1,022 research outputs found
Stability Conditions for Cluster Synchronization in Networks of Heterogeneous Kuramoto Oscillators
In this paper we study cluster synchronization in networks of oscillators
with heterogenous Kuramoto dynamics, where multiple groups of oscillators with
identical phases coexist in a connected network. Cluster synchronization is at
the basis of several biological and technological processes; yet the underlying
mechanisms to enable cluster synchronization of Kuramoto oscillators have
remained elusive. In this paper we derive quantitative conditions on the
network weights, cluster configuration, and oscillators' natural frequency that
ensure asymptotic stability of the cluster synchronization manifold; that is,
the ability to recover the desired cluster synchronization configuration
following a perturbation of the oscillators' states. Qualitatively, our results
show that cluster synchronization is stable when the intra-cluster coupling is
sufficiently stronger than the inter-cluster coupling, the natural frequencies
of the oscillators in distinct clusters are sufficiently different, or, in the
case of two clusters, when the intra-cluster dynamics is homogeneous. We
illustrate and validate the effectiveness of our theoretical results via
numerical studies.Comment: To apper in IEEE Transactions on Control of Network System
A Framework to Control Functional Connectivity in the Human Brain
In this paper, we propose a framework to control brain-wide functional
connectivity by selectively acting on the brain's structure and parameters.
Functional connectivity, which measures the degree of correlation between
neural activities in different brain regions, can be used to distinguish
between healthy and certain diseased brain dynamics and, possibly, as a control
parameter to restore healthy functions. In this work, we use a collection of
interconnected Kuramoto oscillators to model oscillatory neural activity, and
show that functional connectivity is essentially regulated by the degree of
synchronization between different clusters of oscillators. Then, we propose a
minimally invasive method to correct the oscillators' interconnections and
frequencies to enforce arbitrary and stable synchronization patterns among the
oscillators and, consequently, a desired pattern of functional connectivity.
Additionally, we show that our synchronization-based framework is robust to
parameter mismatches and numerical inaccuracies, and validate it using a
realistic neurovascular model to simulate neural activity and functional
connectivity in the human brain.Comment: To appear in the proceedings of the 58th IEEE Conference on Decision
and Contro
Cluster Synchronization of Kuramoto Oscillators and Brain Functional Connectivity
The recent progress of functional magnetic resonance imaging techniques has
unveiled that human brains exhibit clustered correlation patterns of their
spontaneous activities. It is important to understand the mechanism of cluster
synchronization phenomena since it may reflect the underlying brain functions
and brain diseases. In this paper, we investigate cluster synchronization
conditions for networks of Kuramoto oscillators. The key analytical tool that
we use is the method of averaging, and we provide a unified framework of
stability analysis for cluster synchronization. The main results show that
cluster synchronization is achieved if (i) the inter-cluster coupling strengths
are sufficiently weak and/or (ii) the natural frequencies are largely different
among clusters. Moreover, we apply our theoretical findings to empirical brain
networks. Discussions on how to understand brain functional connectivity and
further directions to investigate neuroscientific questions are provided
Synchronization Patterns in Networks of Kuramoto Oscillators: A Geometric Approach for Analysis and Control
Synchronization is crucial for the correct functionality of many natural and
man-made complex systems. In this work we characterize the formation of
synchronization patterns in networks of Kuramoto oscillators. Specifically, we
reveal conditions on the network weights and structure and on the oscillators'
natural frequencies that allow the phases of a group of oscillators to evolve
cohesively, yet independently from the phases of oscillators in different
clusters. Our conditions are applicable to general directed and weighted
networks of heterogeneous oscillators. Surprisingly, although the oscillators
exhibit nonlinear dynamics, our approach relies entirely on tools from linear
algebra and graph theory. Further, we develop a control mechanism to determine
the smallest (as measured by the Frobenius norm) network perturbation to ensure
the formation of a desired synchronization pattern. Our procedure allows us to
constrain the set of edges that can be modified, thus enforcing the sparsity
structure of the network perturbation. The results are validated through a set
of numerical examples
Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators
A chimera state is a spatio-temporal pattern in a network of identical
coupled oscillators in which synchronous and asynchronous oscillation coexist.
This state of broken symmetry, which usually coexists with a stable spatially
symmetric state, has intrigued the nonlinear dynamics community since its
discovery in the early 2000s. Recent experiments have led to increasing
interest in the origin and dynamics of these states. Here we review the history
of research on chimera states and highlight major advances in understanding
their behaviour.Comment: 26 pages, 3 figure
Synchronization in complex networks
Synchronization processes in populations of locally interacting elements are
in the focus of intense research in physical, biological, chemical,
technological and social systems. The many efforts devoted to understand
synchronization phenomena in natural systems take now advantage of the recent
theory of complex networks. In this review, we report the advances in the
comprehension of synchronization phenomena when oscillating elements are
constrained to interact in a complex network topology. We also overview the new
emergent features coming out from the interplay between the structure and the
function of the underlying pattern of connections. Extensive numerical work as
well as analytical approaches to the problem are presented. Finally, we review
several applications of synchronization in complex networks to different
disciplines: biological systems and neuroscience, engineering and computer
science, and economy and social sciences.Comment: Final version published in Physics Reports. More information
available at http://synchronets.googlepages.com
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