Author(s): Yoji Kawamura and Remi Tsubaki
We formulate a theory for the phase reduction of a beating flagellum. The theory enables us to describe the dynamics of a beating flagellum in a systematic manner using a single variable called the phase. The theory can also be considered as a phase reduction method for the limit-cycle solutions in ...
[Phys. Rev. E 97, 022212] Published Mon Feb 12, 2018
Complex spatiotemporal patterns, called chimera states, consist of coexisting coherent and incoherent domains and can be observed in networks of coupled oscillators. The interplay of synchrony and asynchrony in complex brain networks is an important aspect in studies of both brain function and disease. We analyse the collective dynamics of FitzHugh-Nagumo neurons in complex networks motivated by its potential application to epileptology and epilepsy surgery. We compare two topologies: an empirical structural neural connectivity derived from diffusion-weighted magnetic resonance imaging and a mathematically constructed network with modular fractal connectivity. We analyse the properties of chimeras and partially synchronized states, and obtain regions of their stability in the parameter planes. Furthermore, we qualitatively simulate the dynamics of epileptic seizures and study the influence of the removal of nodes on the network synchronizability, which can be useful for applications to epileptic surgery.
Synchronization processes are ubiquitous despite the many connectivity patterns that complex systems can show. Usually, the emergence of synchrony is a macroscopic observable, however, the microscopic details of the system, as e.g. the underlying network of interactions, is many times partially or totally unknown. We already know that different interaction structures can give rise to a common functionality, understood as a common macroscopic observable. Building upon this fact, here, we propose network transformations that keep the collective behavior of a large system of Kuramoto oscillators functionally invariant. We derive a method based on information theory principles, that allows us to adjust the weights of the structural interactions to map random homogeneous -in degree- networks into random heterogeneous networks and vice-versa, keeping synchronization values invariant. The results of the proposed transformations reveal an interesting principle; heterogeneous networks can be mapped to homogeneous ones with local information, but the reverse process needs to exploit higher-order information. The formalism provides new analytical insight to tackle real complex scenarios when dealing with uncertainty in the measurements of the underlying connectivity structure.
In this paper, we study convergence of coupled dynamical systems on convergent sequences of graphs to a continuum limit. We show that the solutions of the initial value problem for the dynamical system on a convergent graph sequence tend to that for the nonlocal diffusion equation on a unit interval, as the graph size tends to infinity. We improve our earlier results in [Arch. Ration. Mech. Anal., 21 (2014), pp. 781--803] and extend them to a larger class of graphs, which includes directed and undirected, sparse and dense, random and deterministic graphs.
There are three main ingredients of our approach. First, we employ a flexible framework for incorporating random graphs into the models of interacting dynamical systems, which fits seamlessly with the derivation of the continuum limit. Next, we prove the averaging principle for approximating a dynamical system on a random graph by its deterministic (averaged) counterpart. The proof covers systems on sparse graphs and yields almost sure convergence on time intervals of order $\log n,$ where $n$ is the number of vertices. Finally, a Galerkin scheme is developed to show convergence of the averaged model to the continuum limit.
The analysis of this paper covers the Kuramoto model of coupled phase oscillators on a variety of graphs including sparse Erd\H{o}s-R{\' e}nyi, small-world, and power law graphs.
In this paper, we study convergence of coupled dynamical systems on convergent sequences of graphs to a continuum limit. We show that the solutions of the initial value problem for the dynamical system on a convergent graph sequence tend to that for the nonlocal diffusion equation on a unit interval, as the graph size tends to infinity. We improve our earlier results in [Arch. Ration. Mech. Anal., 21 (2014), pp. 781--803] and extend them to a larger class of graphs, which includes directed and undirected, sparse and dense, random and deterministic graphs.
There are three main ingredients of our approach. First, we employ a flexible framework for incorporating random graphs into the models of interacting dynamical systems, which fits seamlessly with the derivation of the continuum limit. Next, we prove the averaging principle for approximating a dynamical system on a random graph by its deterministic (averaged) counterpart. The proof covers systems on sparse graphs and yields almost sure convergence on time intervals of order $\log n,$ where $n$ is the number of vertices. Finally, a Galerkin scheme is developed to show convergence of the averaged model to the continuum limit.
The analysis of this paper covers the Kuramoto model of coupled phase oscillators on a variety of graphs including sparse Erd\H{o}s-R{\' e}nyi, small-world, and power law graphs.
Isospectral transformations (IT) of matrices and networks allow for compression of either object while keeping all the information about their eigenvalues and eigenvectors.We analyze here what happens to generalized eigenvectors under isospectral transformations and to what extent the initial network can be reconstructed from its compressed image under IT. We also generalize and essentially simplify the proof that eigenvectors are invariant under isospectral transformations and generalize and clarify the notion of spectral equivalence of networks.
Accessing the network through which a propagation dynamics diffuse is essential for understanding and controlling it. In a few cases, such information is available through direct experiments or thanks to the very nature of propagation data. In a majority of cases however, available information about the network is indirect and comes from partial observations of the dynamics, rendering the network reconstruction a fundamental inverse problem. Here we show that it is possible to reconstruct the whole structure of an interaction network and to simultaneously infer the complete time course of activation spreading, relying just on single epoch (i.e. snapshot) or time-scattered observations of a small number of activity cascades. The method that we present is built on a Belief Propagation approximation, that has shown impressive accuracy in a wide variety of relevant cases, and is able to infer interactions in presence of incomplete time-series data by providing a detailed modeling of the posterior distribution of trajectories conditioned to the observations. Furthermore, we show by experiments that the information content of full cascades is relatively smaller than that of sparse observations or single snapshots.
We present a necessary and sufficient condition for a random product of maps on a compact metric space to be (strongly) synchronizing on average.
Author(s): Tatsuro Kawamoto and Yoshiyuki Kabashima
We conduct a comparative analysis on various estimates of the number of clusters in community detection. An exhaustive comparison requires testing of all possible combinations of frameworks, algorithms, and assessment criteria. In this study we focus on the framework based on a stochastic block mode...[Phys. Rev. E] Published Tue Feb 06, 2018
Terrorism instills fear in the minds of people and takes away the freedom of individuals to act as they will. Terrorism has turned out to be an international menace in the global community; every nation is getting affected, directly or indirectly. Here, we study the terrorist attack incidents which occurred in the last half century across the globe from the open source, Global terrorism database, and develop a view on their spatio-temporal dynamics. We construct a complex network of global terrorism and study its growth dynamics, along with the statistical properties of the network, which are quite intriguing. Normally, each nation pursues its own vision of international security based upon its mandate and particular notions of politics and its policies to counter the threat of terrorism that could naturally include the use of tactical measures and strategic negotiations, or even physical power. We study the resilience of the network against targeted attacks and random failures, which could guide the counter-terrorist outfits in designing strategies to fight terrorism. We then use a disparity filter method to isolate the backbone of the giant component, and identify the terror hubs and vulnerable motifs of global terrorism. We also examine the evolution of the hubs and motifs in a few exemplary cases like Afghanistan, Colombia, Israel, Peru and United Kingdom. The dynamics of the terror hubs and the vulnerable motifs that we discover in the network backbone can provide deep insight on their formations and spreading, and thereby help in contending terrorism or making public policies that can check their spread.
The dynamics of networks of neuronal cultures has been recently shown to be strongly dependent on the network geometry and in particular on their dimensionality. However, this phenomenon has been so far mostly unexplored from the theoretical point of view. Here we reveal the rich interplay between network geometry and synchronization of coupled oscillators in the context of a simplicial complex model of manifolds called Complex Network Manifold. The networks generated by this model combine small world properties (infinite Hausdorff dimension) and a high modular structure with finite and tunable spectral dimension. We show that the networks display frustrated synchronization for a wide range of the coupling strength of the oscillators, and that the synchronization properties are directly affected by the spectral dimension of the network.
Recently, eigenvector localization of complex network has seen a spurt in activities due to its versatile applicability in many different areas which includes networks centrality measure, spectral partitioning, development of approximation algorithms and disease spreading phenomenon. For a network, an eigenvector is said to be localized when most of its components are near to zero, with few taking very high values. Here, we develop three different randomized algorithms, which by using edge rewiring method, can evolve a random network having a delocalized principal eigenvector to a network having a highly localized principal eigenvector. We discuss drawbacks and advantages of these algorithms. Additionally, we show that the construction of such networks corresponding to the highly localized principal eigenvector is a non-convex optimization problem when the objective function is the inverse participation ratio.
Starting from a microscopic model for a system of neurons evolving in time which individually follow a stochastic integrate-and-fire type model, we study a mean-field limit of the system. Our model is described by a system of SDEs with discontinuous coefficients for the action potential of each neuron and takes into account the (random) spatial configuration of neurons allowing the interaction to depend on it. In the limit as the number of particles tends to infinity, we obtain a nonlinear Fokker-Planck type PDE in two variables, with derivatives only with respect to one variable and discontinuous coefficients. We also study strong well-posedness of the system of SDEs and prove the existence and uniqueness of a weak measure-valued solution to the PDE, obtained as the limit of the laws of the empirical measures for the system of particles.
Author(s): Chuang Ma, Han-Shuang Chen, Ying-Cheng Lai, and Hai-Feng Zhang
Complex networks hosting binary-state dynamics arise in a variety of contexts. In spite of previous works, to fully reconstruct the network structure from observed binary data remains challenging. We articulate a statistical inference based approach to this problem. In particular, exploiting the exp...
[Phys. Rev. E 97, 022301] Published Mon Feb 05, 2018
The Kuramoto model has become a paradigm to describe the dynamics of nonlinear oscillator under the influence of external perturbations, both deterministic and stochastic. It is based on the idea to describe the oscillator dynamics by a scalar differential equation, that defines the time evolution for the phase of the oscillator. Starting from a phase and amplitude description of noisy oscillators, we discuss the reduction to a phase oscillator model, analogous to the Kuramoto model. The model derived shows that the phase noise problem is a drift-diffusion process. Even in the case where the expected amplitude remains unchanged, the unavoidable amplitude fluctuations do change the expected frequency, and the frequency shift depends on the amplitude variance. We discuss different degrees of approximation, yielding increasingly accurate phase reduced descriptions of noisy oscillators.
One of the main challenges of modern physics is to provide a systematic understanding of systems far from equilibrium exhibiting emergent behavior. Prominent examples of such complex systems include, but are not limited to the cardiac electrical system, the brain, the power grid, social systems, material failure and earthquakes, and the climate system. Due to the technological advances over the last decade, the amount of observations and data available to characterize complex systems and their dynamics, as well as the capability to process that data, has increased substantially. The present issue discusses a cross section of the current research on complex systems, with a focus on novel experimental and data-driven approaches to complex systems that provide the necessary platform to model the behavior of such systems.
Author(s): Yoji Kawamura and Remi Tsubaki
We formulate a theory for the phase reduction of a beating flagellum. The theory enables us to describe the dynamics of a beating flagellum in a systematic manner using a single variable called the phase. The theory can also be considered as a phase reduction method for the limit-cycle solutions in ...[Phys. Rev. E] Published Wed Jan 31, 2018
The Kuramoto model (KM) of coupled phase oscillators on scale free graphs is analyzed in this work. The W-random graph model is used to define a convergent family of sparse graphs with power law degree distribution. For the KM on this family of graphs, we derive the mean field description of the system's dynamics in the limit as the size of the network tends to infinity. The mean field equation is used to study two problems: synchronization in the coupled system with randomly distributed intrinsic frequencies and existence and bifurcations of chimera states in the KM with repulsive coupling. The analysis of both problems highlights the role of the scale free network organization in shaping dynamics of the coupled system. The analytical results are complemented with the results of numerical simulations.
We investigate a time-delayed epidemic model for multi-strain diseases with temporary immunity. In the absence of cross-immunity between strains, dynamics of each individual strain exhibits emergence and anni- hilation of limit cycles due to a Hopf bifurcation of the endemic equilibrium, and a saddle-node bifurcation of limit cycles depending on the time delay associated with duration of temporary immunity. Effects of all-to-all and non-local coupling topologies are systematically investigated by means of numerical simulations, and they suggest that cross-immunity is able to induce a diverse range of complex dynamical behaviors and synchro- nization patterns, including discrete traveling waves, solitary states, and amplitude chimeras. Interestingly, chimera states are observed for narrower cross-immunity kernels, which can have profound implications for understanding the dynamics of multi-strain diseases.
Author(s): Philipp M. Holl and Friedemann Reinhard
The Wi-Fi signals that provide internet access can also produce images of the transmitter’s 3D surroundings, even through walls.
[Phys. Rev. Lett. 118, 183901] Published Fri May 05, 2017
Author(s): Amikam Patron, Reuven Cohen, Daqing Li, and Shlomo Havlin
Strengthening or destroying a network is a very important issue in designing resilient networks or in planning attacks against networks, including planning strategies to immunize a network against diseases, viruses, etc. Here we develop a method for strengthening or destroying a random network with …
[Phys. Rev. E 95, 052305] Published Fri May 05, 2017
Author(s): Janina Hesse, Jan-Hendrik Schleimer, and Susanne Schreiber
Prominent changes in neuronal dynamics have previously been attributed to a specific switch in onset bifurcation, the Bogdanov-Takens (BT) point. This study unveils another, relevant and so far underestimated transition point: the saddle-node-loop bifurcation, which can be reached by several paramet…
[Phys. Rev. E 95, 052203] Published Wed May 03, 2017
Chimera states, which consist of coexisting domains of spatially coherent and incoherent dynamics, have been widely found in nonlocally coupled oscillatory systems. We demonstrate for the first time that chimera states can emerge from excitable systems under nonlocal coupling in which isolated units only allow for the equilibrium. We theoretically reveal that nonlocal coupling induced collective oscillation is behind the occurrence of the chimera states. We find two different types of chimera states, phase-chimera state and excitability-chimera states, depending on the coupling strength. At weak coupling strength where collective oscillation is localized around the unstable homogeneous equilibrium, the chimera states are similar to the ones in nonlocally coupled phase oscillators. For the chimera states at strong coupling strength, the dynamics of both coherent units and incoherent units shift back and forth between low amplitude oscillation induced by collective oscillation and high amplitude oscillation induced by excitability of local units.
Cell diversity and network dynamics in photosensitive human brain organoids
Nature 545, 7652 (2017). doi:10.1038/nature22047
Authors: Giorgia Quadrato, Tuan Nguyen, Evan Z. Macosko, John L. Sherwood, Sung Min Yang, Daniel R. Berger, Natalie Maria, Jorg Scholvin, Melissa Goldman, Justin P. Kinney, Edward S. Boyden, Jeff W. Lichtman, Ziv M. Williams, Steven A. McCarroll & Paola Arlotta
In vitro models of the developing brain such as three-dimensional brain organoids offer an unprecedented opportunity to study aspects of human brain development and disease. However, the cells generated within organoids and the extent to which they recapitulate the regional complexity, cellular diversity and
We introduce and implement an importance-sampling Monte Carlo algorithm to study systems of globally-coupled oscillators. Our computational method efficiently obtains estimates of the tails of the distribution of various measures of dynamical trajectories corresponding to states occurring with (exponentially) small probabilities. We demonstrate the general validity of our results by applying the method to two contrasting cases: the driven-dissipative Kuramoto model, a paradigm in the study of spontaneous synchronization; and the conservative Hamiltonian mean-field model, a prototypical system of long-range interactions. We present results for the distribution of the finite-time Lyapunov exponent and a time-averaged order parameter. Among other features, our results show most notably that the distributions exhibit a vanishing standard deviation but a skewness that is increasing in magnitude with the number of oscillators, implying that non-trivial asymmetries and states yielding rare/atypical values of the observables persist even for a large number of oscillators.
We illustrate a counter-intuitive effect of an additive stochastic force, which acts independently on each element of an ensemble of globally coupled oscillators. We show numerically and semi-analytically that a very small white noise is able to stabilize an otherwise linearly unstable collective periodic regime. Microscopic simulations reveal two noise-induced bifurcations of different nature towards self-consistent partial synchrony. We develop a macroscopic treatment solving the corresponding nonlinear Fokker-Planck equation by means of a perturbative approach. The associated linear stability analysis confirms the results anticipated by the numerics. We also argue about the generality of the phenomenon.