171 research outputs found
Controlling Chimeras
Coupled phase oscillators model a variety of dynamical phenomena in nature
and technological applications. Non-local coupling gives rise to chimera states
which are characterized by a distinct part of phase-synchronized oscillators
while the remaining ones move incoherently. Here, we apply the idea of control
to chimera states: using gradient dynamics to exploit drift of a chimera, it
will attain any desired target position. Through control, chimera states become
functionally relevant; for example, the controlled position of localized
synchrony may encode information and perform computations. Since functional
aspects are crucial in (neuro-)biology and technology, the localized
synchronization of a chimera state becomes accessible to develop novel
applications. Based on gradient dynamics, our control strategy applies to any
suitable observable and can be generalized to arbitrary dimensions. Thus, the
applicability of chimera control goes beyond chimera states in non-locally
coupled systems
Basins of Attraction for Chimera States
Chimera states---curious symmetry-broken states in systems of identical
coupled oscillators---typically occur only for certain initial conditions. Here
we analyze their basins of attraction in a simple system comprised of two
populations. Using perturbative analysis and numerical simulation we evaluate
asymptotic states and associated destination maps, and demonstrate that basins
form a complex twisting structure in phase space. Understanding the basins'
precise nature may help in the development of control methods to switch between
chimera patterns, with possible technological and neural system applications.Comment: Please see Ancillary files for the 4 supplementary videos including
description (PDF
Intermittent chaotic chimeras for coupled rotators
Two symmetrically coupled populations of N oscillators with inertia
display chaotic solutions with broken symmetry similar to experimental
observations with mechanical pendula. In particular, we report the first
evidence of intermittent chaotic chimeras, where one population is synchronized
and the other jumps erratically between laminar and turbulent phases. These
states have finite life-times diverging as a power-law with N and m. Lyapunov
analyses reveal chaotic properties in quantitative agreement with theoretical
predictions for globally coupled dissipative systems.Comment: 6 pages, 5 figures SUbmitted to Physical Review E, as Rapid
Communicatio
Solvable Model of Spiral Wave Chimeras
Spiral waves are ubiquitous in two-dimensional systems of chemical or
biological oscillators coupled locally by diffusion. At the center of such
spirals is a phase singularity, a topological defect where the oscillator
amplitude drops to zero. But if the coupling is nonlocal, a new kind of spiral
can occur, with a circular core consisting of desynchronized oscillators
running at full amplitude. Here we provide the first analytical description of
such a spiral wave chimera, and use perturbation theory to calculate its
rotation speed and the size of its incoherent core.Comment: 4 pages, 4 figures; added reference, figure, further numerical test
Directed Flow of Information in Chimera States
We investigated interactions within chimera states in a phase oscillator
network with two coupled subpopulations. To quantify interactions within and
between these subpopulations, we estimated the corresponding (delayed) mutual
information that -- in general -- quantifies the capacity or the maximum rate
at which information can be transferred to recover a sender's information at
the receiver with a vanishingly low error probability. After verifying their
equivalence with estimates based on the continuous phase data, we determined
the mutual information using the time points at which the individual phases
passed through their respective Poincar\'{e} sections. This stroboscopic view
on the dynamics may resemble, e.g., neural spike times, that are common
observables in the study of neuronal information transfer. This discretization
also increased processing speed significantly, rendering it particularly
suitable for a fine-grained analysis of the effects of experimental and model
parameters. In our model, the delayed mutual information within each
subpopulation peaked at zero delay, whereas between the subpopulations it was
always maximal at non-zero delay, irrespective of parameter choices. We
observed that the delayed mutual information of the desynchronized
subpopulation preceded the synchronized subpopulation. Put differently, the
oscillators of the desynchronized subpopulation were 'driving' the ones in the
synchronized subpopulation. These findings were also observed when estimating
mutual information of the full phase trajectories. We can thus conclude that
the delayed mutual information of discrete time points allows for inferring a
functional directed flow of information between subpopulations of coupled phase
oscillators
Bistable Chimera Attractors on a Triangular Network of Oscillator Populations
We study a triangular network of three populations of coupled phase
oscillators with identical frequencies. The populations interact nonlocally, in
the sense that all oscillators are coupled to one another, but more weakly to
those in neighboring populations than to those in their own population. This
triangular network is the simplest discretization of a continuous ring of
oscillators. Yet it displays an unexpectedly different behavior: in contrast to
the lone stable chimera observed in continuous rings of oscillators, we find
that this system exhibits \emph{two coexisting stable chimeras}. Both chimeras
are, as usual, born through a saddle node bifurcation. As the coupling becomes
increasingly local in nature they lose stability through a Hopf bifurcation,
giving rise to breathing chimeras, which in turn get destroyed through a
homoclinic bifurcation. Remarkably, one of the chimeras reemerges by a reversal
of this scenario as we further increase the locality of the coupling, until it
is annihilated through another saddle node bifurcation.Comment: 12 pages, 5 figure
Integrability of a globally coupled complex Riccati array: quadratic integrate-and-fire neurons, phase oscillators and all in between
We present an exact dimensionality reduction for dynamics of an arbitrary
array of globally coupled complex-valued Riccati equations. It generalizes the
Watanabe-Strogatz theory [Phys. Rev. Lett. 70, 2391 (1993)] for sinusoidally
coupled phase oscillators and seamlessly includes quadratic integrate-and-fire
neurons represented by the special case of real-valued Riccati equations. It
provides a low dimensional description to a wide new class of complex dynamical
systems, warranting their rigorous analysis and thus providing deep insights
into their collective dynamics. This result represents a significant
advancement in our comprehending of coupled oscillatory systems and opens up
many new avenues of research.Comment: 6 pages, 4 figure
Birth and destruction of collective oscillations in a network of two populations of coupled type 1 neurons
We study the macroscopic dynamics of large networks of excitable type 1
neurons composed of two populations interacting with disparate but symmetric
intra- and inter-population coupling strengths. This nonuniform coupling scheme
facilitates symmetric equilibria, where both populations display identical
firing activity, characterized by either quiescent or spiking behavior, or
asymmetric equilibria, where the firing activity of one population exhibits
quiescent but the other exhibits spiking behavior. Oscillations in the firing
rate are possible if neurons emit pulses with non-zero width but are otherwise
quenched. Here, we explore how collective oscillations emerge for two
statistically identical neuron populations in the limit of an infinite number
of neurons. A detailed analysis reveals how collective oscillations are born
and destroyed in various bifurcation scenarios and how they are organized
around higher codimension bifurcation points. Since both symmetric and
asymmetric equilibria display bistable behavior, a large configuration space
with steady and oscillatory behavior is available. Switching between
configurations of neural activity is relevant in functional processes such as
working memory and the onset of collective oscillations in motor control
The genotype-phenotype relationship in multicellular pattern-generating models - the neglected role of pattern descriptors
Background: A deep understanding of what causes the phenotypic variation arising from biological patterning
processes, cannot be claimed before we are able to recreate this variation by mathematical models capable of
generating genotype-phenotype maps in a causally cohesive way. However, the concept of pattern in a
multicellular context implies that what matters is not the state of every single cell, but certain emergent qualities
of the total cell aggregate. Thus, in order to set up a genotype-phenotype map in such a spatiotemporal pattern
setting one is actually forced to establish new pattern descriptors and derive their relations to parameters of the
original model. A pattern descriptor is a variable that describes and quantifies a certain qualitative feature of the
pattern, for example the degree to which certain macroscopic structures are present. There is today no general
procedure for how to relate a set of patterns and their characteristic features to the functional relationships,
parameter values and initial values of an original pattern-generating model. Here we present a new, generic
approach for explorative analysis of complex patterning models which focuses on the essential pattern features
and their relations to the model parameters. The approach is illustrated on an existing model for Delta-Notch
lateral inhibition over a two-dimensional lattice.
Results: By combining computer simulations according to a succession of statistical experimental designs,
computer graphics, automatic image analysis, human sensory descriptive analysis and multivariate data modelling,
we derive a pattern descriptor model of those macroscopic, emergent aspects of the patterns that we consider
of interest. The pattern descriptor model relates the values of the new, dedicated pattern descriptors to the
parameter values of the original model, for example by predicting the parameter values leading to particular
patterns, and provides insights that would have been hard to obtain by traditional methods.
Conclusion: The results suggest that our approach may qualify as a general procedure for how to discover and
relate relevant features and characteristics of emergent patterns to the functional relationships, parameter values
and initial values of an underlying pattern-generating mathematical model
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