3,178 research outputs found
Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function
Synchronization is central to many complex systems in engineering physics
(e.g., the power-grid, Josephson junction circuits, and electro-chemical
oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms).
Despite these widespread applications---for which proper functionality depends
sensitively on the extent of synchronization---there remains a lack of
understanding for how systems evolve and adapt to enhance or inhibit
synchronization. We study how network modifications affect the synchronization
properties of network-coupled dynamical systems that have heterogeneous node
dynamics (e.g., phase oscillators with non-identical frequencies), which is
often the case for real-world systems. Our approach relies on a synchrony
alignment function (SAF) that quantifies the interplay between heterogeneity of
the network and of the oscillators and provides an objective measure for a
system's ability to synchronize. We conduct a spectral perturbation analysis of
the SAF for structural network modifications including the addition and removal
of edges, which subsequently ranks the edges according to their importance to
synchronization. Based on this analysis, we develop gradient-descent algorithms
to efficiently solve optimization problems that aim to maximize phase
synchronization via network modifications. We support these and other results
with numerical experiments.Comment: 25 pages, 6 figure
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
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
Koopman Operator and its Approximations for Systems with Symmetries
Nonlinear dynamical systems with symmetries exhibit a rich variety of
behaviors, including complex attractor-basin portraits and enhanced and
suppressed bifurcations. Symmetry arguments provide a way to study these
collective behaviors and to simplify their analysis. The Koopman operator is an
infinite dimensional linear operator that fully captures a system's nonlinear
dynamics through the linear evolution of functions of the state space.
Importantly, in contrast with local linearization, it preserves a system's
global nonlinear features. We demonstrate how the presence of symmetries
affects the Koopman operator structure and its spectral properties. In fact, we
show that symmetry considerations can also simplify finding the Koopman
operator approximations using the extended and kernel dynamic mode
decomposition methods (EDMD and kernel DMD). Specifically, representation
theory allows us to demonstrate that an isotypic component basis induces block
diagonal structure in operator approximations, revealing hidden organization.
Practically, if the data is symmetric, the EDMD and kernel DMD methods can be
modified to give more efficient computation of the Koopman operator
approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out
the development, we discuss the effect of measurement noise
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
Synchronization of Heterogeneous Oscillators Under Network Modifications: Perturbation and Optimization of the Synchrony Alignment Function
Synchronization is central to many complex systems in engineering physics (e.g., the power-grid, Josephson junction circuits, and electro-chemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these widespread applications—for which proper functionality depends sensitively on the extent of synchronization—there remains a lack of understanding for how systems can best evolve and adapt to enhance or inhibit synchronization. We study how network modifications affect the synchronization properties of network-coupled dynamical systems that have heterogeneous node dynamics (e.g., phase oscillators with non-identical frequencies), which is often the case for real-world systems. Our approach relies on a synchrony alignment function (SAF) that quantifies the interplay between heterogeneity of the network and of the oscillators and provides an objective measure for a system’s ability to synchronize. We conduct a spectral perturbation analysis of the SAF for structural network modifications including the addition and removal of edges, which subsequently ranks the edges according to their importance to synchronization. Based on this analysis, we develop gradient-descent algorithms to efficiently solve optimization problems that aim to maximize phase synchronization via network modifications. We support these and other results with numerical experiments
- …