support of epileptic seizure treatments in practice. In this study, we introduce a novel dynamic learning method that first infers a time-varying network constituted by multivariate EEG signals, which represents the overall dynamics of the brain network, and subsequently quantifies its topological property using graph theory. We demonstrate the efficacy of our learning method to detect relatively strong synchronization (characterized by the algebraic connectivity metric) caused by abnormal neuronal firing during a seizure onset. The computational results for a realistic scalp EEG database show a detection rate of 93.6% and a false positive rate of 0.16 per hour (FP/h); furthermore, our method observes potential pre-seizure phenomena in some cases.

Synchronizability of directed networks: The power of non-existent ties

The understanding of how synchronization in directed networks is influenced by structural changes in network topology is far from complete. While the addition of an edge always promotes synchronization in a wide class of undirected networks, this addition may impede synchronization in directed networks. In this paper, we develop the augmented graph stability method, which allows for explicitly connecting the stability of synchronization to changes in network topology. The transformation of a directed network into a symmetrized-and-augmented undirected network is the central component of this new method. This transformation is executed by symmetrizing and weighting the underlying connection graph and adding new undirected edges with consideration made for the mean degree imbalance of each pair of nodes. These new edges represent "non-existent ties" in the original directed network and often control the location of critical nodes whose directed connections can be altered to manipulate the stability of synchronization in a desired way. In particular, we show that the addition of small-world shortcuts to directed networks, which makes "non-existent ties" disappear, can worsen the synchronizability, thereby revealing a destructive role of small-world connections in directed networks. An extension of our method may open the door to studying synchronization in directed multilayer networks, which cannot be effectively handled by the eigenvalue-based methods.

State-dependent effective interactions in oscillator networks through coupling functions with dead zones

The dynamics of networks of interacting dynamical systems depend on the nature of the coupling between individual units. We explore networks of oscillatory units with coupling functions that have 'dead zones', that is the coupling functions are zero on sets with interior. For such networks, it is convenient to look at the effective interactions between units rather than the (fixed) structural connectivity to understand the network dynamics. For example, oscillators may effectively decouple in particular phase configurations. Along trajectories, the effective interactions are not necessarily static, but the effective coupling may evolve in time. Here, we formalize the concepts of dead zones and effective interactions. We elucidate how the coupling function shapes the possible effective interaction schemes and how they evolve in time. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Inferring dynamic topology for decoding spatiotemporal structures in complex heterogeneous networks

Inferring connections forms a critical step toward understanding large and diverse complex networks. To date, reliable and efficient methods for the reconstruction of network topology from measurement data remain a challenge due to the high complexity and nonlinearity of the system dynamics. These obstacles also form a bottleneck for analyzing and controlling the dynamic structures (e.g., synchrony) and collective behavior in such complex networks. The novel contribution of this work is to develop a unified data-driven approach to reliably and efficiently reveal the dynamic topology of complex networks in different scales—from cells to societies. The developed technique provides guidelines for the refinement of experimental designs toward a comprehensive understanding of complex heterogeneous networks.

Extracting complex interactions (i.e., dynamic topologies) has been an essential, but difficult, step toward understanding large, complex, and diverse systems including biological, financial, and electrical networks. However, reliable and efficient methods for the recovery or estimation of network topology remain a challenge due to the tremendous scale of emerging systems (e.g., brain and social networks) and the inherent nonlinearity within and between individual units. We develop a unified, data-driven approach to efficiently infer connections of networks (ICON). We apply ICON to determine topology of networks of oscillators with different periodicities, degree nodes, coupling functions, and time scales, arising in silico, and in electrochemistry, neuronal networks, and groups of mice. This method enables the formulation of these large-scale, nonlinear estimation problems as a linear inverse problem that can be solved using parallel computing. Working with data from networks, ICON is robust and versatile enough to reliably reveal full and partial resonance among fast chemical oscillators, coherent circadian rhythms among hundreds of cells, and functional connectivity mediating social synchronization of circadian rhythmicity among mice over weeks.

Emergence of synchronization and regularity in firing patterns in time-varying neural hypernetworks

We study synchronization of dynamical systems coupled in time-varying network architectures, composed of two or more network topologies, corresponding to different interaction schemes. As a representative example of this class of time-varying hypernetworks, we consider coupled Hindmarsh-Rose neurons, involving two distinct types of networks, mimicking interactions that occur through the electrical gap junctions and the chemical synapses. Specifically, we consider the connections corresponding to the electrical gap junctions to form a small-world network, while the chemical synaptic interactions form a unidirectional random network. Further, all the connections in the hypernetwork are allowed to change in time, modeling a more realistic neurobiological scenario. We model this time variation by rewiring the links stochastically with a characteristic rewiring frequency f. We find that the coupling strength necessary to achieve complete neuronal synchrony is lower when the links are switched rapidly. Further, the average time required to reach the synchronized state decreases as synaptic coupling strength and/or rewiring frequency increases. To quantify the local stability of complete synchronous state we use the Master Stability Function approach, and for global stability we employ the concept of basin stability. The analytically derived necessary condition for synchrony is in excellent agreement with numerical results. Further we investigate the resilience of the synchronous states with respect to increasing network size, and we find that synchrony can be maintained up to larger network sizes by increasing either synaptic strength or rewiring frequency. Last, we find that time-varying links not only promote complete synchronization, but also have the capacity to change the local dynamics of each single neuron. Specifically, in a window of rewiring frequency and synaptic coupling strength, we observe that the spiking behavior becomes more regular.

Introduction to focus issue: Patterns of network synchronization

The study of synchronization of coupled systems is currently undergoing a major surge fueled by recent discoveries of new forms of collective dynamics and the development of techniques to characterize a myriad of new patterns of network synchronization. This includes chimera states, phenomena determined by symmetry, remote synchronization, and asymmetry-induced synchronization. This Focus Issue presents a selection of contributions at the forefront of these developments, to which this introduction is intended to offer an up-to-date foundation.

Enhancing synchronizability of diffusively coupled dynamical networks: a survey

In this paper, we review the literature on enhancing synchronizability of diffusively coupled dynamical networks with identical nodes. The last decade has witnessed intensive investigations on the collective behavior over complex networks and synchronization of dynamical systems is the most common form of collective behavior. For many applications, it is desired that the synchronizability-the ability of networks in synchronizing activity of their individual dynamical units-is enhanced. There are a number of methods for improving the synchronization properties of dynamical networks through structural perturbation. In this paper, we survey such methods including adding/removing nodes and/or edges, rewiring the links, and graph weighting. These methods often try to enhance the synchronizability through minimizing the eigenratio of the Laplacian matrix of the connection graph-a synchronizability measure based on the master-stability-function formalism. We also assess the performance of the methods by numerical simulations on a number of real-world networks as well as those generated through models such as preferential attachment, Watts-Strogatz, and Erdos-Rényi.

Failure tolerance of spike phase synchronization in coupled neural networks

Neuronal synchronization plays an important role in the various functionality of nervous system such as binding, cognition, information processing, and computation. In this paper, we investigated how random and intentional failures in the nodes of a network influence its phase synchronization properties. We considered both artificially constructed networks using models such as preferential attachment, Watts-Strogatz, and Erdős-Rényi as well as a number of real neuronal networks. The failure strategy was either random or intentional based on properties of the nodes such as degree, clustering coefficient, betweenness centrality, and vulnerability. Hindmarsh-Rose model was considered as the mathematical model for the individual neurons, and the phase synchronization of the spike trains was monitored as a function of the percentage∕number of removed nodes. The numerical simulations were supplemented by considering coupled non-identical Kuramoto oscillators. Failures based on the clustering coefficient, i.e., removing the nodes with high values of the clustering coefficient, had the least effect on the spike synchrony in all of the networks. This was followed by errors where the nodes were removed randomly. However, the behavior of the other three attack strategies was not uniform across the networks, and different strategies were the most influential in different network structure.

Synchrony in tritrophic food chain metacommunities

The synchronous behaviour of interacting communities is studied in this paper. Each community is described by a tritrophic food chain model, and the communities interact through a network with arbitrary topology, composed of patches and migration corridors. The analysis of the local synchronization properties (via the master stability function approach) shows that, if only one species can migrate, the dispersal of the consumer (i.e., the intermediate trophic level) is the most effective mechanism for promoting synchronization. When analysing the effects of the variations of demographic parameters, it is found that factors that stabilize the single community also tend to favour synchronization. Global synchronization is finally analysed by means of the connection graph method, yielding a lower bound on the value of the dispersion rate that guarantees the synchronization of the metacommunity for a given network topology.

Synchronization in asymmetrically coupled networks with node balance

We study global stability of synchronization in asymmetrically connected networks of limit-cycle or chaotic oscillators. We extend the connection graph stability method to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums. We obtain the same upper bound for synchronization in asymmetrically connected networks as in the network with a symmetrized matrix, provided that the condition of node balance is satisfied. In terms of graphs, the symmetrization operation amounts to replacing each directed edge by an undirected edge of half the coupling strength. It should be stressed that without node balance this property in general does not hold.

Synchronizing weighted complex networks

Real networks often consist of local units, which interact with each other via asymmetric and heterogeneous connections. In this work, we explore the constructive role played by such a directed and weighted wiring for the synchronization of networks of coupled dynamical systems. The stability condition for the synchronous state is obtained from the spectrum of the respective coupling matrices. In particular, we consider a coupling scheme in which the relative importance of a link depends on the number of shortest paths through it. We illustrate our findings for networks with different topologies: scale free, small world, and random wirings.

Synchronization in coupled chaotic oscillators with a no-flux boundary condition

We investigate the synchronization of coupled chaotic oscillators with a no-flux boundary condition. We find that the spectrum of the coupling matrix is divided into two parts, the isolated part with a zero eigenvalue and the continuous one with the other N-1 eigenvalues falling onto a line. Based on the eigenvalue analysis, the stability of the synchronization in a coupled Lorenz system is explored thoroughly in the parameter space of the size of the system, the diffusion, and gradient coupling constants.

Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays

We show that the stability surface that governs the synchronization of a large class of arrays of identical oscillators can be probed with a simple array of just three identical oscillators. Experimentally this implies that it may be possible to probe the synchronization conditions of many arrays all at the same time. In the process of developing a theory of the three-oscillator probe, we also show that several regimes of asymptotic coupling can be derived for the array classes, including the case of large imaginary coupling, which apparently has not been explored.