Synchronization in Interacting Populations of
Heterogeneous Oscillators with Time-Varying Coupling
Paul So, Bernard C. Cotton, and Ernest Barreto
Abstract
In many networks of interest (including technological,
biological,
and social networks), the connectivity between the interacting elements
is not static, but changes in time. Furthermore, the
elements themselves are often not identical, but rather display a
variety of behaviors, and may
come in different classes. Here, we investigate the dynamics of such
systems. Specifically, we
study a network of two large interacting heterogeneous populations of
limit-cycle oscillators whose
connectivity switches between two fixed arrangements at a particular
frequency. We show that
for sufficiently high switching frequency, this system behaves as if
the connectivity were static and
equal to the time-average of the switching connectivity. We also
examine the mechanisms by which
this fast-switching limit is approached in several non-intuitive cases.
The results illuminate novel
mechanisms by which synchronization can arise or be thwarted in large
populations of coupled
oscillators with non-static coupling.
This paper appears in Chaos 18, 037114 (2008).
A PDF
file of this paper is available.
Synchronization in Networks of Networks: The
Onset of Coherent
Collective Behavior in Interacting Populations of Heterogeneous
Oscillators
Ernest Barreto, Brian Hunt, Edward Ott, and Paul So
Abstract
The onset of synchronization in networks of networks is
investigated.
Specifically, we consider networks of interacting
phase oscillators in which the set of
oscillators is composed of several distinct populations. The
oscillators
in a given population are heterogeneous in that their natural
frequencies
are drawn from a given distribution, and each population has
its own such distribution. The coupling among the oscillators is
global, however,
we permit the coupling strengths between the members of different
populations to
be separately specified.
We determine the critical condition for the onset
of coherent collective behavior, and develop the illustrative case in
which the oscillator frequencies are drawn from a set of
(possibly different) Cauchy-Lorentz distributions.
One motivation is drawn from neurobiology, in which the collective
dynamics of
several interacting populations of oscillators (such as excitatory and
inhibitory
neurons and glia) are of interest.
This paper appears in Physical Review E, Vol. 77, 036107 (2008).
A PDF
file of this paper is available.
Interneuron and Pyramidal Cell Interplay
During In Vitro Seizure-like Events
Jokubas Ziburkus, John R. Cressman, Ernest Barreto, and
Steven J. Schiff
Abstract
Excitatory and inhibitory (EI) interactions shape network
activity. However, little
is known about the EI interactions in pathological conditions such as
epilepsy. To
investigate EI interactions during seizure-like events (SLEs), we
performed
simultaneous dual and triple whole cell and extracellular recordings in
pyramidal
cells and oriens interneurons in rat hippocampal CA1. We describe a
novel pattern
of interleaving EI activity during spontaneous in vitro SLEs generated
by the
potassium channel blocker 4-aminopyridine in the presence of decreased
magnesium.
Interneuron activity was increased during interictal periods. During
ictal discharges,
interneurons entered into long-lasting depolarization block (DB) with
suppression of
spike generation; simultaneously, pyramidal cells produced spike trains
with
increased frequency (6-14 Hz) and correlation. After this period of
runaway excitation,
interneuron postictal spiking resumed and pyramidal cells became
progressively
quiescent. We performed correlation measures of cell-pair interactions
using either
the spikes alone or the subthreshold postsynaptic interspike signals.
EE spike correlation
was notably increased during interneuron DB, whereas subthreshold EE
correlation
decreased. EI spike correlations increased at the end of SLEs, whereas
II subthreshold
correlations increased during DB. Our findings underscore the
importance of complex
cell-type-specific neuronal interactions in the formation of seizure
patterns.
This paper appears in the Journal of Neurophysiology, Vol. 95, pp.
3948-3954 (2006).
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Dynamics of Noninvertibility in Delay
Equations
Evelyn Sander, Ernest Barreto, S. Schiff, and P. So
Abstract
Models with a time delay often occur, since there is a
naturally occurring
delay in the transmission of information. A model with a delay can be
noninvertible,
which in turn leads to qualitative differences between the dynamical
properties of a
delay equation and the familiar case of an ordinary differential
equation. We give
specifc conditions for the existence of noninvertible solutions in
delay equations, and
describe the consequences of noninvertibility.
This paper appears in Discrete and Continuous Dynamical Systems,
Supplementary Volume, pp. 768-777 (August 2005).
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
A Model of the Effects of Applied Electric
Fields on Neuronal Synchronization
Eun-Hyoung Park, Ernest Barreto, Bruce J. Gluckman, Steven
J. Schiff, and Paul So
Abstract
We examine the effects of applied electric fields on
neuronal synchronization. Twocompartment
model neurons were synaptically coupled and embedded within a resistive
array,
thus allowing the neurons to interact both chemically and electrically.
In addition, an external
electric field was imposed on the array. The effects of this field were
found to be nontrivial,
giving rise to domains of synchrony and asynchrony as a function of the
heterogeneity among
the neurons. A simple phase oscillator reduction was successful in
qualitatively reproducing
these domains. The findings form several readily testable experimental
predictions, and the
model can be extended to a larger scale in which the effects of
electric fields on seizure activity
may be simulated.
This paper appears in the Journal of Computational Neuroscience, Vol.
19 #1, pp. 53-70 (2005).
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Topology of Windows in the High-dimensional
Parameter Space of Chaotic
Maps
Murilo S. Baptista, Celso Grebogi, and Ernest Barreto
Abstract
Periodicity is ubiquitous in nature. In this work, we
analyze the
dynamical
reasons for which periodic windows that appear in parameter space
diagrams
have different shapes and structures. For that, we make use of a
dynamical
quantity called spine -- the skeleton of the window -- in order to
explain
a conjecture that describes the presence of periodic windows in the
parameter space of high-dimensional chaotic systems.
This paper appears in the International Journal of
Bifurcation and
Chaos, Vol 13 #9, p. 2681-2688 (2003).
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Electric Field Modulation of Synchronization
in Neural Networks
Eun-Hyoung Park, Paul So, Ernest Barreto, Bruce J.
Gluckman, and Steven
J. Schiff
Abstract
Motivated by the observation that applied electric fields
modulate
hippocampal seizures,
and that seizures may be asynchronous, we modeled synaptically-coupled
2-compartment
hippocampal pyramidal neurons embedded within an electrically resistive
lattice in
order to examine network synchronization properties under the influence
of
externally applied electric fields. Excitatory electric fields were
shown to
synchronize or desynchronize the network depending on the natural
frequency
mismatch between the neurons. Such frequency mismatch was found to
decrease
as a function of increasing electric field amplitude. These findings
provide
testable hypotheses for future seizure control experiments.
This paper appears in Neurocomputing 52-54, pp. 169-175
(2003).
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
The Geometry of Chaos Synchronization
Ernest Barreto, Kresimir Josic, Carlos J. Morales, Evelyn
Sander, and
Paul So
Abstract
Chaos synchronization in coupled systems is often
characterized by a
map $\phi$ between the states of the components. In noninvertible
systems, or in systems without inherent symmetries, the synchronization
set -- by which we mean graph($\phi$) -- can be extremely complicated.
We identify, describe, and give examples of several different
complications that can arise, and we link each to inherent properties
of the underlying dynamics. We also discuss how these features can be
quantified, and how they interfere with standard methods for detecting
synchronization in measured data.
This paper appears in Chaos 13 #1, pp. 151-164 (2003).
A PDF file
of this paper is
available.
The Onset of Synchronization in Systems of
Globally Coupled Chaotic and
Periodic Oscillators
Edward Ott, Paul So, Ernest Barreto, and Thomas Antonsen
Abstract
A general stability analysis is presented for the
determination of the
transition from
incoherent to coherent behavior in an ensemble of globally coupled,
heterogeneous,
continuous-time dynamical systems. The formalism allows for the
simultaneous presence
of ensemble members exhibiting chaotic and periodic behavior, and, in a
special case,
yields the Kuramoto model for globally coupled periodic oscillators
described by a phase.
Numerical experiments using different types of ensembles of Lorenz
equations
with a distribution of parameters are presented.
This paper appears in Physica D Vol. 173, pp. 29-51 (1
December 2002).
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Limits to the Experimental Detection of
Nonlinear Synchrony
Paul So, Ernest Barreto, Kresimir Josic, Evelyn Sander, and
Steven J.
Schiff
Abstract
Chaos synchronization is often characterized by the
existence of
a continuous function between the states of the components. However,
in coupled systems without inherent symmetries, the synchronization
set can be extremely complicated. We describe and illustrate three
typical
complications that can arise, and we discuss how existing methods for
detecting synchronization will be hampered by the presence of these
features.
This paper appears in Physical Review E, Vol. 65, 046225
(2002).
A PDF
file of this paper is
available.
The Breakdown of Synchronization in Systems
of Non-identical Chaotic
Oscillators: Theory and Experiment
Jennifer Chubb, Ernest Barreto, Paul So, and Bruce J.
Gluckman
Abstract
The synchronization of chaotic systems has received a great
deal of
attention.
However, most of the literature has focused on systems that possess
invariant
manifolds that persist as the coupling is varied. In this paper, we
describe
the process whereby synchronization is lost in systems of non-identical
coupled
chaotic oscillators without special symmetries. We qualitatively and
quantitatively
analyze such systems in terms of the evolution of the unstable periodic
orbit
structure. Our results are illustrated with data from physical
experiments.
This paper appears in the International Journal of
Bifurcation and
Chaos, Vol 11 #10, p. 2705-2713 (2001).
For further information, click
here to send me an email
at ebarreto_AT_gmu_DOT_edu.
Mechanisms for the Development of Unstable
Dimension Variability
and the Breakdown of Shadowing in Coupled Chaotic Systems
Ernest Barreto and Paul So
Abstract
A chaotic attractor containing unstable periodic orbits with
different
numbers of
unstable directions is said to exhibit unstable dimension variability
(UDV). We
present a general mechanism for the progressive development of UDV in
uni- and
bidirectionally coupled systems of chaotic elements. Our results are
applicable
to systems of dissimilar elements without invariant manifolds. We also
quantify
the severity of UDV to identify coupling ranges where the shadowability
and
modelability of such systems are significantly compromised.
This paper appears in Physical Review Letters, 85
#12, p.
2490-2493 (18 September 2000).
A PDF
file of this paper is
available.
From Generalized Synchrony to Topological
Decoherence:
Emergent Sets in Coupled Chaotic Systems
Ernest Barreto, Paul So, Bruce J. Gluckman, and Steven J.
Schiff
Abstract
We consider the evolution of the unstable periodic orbit
structure of
coupled
chaotic systems. This involves the creation of a complicated set
outside of the
synchronization manifold (the emergent set). We quantitatively identify
a critical
transition point in its development (the decoherence transition). For
asymmetric
systems we also describe a migration of unstable periodic orbits that
is of central
importance in understanding these systems. Our framework provides an
experimentally
measurable transition, even in situations where previously described
bifurcation
structures are inapplicable.
This paper appears in Physical Review Letters 84
#8, p.
1689-1692 (21 February 2000).
A PDF
file of this
paper is available.
Box-Counting Dimension Without Boxes:
Computing D_0 from Average Expansion Rates
Paul So, Ernest Barreto and Brian R. Hunt
Abstract
We propose an iterative scheme for calculating the
box-counting
(capacity) dimension of a chaotic attractor in terms of its average
expansion rates. Similar to the Kaplan-Yorke conjecture for the
information dimension, this scheme provides a connection between a
geometric property of a strange set and its underlying dynamical
properties. Our conjecture is demonstrated analytically with an
exactly solvable two dimensional hyperbolic map and numerically with a
more complicated higher dimensional non-hyperbolic map.
This paper appears in Physical Review E 60
#1, p. 378-385
(July 1999).
A PDF
file of this
paper is available.
From High Dimensional Chaos to Stable
Periodic Orbits:
The Structure of Parameter Space
Ernest Barreto, Brian R. Hunt, Celso Grebogi, and James A.
Yorke
Abstract
Regions in the parameter space of chaotic systems that
correspond to
stable behavior are often referred to as windows. In this Letter, we
elucidate the occurrence of such regions in higher dimensional chaotic
systems. We describe the fundamental structure of these windows, and
also indicate under what circumstances one can expect to find
them. These results are applicable to systems that exhibit several
positive Lyapunov exponents, and are of importance to both the
theoretical and the experimental understanding of dynamical systems.
This paper appears in Physical Review Letters 78
#24, p.
4561-4564 (16 June 1997).
A PDF
file of this paper is
available.
Multiparameter
Control of Chaos
Ernest Barreto and Celso Grebogi
Abstract
Controlling chaos by using more than one available control
parameter
is presented as an experimentally feasable way to reduce the transient
times that precede stabilization and improve performance in the
presence of noise. We demonstrate these advantages by applying our
method to a numerical model of a physical system.
This paper appears in Physical Review E 52
#4, p.
3553-3557 (October 1995).
A PDF file of
this paper is
available.
Efficient Switching Between Controlled
Unstable
Periodic Orbits in Higher Dimensional Chaotic Systems
Ernest Barreto, Eric J. Kostelich, Celso Grebogi, Edward
Ott, and
James A. Yorke
Abstract
We develop an efficient targeting technique and demonstrate
that when
used with an unstable periodic orbit stabilization method, fast and
efficient switching between controlled periodic orbits is possible.
This technique is particularly relevant to cases of higher attractor
dimension. We present a numerical example and report an improvement
of up to 4 orders of magnitude in the switching time over the case
with no targeting.
This paper appears in Physical Review E 51
#5, p.
4169-4172 (May 1995).
A PDF file of this
paper is
available.
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