List of poster contributions

We study the evolution of heterogeneous networks of oscillators subject to a state-dependent interconnection rule.
We find that heterogeneity in the node dynamics is key in organizing the architecture of the functional emerging networks.
We demonstrate that increasing heterogeneity among the nodes in state-dependent networks of phase oscillators causes a
differentiation in the activation probabilities of the links. This, in turn, yields the formation of  hubs associated
to nodes with larger distances from the average frequency of the ensemble. Our generic local evolutionary strategy can
be used to solve a wide range of synchronization and control problems.

We have established the impact of delays on a population of coupled phase oscillators affected by
external forcing. Thus, we have proposed a generalization of the Kuramoto model that can be used for interpretation
of the complex mean-field behavior of nonautonomous systems that have non-negligible signal transmission delays.
Here, the observed dynamics is dependent on three independent time scales: the original system, the external
forcing and the value of the delays. These can create the low-dimensional echo effects. The first harmonic
approximation for these systems has also been described.

In this work we study the role of the homoclinic and heteroclinic bifurcations in small neuron networks and 
to study rhythmogenesis in Central Pattern Generators (CPG).

The heteroclinic cycles in neuron networks (CPGs) create slow switching oscillations that achieve the level of robustness and stability
observed in nature. This fact creates a typical behavior of a robust ``trembling'' bursting synchronization. 
Neuron networks are characterized by its basic elements that are the neurons (the single neuron models).
To study biologically plausible CPG models (in our case a CPG model of 3 leech heart neurons) it is necessary to use specially adapted techniques
to take into account that the equations are grouped by each neuron. A detailed bifurcation analysis of the single neuron model provides the relevant regions
of bursting and chaotic behaviour that can be studied later in the complete network. 
The combination of all these techniques permits to obtain ``continuation-like''
results that detail the complete bifurcation process.

This work has been done in collaboration with M.A. Martinez, M. Rodriguez, S. Serrano, A. Shilnikov.

The control of the synchrony in power grid networks is essential for avoiding catastrophic scenarios.
Motivated by recent works on transient stability analysis of power-grid networks via dynamical systems theory,
a complementary study is presented. The stability of a post-fault equilibrium for a specific power-grid model
is studied using Lyapunov stability analysis. The considered system is modeled as a lattice of coupled second
order oscillators; the network is reduced to a fully connected network by eliminating all zero injection current
nodes (i.e. keeping only the generator nodes).
Early studies derived the Hamiltonian for the purpose of synchronization, transient stability, and the
region-of-attraction estimation. This implies a very strong simplification of the system, since the transfer
conductance between generators has to be considered negligible to fulfill the Hamiltonian conditions.
In this work, a different numerical approach is discussed, which consists in following the time deformation of
infinitesimal volume elements inside and outside the fixed-point vicinity. The volume elements are spanned
either by the Gram-Schmidt vectors or by the covariant Lyapunov vectors. For the synchronized case
(i.e. at the fixed-point) numerical results are compared to values obtained according to the Floquet theory.
We found that the numerical and theoretical values are well in line demonstrating a good accuracy of our algorithm.

Many popular neural mass models for describing cortical population dynamics, such as Wilson-Cowan
or Jansen-Rit, track an average activity without recourse to describing the degree of synchronisation or
coherence within a population.  This could of course be tracked within a large-scale model of synaptically
interacting conductance based neurons, though at the expense of analytical tractability.  Thus it is of
interest to seek levels of description that provide a bridge between microscopic single neuron dynamics
and coarse grained neural mass models, while preserving some notion of within-population coherence.  

For a theta neuron choice of microscopic dynamics we can make use of the Ott-Antonsen ansatz to find an exact mean
field description of the population dynamics on a reduced invariant manifold.  We consider an all-to-all coupled
network of such neurons, allowing us to track the complex valued Kuramoto order parameter z as follows:
d/dt z = -i ((z-1)2)/(2) + ((z+1)2)/(2) [-δ + iη0+i Vsyn G] - (-z2-1)/(2) G,
(1 + 1/α d/dt)2
(1 + 1/α d/dt)2 · G = H (z, ‾z)
where
H(z, ‾z) = 1 +z/(1-z) + ‾z/(1-z),
η0 and Δ are the centre and width of the Lorentzian distribution from which we draw our constant
background drive, k is the synaptic coupling strength, Vsyn is the synaptic reversal potential.

Simulations for a population of 500 theta neurons were carried out and the results were compared to the
reduced mean field approximation. Upon comparing the Kuramoto order parameter for the population of 500
neurons and the mean field model, it was found that even with 500 neurons the reduced model was an excellent
approximation of the average population dynamics.

Extensive bifurcation analysis has been carried out for a single population and for a two population
excitatory-inhibitory structure. As expected the addition of a second population allows for richer dynamics. 

A similar approach can be taken for a population of quadratic integrate-and-fire neurons. In this
framework the mean field variable tracks the population firing rate. A simple transformation exists
which allows us to switch between these two representations.

This work involves collaboration with colleagues in the Sir Peter Mansfield Magnetic Resonance Centre at Nottingham to use this new framework to understand the generation of beta-rhythms seen in motor cortex, and deliver a suitable model for understanding so-called beta-rebound. This is readily observed in MEG recordings whereby the initiation of hand movement causes a drop in the beta power band attributed to a loss of network synchrony.

We report parabolic bursting in a globally coupled network of mixed population of
oscillatory and excitable Josephson junctions. The resistive-capacitive shunted junction (RCSJ)
model of  the superconducitng device is considered for this study. We focus on the parameter
regime of the junction where its dynamics is governed by the saddle-node  on invariant circle
(SNIC) bifurcation. In this SNIC regime, the bursting appears in a broad parameter space of
the ensemble of mixed junctions. For a coupling value above a threshold, the network splits
into two synchronized clusters when a reductionism approach is use to reproduce the bursting
behavior of the large network. The excitable junctions effectively induces a slow dynamics on
the oscillatory units to generate the bursting behavior, which is a generic property of a globally
coupled network with a mixed population of dynamical nodes. We confirm this behavior in another
SNIC model of Morris-Lecar system.

The large deviation theory is applied to compute the survival probability in a quantum stateafter a disordered sequence of projective measurements. 

Phase response curves (PRCs) have become an indispensable tool in understanding the entrainment and
synchronization of biological oscillators. However, biological oscillators are often found in large coupled
heterogeneous systems and the variable of physiological importance is the collective rhythm resulting from an
aggregation of the individual oscillations. To study this phenomena we consider phase resetting of the collective
rhythm for large ensembles of globally coupled Sakagucki-Kuramoto oscillators. Making use of Ott-Antonsen theory
we derive an asymptotically valid analytic formula for the collective PRC. A result of this analysis is a
characteristic scaling for the change in the amplitude and entrainment points for the collective PRC compared
to the individual oscillator PRC. We support the analytical findings with numerical evidence and demonstrate
the applicability of the theory to large ensembles of coupled neuronal oscillators.

Different types of chimera states in the Kuramoto model with inertia are presented. They arise on
the route from coherence, via so-called solitary states, to the rotating waves.
We identify the wide region in parameter space, in which a new type of chimera state, imperfect chimera state,
which is characterized by a certain number of oscillators escaped from synchronized chimera's cluster, appears. 
We reveal that imperfect chimera states represent characteristic spatio-temporal patterns at the transition from
coherence to incoherence.

We analyzed the synchronization and stability of networked Kuramoto oscillators
with inertia. In the presence of a correlation between the oscillators’ natural frequencies
and the network’s degree, it is observed that clusters of nodes with the same degree join
the synchronous component successively, starting with small degrees [1-3].

The basin stability of the synchronization of the second-order Kuramoto model is investigated
via perturbing nodes separately. The novel phenomenon uncovered by basin stability consists
in two first-order transitions occur successively in complex networks: from a global
instability to a local stability and from a local to a global stability [4].

References:

[1]. Ji, P., Peron, T. K. D., Menck, P. J., Rodrigues, F. A., & Kurths, J. (2013).
Cluster explosive synchronization in complex networks. Physical Review Letters, 110, 218701.
[2]. Ji, P., Peron, T. K. D., Rodrigues, F. A., & Kurths, J. (2014). Analysis of
cluster explosive synchronization in complex networks. Physical Review E, 90(6), 062810.
[3]. Peron, T. K. D., Ji, P., Rodrigues, F. A., & Kurths, J. (2015). Effects of
assortative mixing in the second-order Kuramoto model. Physical Review E, 92(6), 052805.
[4]. Ji, P., Lu, W, & Kurths, J. (2015). Onset and suffusing transitions towards
synchronization in complex networks . Europhysics Letters, 109(6), 60005. 

We consider arrays of coupled optomechanical cells, each of which consists of a laser-driven
optical mode interacting with a mechanical (vibrational) mode. The mechanical modes can settle into stable
finite-amplitude oscillations. We study the collective classical nonlinear dynamics of the phases of these
oscillators, which is described by a certain extension of the well-known Kuramoto model. When including
noise in our model, we find connections to the physics of surface growth. Besides, in two-dimensional
arrays, we find that spiral structures and their dynamics play an important role.

Perceptual Grouping describes the formation
of objects from primitives, e.g. a rectangle
is not perceived as simply four unrelated lines.
We present an approach based on machine learning
techniques and Kuramoto oscillators which
can find meaningful groups in low level
features. The basic idea is to represent
features from an input domain in a one to one
relation with coupled Kuramoto oscillators. From
the similarity of these features, attracting and
repelling couplings among the oscillators
are learned to drive oscillators representing
similar features towards the same phase while
separating from oscillators representing dissimilar 
features. By employing a frequency update
based on phase similarity and the learned couplings,
oscillators representing groups of similar features
are gathered at the same frequency, thus allowing
an easy evaluation of the grouping result.

We ask what happens when two systems having a nonequilibrium steady state
are kept in contact and allowed to exchange a quantity, say mass, which
is conserved in the combined system. Will the systems eventually evolve
to a new stationary state where certain intensive thermodynamic variable,
like equilibrium chemical potential, equalizes following zeroth law of
thermodynamics and, if so, under what conditions is it possible? We argue
that an equilibrium-like thermodynamic structure can be extended to
nonequilibrium steady state in general, provided the systems have
short-ranged spatial correlations and they interact weakly to exchange
mass with rates satisfying  a balance condition - reminiscent of detailed
balance condition in equilibrium. This proposition  is proved and
demonstrated in various conserved-mass transport processes having nonzero
spatial correlations. We emphasize that the balance condition ensures not
only equalization of an intensive  thermodynamic variable but also
ensemble equivalence, which is crucial to construct a consistent
nonequilibrium thermodynamics.

We investigate the dynamics of a large network of quadratic
integrate and fire neurons. The neurons interact all-to-all via
synaptic, time delayed excitation or inhibition. Time delays in
synaptic connectivity induce collective synchronization or
asynchronous states in for both excitatory and inhibitory networks.
However, for inhibitory networks, and for certain time delays, the system sets into a novel, partially
synchronized state. Using the recently uncovered Lorentzian Ansatz, we obtain the exact firing rate equations
for a spiking neuron network with time delays, which allow us to find exact formulas for the stability boundaries
of the network's macroscopic states. The resulting phase diagram for excitatory neurons closely resembles that of
the Kuramoto model of coupled oscillators, with stereotyped regions of incoherence and synchronization alternating
as time delay in increased. In contrast, the phase diagram for inhibitory networks differs from that of the Kuramoto model,
indicating the presence of partially synchronous states.

Despite its history of synchronization and the practical importance of injection-locking to date,
there have been an open problem on fundamental limit of an efficient injection-locking for a given oscillator.
In this work, we elucidate a hidden mechanism governing the synchronization limit under weak forcings, which is
related to a widely known inequality; Hölderʼs inequality. This mechanism enables us to understand how and why the
efficient injection-locking is realized; a general theory of synchronization limit is constructed where the maximization
of the synchronization range or the stability of synchronization for general forcings including pulse trains,
and a fundamental limit of general m : n phase locking, are clarified systematically. These synchronization
limits and their utility are systematically verified in the Hodgkin–Huxley neuron model as an example.

References 
[1] H.-A. Tanaka, Physica D, Vol. 288, 1 (2014).
[2] H.-A. Tanaka, J. Phys. A: Math. Theor., Vol 47, 402002 (2014).

Universal scaling laws form one of the essential issues in physics. A nonstandard
scaling law or a breakdown of a standard scaling law, on the other hand, often leads to the finding of a new universality
class in physical systems. We show that a statistical quantity related to fluctuations follows a nonstandard scaling law
in terms of the system size in a synchronized state of globally coupled nonidentical oscillators, by numerical simulations
of several different models. The conditions required for the unusual scaling law will also be discussed.

References
[1] I. Nishikawa, G. Tanaka, T. Horita, and K. Aihara, Chaos, Vol. 22, 013133 (2012).
[2] I. Nishikawa, G. Tanaka, and K. Aihara, Phys. Rev. E, Vol. 88, 024102 (2013).

We consider models of identical pulse-coupled oscillators with global interactions. Previous work showed that under certain conditions such systems always end up in sync, but did not quantify how small clusters of synchronized oscillators progressively coalesce into larger ones. Using tools from the study of aggregation phenomena, we obtain exact results for the time-dependent distribution of cluster sizes as the system evolves from disorder to synchrony. 

We report finite-size numerical investigations and mean-field analysis of a Kuramoto model with inertia
for fully coupled and diluted systems. In particular, we examine for a Gaussian distribution of the frequencies for
the transition from incoherence to coherence for increasingly large system size and inertia. For sufficiently large
inertia the transition is hysteretic, and within the hysteretic region clusters of locked oscillators of various sizes
and different levels of synchronization coexist. A modification of the mean-field theory developed by Tanaka, Lichtenberg,
and Oishi [Physica D 100, 279 (1997)] allows us to derive the synchronization curve associated
to each of these clusters. We have also investigated numerically the limits of existence of the coherent and of the
incoherent solutions. The minimal coupling required to observe the coherent state is largely independent of the system
size, and it saturates to a constant value already for moderately large inertia values. The incoherent state is observable
up to a critical coupling whose value saturates for large inertia and for finite system sizes, while in the thermodinamic
limit this critical value diverges proportionally to the mass. By increasing the inertia the transition becomes more complex,
and the synchronization occurs via the emergence of clusters of whirling
oscillators. The presence of these groups of coherently drifting oscillators induces oscillations in the order parameter.
We have shown that the transition remains hysteretic even for randomly diluted networks up to a level
of connectivity corresponding to a few links per oscillator. Finally, an application to the Italian high-voltage power grid is reported, which reveals the emergence of quasiperiodic oscillations in the order parameter due to
the simultaneous presence of many competing whirling clusters.

In general, synchronization is understood as the phenomenon that
keeps things ‘happening at the same time’. There exists a more exquisite
definition of synchronization due to Christiaan Huygens, a Dutch scientist
who around 1665 discovered that a pair of pendulum clocks, placed on a
flexible structure, exhibited synchronized motion due to the imperceptible
vibrations of the coupling structure. Huygens referred to this phenomenon
as the ‘sympathy’ of two clocks.

In the words of Huygens, it can be said that we live in a sympathetic
world. For example, frogs make alternated mating calls, neurons and
pacemaker cells fire in unison, and many other examples that are
perceptible to everyone. In fact, the ‘sympathy’ described by Huygens, has
been observed in different areas like physics, mathematics, nature, and
engineering.

In this poster, a modern version of the classical experiment on
synchronization of pendulum clocks is presented. In particular, an
improved model (regarding current models in the literature) is discussed.
Furthermore, a rigorous analysis of the synchronous motion is conducted by
using the theory of Piece-wise linear systems. Finally, the theoretical
results are illustrated by experiments using two monumental clocks, which
were provided by the clocks company ‘Relojes Centenario’
(http://centenario.org.mx/).

Systems modeled by delay differential equations (DDE) may exhibit
very complex behavior, because of their infinite dimensional 
phase space. Recently, a new class of nonlinear DDE involving a second order 
differential model, has attracted a growing interest. Their model 
is derived from physical optoelectronic setup (GOE98,UDA02) and they also
provide qualitatively new solutions compared to conventional scalar DDE
(Ikeda or Mackey-Glass dynamics).  An example of such solutions, so-called
virtual chimera states discovered recently in (LPM13), is
emphasized in this contribution for a new optoelectronic setup. They are
characterized by spontaneous symmetry breaking when observed in a space-time
representation of DDE (AGLM92). The dynamical system of concern is
governed by an Ikeda-like model comprising an integral term:
varε d/(dt)x = -x - δ∫t0t
x(xi),dxi+βF[x(t-τ)], ref(1)
where varε and δ are typically small bifurcation
parameters, β is the
third one, and 
F is some function describing the nonlinear delayed feedback
of the dynamics. In our optoelectronic setup, F is provided
by the wavelength-to-intensity nonlinear transformation obtained
from a Fabry-P'erot interferometer,
i.e. F[x]=[1+m,sin2(x+Φ0)]-1, m=8, Φ0=-0.4. 

We report on the particular bifurcation scenario for the chimera states
appearance as β increases, observed both numerically for the model
(ref(1)) and experimentally for the setup. 
On the other hand, with increase of ratio δ / varε, in the 
system-which exhibits multistability-a higher number of chaotic "heads" may
appear.
We explore the probability of different chimera solution occurrences in the
δ-varε bifurcation region. It is crucial to admit the close
correspondence between numerical and experimental results.



    (GOE98) J.-P. Goedgebuer, L. Larger, H. Porte
        Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laserdiode
        Phys. Rev. Lett., Vol.80, No.10, pp. 2249-2252 (1998).
   (UDA02) V. S. Udaltsov, L. Larger, J.-P. Goedgebuer, M. W.  Lee, É. Genin and W. T. Rhodes
        Chaotic bandpass communication system
       IEEE Trans. On Circuits And Systems, Vol.49, No.7, pp.  1006-1009 (2002).
    (LPM13) L.Larger, B.Penkovsky, Y.Maistrenko
        Virtual chimera states for delayed-feedback systems
      Phys. Rev. Lett. 111, 054103 (2013).
    (AGLM92) F. T. Arecchi, G. Giacomelli, A. Lapucci, R. Meucci
        Two-dimensional representation of a delayed dynamical system
        Phys. Rev. A 45, R4225 (1992).
    

Synchronization is an ubiquitous phenomenon observed in a wide range of different complex systems. It consists
in the dynamics of adjustment of rhythms and examples of its emergence can be found in the cells of the heart,
superconductors, neutrino oscillations, chemical reactions and many others. The most successful model that describes
the emergence of such behavior in complex systems is arguably the paradigmatic Kuramoto model. Until 2011, only
continuous synchronization transitions were known to occur in networks of first-order Kuramoto oscillators. However,
Gomez-Gardeñez et al. [1] reported the first observation of discontinuous phase synchronization transitions in scale-free
networks, triggering further works on the subject. Gomez-Gardeñez et al. [1] considered a new kind of interplay
between the connectivity pattern and the dynamics. More specifically, the authors considered the natural frequencies
of the oscillators to be positively correlated with the degree distribution of the network by assigning to each node
its own degree as its natural frequency, rather than drawing it from a given symmetric distribution independent of
network structure, as performed in previous works. In this work, we substantially extend the first-order Kuramoto
model used in [1] to a second-order Kuramoto model that we modify in order to analyze global synchronization
considering the natural frequency of each node proportional to its degree [2, 3]. In this model, we find a discontinuous
phase transition in which small degree nodes join the synchronous component simultaneously, whereas
other nodes synchronize successively according to their degrees [2, 3] (in contrast to [1] where all the nodes join the
synchronous component abruptly). Finally, we also show here that one is able to control such hysteretic behaviour of
the second-order Kuramoto model by tuning the assortative properties of the network, a phenomenon that was not
investigated before.


References: 

[1] J. Gomez-Gardeñes, S. Gomez, A. Arenas, and Y. Moreno, Phys. Rev. Lett. 106, 128701 (2011).
[2] P. Ji, T. K. D. Peron, P. J. Menck, F. A. Rodrigues, and J. Kurths, Phys. Rev. Lett. 110, 218701 (2013).
[3] P. Ji, T. K. D. Peron, F. A. Rodrigues, and J. Kurths, Physical Review E 90, 062810 (2014).

We investigate interacting phase oscillators whose mean field is at a different frequency from the mean or mode of their natural frequencies. The associated asymmetries lead to a macroscopic traveling wave. We show that the mean-ensemble frequency of such systems differs from their entrainment frequency. In some scenarios these frequencies take values that, counterintuitively, lie beyond the limits of the natural frequencies. The results indicate that a clear distinction should be drawn between the two variables describing the macroscopic dynamics of cooperative systems. This has important implications for real systems where a nontrivial distribution of parameters is common.

Synchronization is a phenomenon representing the onset of a collective behaviour presented in many systems, such as: neurons, fireflies, chemical processes and social dynamics. Synchronization process corresponds to the adjustment of rhythms due to a weak interaction of self-sustained oscillators. This dynamical process can be described by the Kuramoto model, which is one of the most studied nowadays, since it is analytically tractable and preserves the non-linearity of the dynamics. In the traditional Kuramoto model, which considers that the natural frequency of the oscillators are distributed according to a symmetric distribution of probability, the order parameter evolve as a second-order phase transition according to the increase of the coupling strength. However, it has been observed recently that discontinuous phase transition can occur in synchronization processes. This phenomenon takes place when the natural frequency of each oscillator is positively correlated with its number of connections. In order to obtain a better understanding of this explosive synchronization, we investigate a particular case: when only a percentage of network nodes have the natural frequency correlated with the degree, i.e. a partial correlation. We have also analyzed the influence of the most connected vertices, textit{hubs}, on the emergence of the explosive synchronization.

Michael Rosenblum1 and Arkady Pikovsky

We analyze quasiperiodic partially synchronous states in an ensemble of Stuart-Landau oscillators
with global nonlinear coupling. We reveal two types of such dynamics: in the rst case the time-averaged frequencies of
oscillators and of the mean eld dier, while in the second case they are equal, but the motion of oscillators is additionally
modulated. We present an example of a bifurcation diagram, where we show transitions between synchronous and partially synchronous states.

From gregarious fireflies flashing in unison at the Malaysian forest riversides to the pacemakers cells collectively
driving our heart beats, periodic collective behaviour is awe inspiring. In this work I introduce a framework to study the emergence
of periodic collective behaviour in any network of diffusively-coupled phase-oscillators. Specifically, we address the onset of the
likely phenomenon known as phase-locking; that is, when the phase difference between the oscillators is constant and the system is
frequency synchronous. Contrary to common belief, our results show that the existence (stability) of phase-locking in these networks
depends on the coupling function parity (form), regardless what the coupling topology is or how heterogeneous the oscillator’s natural
frequencies are. To illustrate the predictive power of our framework, we reveal the existence of non-trivial phase-locked states and
their stability in generic networks of phase-oscillators.

The exploration of networks of delay-coupled oscillators has a large number of applications in nature, for example neuron
and laser dynamics. Based on a finite system of Kuramoto oscillators with time-delayed coupling, we impose a thermodynamic limit for an
assortative scale free network with positive frequency degree correlations in order to investigate the transition towards synchronisation.
Using the ansatz of Ott and Anderson, we investigate rotating wave solutions to the resulting Fourier mode equations. We numerically relate
their corresponding stability properties to previous findings of explosive synchronisation in similar settings.

Based on the works of Ott and Antonsen (2009) and Pikovsky and Komarov (2011), we consider external
driving of an ensemble of two clusters of coupled limit cycle oscillators with non-resonant interaction.
We present the results of bifurcation analysis, as well as numerical simulations for different parameters.
There exist two modes of synchronization in which either first, or the second cluster dominates.
Hard transitions between the two modes are observed with variation of the driving power and frequency.
In particular, we consider the effect of bifurcation delay when the system is driven by a frquency-modulated signal.

Synchronization plays a crucial role in power system dynamics, since a synchronized state with
constant voltages, system frequency and steady power transfer is the intended mode of grid operation. 
Progressive decentralization and integration of renewables featuring fluctuating feed-in raise novel questions of power system stability and design.
Networks of synchronous generators and motors, which transform mechanical power into electrical power and vice versa,
can be thought of simplified power grid models. They can be described by Kuramoto-like models, i. e. modifications of
the Kuramoto model, whose level of detail depends on the specific issue of interest. Basic questions of power system
stability and dynamics and their interaction with topological characteristics can be addressed to these models.
We motivate and present results on different levels of grid modeling. 

The aphorism “neurons that fire together, wire together” (Donald Hebb) broadly
means that a repetitive experience of any kind, strengthens the connection between
the corresponding brain cells or the neurons. The Chimera states for an all-to-all
uniformly wired system of identical nonisochronous limit-cycle oscillators (Phys.
Rev. Lett. 112, 144101, 2014) depicts the scenario where “neurons wired together, do not necessarily fire together”:
some do synchronize and some do not. The structural or the anatomical connectivity of the system does not necessarily
lead to the functional connectivity. Although the brain neurons are not all-to-all coupled in
reality, the glial cells do exhibit all-to-all or global coupling via gap junctions. It is
now believed that the role of glial cells may be as critical to the brain dynamics as that of the neurons.
The Chimera state in our model is self-organized and is maintained by a dynamically
self-sustaining mean-field or the order parameter. The nonisochronicity (variously
known as shear, nonlinear dispersion, amplitude-phase coupling etc.) of the oscillators
is at the root of the occurrence of Chimera states in the globally coupled system. The
mean-field exhibits low-frequency temporal fluctuations, which decrease in intensity
in the presence of stimuli. Our model system interestingly exhibits a repertoire of
metastable Chimera states with or without any external stimuli. We investigate the
emergence of metastable chimera states by examining the bifurcation scenario of the
differential equation governing the dynamics of the order parameter.
It is widely believed that the brain is a self-organized metastable system We discuss
the relevance of our model (exhibiting self-organized metastable chimera states) to
explain a number of experimentally observed features of the brain dynamics. In
particular, the order parameter in our model replicates the low-frequency fluctuations
(<0.1 Hz) observed in blood-oxygen-level dependent (BOLD) functional magnetic
resonance imaging (fMRI) signals of the brain in the absence of any stimulation. The
order parameter also shows respiration with periodicity around 0.25 Hz, suggesting its
neuronal origin. Freeman in one of his papers describes consciousness as an order
parameter. The fluctuation level in the BOLD signals decreases in the presence of
stimulation as shown by the model. The structural connectivity in the brain is
relatively static at the time scales of perception and action associated with cognition.
In our model, the metastability of the brain arises naturally without taking recourse to
any changes in the structural connectivity.
Our simple model of globally coupled limit-cycle oscillators shows that the brain
dynamics is complex but not complicated.
Essentially, all models are wrong, but some are useful.
--- George E. P. Box

We study two intertwined globally coupled networks of noisy Kuramoto phase oscillators that
have the same natural frequency, but differ in their perception of the mean field and their contribution
to it. These traits are given asymmetrically by in- and out-coupling strengths which can be both positive
and negative. We uncover in this network of networks intriguing patterns of discordance. The ensemble splits
into two clusters separated by a constant phase lag. If it differs from π ("π-state"), a traveling wave (TW)
is induced. Two distinct routes to TW solutions are observed, namely either through one-cluster or through
π-states. Remarkably, abrupt, hysteretic and reentrant synchronization arise naturally thereby. Bifurcation
diagrams are obtained with the help of a reduced system. Finally, we discuss discordant synchronization
in coupled chaotic noise-perturbed Rössler systems.

Interacting oscillatory systems abound in nature and science attempts to understand them as much
as possible. Often, the interest is not only to understand the structure of systems, but also the functions
and mechanisms that define and connect them.
First, we developed a method for time-evolving dynamical Bayesian inference of coupled systems in presence
of noise [1]. The method was designed to infer multivariate phase dynamics of interacting oscillators.
We applied it to better understand the human cardio-respiratory interactions under time-varying conditions.
The reconstructed cardio-respiratory coupling functions demonstrated in detail how the heart oscillations
change due to the influence from the lungs. Similarly, recently we inferred the cross-frequency coupling
functions of multivariate neuronal oscillations [2]. Moreover, we found that the biological i.e. cardio-respiratory
coupling functions are time-varying processes for themselves.
Then we used this knowledge of time-varying coupling functions and applied it to build new improved secure
encryption protocol [3, 4]. We encode the information signals as modulating scales of plurality of coupling
functions between two dynamical systems at the transmitter. Two signals, one from each system, are transmitted
and the dynamical systems are reconstructed at the receiver through complete synchronization. Knowing the exact
form of coupling functions in use (eventually forming the encryption key) we were able to decrypt the information
signals by use of the time-evolving dynamical Bayesian inference. In this way, we showed that the coupling function
protocol has unbounded encryption possibilities, allows multiplexing inherently and is extremely noise resistant.
We presented the encryption protocol on amplitude state model consisting of coupled chaotic Lorenz and Rossler
systems forming a coupled pair in the transmitter (and in the receiver) subject to channel noise.
We have connected two seemingly very different areas – biology and communications – because we exploited
the universality of the theoretical construct of coupling functions. Needless to say, the methodology and
the theoretical concepts have wide implications for interacting oscillators in general.
References:
[1] Stankovski, T., Duggento, A., McClintock, P. V., & Stefanovska, A. (2012). Inference of time-evolving coupled dynamical systems in the presence of noise. Physical Review Letters, 109(2),024101. 
[2] Stankovski, T., Ticcinelli, V., McClintock, P. V., & Stefanovska, A. (2015). Coupling functions in networks of oscillators. New Journal of Physics, 17(3), 035002.
[3] Stankovski, T., McClintock, P. V., & Stefanovska, A. (2014). Coupling functions enable secure communications. Physical Review X, 4(1),011026.
[4] Subject to patent application No. WO2015019054A1.

Robert Ton1,2, Gustavo Deco2,3, Andreas Daffertshofer1

We study the dynamics of phase reductions based on networks of two seminal types of neural mass
models: Wilson-Cowan [7] and Freeman [2]. Proper balance between attracting and repelling forces
within these networks ensures their oscillatory behaviour. By using a combination of the rotating wave
approximation and slowly varying amplitude approximation, we could derive the phase dynamics φWC
[1] and φF
F [6] corresponding to the Wilson-Cowan and Freeman model respectively. The phase dynamics
yield
d/dt φk = ωk + ∑l=1N Dkl sin(φl - φk + Δkl)  

and reveals similarities with the seminal Kuramoto model, albeit with a heterogeneous coupling Dkl
and a delay-dependent phase shifts δkl. Irrespective of the underlying neural mass model, the phase
oscillator dynamics (1) capture the spatial correlation structure of brain activity.

The brain is believed to be in a permanently critical state, a state that is considered optimal in
terms of network performance [5]. Empirical evidence for this critical state consists of the observation of
power-laws in various aspects of brain activity, one of those being R(t) obtained from the Hilbert phases
of source-reconstructed MEG recordings [4].
Therefore we analysed the auto-correlation structure of the Kuramoto order parameter R(t) =
R(t) = 1/N | ∑k=1N eiφk(t)|
by means of a detrended 
fluctuation analysis [3]. In this aspect d/dt φF
and d/dt φWCdid
differ, because only the first displayed power-law scaling in the R(t) auto-correlation structure.
The Freeman and WC based phase dynamics differ in their topology: d/dt φWC
contains excitatory-inhibitory (E-I) phase interaction whereas  d/dt φWC
WC does not. This is due to the higher dimensionality of
the underlying voltage dynamics (Freeman) compared to the firing rate dynamics in the Wilson-Cowan
model. To test whether these topological difference caused the observed power-law scaling in R(t), we
compared conventional Kuramoto network behaviour with that originating from a Kuramoto model with
E-I interactions. Excitatory nodes were uniformly coupled in both cases.
Despite being located in its critical regime, the conventional Kuramoto model did not show R(t)
power-law scaling in contrast to the model containing E-I interactions. The values of the Hurst (H)
indicate persistent behaviour, resembling the behaviour observed in d/dt φF
F and in brain activity [4].
We conclude that spatial correlation structure is mainly determined by the brain's structural con-
nectivity through Dkl and is relatively independent of the individual node dynamics. However, E - I
phase interactions are vital in producing the critical dynamics observed in brain recordings. This further
corroborates the importance of balance in the cortex to generate critical dynamics [5].

References
[1] A. Daffertshofer and B.C.M. van Wijk. On the in
uence of amplitude on the connectivity between phases. Frontiers
in Neuroinformatics, 5(00006), 2011.
[2] W.J. Freeman. Simulation of chaotic eeg patterns with a dynamic model of the olfactory system. Biological Cybernetics,
56:139{150, 1987.
[3] C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, and A.L. Goldberger. Mosaic organization of dna
nucleotides. Physical Review E, 49(2):1685, 1994.
[4] Ton R., Deco G, Kringelbach ML, Woolrich M, Breakspear M and Daffershofer A. Criticality in the resting brain. In preparation.
[5] W.L. Shew and D. Plenz. The functional benefits of criticality in the cortex. The Neuroscientist, 19(1):88{100, 2013.
[6] R. Ton, G. Deco, and A. Daffertshofer. Structure-function discrepancy: Inhomogeneity and delays in synchronized
neural networks. Accepted in PLOS Computational Biology, 2014.
[7] H.R. Wilson and J.D. Cowan. Excitatory and inhibitory interactions in localized populations of model neurons. Bio-
physical journal, 12(1):1, 1972.

Synchronization phenomena are observed in a wide variety of systems ranging from synchronizing fireflies through firing neurons to electrical power grids [1,2]. Recently it has been demonstrated that permutation symmetries of the underlying oscillator networks are of fundamental importance for zero-lag synchronization patterns [3]. In this contribution, we address the question: What role do network symmetries play, when the frequency detuning of the individual oscillators is too large to allow for zero-lag synchronization? Experiments and simulations on networks of discrete chemical relaxation oscillators [4] reveal transitions from incoherence through partial synchronization to phase waves following symmetry clusters as a function of coupling strength.

Kris Vasudevan, Michael Cavers and Dennis Coombe
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. 

We study the steady state solutions of the 1D and 2D spatially extended Kuramoto model with
nearest-neighbor coupling for seven system sizes. Past the critical coupling σc, different possible steady
states can appear including phase coherence (synchronization), and phase locking (traveling wave states).
We characterize the terminal behavior of a space of initial conditions for a variety of boundary conditions:
periodic, zero-flux and free.

As the size of networks requiring synchronous clocking increases steadily, e.g., antenna arrays
in massive MIMO systems, clock distribution based on state-of-the-art concepts like master-slave clock tree
architectures becomes challenging. Alternatively, robust synchronous clocking can be achieved by mutual
coupling of oscillators, which can lead to self-organized synchronization as known, e.g., from biological systems.
Here we show that mutually coupled   digital phase-locked loops (DPLLs) can enable synchronous clocking in
large-scale systems with transmission delay. Using a modified Kuramoto phase oscillator model of coupled DPLLs
including signal filtering and signal transmission delays, we show how the collective frequency and the time
scales of synchronization depend on system specifications. To test our theoretical predictions, we designed
and carried out experiments, thereby providing a proof-of-principle that mutually delay-coupled DPLLs can
provide robust self-organized synchronous clocking.

It has been found that contrarian oscillators usually take a negative role in the collective
behaviors formed by conformist oscillators. However, experiments revealed that it is also possible to
achieve a strong coherence even when there are contrarians in the system such as neuron networks with
both excitable and inhibitory neurons. To understand the underlying mechanism of this abnormal phenomenon,
we here consider a complex network of coupled Kuramoto oscillators with mixed positive and negative couplings
and present an efficient approach, i.e., tit-for-tat strategy, to suppress the negative role of contrarian
oscillators in synchronization and thus increase the order parameter of synchronization. Two classes of
contrarian oscillators are numerically studied and a brief theoretical analysis is provided to explain
the numerical results.