Attraction of spiral waves by localized inhomogeneities with small-world connections in excitable media

Dynamics of spiral waves rotating around an obstacle and the existence of a minimal obstacle

Pinning of vortices by obstacles plays an important role in various systems. In the heart, anatomical reentry is created when a vortex, also known as the spiral wave, is pinned to an anatomical obstacle, leading to a class of physiologically very important arrhythmias. Previous analyses of its dynamics and instability provide fine estimates in some special circumstances, such as large obstacles or weak excitabilities. Here, to expand theoretical analyses to all circumstances, we propose a general theory whose results quantitatively agree with direct numerical simulations. In particular, when obstacles are small and pinned spiral waves are destabilized, an accurate explanation of the instability in two-dimensional media is provided by the usage of a mapping rule and dimension reduction. The implications of our results are to better understand the mechanism of arrhythmia and thus improve its early prevention.

Removal of pinned scroll waves in cardiac tissues by electric fields in a generic model of three-dimensional excitable media

Spirals or scroll waves pinned to heterogeneities in cardiac tissues may cause lethal arrhythmias. To unpin these life-threatening spiral waves, methods of wave emission from heterogeneities (WEH) induced by low-voltage pulsed DC electric fields (PDCEFs) and circularly polarized electric fields (CPEFs) have been used in two-dimensional (2D) cardiac tissues. Nevertheless, the unpinning of scroll waves in three-dimensional (3D) cardiac systems is much more difficult than that of spiral waves in 2D cardiac systems, and there are few reports on the removal of pinned scroll waves in 3D cardiac tissues by electric fields. In this article, we investigate in detail the removal of pinned scroll waves in a generic model of 3D excitable media using PDCEF, AC electric field (ACEF) and CPEF, respectively. We find that spherical waves can be induced from the heterogeneities by these electric fields in initially quiescent excitable media. However, only CPEF can induce spherical waves with frequencies higher than that of the pinned scroll wave. Such higher-frequency spherical waves induced by CPEF can be used to drive the pinned scroll wave out of the cardiac systems. We hope this remarkable ability of CPEF can provide a better alternative to terminate arrhythmias caused by pinned scroll waves.

Mechanism of unpinning spirals by a series of stimuli

Antitachycardia pacing (ATP) is widely used to terminate tachycardia before it proceeds to lethal fibrillation. The important prerequisite for successful ATP is unpinning of the spirals anchored to the obstacle by a series of stimuli. Here, to understand the mechanism of unpinning spirals by ATP, we propose a theoretical explanation based on a nonlinear eikonal relation and a kinematical model. The theoretical results are quantitatively consistent with the numerical simulations in both weak and high excitabilities.

Topology identification of uncertain nonlinearly coupled complex networks with delays based on anticipatory synchronization

This paper presents an adaptive anticipatory synchronization based method for simultaneous identification of topology and parameters of uncertain nonlinearly coupled complex dynamical networks with time delays. An adaptive controller is proposed, based on Lyapunov stability theorem and Barbǎlat's Lemma, to guarantee the stability of the anticipatory synchronization manifold between drive and response networks. Meanwhile, not only the identification criteria of network topology and system parameters are obtained but also the anticipatory time is identified. Numerical simulation results illustrate the effectiveness of the proposed method.

Oscillation propagation in neural networks with different topologies

In light of the issue of oscillation propagation in neural networks, various topologies of FitzHugh-Nagumo neuron populations are investigated. External Gaussian white noise is injected into the first neuron only. Before the oscillation spreads to the other neurons in the network, some of the inherent stochasticity within the noise-induced oscillation of the first neuron is filtered out due to the neuron's nonlinear dynamics. Both the temporal and the spatial coherence of the evoked activity's propagation are analyzed in conjunction with the network topology randomness p, the coupling strength between neurons g, and the noise amplitude D. The temporal periodicity of the global neural network presents a typical coherence biresonance (CBR) characteristic with regard to the noise intensity. The network topology randomness exerts different influences on the resonance effects for different coupling strength regimes. At an intermediate coupling strength, the random shortcuts reinforce the interactions between the neurons, and then more stochasticity in the firings of the first neuron spreads within the network. Consequently, CBR is decreased with the increase of the network topology randomness. At a large coupling strength, the random shortcuts assist the nonlinearity in impairing the stochastic components, and consequently help to enhance the resonance effects, which differed significantly from previous related work. However, the degree of the spatial synchronization of the systems increases monotonically as the network topology randomness increases at any coupling strength.

Inferring local dynamics and connectivity of spatially extended systems with long-range links based on steady-state stabilization

A method is presented for system identification of spatially extended systems with structural inhomogeneities of local dynamics and additional long-range links. The proposed identification procedure is based on steady-state stabilization and is illustrated with an inhomogeneous two-dimensional grid of coupled FitzHugh-Nagumo models.

Spontaneous suppression of spiral turbulence based on feedback strategy

Control of defect-mediated turbulence or spiral turbulence has been a subject of pattern dynamics study in the last two decades. As a result, many control strategies has been proposed in theory and tested in computer simulation. Here we propose a self-regulation system that can spontaneously eliminate topological defects in the reaction-diffusion system that suffers from the Doppler instability. The core of the self-regulation scheme is a feedback introduction of local inhomogeneities around spiral tips. We used modified FitzHugh-Nagumo model in computer simulations and light-sensitive Belousov-Zhabotinsky reaction in experiments to test the functionality of the proposed system. We found that if the inhomogeneities were induced on the spiral tips once they were identified, the spiral can be exterminated thus the spiral turbulence can be controlled. Compared with the studies on anatomic obstacles in heart, this self-regulation model may provide a possible mechanism for spontaneous termination of cardiac fibrillation.

Stabilizing spiral waves by noninvasive structural perturbations

We suggest a promising engineering approach to structural perturbation that in principle generates arbitrary additional connections artificially. We show that this structural perturbation method can be applied to stabilize spiral waves noninvasively. Furthermore, the stabilization performance is improved dramatically using proper delay for each additional connection to be created. This structural perturbation method with proper parameters can also be considered as a noninvasive adaptive pinning control that obtains better control performance than the typical constant pinning control. Remarkably, we numerically illustrate that a few additional connections (i.e., small structural perturbation) may result in stabilization of spiral waves. All methods suggested are motivated and illustrated with a FitzHugh-Nagumo model.

Stochastic resonance on weakly paced scale-free networks

We study the impact of additive Gaussian noise and weak periodic forcing on the dynamics of a scale-free network of bistable overdamped oscillators. The periodic forcing is introduced to a single oscillator and therefore acts as a pacemaker trying to impose its rhythm on the whole ensemble. We show that an intermediate intensity of temporally and spatially uncorrelated noise is able to optimally assist the pacemaker in achieving this goal, thus providing evidence for stochastic resonance on weakly paced scale-free networks. Because of the inherent degree inhomogeneity of individual oscillators forming the scale-free network, the placement of the pacemaker within the network is thereby crucial. As two extremes, we consider separately the introduction of the pacemaker to the oscillator with the highest degree and to one of the oscillators having the lowest degree. In both cases the coupling strength plays a crucial role, since it determines to what extent the whole network will follow the pacemaker on the expense of a weaker correlation between the pacemaker and the units that are directly linked with the paced oscillator. Higher coupling strengths facilitate the global outreach of the pacemaker, but require higher noise intensities for the optimal response. In contrast, lower coupling strengths and comparatively low noise intensities localize the optimal response to immediate neighbors of the paced oscillator. If the pacemaker is introduced to the main hub, the transition between the locally and globally optimal responses is characterized by a double resonance that postulates the existence of an optimal coupling strength for the transmission of weak rhythmic activity across scale-free networks. We corroborate the importance of the inhomogeneous structure of scale-free networks by additionally considering regular networks of oscillators with different degrees of coupling.

Control of scroll wave turbulence in a three-dimensional reaction-diffusion system with gradient

In this paper, we summarize our recent experimental and theoretical works on observation and control of scroll wave (SW) turbulence. The experiments were conducted in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system with chemical concentration gradients in one dimension. A spatially homogeneous external forcing was used in the experiments as a control; it was realized by illuminating white light on the light sensitive reaction medium. We observed that, in the oscillatory regime of the system, SW can appear automatically in the gradient system, which will be led to spatiotemporal chaos under certain conditions. A suitable periodic forcing may stabilize inherent turbulence of SW. The mechanism of the transition to SW turbulence is due to the phase twist of SW in the presence of chemical gradients, while modulating the phase twist with a proper periodic forcing can delay this transition. Using the FitzHugh-Nagumo model with an external periodic forcing, we confirmed the control mechanism with numerical simulation. Moreover, we also show in the simulation that adding temporal external noise to the system may have the same control effect. During this process, we observed a new state called "intermittent turbulence," which may undergo a transition into a new type of SW collapse when the noise intensity is further increased. The intermittent state and the collapse could be explained by a random process.

Control of spiral turbulence by periodic forcing in a reaction-diffusion system with gradients

We report an experimental result on successfully controlling spiral turbulence in a reaction-diffusion system. The control is realized by periodic forcing in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system, which has chemical concentration gradients in the third dimension. We observe that, in the oscillatory regime of the system, a suitable periodic forcing may stabilize scroll waves (SWs), which otherwise undergo a transition to spiral turbulence. Relating the spiral phase shift due to gradients and the forcing frequency, the mechanism of the control can be well understood by modulating the phase twist of SWs. We use the FitzHugh-Nagumo model to demonstrate this mechanism.

Control of spiking regularity in a noisy complex neural network

The effects of spatiotemporally correlated noise on the regularity of spiking oscillations are studied in a network composed of Fitz-Hugh-Nagumo neurons. The spiking regularity of the neural network becomes the best at a moderate noise intensity, indicating the occurrence of coherence resonance (CR). The CR in a Watts-Strogatz small-world network is further improved by adding a small fraction of long-range connections. Given a set of temporal correlation constant tau and spatial correlation length lambda of the noise, there exists an optimal network topology randomness, at which the spiking oscillations show the best regularity. The optimal randomness of the network topology at different tau and lambda varies in a narrow range. Changing lambda does not affect the optimal tau for achieving the most regular spike train, whereas varying tau, the best spiking regularity emerges at different optimal lambda.

Stochastic resonance on excitable small-world networks via a pacemaker

We show that the correlation between the frequency of subthreshold pacemaker activity and the response of an excitable array is resonantly dependent on the intensity of additive spatiotemporal noise. Thereby, the effect of the underlying network, defining the interactions among excitable units, largely depends on the coupling strength. Only for intermediate coupling strengths is the small world property able to enhance the stochastic resonance, whereas for smaller and larger couplings the impact of the transition from diffusive to random networks is less profound. Thus, the optimal interplay between a localized source of weak rhythmic activity and the response of the whole array demands a delicate balance between the strength of excitation transfer and the effectiveness of the network structure to support it.

Spiking regularity in a noisy small-world neuronal network

The regularity of spiking oscillations is studied in the networks with different topological structures. The network is composed of coupled Fitz-Hugh-Nagumo neurons driven by colored noise. The investigation illustrates that the spike train in both the regular and the Watts-Strogatz small-world neuronal networks can show the best regularity at a moderate noise intensity, indicating the existence of coherence resonance. Moreover, the temporal coherence of the spike train in the small-world network is superior to that in a regular network due to the increase of the randomness of the network topology. Besides the noise intensity, the spiking regularity can be optimized by tuning the randomness of the network topological structure or by tuning the correlation time of the colored noise. In particular, under the cooperation of the small-world topology and the correlation time, the spike train with good regularity could sustain a large magnitude of the local noise.

Sparse repulsive coupling enhances synchronization in complex networks

Through the last years, different strategies to enhance synchronization in complex networks have been proposed. In this work, we show that synchronization of nonidentical dynamical units that are attractively coupled in a small-world network is strongly improved by just making phase-repulsive a tiny fraction of the couplings. By a purely topological analysis that does not depend on the dynamical model, we link the emerging dynamical behavior with the structural properties of the sparsely coupled repulsive network.