Splay State Research Articles

The collective dynamics of NF − κB in cellular ensembles

The transcription factor NF − κB is a crucial component in inflammatory signalling. Its dynamics is known to be oscillatory and has been extensively studied. Using a recently developed model of NF − κB regulation, we examine the collective dynamics of a network of NF − κB oscillators that are coupled exogenously by a common drive (in this case a periodically varying cytokine signal corresponding to the TNF molecule concentration). There is multistability owing to the overlapping of Arnol’d tongues in each of the oscillators, and thus the collective dynamics exhibit a variety of complex dynamical states. We also study the case of a globally (mean field) coupled network and observe that the ensemble can display global synchronisation, cluster synchronisation and splay states. In addition, there can be dynamical chimeras, namely coexisting synchronised and desynchronized clusters. The basins of attraction of these different collective states are studied and the parametric dependence in the basin uncertainty is examined.

The human platelet antigen-1b (Pro33) variant of αIIbβ3 allosterically shifts the dynamic conformational equilibrium of this integrin toward the active state

Integrins are heterodimeric cell-adhesion receptors comprising α and β subunits that transmit signals allosterically in both directions across the membrane by binding to intra- and extracellular components. The human platelet antigen-1 (HPA-1) polymorphism in αIIbβ3 arises from a Leu → Pro exchange at residue 33 in the genu of the β3 subunit, resulting in Leu33 (HPA-1a) or Pro33 (HPA-1b) isoforms. Although clinical investigations have provided conflicting results, some studies have suggested that Pro33 platelets exhibit increased thrombogenicity. Under flow-dynamic conditions, the Pro33 variant displays prothrombotic properties, characterized by increased platelet adhesion, aggregate/thrombus formation, and outside-in signaling. However, the molecular events underlying this prothrombotic phenotype have remained elusive. As residue 33 is located >80 Å away from extracellular binding sites or transmembrane domains, we hypothesized that the Leu → Pro exchange allosterically shifts the dynamic conformational equilibrium of αIIbβ3 toward an active state. Multiple microsecond-long, all-atom molecular dynamics simulations of the ectodomain of the Leu33 and Pro33 isoforms provided evidence that the Leu → Pro exchange weakens interdomain interactions at the genu and alters the structural dynamics of the integrin to a more unbent and splayed state. Using FRET analysis of fluorescent proteins fused with αIIbβ3 in transfected HEK293 cells, we found that the Pro33 variant in its resting state displays a lower energy transfer than the Leu33 isoform. This finding indicated a larger spatial separation of the cytoplasmic tails in the Pro33 variant. Together, our results indicate that the Leu → Pro exchange allosterically shifts the dynamic conformational equilibrium of αIIbβ3 to a structural state closer to the active one, promoting the fully active state and fostering the prothrombotic phenotype of Pro33 platelets.

Performance enhancement using a stable low-pretilt molecular configuration and a novel driving method for optically compensated bend liquid crystal devices.

A stable low-pretilt molecular configuration (SLPMC) is successfully developed in optically compensated bend (OCB) liquid crystal (LC) devices by simultaneously employing the curing voltage and surface-anchored crosslinking monomer during the polymerization process. For the SLPMC OCB cell with the low-bend state, the warm-up voltage making the LC molecules reorient from the splay to the bend state is annihilated, and the transient twist state occurring as the driven LC molecules recover from the bend to the splay state is also eliminated. In addition, with the novel driving method selecting the specific driving point, the proposed SLPMC OCB cell not only exhibits a good response performance, but also outputs a higher light transmittance, which is superior to the conventional OCB and no-bias-bend cells. This Letter demonstrates an effective SLPMC fabrication method, and points out the significant contributions of SLPMC on the electro-optical properties, which will benefit and enhance the performance design in OCB-based applications.

Spatial splay states in coupled map lattices and Josephson junction arrays

Spatial splay states in coupled map lattices and Josephson junction arrays

Symmetry breaking in two interacting populations of quadratic integrate-and-fire neurons.

We analyze the dynamics of two coupled identical populations of quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The populations are heterogeneous; they include both inherently spiking and excitable neurons. The coupling within and between the populations is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rates and the mean membrane potentials in both populations. The reduced equations are exact in the infinite-size limit. The bifurcation analysis of the equations reveals a rich variety of nonsymmetric patterns, including a splay state, antiphase periodic oscillations, chimera-like states, and chaotic oscillations as well as bistabilities between various states. The validity of the reduced equations is confirmed by direct numerical simulations of the finite-size networks.

Robust Almost Global Splay State Stabilization of Pulse Coupled Oscillators

This technical note deals with the problem of asymptotically stabilizing the splay state configuration of a network of identical pulse coupled oscillators through the design of the their phase response function. The network of pulse coupled oscillators is modeled as a hybrid system. The design of the phase response function is performed to achieve almost global asymptotic stability of a set wherein oscillators' phases are evenly distributed on the unit circle. To establish such a result, a novel Lyapunov function is proposed. Robustness with respect to frequency perturbation is assessed. Finally, the effectiveness of the proposed methodology is shown in an example.

Cluster synchronization in networks of identical oscillators with α-function pulse coupling.

We study a network of N identical leaky integrate-and-fire model neurons coupled by α-function pulses, weighted by a coupling parameter K. Studies of the dynamics of this system have mostly focused on the stability of the fully synchronized and the fully asynchronous splay states, which naturally depends on the sign of K, i.e., excitation vs inhibition. We find that there is also a rich set of attractors consisting of clusters of fully synchronized oscillators, such as fixed (N-1,1) states, which have synchronized clusters of sizes N-1 and 1, as well as splay states of clusters with equal sizes greater than 1. Additionally, we find limit cycles that clarify the stability of previously observed quasiperiodic behavior. Our framework exploits the neutrality of the dynamics for K=0 which allows us to implement a dimensional reduction strategy that simplifies the dynamics to a continuous flow on a codimension 3 subspace with the sign of K determining the flow direction. This reduction framework naturally incorporates a hierarchy of partially synchronized subspaces in which the new attracting states lie. Using high-precision numerical simulations, we describe completely the sequence of bifurcations and the stability of all fixed points and limit cycles for N=2-4. The set of possible attracting states can be used to distinguish different classes of neuron models. For instance from our previous work [Chaos 24, 013114 (2014)CHAOEH1054-150010.1063/1.4858458] we know that of the types of partially synchronized states discussed here, only the (N-1,1) states can be stable in systems of identical coupled sinusoidal (i.e., Kuramoto type) oscillators, such as θ-neuron models. Upon introducing a small variation in individual neuron parameters, the attracting fixed points we discuss here generalize to equivalent fixed points in which neurons need not fire coincidently.

Spatial splay states and splay chimera states in coupled map lattices.

We study the existence and stability of splay states in the coupled sine circle map lattice system using analytic and numerical techniques. The splay states are observed for very low values of the nonlinearity parameter, i.e., for maps which deviate very slightly from the shift map case. We also observe that depending on the parameters of the system the splay state bifurcates to a mixed or chimera splay state consisting of a mixture of splay and synchronized states, together with kinks in the phases of some of the maps and then to a stable globally synchronized state. We show that these pure states and the mixed states are all temporally chaotic for our systems, and we explore the stability of these states to perturbations. Our studies may provide pointers to the behavior of systems in diverse application contexts such as Josephson junction arrays and chemical oscillations.

Dynamic Tuning and Memory Switching of Defect Modes in a Hybrid Photonic Structure

We propose a memorable and electrically tunable photonic device by infiltrating a dual-mode chiral-doped dual-frequency liquid crystal (LC) as the central defect layer in a one-dimensional photonic crystal (PC). According to the transmission properties of this structure, the wavelength tunability of defect modes is obtained by manipulating the LC layer in the dynamic mode due to the electrically controlled birefringence effect. Moreover, the switching between two memorable states, the splay and π-twist states, creates two distinct sets of defect modes at null voltage. The spectral characteristics of this device ensure its potential application as an energy-efficient multichannel wavelength filter.

A minimal model of self-consistent partial synchrony

We show that self-consistent partial synchrony in globally coupled oscillatory ensembles is a general phenomenon. We analyze in detail appearance and stability properties of this state in possibly the simplest setup of a biharmonic Kuramoto–Daido phase model as well as demonstrate the effect in limit-cycle relaxational Rayleigh oscillators. Such a regime extends the notion of splay state from a uniform distribution of phases to an oscillating one. Suitable collective observables such as the Kuramoto order parameter allow detecting the presence of an inhomogeneous distribution. The characteristic and most peculiar property of self-consistent partial synchrony is the difference between the frequency of single units and that of the macroscopic field.

Chimera states in two populations with heterogeneous phase-lag.

The simplest network of coupled phase-oscillators exhibiting chimera states is given by two populations with disparate intra- and inter-population coupling strengths. We explore the effects of heterogeneous coupling phase-lags between the two populations. Such heterogeneity arises naturally in various settings, for example, as an approximation to transmission delays, excitatory-inhibitory interactions, or as amplitude and phase responses of oscillators with electrical or mechanical coupling. We find that breaking the phase-lag symmetry results in a variety of states with uniform and non-uniform synchronization, including in-phase and anti-phase synchrony, full incoherence (splay state), chimera states with phase separation of 0 or π between populations, and states where both populations remain desynchronized. These desynchronized states exhibit stable, oscillatory, and even chaotic dynamics. Moreover, we identify the bifurcations through which chimeras emerge. Stable chimera states and desynchronized solutions, which do not arise for homogeneous phase-lag parameters, emerge as a result of competition between synchronized in-phase, anti-phase equilibria, and fully incoherent states when the phase-lags are near ±π2 (cosine coupling). These findings elucidate previous experimental results involving a network of mechanical oscillators and provide further insight into the breakdown of synchrony in biological systems.

Bistable chiral nematic liquid crystal device switchable between the transmissive and reflective modes

Proposed herein is a single-panel liquid crystal device that is switchable between the transmissive and reflective modes without subpixel division. Dye-doped liquid crystals were used to eliminate the image inversion caused by a reflective polarizer. The splay and π-twist states of the bistable chiral splay nematic mode, which has two stable states, were employed for a good dark state in both the transmissive and reflective modes. Operation in the transmissive mode can be achieved by applying an in-plane field, whereas a vertical field is applied for operation in the reflective mode. It is expected that the proposed device can be used as a highly efficient transflective device for outdoor as well as indoor environments.

Are Multiphase Competition and Order by Disorder the Keys to Understanding Yb(2)Ti(2)O(7)?

If magnetic frustration is most commonly known for undermining long-range order, as famously illustrated by spin liquids, the ability of matter to develop new collective mechanisms in order to fight frustration is perhaps no less fascinating, providing an avenue for the exploration and discovery of unconventional behaviors. Here, we study a realistic minimal model where a number of such mechanisms converge, which, incidentally, pertain to the perplexing quantum spin ice candidate Yb(2)Ti(2)O(7). Specifically, we explain how thermal and quantum fluctuations, optimized by order-by-disorder selection, conspire to expand the stability region of a degenerate continuous U(1) manifold against the classical splayed ferromagnetic ground state that is displayed by the sister compound Yb(2)Ti(2)O(7). The resulting competition gives rise to multiple phase transitions, in striking similitude with recent experiments on Yb(2)Ti(2)O(7) [Lhotel et al., Phys. Rev. B 89, 224419 (2014)]. By combining a gamut of numerical techniques, we obtain compelling evidence that such multiphase competition is a natural engine for the substantial sample-to-sample variability observed in Yb(2)Ti(2)O(7) and is the missing key to ultimately understand the intrinsic properties of this material. As a corollary, our work offers a pertinent illustration of the influence of chemical pressure in rare-earth pyrochlores.

An entropy based clustering order parameter for finite ensembles of oscillators.

Based on the entropy concept, we define a new clustering order parameter, denoted by c, feasible for finite systems of interacting oscillators. Unlike the generalized synchronization order parameters of the Kuramoto type, this new order parameter singles out the splay state as the unique state with c = 0, thus yielding a positive value whenever there is some kind of cluster formation in the system. It is therefore proposed to be monitored alongside the Kuramoto order parameters as a means to quantify the overall amount of clustering in the system.

Chimera states in systems of nonlocal nonidentical phase-coupled oscillators.

Chimera states consisting of domains of coherently and incoherently oscillating nonlocally coupled phase oscillators in systems with spatial inhomogeneity are studied. The inhomogeneity is introduced through the dependence of the oscillator frequency on its location. Two types of spatial inhomogeneity, localized and spatially periodic, are considered and their effects on the existence and properties of multicluster and traveling chimera states are explored. The inhomogeneity is found to break up splay states, to pin the chimera states to specific locations, and to trap traveling chimeras. Many of these states can be studied by constructing an evolution equation for a complex order parameter. Solutions of this equation are in good agreement with the results of numerical simulations.

Periodic States in Chaotic Rössler Oscillators with On-Off Coupling

It has been revealed that two coupled chaotic Rössler oscillators can transit to a period-5 splay state in an extremely weak coupling region [Zhan et al., Phys. Rev. Lett. 86 (2001) 1510]. Here we show that with further increase of coupling, two other coupling regions exist that may induce the period-4 splay state and period-5 synchronous state. Using an on-off coupling strategy, we find that the coupling regions for inducing period-5 states can be significantly extended and the extending effect is regulated by the match of two time scales: one is of the on-off coupling and the other is of the individual Rössler oscillator. We then compare the sensitivity of the periodic states with the initial conditions of the oscillators. We also analyze the mechanism behind these two new types of periodic states.

Dynamical behaviors in time-delay systems with delayed feedback and digitized coupling

We consider a network of delay dynamical systems connected in a ring via unidirectional positive feedback with constant delay in coupling. For the specific case of Mackey–Glass systems on the ring topology, we capture the phenomena of amplitude death, isochronous synchronization and phase-flip bifurcation as the relevant parameters are tuned. Using linear stability analysis and Master Stability Function approach, we predict the region of amplitude death and synchronized states respectively in the parameter space and study the nature of transitions between the different states. For a large number of systems in the same dynamical configuration, we observe splay states, mixed splay states and phase locked clusters. We extend the study to the case of digitized coupling and observe that these emergent states still persist. However, the sampling and quantization reduce the regions of amplitude death and induce phase-flip bifurcation.

Synchronization of networks of oscillators with distributed delay coupling.

This paper studies the stability of synchronized states in networks, where couplings between nodes are characterized by some distributed time delay, and develops a generalized master stability function approach. Using a generic example of Stuart-Landau oscillators, it is shown how the stability of synchronized solutions in networks with distributed delay coupling can be determined through a semi-analytic computation of Floquet exponents. The analysis of stability of fully synchronized and of cluster or splay states is illustrated for several practically important choices of delay distributions and network topologies.

Phase-locked regimes in delay-coupled oscillator networks.

For an ensemble of globally coupled oscillators with time-delayed interactions, an explicit relation for the frequency of synchronized dynamics corresponding to different phase behaviors is obtained. One class of solutions corresponds to globally synchronized in-phase oscillations. The other class of solutions have mixed phases, and these can be either randomly distributed or can be a splay state, namely with phases distributed uniformly on a circle. In the strong coupling limit and for larger networks, the in-phase synchronized configuration alone remains. Upon variation of the coupling strength or the size of the system, the frequency can change discontinuously, when there is a transition from one class of solutions to another. This can be from the in-phase state to a mixed-phase state, but can also occur between two in-phase configurations of different frequency. Analytical and numerical results are presented for coupled Landau-Stuart oscillators, while numerical results are shown for Rössler and FitzHugh-Nagumo systems.

Splay states in networks of identical integrate-and-fire neurons

We develop an analytic framework to investigate the stability of splay states in infinite networks of identical integrate-and-fire neurons coupled through synaptic pulses. More specifically we perform a linear stability analysis of the splay state probability distribution whose dynamics is governed by an appropriate Fokker Planck equation. For exponentially decaying synaptic pulses the splay state is unstable for excitation and stable for inhibition. For excitatory alpha-function pulses the splay state becomes stable for sufficiently large decay times and we find an analytic expression for the boundary of stability. This large decay time stability is analogous to the stability of synchronous states for inhibition studied by van Vreeswijk, Abbott and Ermentrout [1]. For inhibitory alpha-function pulses the splay state is unstable, but for smaller decay times (when there is no stable synchronous state) the splay state exhibits a remarkable attracting meta-stable transient. We complement our analytic framework with numerical simulations on finite networks.