Abstracts of talks

A chimera state in a coupled oscillator system is a dynamical state that combines regions of coherence
(or synchrony) with regions if incoherence (or asynchrony). However exactly how one defines these terms affects which
states can be identified as chimeras and which not, especially for small groups of phase oscillators. In this talk I
will discuss some joint work with O. Burylko (Kiev) where we propose a definition of a weak chimera based on partial
frequency synchrony. This allows one to explore the existence and stability of weak chimeras in small networks of
phase oscillators. In particular we find that the usual coupling considered (of Kuramoto-Sakaguchi type) for investigating
chimeras leads to degenerate sets of neutrally stable chimeras, while more general coupling unfolds this degeneracy.
I will also discuss work on chaotic weak chimera states from recent work with C. Bick (Exeter).

Under certain conditions, the collective behavior of a large globally-coupled heterogeneous network of
coupled oscillators, as quantified by the macroscopic mean field or order parameter, can exhibit low-dimensional chaotic
behavior. Recent advances describe how a small set of reduced ordinary differential equations can be derived that
captures this mean field behavior. Here, we show that chaos control algorithms designed using the reduced equations
can be successfully applied to imperfect realizations of the full network. To systematically study the effectiveness
of this technique, we measure the quality of control as we relax conditions that are required for the strict accuracy
of the reduced equations, and hence, the controller. Although the effects are network-dependent, we show that the
method is effective for surprisingly small networks, for modest departures from global coupling, and even with mild
inaccuracy in the estimate of network heterogeneity.

Phase oscillators are interacting oscillatory units whose state is solely determined by a phase variable
taking values on the circle. Probably the most widely studied system of interacting phase oscillators
is the Kuramoto model; here, the interaction between two oscillators is given by the sine of the phase difference.
What happens if the interaction between pairs of oscillators is more general, for example if
the interaction contains more than a single Fourier component? We discuss the impact of generalized coupling on phase
oscillator dynamics. Moreover, we show how generalized couplings can be useful in applications; for example, it allows
to control spatially localized states in non-locally coupled oscillator systems.

Eukaryotic cilia and flagella are biological chemo-mechanical oscillators capable of generating a variety
of large scale coordinated motions, commonly referred to as Metachronal Waves. Despite recent progress in our
understanding of the basic mechanisms responsible for phase locking of isolated pairs of cilia/flagella, it has
become clear that pair synchronization is a necessary but not sufficient requirement for coordination of a group,
chiefly due to the effect of long range interactions. Here we explore experimentally and with simulations the
behaviour of a group of hydrodynamically coupled oscillators rotating above a no-slip plane, intended as a minimal
model for a ciliated surface. We show that, as the oscillators' distance from the wall increases, their global
synchronization state undergoes a transition from a symplectic Metachronal Wave to a chevron-like profile, and
discuss the origin of this behaviour. The transition between these two regimes is not sharp, but encompasses a
range of distances from the wall within which the system displays mixed behaviour reminiscent of chimera states. 

A method is presented that allows to compute the inhomogeneous phase space distribution in the
nonequilibrium stationary state of a wide class of
mean-field systems involving rotators subject to quenched disordered external
drive and dissipation. The method involves a series expansion of the 
stationary distribution in inverse of the damping coefficient; the expansion
coefficients satisfy recursion relations whose solution requires computing
a sparse matrix, making numerical evaluation simple and efficient. We illustrate our method for the paradigmatic Kuramoto model of spontaneous
collective synchronization and for its two mode generalization, in presence
of noise and inertia, and demonstrate an excellent agreement between simulations and theory for the phase space distribution. 

We revisit the Kuramoto model to explore the finite-size scaling (FSS) of the order parameter and
its dynamic fluctuations near the onset of the synchronization transition, paying particular attention to effects
induced by the randomness of the intrinsic frequencies of oscillators. For a population of size N, we study
two ways of sampling the intrinsic frequencies according to the same given unimodal distribution g(ω).
In the ‘random’ case, frequencies are generated independently in accordance with g(ω), which gives rise to
oscillator number fluctuation within any given frequency interval. In the ‘regular’ case, the N frequencies
are generated in a deterministic manner that minimizes the oscillator number fluctuations, leading to quasi-uniformly
spaced frequencies in the population. We find that the two samplings yield substantially different finite-size properties
with clearly distinct scaling exponents. Moreover, the hyperscaling relation between the order parameter and its
fluctuations is valid in the regular case, but is violated in the random case. In this last case, a self-consistent
mean-field theory that completely ignores dynamic fluctuations correctly predicts the FSS exponent of the order
parameter but not its critical amplitude.

The Kuramoto model is a system of ordinary differential equations for describing
synchronization phenomena defined as a coupled phase oscillators. 
In this talk, an infinite dimensional Kuramoto model is considered, and 
Kuramoto's conjecture on a bifurcation diagram of the system will be proved.
A linear operator obtained from the infinite dimensional Kuramoto model has 
the continuous spectrum on the  imaginary axis, so that the usual spectrum 
does not determine the dynamics. To handle such continuous spectra, a new spectral theory of linear operators  based on Gelfand triplets is developed.
In particular, a generalized eigenvalue (resonance) is defined. It is proved that 
a generalized eigenvalue determines the stability and bifurcation of the system.

In Kuramoto model, the natural frequency distribution gω of oscillators plays an important
role in determination the phase transition’s types, properties and its universality class. It is known that Kuramoto
model with unimodal and symmetric natural frequency distribution (such as Gaussian or Cauchy distribution) exhibits
a second-order phase transition with critical exponent β= 1/2 whereas uniform distribution or bimodal and symmetric
distribution make it a first-order phase transition. Here we generalize the unimodal distribution and show that this
generalization changes the critical exponent β. Therefore, we need to use the term ‘unimodality’ more precisely.
As a result, the finite-size exponent ‾{ν} and dynamic exponent z are also changed to other values than
the known values. We derive those exponents analytically and confirm them numerically.

Although the Kuramoto model has been around for four decays now, there has been relatively little work
demonstrating well controlled experimental realizations of this model. Here we experimentally investigate the phase
relations of a large network of coupled lasers in a degenerate cavity [1-3]. The laser network obeys the Kuramoto
model under the assumption of equal intensities of the lasers, which is a good approximation for coupling strengths
that are not too close to the critical coupling. By controlling the ratio between coupling strength and quenched
disorder, we examine synchronization for short range and long range coupling [4]. Furthermore, we compare the effect
of different topological structures on synchronization of the entire network array [5], and observe large scale
geometric frustration for a Kagome lattice network [6]. Our experimental configuration also allows us to introduce
time delayed coupling between lasers. We show that such networks form at most two synchronized clusters, and we
establish fundamental rules which govern the synchronization state of networks with homogeneous [7] and heterogeneous [8] time delayed coupling. 

[1] M. Nixon et al. Real-time wavefront shaping through scattering media by all-optical feedback , Nat. Photon. 7, 919 (2013).
[2] M. Nixon et al. Efficient method for controlling the spatial coherence of a laser, Opt. Lett. 38, 3858 (2013).
[3] R. Chriki et al. Manipulating the spatial coherence of a laser source, submitted (2015).
[4] E. Ronen, Demonstration of statistical mechanics phase transitions with arrays of thousands of coherent lasers, OSA conference on lasers and electro-optics (CLEO), QWF7 (2011). 
[5] R. Chriki et al. small world networks of coupled lasers, unpublished (2015).
[6] M. Nixon et al. Observing geometric frustration with thousands of coupled lasers, Phys. Rev. Lett. 110, 184102 (2013).
[7] M. Nixon et al. Synchronized cluster formation in coupled laser networks, Phys. Rev. Lett. 106, 223901 (2011).
[8] M. Nixon et al. Controlling synchronization in large laser networks, Phys. Rev. Lett. 108, 214101 (2012). 

The interplay between structural and functional brain networks has become a popular research topic in the
past years. Topologies of structural and functional networks may disagree. Several modeling studies tried to address this
issue more systematically as it can give insight into the way structural connectivity facilitates but also constrains
information exchange. For instance, in a combined neural mass and graph theoretical model of the electroencephalogram,
it was found that patterns of functional connectivity are influenced by, but not identical to those of the corresponding
structural level [3]. Functional connectivity is often been defined through the synchronization between activities at
different nodes. These activities are considered to stem from meso-scale neural populations that oscillate at certain
frequencies with certain amplitudes. The amplitude of a neural population reflects the degree of synchronization of its members. 
To show how amplitude affects the phase dynamics in neural networks we approximate the dynamics of the node by defining
neural populations as self-sustaining, weakly non-linear oscillators (either Wilson-Cowan or Freeman models). This description
allows for deducing the corresponding phase dynamics proper [2,4]. Dependent on the type of oscillator, the phase dynamics is
influenced by the amplitudes of the individual oscillators, which can be shown analytically. This will be illustrated using a
network of neural oscillators j = 1, 2,..., N. Each oscillator is characterized by its amplitude Aj and phase φj.
The latter has a dynamics of a Kuramoto-like network [1]. Given a certain (structural) connectivity between the oscillators
denoted by Cij we discuss in detail how the connectivity between phases, Dij explicitly depends on the oscillators
amplitudes, i.e., Dij = Dij(A1, A2,..., AN). It will be shown that phase dynamics and, hence, synchrony
patterns should always be analyzed in conjunction with the corresponding amplitude changes [2,4]. This may have profound
impact when linking neuro-anatomical and -imaging studies to modeling. 

[1] Breakspear M, Heitmann S, Daffertshofer A (2010) Generative models of cortical oscillations: From Kuramoto to the nonlinear Fokker–Planck equation, Frontiers in Human Neuroscience, doi:10.3389/fnhum.2010.00190.
[2] Daffertshofer A, van Wijk, BCM (2011). On the influence of amplitude on the connectivity between phases. Frontiers in Neuroinformatics, 5, art. 6. doi: 10.3389/fninf.2011.00006.
[3] Ponten SC, Daffertshofer A, Hillebrand A, Stam, CJ (2010) The relationship between structural and functional connectivity: Graph theoretical analysis of an EEG neural mass model. NeuroImage, 52(3):985-994.
[4] Ton R, Deco G, Daffertshofer A (2014). Structure-function discrepancy: inhomogeneity and delays in synchronized neural networks. PLoS Computational Biology, 10(7): e1003736.

Dynamic activity exhibited by large populations of coupled oscillators plays an important role in many fields
of science and technology. A crucial issue is to examine its robustness against the deterioration of element oscillators
caused by aging, accidents, and so on. This presentation will address theoretical studies of this issue, focusing on such
a case that the deterioration makes "active" (self-oscillatory) elements turn "inactive" (not self-oscillatory). There are
two typical scenarios for this qualitative change: One is a Hopf bifurcation and the other an SNIC (saddle-node bifurcation
on an invariant circle). This talk begins with a brief overview of the studies done so far for the former scenario and then
treat the latter. In both scenarios, the crucial phenomenon is the aging transition, which means a transition of the whole
system from the dynamic phase to the static occurring as the ratio of inactive oscillators exceeds a critical value. Such a
transition should lead to a fatal situation, whether the system is biological or technological. This is why it is crucial.
Other phenomena may also be touched upon, which are induced by "aging" defined as the increase of inactive elements.

We observe chimeralike states in networks of dynamical systems using a type of global coupling consisting of
two components: attractive and repulsive mean-field feedback. We identify existence of two types of chimeralike states
in a  Liénard system; in one type, both the coherent and the noncoherent populations are in chaotic states and,
in the other type, the noncoherent population is in a periodic state while the coherent population is in periodic or
chaotic and even be quasiperiodic. We locate the coupling parameter regimes of the two types of chimerlike states in a
phase diagram. We study other bistable systems, a forced van der Pol-Duffing system and the Josephson junction model to
investigate generality of the coupling configuration in creating chimeralike states. We find chaos-chaos chimeralike states
in the network of bistable van der Pol-Duffing system,  period-period chimeralike states in the network of Josephson junction
model in the bistable regime. Furthermore, we apply the coupling to a network of chaotic Rössler system where we find the
chaos-chaos chimeralike states.

We study the mean-field limit of the Kuramoto model of globally coupled oscillators. By studying the evolution
in Fourier space and understanding the domain of dependence, we show a global stability result. Moreover, we can identify
function norms to show damping of the order parameter for velocity distributions and perturbations in Wn,1 for n>1.
Finally, for sufficiently regular velocity distributions we can identify exponential decay in the stable case and otherwise identify
finitely many eigenmodes. For these eigenmodes we can show a center-unstable manifold reduction, which gives a rigorous tool to
obtain the bifurcation behaviour. The damping is similar to Landau damping for the Vlasov equation. 

The Kuramoto model is the archetype of heterogeneous systems of (globally) coupled oscillators with dissipative
dynamics. In this model, the order parameter that quantifies the population synchrony decays to 0 in time, as long as the
interaction strength remains small (so that the uniformly distributed stationary solution remains stable).
While this phenomenon has been identified since the first studies of the model, its proof remained to be provided (most studies
in the literature are limited to the linearized dynamics).
The goal of this talk is to present rigorous results on the nonlinear dynamics of the Kuramoto model, and in particular, a proof
of damping of the order parameter in the weak coupling regime. Time permitting, I'll talk about proofs which closely follow recent
approaches to Landau damping in the Vlasov equation and in the Vlasov-HMF model.

Joint work with D. Gérard-Varet and G. Giacomin

Daily (circadian) rhythms are generated in mammals by &tiled;10,000 coupled oscillators in a region of the brain called the
suprachiasmatic nucleus (SCN). I will describe how we generated a detailed and accurate model of the SCN. Special techniques used
include GPU computing, the Ott-Antonsen ansatz and an efficient computational method based on a population density approach.
This model leads to predictions verified by many experimental labs leading to the discovery, for example, of unique forms of
communication among the oscillators in the SCN or how both noise and coupling may be needed to generate rhythms. I will also
briefly describe our efforts to simulate models of circadian timekeeping on smartphones with an app ENTRAIN. This app has been
used by over 100,000 individuals in over 100 counties to minimize jet lag.

We investigate the universality class in synchronization using the extended finite-size scaling (FSS) form, where we
propose a systematic analysis of dynamic scaling in the time evolution of the phase order parameter for coupled oscillators with
non-identical natural frequencies in terms of the Kuramoto model. In particular, we focus on a comprehensive view of phase
synchronization for various setups in the context of dynamic scaling, which determines critical exponents and finds the critical
coupling strength. Finally, we discuss how the sampling of natural frequencies and thermal noise affect dynamic scaling, which is numerically confirmed. 

O. D`Huys, T. Jüngling, and W. Kinzel Phys. Rev. E 90, 032918 (2014): 

Networks with coupling delays play an important role in various systems, such as coupled semiconductor lasers, traffic dynamics,
communication networks, genetic transcription circuits or the brain. A well established effect of a delay is to induce multistability:
In oscillatory systems a delay gives rises to coexistent periodic orbits with different frequencies [1] and oscillation patterns.
Such coexistent patterns could be related to memory storage, especially in neural networks [2]. Adding noise to the dynamics,
the network switches stochastically between these orbits. Following the approach of M/oslash;rk et al. [3], we construct a potential
for phase oscillators. This allows us to analytically compute the distribution of frequencies visited by the network and the
corresponding residence times. We find some surprising stochastic effects: While without noise, the range of locking frequencies
scales with the coupling strength, in the stochastic system it scales inversely with the squareroot of roundtrip delay, and does
not depend on the coupling strength. In contrast, the robustness of the orbits, measured by the average residence time, is mainly
determined by the coupling strength, while the effect of the delay is limited. The role of the network topology is shown to play
am important role: while in unidirectional rings the oscillators spend equally much time in all the possible phase configurations
allowed by the topology, a globally coupled network shows a clear tendency to in-phase synchrony. Finally, we demonstrate the
generality of our results with delay-coupled FitzHugh- Nagumo oscillators. 

[1] S. Yanchuk and P. Perlikowski, Phys. Rev. E 79, 046221 (2009).
[2] J. Foss ,F. Moss and J. Milton, Phys. Rev. E 55, 4536 (1997).
[3] J. Mørk, B. Semkow and B. Tromborg, Electronics Letters 26, 609 (1990).

Electrochemical reactions can exhibit oscillations whose dynamics depend on many variables that include electrode
potential and chemical concentrations. When the reactions take place on electrode arrays and the interactions between the reaction
units are weak, the features of the dynamics are still complex but can be interpreted with a single variable (the phase of the oscillations)
within the framework of the Kuramoto model. 
In the contribution we review how experiment-based phase models can be used for description and design of complex synchronization structures.
The results are organized around two central concepts: (i) construction of slightly heterogeneous networks of chemical oscillations,
in which synchronization structures are recorded due to weak, inherent or imposed coupling between the elements, and (ii) extracting
Kuramoto model parameters from direct experiments with chemical reactions. The performance of traditional kinetic modeling is compared
to that of the experiment-based Kuramoto modeling technique; the advantages of each approach are outlined.  

References: 
1.	Kiss IZ, Zhai Y, Hudson JL (2002) Emerging coherence in a population of chemical oscillators. Science 296:1676–1678.
2.	Kiss IZ, Rusin CG, Kori H, Hudson JL (2007) Engineering complex dynamical structures: sequential patterns and desynchronization. Science 316:1886–1889.

Chimera states consisting of domains of coherently and incoherently oscillating nonlocally-coupled phase oscillators on a
ring are studied. Systems of both identical and heterogeneous oscillators are considered. In the former several classes of chimera states
have been found: (a) stationary multi-cluster states with evenly or unevenly distributed coherent clusters, and (b) traveling chimera states.
Single coherent clusters traveling with a constant speed across the system are also found. In the presence of spatial heterogeneity in the
oscillator frequencies these traveling states undergo a variety of pinning and depinning transitions. In this talk I will describe these results,
provide a self-consistent continuum description of many of these states, and will use this description to study transitions between them.

J. Xie, E. Knobloch and H.-C. Kao. Multicluster and traveling chimera 
states in nonlocal phase-coupled oscillators. Phys. Rev. E 90, 022919 (2014)

Synchronization often plays a vital role in chemical and biological
systems. The theoretical frameworks of synchronization are necessary
for the understanding and control of synchronization dynamics. In
particular, they have been well developed in two parameter regimes.
One is in the vicinity of a Hopf bifurcation, in which complex
amplitude equations such as Ginzburg-Landau and Stuart-Landau
equations are available. The other one is weak coupling among
oscillators, in which phase approximation is applicable. Although
those methods are very powerful, they can not treat all the rich
dynamics that appear in such and other regimes. One example is
clustering behavior, which has been observed near a Hopf bifurcation
in an electrochemical system [1]. Another example is a reentrant
synchronization transition that occurs in an noisy oscillator strongly
coupled to a pacemaker [2]. Conventional theories fail to predict
these examples. In this presentation, I will present recent progresses
on the development of theoretical frameworks. I will also discuss the
mechanism of jet lag [3] in light of such frameworks.


1. H. Kori, Y. Kuramoto, S. Jain, I.Z. Kiss, J. Hudson: “Clustering in
Globally Coupled Oscillators Near a Hopf Bifurcation: Theory and
Experiments”, Phys. Rev. E 89, 062906 (2014).
2. Y. Kobayashi, H. Kori: “Reentrant transition in coupled noisy
oscillators”, Phys. Rev. E 91, 012901 (2015).
3. Y. Yamaguchi et al., “Mice Genetically Deficient in Vasopressin V1a
and V1b Receptors Are Resistant to Jet Lag”, Science 342, 85 (2013).

Neural field models are nonlocal PDEs used to describe macroscopic dynamics of the cortex. They are normally derived
under the assumption that connections between 
neurons are synaptic rather than via gap junctions. I will show how to derive
a new type of neural field model from a network of quadratic integrate and fire neurons with both synaptic and gap junction connectivity.

Chimera state is a remarkable pattern in networks of nonlocally coupled oscillators, that displays a self-organized
behavior of spatially separated domains of coherence and incoherence.  It was found in 2002 by Kuramoto and Battogtock
[1] as a spontaneous symmetry breaking for one-dimensional complex Ginzburg-Landau equation and for its phase approximation,
the Kuramoto model.  Currently, this is an area of intense theoretical and experimental research, for now, chimeras have been
found in many systems from various fields.
In this talk, a diversity of chimera states for three-dimensional Kuramoto model is reported. Systematic analysis of the spatiotemporal
dynamics on the range and strength of coupling shows that there are two principal classes of the three-dimensional chimera patterns:
oscillating and spirally rotating. Characteristic examples from the first class include incoherent as well as coherent balls, tubes,
crosses, and layers in coherent or incoherent surrounding; the second class includes scroll waves with incoherent rolls of different
modality and dynamics. Numerical simulations started from random initial conditions indicate that the states are stable over the integration time.
The presentation is illustrated by 3-Dim videos of the complex spatiotemporal chimera dynamics. 

[1] Kuramoto, Y. & Battogtokh, D. [2002] "Coexistence of coherence and incoherence in nonlocally coupled phase oscillators," Nonlinear Phenom. Complex Syst. 5, 380-385.

 

We consider classical nonlinear oscillators like rotators and Kuramoto oscillators on hexagonal lattices of
small or intermediate size. When the coupling between these elements is repulsive and the bonds are frustrated, we observe
coexisting states, each one with its own basin of attraction. For special lattices sizes the multiplicity of stationary states
gets extremely rich. When disorder is introduced into the system by additive or multiplicative Gaussian noise, we observe a
noise-driven migration of oscillator phases in a rather rough potential landscape. Upon this migration, a multitude of different
escape times from one metastable state to the next is generated [1]. Based on these observations, it does not come as a surprise
that the set of oscillators shows physical aging. Physical aging is characterized by non-exponential relaxation after a perturbation,
breaking of time-translation invariance, and dynamical scaling. When our system of oscillators is quenched from the regime of a
unique fixed point toward the regime of multistable limit-cycle solutions, the autocorrelation functions depend on the waiting
time after the quench, so that time translation invariance is broken, and dynamical scaling is observed for a certain range of
time scales [2]. We point to open questions concerning the relation between physical and biological aging.

[1] Ionita, D.Labavic, M.Zaks, and H.Meyer-Ortmanns, Eur. Phys.J.B 86(12), 511 (2013).
[2] F.Ionita, H.Meyer-Ortmanns, Phys.Rev.Lett.112, 094101 (2014).

Systems of identical sinusoidally coupled phase oscillators have a natural action of the three-dimensional
group consisting of the Mobius transformations which preserve the unit disc.  The dynamics of these systems are constrained
to lie in the (at most) three-dimensional group orbits under this action. We will discuss how this group-theoretic reduction
explains many of the dynamic phenomena particular to Kuramoto oscillator networks, such as the Strogatz-Watanabe constants
of motion, the neutral stability of splay states in Josephson junction networks, the special role of Poisson densities in
the Ott-Antonsen ansatz and the neutral stability of partially locked states in the infinite-N Kuramoto model.
We will also present a complete classification of attractors for systems of coupled identical Kuramoto oscillators.

Some generalizations of the phase reduction theory to collective rhythmic dynamics in reaction-diffusion systems
and networks of coupled dynamical elements are considered. The adjoint equation for the ordinary limit-cycle oscillators can
naturally be generalized to collective rhythmic dynamics in such high-dimensional systems and yields their phase sensitivity
functions to weak perturbations. Using the generalized theory, we analyze synchronization dynamics of rhythmic spatiotemporal
patterns and collective network oscillations. 

Chimera states are self-organized patterns of coherence and incoherence
observed in systems of nonlocally coupled oscillators.
Because of different types of macroscopic instabilities and sudden collapse for small system sizes, their appearance is very
restricted both in the phase and parameter space.
In this talk we will describe some control techniques which permit us to circumvent these limitations and observe new types of
chimera states that are inaccessible in conventional simulations or experiments.
This is joint work with M. Wolfrum and J. Sieber.

It has previously been found that large mean field coupled systems of identical Landau-Stuart oscillators can have
clumped state attractors (in which each oscillator is in one of a small number of groups for which all oscillators in that group
have the same state), as well as attractors for which all oscillators have distinct states. In the latter case the macroscopic mean
field dynamics is chaotic, and we show that the dynamics is "extensive" in the sence that the attractor dimension and number of
positive Lyapunov exponents scales linearly with the number of oscillators. In addition, we study the dynamical character of
transitions between low-dimensional clumped states and extensively chaotic states with parameter variations. We find that in
the clumped state, as a system parameter is continuously varied toward a transition, the oscillator population distribution between
clumps continuously evolves to maintain marginal stability. This behavior is used to explain the observed explosive transition
to extensive chaos. It is argued that results similar to those above for our particular system should also occur in other mean
field coupled systems. Beyond that, we discuss the possibility that our basic observation of transitions between extensive chaos
and low dimensional (nonextensive) dynamics may be relevant to such questions as the onset of turbulence in fluids.

*Wai Lim Ku, Michelle Girvan and Edward Ott, arXiv:1412.3803 

In 1967, Arthur Winfree proposed a mathematical model consisting on an ensemble of pulse-coupled oscillators, which
successfully replicated collective synchronization. The model, in contrast to the simpler Kuramoto model, has remained largely unexplored.
I review on recent progress in the mathematical description of the Winfree model that permits to uncover the synchronizing effect
of narrow pulses, among other effects.

Moreover, I show how the 'Ott-Antonsen ansatz', used to analyze the Winfree model, can be transformed into a 'Lorentzian ansatz' permitting
to investigate ensembles of pulse-coupled quadratic integrate-and-fire neurons. With this approach a low-dimensional firing-rate model is
rigorously derived, in contrast to the heuristic firing-rate models customarily used in neuroscience. 

Modeling of large-scale brain dynamics in time and space begins with data about the brain structure and the connectivity
between brain areas, which are then mathematically described by attractors. Experimental data suggests that the lengths of connection
routes between brain areas follow a multimodal distribution.
We analyse synchronization in three conceptual networks of phase oscillators structured by time-delays: (i) randomly and (ii) in clusters.
In the latter case, functional clusters of (i) phase-shifted; (ii) anti-phase; or (iii) oscillators with non-stationary order parameters occur. 
Synchronization patterns like traveling or standing waves, and multistability,  can be controlled by varying the spatial
distribution of the time delays in the network. Stability boundaries are obtained analytically and all results are confirmed
numerically. The findings are discussed with respect to modeling large-scale brain dynamics.

In this talk I present a technique to study synchronization in large networks of coupled oscillators, combining
a mean field approximation with the dimension reduction method of Ott and Antonsen. The formulation is illustrated on a network
Kuramoto problem in which oscillators with similar frequencies are more likely to connect to each other (frequency assortativity).
We find that frequency assortativity can produce chaos in the macroscopic synchronization dynamics. Using the mean field approach,
we show that the network system can be described by a small number of ordinary differential equations. Using this simpler system,
we characterize the chaotic dynamics using Lyapunov exponents, bifurcation diagrams, and time-delay embeddings. We find that the
chaotic dynamics result from the interaction of various groups of synchronized oscillators, or ‘meta-oscillators’.

We report the emergence of chimera like states in an assembly of identical nonlinear oscillators through all-to-all
global mean field interaction through cross planer linking scheme. The rotational symmetry breaking of the coupling term seems to
be responsible for exhibiting such a collective behavior displaying characteristic division into coexisting coherent and incoherent
populations. This spontaneous splitting of the oscillators into two subpopulations of coherent and incoherent oscillators was first
discovered by Kuramoto and Battogtokh for a system of phase oscillators that was later named named a chimera state.
We also examined another network of bistable lienard oscillators with all-to-all coupling a kind of linking through attractive and
repulsive feedback.  In this case also we are successful to identify the region of parameter space where chimera like states appears.

I put forward a general framework in which I will discuss in a unified way known results with more recent developments
obtained for a generalized Kuramoto model that includes inertial effects and noise. I describe the model from a different perspective,
emphasizing the equilibrium and out-of-equilibrium aspects of its dynamics from a statistical physics point of view: i) I discuss the
phase diagram for unimodal frequency distributions; ii) I analyze the dynamics on a lattice where the coupling decays algebraically
with separation between lattice sites, and discuss for specific cases how the long-time transition to synchrony is essentially
governed by the dynamics of the mean-field mode.

Chimera states, representing a spontaneous breakup of a population of identical oscillators that are
identically coupled, into subpopulations displaying synchronized and desynchronized behavior, have
traditionally been found to exist in weakly coupled systems and with some form of nonlocal coupling
between the oscillators. Here we show that neither the weak-coupling approximation nor nonlocal coupling
are essential conditions for their existence. We obtain, for the first time, amplitude-mediated chimera states in a system of
globally coupled complex Ginzburg-Landau oscillators. We delineate the dynamical origins for the formation of such states from
a bifurcation analysis of a reduced model equation and also discuss the practical implications of our discovery of this broader
class of chimera states.

We identify and describe the key qualitative rhythmic states in various 3-cell network motifs of a multifunctional
central pattern generator (CPG). Such CPGs are neural microcircuits of cells whose synergetic interactions produce multiple
states with distinct phase-locked patterns of bursting activity. To study biologically plausible CPG models, we develop a
suite of computational tools that reduce the problem of stability and existence of rhythmic patterns in networks to the
bifurcation analysis of fixed points and invariant curves of a Poincaré return maps for phase lags between cells. We explore
different functional possibilities for motifs involving symmetry breaking and heterogeneity. This is achieved by varying coupling
properties of the synapses between the cells and studying the qualitative changes in the structure of the corresponding return maps.
Our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of rhythmic
patterns generated by various CPGs in the context of motor control such as gait-switching in locomotion. Our approach is
applicable to a wide range of biological phenomena beyond motor control.

We have studied heterogeneous populations of chemical oscillators to characterize different types of synchronization
behavior. The formation of phase clusters in stirred suspensions of Belousov-Zhabotinsky oscillators is described, where
the (global) coupling occurs through the medium. We then describe the formation of phase clusters and chimera states in
populations of photosensitive oscillators. The nonlocal coupling occurs via illumination intensity that is dependent on
the state of each oscillator. Finally, we describe studies of phase-lag synchronization in networks of photochemically
coupled oscillators, where the influence of permutation symmetries is explored. 
References:
A. F. Taylor et al., Angewandte Chemie Int. Ed. 50, 10161 (2011); M. R. Tinsley et al., Nature Physics 8, 662 (2012); S. Nkomo et al., Phys. Rev. Lett. 110, 244102 (2013).

The control of complex systems and network-coupled dynamical processes is an important topic of research at
the intersection of mathematics, physics, engineering, biology, chemistry, and the social sciences. The control of coupled
oscillator networks is particularly important and has applications ranging from the development to non-invasive treatments
for terminating arrhythmic cardiac behavior to the design of effective and efficient rescue methods for the power grid.
In this talk I will consider the control of networks of Kuramoto oscillators with the goal of attaining a fully synchronized
state, a.k.a. consensus, by applying control to as few oscillators as possible. 

Our method is based on identifying a target synchronized state and inspecting its Jacobian matrix. By studying its spectrum,
we identify precisely the oscillators that require control and how strong a control gain is required to attain consensus.
Denoting oscillators the require and do not require control as driver nodes and free nodes, respectively, we find that driver
nodes tend to balance a large ratio (in absolute value) of natural frequencies to nodal degree. Finally, we compare the amount of
control required (i.e., the fraction of driver nodes in a given network) in different network models and find that, surprisingly,
the fraction of driver nodes in a network depends little on the network topology or on the degree distribution, but rather on the
mean degree of the network along with the coupling strength and the natural frequency distribution.

Per Sebastian Skardal and Alex Arenas, "Control of Coupled Oscillator Networks." Submitted, arxiv:1501.00612 (2015).

The Theta neuron is the normal form for all neurons exhibiting type-I excitability and a large densely coupled network
of them can be well described by a recent mean field reduction technique originally developed for the Kuramoto network.
The mean field dynamical equation can be fully analyzed to provide a comprehensive categorization of all possible asymptotic
collective states observable from the network. Using the Theta neuron network as our basic population unit and a good set of
tools for characterizing and manipulating the dynamically invariant sets contained within the macroscopic mean field, I will
detail our initial efforts in building a “canonical” brain model as more neurophysiological details and hierarchical structure
are included in the theta neuron network.  We hope to expand our understanding of the basic repertoire of possible collective
behaviors of larger and larger neuronal assemblies in the brain and how information flows among them.

B. Sonnenschein1 2  and L. Schimansky-Geier1 2
1Institute of Physics, Humboldt University at Berlin, Newtonstr 15, D-12489 Berlin
²Bernsteincenter for Computational Neuroscience,  Philippstr 12, D-10115 Berlin
E-mail: sonne@physik.hu-berlin.de, alsg@physik.hu-berlin.de
   We study the synchronization of stochastic dynamical (oscillatory and excitable) elements at nodes of complex networks.
   The various elements at the nodes  differ by parameters as frequencies and excitability specified by probability distributions
   and correlations between them.  The complexity of the network is defined via the degree distribution of its nodes. 
   At first, probability distribution densities for families of elements with the same degree will be introduced and  we derive a
   system of nonlinear Fokker-Planck equation for these pdf's. A Gaussian approximation yields ODE's for the first two moments of
   the dynamics of elements with the same degree. The onset of synchronization is discussed as bifurcations of ODE's for the moments
   and the order parameters of the complex networks. We discuss how  correlations between the node distribution  and other parameter
   distribution yield modifications of the bifurcation senario.  
   Finally, we discuss the validity of the Gaussian approach to synchronization on complex networks by the comparison of numerical
   simulations and analytical findings. Good agreement is obtained for homogeneous and dense random networks.  

References

[1]	B.Sonnenschein and L. Schimansky-Geier, Phys. Rev. E 85 051116 (2012).
[2]	B. Sonnenschein and L. Schimansky-Geier, Phys. Rev. E 88 052111 (2013).
[3]	B. Sonnenschein, F. Sagues, and L. Schimansky-Geier, Eur. Phys. Journ B. 86, 12 (2013).
[4]	M. A. Zaks,  A. B. Neiman, S. Feistel, and L. Schimansky-Geier, Phys. Rev. E 68, 066206 (2003). 
[5]	B. Sonnenschein, M.A. Zaks, A.B. Neiman, and L. Schimansky-Geier, Eur. Phys. J. Special Topics,  222,  2517–2529 (2013).
[6]	 B. Sonnenschein, T.K.D.M. Peron, A.J. Rodrigues, J. Kurths, ans L. Schimansky-Geier, EuroPhys.J B (/, 162 (2014).
[7]	B. Sonnenschein, T.K.D.M. Peron, A.J. Rodrigues, J. Kurths, ans L. Schimansky-Geier, DOI: 10.1103/PhysRevE00.002900.

We will discuss whether it is possible to identify the presence of a non-zero mean field and synchronization in a
network by observing only a single oscillator. 

We've learned a lot about the Kuramoto model over the past 40 years, but several nice problems remain unsolved.
I will discuss a few of these problems in the hope of stimulating fresh ideas about them. 

We study optimal synchronization in networks of heterogeneous phase oscillators. Our main result is the derivation of
a "synchrony alignment function" that encodes the interplay between network structure and oscillators' frequencies and can be readily
optimized. We highlight its utility in two general problems: constrained frequency allocation and network design. In general, we find
that synchronization is promoted by strong alignments between frequencies and the dominant Laplacian eigenvectors, as well as a matching
between the heterogeneity of frequencies and network structure.

The Kuramoto model constitutes a paradigmatic model for the dissipative collective dynamics of coupled oscillators,
characterizing in particular the emergence of synchrony (phase locking). Here we present a classical Hamiltonian (and thus conservative)
system with 2N state variables that in its action-angle representation exactly yields Kuramoto dynamics on N-dimensional invariant manifolds.
We show that locking of the phase of one oscillator on a Kuramoto manifold to the average phase emerges where the transverse Hamiltonian
action dynamics of that specific oscillator becomes unstable. Moreover, the inverse participation ratio of the Hamiltonian dynamics
perturbed off the manifold indicates the global synchronization transition point for finite N more precisely than the standard
Kuramoto order parameter. The uncovered Kuramoto dynamics in Hamiltonian systems thus distinctly links dissipative to conservative dynamics.

If time permits, direct links to Bose Einstein condensation both on the semiclassical and the quantum level are discussed.

Witthaut & Timme, Phys. Rev. E. 90:032917  (2014).
http://dx.doi.org/10.1103/PhysRevE.90.032917  

Two symmetrically coupled populations of N Kuramoto oscillators with
inertia m 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.

Non-autonomous dynamics of auto-oscillators under the action of weak perturbations is usually described by the
“phase model”, where the only dynamical variable is the phase describing the auto-oscillator position on a stable limit cycle
(or orbit) [1]. This description is based on the fact that the phase perturbations are marginally stable, while perturbations
in all other directions in the oscillator’s phase space decay in time. We demonstrate that the accuracy of this approach may
be insufficient for a wide class of quasi-Hamiltonian auto-oscillators, dynamics of which is described by Hamiltonian equations
with small non-conservative terms that stabilize one particular limit cycle among many possible periodic Hamiltonian orbits.
In addition to the marginally stable phase, reduced dynamics of quasi-Hamiltonian oscillators should include additional dynamical
variable - action - which is related to the auto-oscillation amplitude. The generalized action-phase model of quasi-Hamiltonian
oscillator has much wider region of applicability than the standard one. In a case of non-isochronous oscillator - oscillator
having frequency dependent on the oscillation amplitude - the action-phase model predicts qualitatively new non-autonomous
regimes and is crucial for correct description of synchronization phenomena in real auto-oscillatory systems.
In the framework of the proposed two-dimensional action-phase model different possible auto-oscillator orbits correspond to
different values of the action variable, while the position of the auto-oscillator on a particular orbit is described  by
a phase variable linearly proportional to time (as in the standard one-dimensional phase model). The action-phase model is
the most useful for the analytical description of the non-autonomous dynamics in complex non-isochronous quasi-Hamiltonian
auto-oscillators where the autonomous dynamics can be analyzed only numerically. When a family of stable periodic orbits of
such a complex auto-oscillator is found numerically, the full dynamical auto-oscillator equations are projected onto the
two dimensional sub-manifold formed by these periodic orbits using the proper projection procedure dictated by the symplectic
structure of the auto-oscillator's phase space. As a result, a system of two equations for the action and phase is obtained.
Using this system the auto-oscillator dynamics under arbitrary small perturbations can be analyzed analytically. The developed model
can describe new qualitative effects, such as hysteretic injection locking, that were observed experimentally [2], but can not
be described by a traditional one-dimensional phase model [1].
The proposed model is illustrated on a particular example of a non-isochronous nano-magnetic vortex auto-oscillator driven by
spin-polarized current [3,4]. Magnetic vortices are characterized by complex magnetization distribution which complicates
analytical treatment of the vortex excitations. We show, that when the stable orbits of a magnetic vortex auto-oscillator
are found numerically, the proposed two-dimensional reduction scheme provides an effective and accurate analytical
description of non-autonomous dynamics (in particular, synchronization) in this complex auto-oscillator.

1. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences  (Cambridge University Press, England, 2001).
2. P. Tabor, V. Tiberkevich, A. Slavin, and S. Urazhdin, Phys. Rev. B 82, 020407 (2010).
3. V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C. Ralph & R. A. Buhrman, Magnetic vortex oscillator driven by d.c. spin-polarized current, Nature Physics 3, 498 - 503 (2007)
4. A. Dussaux, B. Georges, J. Grollier, V. Cros, A.V. Khvalkovskiy, A. Fukushima, M. Konoto, H. Kubota, K. Yakushiji,
S. Yuasa, K.A. Zvezdin, K. Ando & A. Fert, Large microwave generation from current-driven magnetic vortex oscillators
in magnetic tunnel junctions, Nature Communications 1, Article number: 8 (2010)

Kris Vasudevan, Michael Cavers and Dennis Coombe
Department of Mathematics and Statistics
University of Calgary
Calgary, Alberta T2N 1N4
Canada
(vasudeva@ucalgary.ca, mcavers@ucalgary.ca; dennis.coombe@cmgl.ca )

Epileptic seizures of different types and of different intensities affect roughly 1% of the global population.  They go through
a cycle of inter-ictal period, pre-ictal, ictal and post-ictal periods.  Understanding the behavior of the brain under different
settings of the above cycle is of paramount importance in seeking mitigation measures.  In this regard, casting this problem
in terms of neuronal dynamics on a complex brain network has drawn considerable interest among physical scientists, mathematicians,
and health scientists.  This involves a mathematical model that accommodates the biophysical behaviour of neurons and assumes
a complex network with vertices marking the neurons and the edges defining the style of interactions between vertices.
This model is tied to a non-linear dynamics of weakly-coupled oscillators (neurons).  The nature of the firing of the neuron,
the duration of quiescence period between firing, and the directionality and strength of firing define the mathematical model.
Added to this is the complexity of the network for dynamical studies.  The complex network here is expressed in terms of graphs.
To examine the complete cycle of ictal periods, it is important that rewiring dynamics of the evolving graph is also taken
into consideration.  In this tutorial, we select three different neuronal models and use them to compare the non-linear
dynamics on evolving graphs.  Also, we explore the synchronization and asynchronization behaviour of oscillators during simulations. 

Synthetic genetic oscillators [1] have become popular since 2000 when a simple ring circuit with unidirectional
repression, called the Repressilator, was engineered [2-Elowitz] and realized inside bacterial cells. The oscillators are
considered to be an effective test bed for assessing ideas about gene expression regulation, as well as possible instruments
for gene therapy. However, the effectiveness of oscillators depends on how they can function collectively, thereby requiring
some coupling method. An almost obvious suggestion was to use natural bacterial quorum sensing (QS) as the method for synchronization [3, 4].
The core of QS is the production of small molecules (autoinducer) which can, first, easily diffuse across the cell membrane and,
second, work to activate target gene transcription. Manipulating the gene positions and QS-sensitive promoters in the genetic
circuits, one can obtain different coupling types and, as a result, different sets of collective modes in bacterial populations.
The effective synchronization of relaxation QS-dependent oscillators  was recently demonstrated in an impressive experiment [5]. 
We use the reduced Repressilator model with the QS mechanism added in a way that autoinducer diffusion provides for repulsive
coupling [6]. This type of coupling of two identical Repressilators leads to the domination of the antiphase limit cycle, the
formation of stable inhomogeneous stationary states and inhomogeneous limit cycles, as well as to the emergence of chaos in a
wide range of parameters. The phase diagram structure and the degree of stable attractor overlapping  in the parameter space
depends strongly on the degree of cooperativity (steepness of nonlinear terms in the ODE system) in the regulation of gene
expression: the higher the cooperativity the larger the parameter area occupied by the chaotic regime. Apart from usual chaos
emerging via torus bifurcation, we found the asymmetric chaotic regime in which the amplitudes of oscillations differ significantly.
This chaos is located far from torus bifurcation and includes the set of “periodic windows” with asymmetric regular oscillations.
The main part of the described dynamics has been reproduced in measurements using the electronic version of the coupled Repressilators [7]
(except the most delicate regimes inside chaos (work in progress)).

[1] – O. Purcell, N. Savery, C. Grierson, and M. di Bernardo, “A comparative analysis of synthetic genetic oscillators”, J. R. Soc. Interface 7, 1503 (2010).
[2] – M. B. Elowitz and S. Leibler, “A synthetic oscillatory network of transcriptional regulators,” Nature 403: 335, 2000.
[3] - D. McMillen, N. Kopell, J. Hasty, and J. J. Collins, “Synchronizing genetic relaxation oscillators by intercell signaling,”  PNAS U.S.A. 99, 679 (2002).
[4] - J. Garcia-Ojalvo, M. Elowitz, and S. H. Strogatz, “Modeling a synthetic multicellular clock: Repressilators coupled by quorum sensing,” PNAS U.S.A. 101, 10955, 2004. 
[5] - Tal Danino, Octavio Mondrago´n-Palomino, Lev Tsimring, Jeff Hasty, A synchronized quorum of genetic clocks, Nature      463, 326-330 (2010).
[6] - E. Ullner, A. Zaikin, E. I. Volkov, and J. Garcia-Ojalvo, “Multistability and Clustering in a Population of Synthetic Genetic Oscillators via Phase-Repulsive Cell-to-Cell Communication,”  Phys. Rev. Lett., vol. 99, p. 148103, 2007. 
[7] - Hellen, E.H., Dana, S. K., Zhurov, B., Volkov, E. Electronic implementation of a Repressilator with Quorum Sensing Feedback, PLoS One, 8(5): e62997, 2013

We show that cavity soliton modelocking dynamics in parametrically driven frequency combs occurs through a
two-part process, the first of which is a phase entrainment process and the second of which is equivalent to synchronization
phenomena that occur in many physical systems as described by the Kuramoto model for coupled oscillators. We present a reduced
set of modal phase equations derived from the Lugiato-Lefever equations that is amenable to an order parameter formulation and
exhibits coherence-coupling feedback. The link between the Kuramoto model and the phase locking of a massively multi-modal
distribution of oscillators constitutes a form of optical syncopation. 

Originally studied for their practical potential in high frequency electronics, Josephson junction arrays are
now known to represent a physical realization of the Kuramoto Model.  Less appreciated is that this connection has helped
resolve major issues confronting a successful Josephson technology.  I will describe these issues -- which involve junction
type, network connectivity, and even fabrication tolerances -- and the central role of the Kuramoto Model in resolving them. 

We analyze the Sakaguchi-Kuramoto model of coupled phase oscillators in a continuum limit, given by a frequency
dependent version of the Ott-Antonsen system. Based on a self-consistency equation, we provide a detailed analysis of partially
synchronized states, their bifurcation from the completely incoherent state and their stability properties.
We use this method to analyze the bifurcations for various types of frequency distributions and explain the appearance of
non-universal synchronization transitions.