Topological constraints on scroll and spiral waves in excitable media

A conservation equation for topological charges of phase singularities (scroll and spiral waves) in excitable media is given. It provides some topological properties of scroll (spiral) waves: for example, the topological charge of the generated or annihilated spiral pair must be opposite. Additionally, we obtain another equation on scroll waves, which shows that singular filaments of scroll waves occur on a set of one-dimensional curves which may be either closed loops or infinite lines.

Desynchronization of cells on the developmental path triggers the formation of spiral waves of cAMP during Dictyostelium aggregation

Whereas it is relatively easy to account for the formation of concentric (target) waves of cAMP in the course of Dictyostelium discoideum aggregation after starvation, the origin of spiral waves remains obscure. We investigate a physiologically plausible mechanism for the spontaneous formation of spiral waves of cAMP in D. discoideum. The scenario relies on the developmental path associated with the continuous changes in the activity of enzymes such as adenylate cyclase and phosphodiesterase observed during the hours that follow starvation. These changes bring the cells successively from a nonexcitable state to an excitable state in which they relay suprathreshold cAMP pulses, and then to autonomous oscillations of cAMP, before the system returns to an excitable state. By analyzing a model for cAMP signaling based on receptor desensitization, we show that the desynchronization of cells on this developmental path triggers the formation of fully developed spirals of cAMP. Developmental paths that do not correspond to the sequence of dynamic transitions no relay-relay-oscillations-relay are less able or fail to give rise to the formation of spirals.

Selection for spiral waves in the social amoebae Dictyostelium

Starving Dictyostelium amoebae emit pulses of the chemoattractant cAMP that are relayed from cell to cell as circular and spiral waves. We have recently modeled spiral wave formation in Dictyostelium. Our model suggests that a secreted protein inhibitor of an extracellular cAMP phosphodiesterase selects for spirals. Herein we test the essential features of this prediction by comparing wave propagation in wild type and inhibitor mutants. We find that mutants rarely form spirals. The territory size of mutant strains is approximately 50 times smaller than wild type, and the mature fruiting bodies are smaller but otherwise normal. These results identify a mechanism for selecting one wave symmetry over another in an excitable system and suggest that the phosphodiesterase inhibitor may be under selection because it helps regulate territory size.

Spiral-Wave Turbulence and Its Control in the Presence of Inhomogeneities in Four Mathematical Models of Cardiac Tissue

Regular electrical activation waves in cardiac tissue lead to the rhythmic contraction and expansion of the heart that ensures blood supply to the whole body. Irregularities in the propagation of these activation waves can result in cardiac arrhythmias, like ventricular tachycardia (VT) and ventricular fibrillation (VF), which are major causes of death in the industrialised world. Indeed there is growing consensus that spiral or scroll waves of electrical activation in cardiac tissue are associated with VT, whereas, when these waves break to yield spiral- or scroll-wave turbulence, VT develops into life-threatening VF: in the absence of medical intervention, this makes the heart incapable of pumping blood and a patient dies in roughly two-and-a-half minutes after the initiation of VF. Thus studies of spiral- and scroll-wave dynamics in cardiac tissue pose important challenges for in vivo and in vitro experimental studies and for in silico numerical studies of mathematical models for cardiac tissue. A major goal here is to develop low-amplitude defibrillation schemes for the elimination of VT and VF, especially in the presence of inhomogeneities that occur commonly in cardiac tissue. We present a detailed and systematic study of spiral- and scroll-wave turbulence and spatiotemporal chaos in four mathematical models for cardiac tissue, namely, the Panfilov, Luo-Rudy phase 1 (LRI), reduced Priebe-Beuckelmann (RPB) models, and the model of ten Tusscher, Noble, Noble, and Panfilov (TNNP). In particular, we use extensive numerical simulations to elucidate the interaction of spiral and scroll waves in these models with conduction and ionic inhomogeneities; we also examine the suppression of spiral- and scroll-wave turbulence by low-amplitude control pulses. Our central qualitative result is that, in all these models, the dynamics of such spiral waves depends very sensitively on such inhomogeneities. We also study two types of control chemes that have been suggested for the control of spiral turbulence, via low amplitude current pulses, in such mathematical models for cardiac tissue; our investigations here are designed to examine the efficacy of such control schemes in the presence of inhomogeneities. We find that a local pulsing scheme does not suppress spiral turbulence in the presence of inhomogeneities; but a scheme that uses control pulses on a spatially extended mesh is more successful in the elimination of spiral turbulence. We discuss the theoretical and experimental implications of our study that have a direct bearing on defibrillation, the control of life-threatening cardiac arrhythmias such as ventricular fibrillation.

Spiral-wave dynamics in ionically realistic mathematical models for human ventricular tissue: the effects of periodic deformation

We carry out an extensive numerical study of the dynamics of spiral waves of electrical activation, in the presence of periodic deformation (PD) in two-dimensional simulation domains, in the biophysically realistic mathematical models of human ventricular tissue due to (a) ten-Tusscher and Panfilov (the TP06 model) and (b) ten-Tusscher, Noble, Noble, and Panfilov (the TNNPO4 model). We first consider simulations in cable-type domains, in which we calculate the conduction velocity theta and the wavelength lambda of a plane wave; we show that PD leads to a periodic, spatial modulation of theta and a temporally periodic modulation of lambda; both these modulations depend on the amplitude and frequency of the PD. We then examine three types of initial conditions for both TP06 and TNNPO4 models and show that the imposition of PD leads to a rich variety of spatiotemporal patterns in the transmembrane potential including states with a single rotating spiral (RS) wave, a spiral-turbulence (ST) state with a single meandering spiral, an ST state with multiple broken spirals, and a state SA in which all spirals are absorbed at the boundaries of our simulation domain. We find, for both TP06 and TNNPO4 models, that spiral-wave dynamics depends sensitively on the amplitude and frequency of PD and the initial condition. We examine how these different types of spiral-wave states can be eliminated in the presence of PD by the application of low-amplitude pulses by square- and rectangular-mesh suppression techniques. We suggest specific experiments that can test the results of our simulations.

Development and transition of spiral wave in the coupled Hindmarsh-Rose neurons in two-dimensional space

The dynamics and the transition of spiral waves in the coupled Hindmarsh-Rose (H-R) neurons in two-dimensional space are investigated in the paper. It is found that the spiral wave can be induced and developed in the coupled HR neurons in two-dimensional space, with appropriate initial values and a parameter region given. However, the spiral wave could encounter instability when the intensity of the external current reaches a threshold value of 1.945. The transition of spiral wave is found to be affected by coupling intensity D and bifurcation parameter r. The spiral wave becomes sparse as the coupling intensity increases, while the spiral wave is eliminated and the whole neuronal system becomes homogeneous as the bifurcation parameter increases to a certain threshold value. Then the coupling action of the four sub-adjacent neurons, which is described by coupling coefficient D', is also considered, and it is found that the spiral wave begins to breakup due to the introduced coupling action from the sub-adjacent neurons (or sites) and together with the coupling action of the nearest-neighbour neurons, which is described by the coupling intensity D.

Travelling waves in a complex Ginzburg-Landau equation with time-delay feedback

One of the simplest ways to create nonlinear oscillations is the Hopf bifurcation. The spatiotemporal dynamics observed in an extended medium with diffusion (e.g., a chemical reaction) undergoing this bifurcation is governed by the complex Ginzburg-Landau equation, one of the best-studied generic models for pattern formation, where besides uniform oscillations, spiral waves, coherent structures and turbulence are found. The presence of time delay terms in this equation changes the pattern formation scenario, and different kind of travelling waves have been reported. In particular, we study the complex Ginzburg-Landau equation that contains local and global time-delay feedback terms. We focus our attention on plane wave solutions in this model. The first novel result is the derivation of the plane wave solution in the presence of time-delay feedback with global and local contributions. The second and more important result of this study consists of a linear stability analysis of plane waves in that model. Evaluation of the eigenvalue equation does not show stabilisation of plane waves for the parameters studied. We discuss these results and compare to results of other models.

The Mathematical modeling of inhomogeneites in ventricular tissue

Cardiac arrhythmias such as ventricular tachycardia (VT) or ventricular fibrillation (VF) are the leading cause of death in the industrialised world. There is a growing consensus that these arrhythmias arise because of the formation of spiral waves of electrical activation in cardiac tissue; unbroken spiral waves are associated with VT and broken ones with VF. Several experimental studies have been carried out to determine the effects of inhomogeneities in cardiac tissue on such arrhythmias. We give a brief overview of such experiments, and then an introduction to partial-differential-equation models for ventricular tissue. We show how different types of inhomogeneities can be included in such models, and then discuss various numerical studies, including our own, of the effects of these inhomogeneities on spiral-wave dynamics. The most remarkable qualitative conclusion of our studies is that the spiral-wave dynamics in such systems depends very sensitively on the positions of these inhomogeneities.

Nonequilibrium arrhythmic states and transitions in a mathematical model for diffuse fibrosis in human cardiac tissue

We present a comprehensive numerical study of spiral-and scroll-wave dynamics in a state-of-the-art mathematical model for human ventricular tissue with fiber rotation, transmural heterogeneity, myocytes, and fibroblasts. Our mathematical model introduces fibroblasts randomly, to mimic diffuse fibrosis, in the ten Tusscher-Noble-Noble-Panfilov (TNNP) model for human ventricular tissue; the passive fibroblasts in our model do not exhibit an action potential in the absence of coupling with myocytes; and we allow for a coupling between nearby myocytes and fibroblasts. Our study of a single myocyte-fibroblast (MF) composite, with a single myocyte coupled to N-f fibroblasts via a gap-junctional conductance G(gap), reveals five qualitatively different responses for this composite. Our investigations of two-dimensional domains with a random distribution of fibroblasts in a myocyte background reveal that, as the percentage P-f of fibroblasts increases, the conduction velocity of a plane wave decreases until there is conduction failure. If we consider spiral-wave dynamics in such a medium we find, in two dimensions, a variety of nonequilibrium states, temporally periodic, quasiperiodic, chaotic, and quiescent, and an intricate sequence of transitions between them; we also study the analogous sequence of transitions for three-dimensional scroll waves in a three-dimensional version of our mathematical model that includes both fiber rotation and transmural heterogeneity. We thus elucidate random-fibrosis-induced nonequilibrium transitions, which lead to conduction block for spiral waves in two dimensions and scroll waves in three dimensions. We explore possible experimental implications of our mathematical and numerical studies for plane-, spiral-, and scroll-wave dynamics in cardiac tissue with fibrosis.

Turbulent electrical activity at sharp-edged inexcitable obstacles in a model for human cardiac tissue

Wave propagation around various geometric expansions, structures, and obstacles in cardiac tissue may result in the formation of unidirectional block of wave propagation and the onset of reentrant arrhythmias in the heart. Therefore, we investigated the conditions under which reentrant spiral waves can be generated by high-frequency stimulation at sharp-edged obstacles in the ten Tusscher-Noble-Noble-Panfilov (TNNP) ionic model for human cardiac tissue. We show that, in a large range of parameters that account for the conductance of major inward and outward ionic currents of the model fast inward Na+ current (INa), L-type slow inward Ca2+ current (I-CaL), slow delayed-rectifier current (I-Ks), rapid delayed-rectifier current (I-Kr), inward rectifier K+ current (I-K1)], the critical period necessary for spiral formation is close to the period of a spiral wave rotating in the same tissue. We also show that there is a minimal size of the obstacle for which formation of spirals is possible; this size is similar to 2.5 cm and decreases with a decrease in the excitability of cardiac tissue. We show that other factors, such as the obstacle thickness and direction of wave propagation in relation to the obstacle, are of secondary importance and affect the conditions for spiral wave initiation only slightly. We also perform studies for obstacle shapes derived from experimental measurements of infarction scars and show that the formation of spiral waves there is facilitated by tissue remodeling around it. Overall, we demonstrate that the formation of reentrant sources around inexcitable obstacles is a potential mechanism for the onset of cardiac arrhythmias in the presence of a fast heart rate.

Dynamics of Vortex-Wave under a Travelling-Wave Modulation

The mecha nism of destabilization is studied for the rotating vortices (scroll waves and spiral waves) in excitable media induced by a parameter modulation in the form of a travelling-wave. It is found that a rigid rotating spiral in the two-dimensional (2D) system undergoes asynchronized drift along a straightline, and a 3D scrolling with its filament closed into a circle can be reoriented only if the direction of wavenumber of a travelling-wave perturbation is parallel to the ring plane. Then, in order to describe the behaviour of the synchronized drift of spiral wave and the reorientation of scrollring, the approximate formulas are given to exhibit qualitative agreements with the observed results.

A Galactic ring of minimum stellar density near the solar orbit radius

We analyse the secular effects of a long-lived Galactic spiral structure on the stellar orbits with mean radii close to the corotation resonance. By test-particle simulations and different spiral potential models with parameters constrained on observations, we verified the formation of a minimum with amplitude ∼30–40 per cent of the background disc stellar density at corotation. Such a minimum is formed by the secular angular momentum transfer between stars and the spiral density wave on both sides of corotation. We demonstrate that the secular loss (gain) of angular momentum and decrease (increase) of mean orbital radius of stars just inside (outside) corotation can counterbalance the opposite trend of exchange of angular momentum shown by stars orbiting the librational points L4/5 at the corotation circle. Such secular processes actually allow steady spiral waves to promote radial migration across corotation. We propose some pieces of observational evidence for the minimum stellar density in the Galactic disc, such as its direct relation to the minimum in the observed rotation curve of the Galaxy at the radius r ∼ 9 kpc (for R0 = 7.5 kpc), as well as its association with a minimum in the distribution of Galactic radii of a sample of open clusters older than 1Gyr. The closeness of the solar orbit adius to the corotation resonance implies that the solar orbit lies inside a ring of minimum surface density (stellar + gas). This also implies a correction to larger values for the estimated total mass of the Galactic disc, and consequently, a greater contribution of the disc componente to the inner rotation curve of the Galaxy.

Directed percolation in a two dimensional stochastic fire-diffuse-fire model

In this paper we establish, from extensive numerical experiments, that the two dimensional stochastic fire-diffuse-fire model belongs to the directed percolation universality class. This model is an idealized model of intracellular calcium release that retains the both the discrete nature of calcium stores and the stochastic nature of release. It is formed from an array of noisy threshold elements that are coupled only by a diffusing signal. The model supports spontaneous release events that can merge to form spreading circular and spiral waves of activity. The critical level of noise required for the system to exhibit a non-equilibrium phase-transition between propagating and non-propagating waves is obtained by an examination of the \textit{local slope} $\delta(t)$ of the survival probability, $\Pi(t) \propto \exp(- \delta(t))$, for a wave to propagate for a time $t$.

Finite Element Modeling of Spiral Frequency Steerable Acoustic Transducers (FSATs) for guided waves based Structural Health Monitoring of plate-like structures

Structural Health Monitoring (SHM) is an emerging area of research associated to improvement of maintainability and the safety of aerospace, civil and mechanical infrastructures by means of monitoring and damage detection. Guided wave structural testing method is an approach for health monitoring of plate-like structures using smart material piezoelectric transducers. Among many kinds of transducers, the ones that have beam steering feature can perform more accurate surface interrogation. A frequency steerable acoustic transducer (FSATs) is capable of beam steering by varying the input frequency and consequently can detect and localize damage in structures. Guided wave inspection is typically performed through phased arrays which feature a large number of piezoelectric transducers, complexity and limitations. To overcome the weight penalty, the complex circuity and maintenance concern associated with wiring a large number of transducers, new FSATs are proposed that present inherent directional capabilities when generating and sensing elastic waves. The first generation of Spiral FSAT has two main limitations. First, waves are excited or sensed in one direction and in the opposite one (180 ̊ ambiguity) and second, just a relatively rude approximation of the desired directivity has been attained. Second generation of Spiral FSAT is proposed to overcome the first generation limitations. The importance of simulation tools becomes higher when a new idea is proposed and starts to be developed. The shaped transducer concept, especially the second generation of spiral FSAT is a novel idea in guided waves based of Structural Health Monitoring systems, hence finding a simulation tool is a necessity to develop various design aspects of this innovative transducer. In this work, the numerical simulation of the 1st and 2nd generations of Spiral FSAT has been conducted to prove the directional capability of excited guided waves through a plate-like structure.

Design and implementation of wideband baluns for Archimedean spiral antennas

The demand for high-speed data services for portable device has become a driving force for development of advanced broadband access technologies. Despite recent advances in broadband wireless technologies, there remain a number of critical issues to be resolved. One of the major concerns is the implementation of compact antennas that can operate in a wide frequency band. Spiral antenna has been used extensively for broadband applications due to its planar structure, wide bandwidth characteristics and circular polarisation. However, the practical implementation of spiral antennas is challenged by its high input characteristic impedance, relatively low gain and the need for balanced feeding structures. Further development of wideband balanced feeding structures for spiral antennas with matching impedance capabilities remain a need. This thesis proposes three wideband feeding systems for spiral antennas which are compatible with wideband array antenna geometries. First, a novel tapered geometry is proposed for a symmetric coplanar waveguide (CPW) to coplanar strip line (CPS) wideband balun. This balun can achieve the unbalanced to balanced transformation while matching the high input impedance of the antenna to a reference impedance of 50 . The discontinuity between CPW and CPS is accommodated by using a radial stub and bond wires. The bandwidth of the balun is improved by appropriately tapering the CPW line instead of using a stepped impedance transformer. Next, the tapered design is applied to an asymmetric CPW to propose a novel asymmetric CPW to CPS wideband balun. The use of asymmetric CPW does away with the discontinuities between CPW and CPS without having to use a radial stub or bond wires. Finally, a tapered microstrip line to parallel striplines balun is proposed. The balun consists of two sections. One section is the parallel striplines which are connected to the antenna, with the impedance of balanced line equal to the antenna input impedance. The other section consists of a microstrip line where the width of the ground plane is gradually reduced to eventually resemble a parallel stripline. The taper accomplishes the mode and impedance transformation. This balun has significantly improved bandwidth characteristics. Characteristics of proposed feeding structures are measured in a back-to-back configuration and compared to simulated results. The simulated and measured results show the tapered microstrip to parallel striplines balun to have more than three octaves of bandwidth. The tapered microstrip line to parallel striplines balun is integrated with a single Archimedean spiral antenna and with an array of spiral antennas. The performance of the integrated structures is simulated with the aid of electromagnetic simulation software, and results are compared to measurements. The back-to-back microstrip to parallel strip balun has a return loss of better than 10 dB over a wide bandwidth from 1.75 to 15 GHz. The performance of the microstrip to parallel strip balun was validated with the spiral antennas. The results show the balun to be an effective mean of feeding network with a low profile and wide bandwidth (2.5 to 15 GHz) for balanced spiral antennas.