Propagation Of Spiral Waves Research Articles

Inter-layer propagation of spiral waves: Effects of time-varying defect blocks and magnetic flows

The variation of magnetic flux with distance and time is relatively common in nature. This paper investigates the inter-layer propagation of spiral waves in bi-layer Hindmarsh-Rose networks under gradient magnetic field and defect blocks motion, with the following conclusions: (1) Magnetic flux slows down the propagation of waves. (2) For defect blocks moving linearly or away from each other, spiral waves appear in the network, while the magnetic flux suppresses the spiral wave formation. (3) For defect blocks close to or merging with each other, the magnetic flow can only convert spiral waves in the network into “eye-like waves”. These findings imply that the propagation of spiral waves is sensitive to the action site of magnetic flux, which may be important for scar tissue (e.g., cardiac tissue with poor postoperative recovery). Moreover, three statistical parameters defined to detect waveform evolution in excitable media show good performance.

Spiral Wave Propagation in Communities with Spatially Correlated Heterogeneity

Many multicellular communities propagate signals in a directed manner via excitable waves. Cell-to-cell heterogeneity is a ubiquitous feature of multicellular communities, but the effects of heterogeneity on wave propagation are still unclear. Here, we use a minimal FitzHugh-Nagumo-type model to investigate excitable wave propagation in a two-dimensional heterogeneous community. The model shows three dynamic regimes in which waves either propagate directionally, die out, or spiral indefinitely, and we characterize how these regimes depend on the heterogeneity parameters. We find that in some parameter regimes, spatial correlations in the heterogeneity enhance directional propagation and suppress spiraling. However, in other regimes, spatial correlations promote spiraling, a surprising feature that we explain by demonstrating that these spirals form by a second, distinct mechanism. Finally, we characterize the dynamics using techniques from percolation theory. Despite the fact that percolation theory does not completely describe the dynamics quantitatively because it neglects the details of the excitable propagation, we find that it accounts for the transitions between the dynamic regimes and the general dependency of the spiral period on the heterogeneity and thus provides important insights. Our results reveal that the spatial structure of cell-to-cell heterogeneity can have important consequences for signal propagation in cellular communities.

Propagation of spiral waves pinned to circular and rectangular obstacles.

We present an investigation of spiral waves pinned to circular and rectangular obstacles with different circumferences in both thin layers of the Belousov-Zhabotinsky reaction and numerical simulations with the Oregonator model. For circular objects, the area always increases with the circumference. In contrast, we varied the circumference of rectangles with equal areas by adjusting their width w and height h. For both obstacle forms, the propagating parameters (i.e., wavelength, wave period, and velocity of pinned spiral waves) increase with the circumference, regardless of the obstacle area. Despite these common features of the parameters, the forms of pinned spiral waves depend on the obstacle shapes. The structures of spiral waves pinned to circles as well as rectangles with the ratio w/h∼1 are similar to Archimedean spirals. When w/h increases, deformations of the spiral shapes are observed. For extremely thin rectangles with w/h≫1, these shapes can be constructed by employing semicircles with different radii which relate to the obstacle width and the core diameter of free spirals.

Origin and Characteristics of High Shannon Entropy at the Pivot of Locally Stable Rotors: Insights from Computational Simulation

BackgroundRotors are postulated to maintain cardiac fibrillation. Despite the importance of bipolar electrograms in clinical electrophysiology, few data exist on the properties of bipolar electrograms at rotor sites. The pivot of a spiral wave is characterized by relative uncertainty of wavefront propagation direction compared to the periphery. The bipolar electrograms used in electrophysiology recording encode information on both direction and timing of approaching wavefronts.ObjectiveTo test the hypothesis that bipolar electrograms from the pivot of rotors have higher Shannon entropy (ShEn) than electrograms recorded at the periphery due to the spatial dynamics of spiral waves.Methods and ResultsWe studied spiral wave propagation in 2-dimensional sheets constructed using a simple cell automaton (FitzHugh-Nagumo), atrial (Courtemanche-Ramirez-Nattel) and ventricular (Luo-Rudy) myocyte cell models and in a geometric model spiral wave. In each system, bipolar electrogram recordings were simulated, and Shannon entropy maps constructed as a measure of electrogram information content. ShEn was consistently highest in the pivoting region associated with the phase singularity of the spiral wave. This property was consistently preserved across; (i) variation of model system (ii) alterations in bipolar electrode spacing, (iii) alternative bipolar electrode orientation (iv) bipolar electrogram filtering and (v) in the presence of rotor meander. Directional activation plots demonstrated that the origin of high ShEn at the pivot was the directional diversity of wavefront propagation observed in this location.ConclusionsThe pivot of the rotor is consistently associated with high Shannon entropy of bipolar electrograms despite differences in action potential model, bipolar electrode spacing, signal filtering and rotor meander. Maximum ShEn is co-located with the pivot for rotors observed in the bipolar electrogram recording mode, and may be an intrinsic property of spiral wave dynamic behaviour.

Numerical study of the death and breakup of spiral wave in the networks of thermosensitive neurons

The experimential evidences confirm that spiral waves are observed in the disinhibited mammalian neocortex. The scheme of ceullar networks is used to simulate the formation and the evolution of spiral wave in the neocortical slices. The regular networks of neurons are constructed in the two-dimensional space, the dynamical properties of thermosensitive neurons is described by temperature factor, and the effect of membrane temperature on the evolution of spiral wave is investigated in detail. A statistical factor of synchronization is defined to measure the critical condition inducing phase transition of spiral wave (death or breakup) by the factor of temperature of membrane. It is confirmed that spiral wave is removed and the whole networks become homogeneous and synchronous completely when the membrane temperature exceeds a certain threshold; the breakup of spiral wave is induced in the presence of weak channel noise being considered. Furthermore, it is suggested that the mechanism of temporary heat stupor could be the blocking of spiral wave propagation in some functional domain.

Line-defect patterns of unstable spiral waves in cardiac tissue

Spiral wave propagation in period-2 excitable media is accompanied by line defects, the locus of points with period-1 oscillations. Here we investigate spiral line defects in cardiac tissue where period-2 behavior has a known arrhythmogenic role. We find that the number of line defects, which is constrained to be an odd integer, is 3 for a freely rotating spiral, with and without meander, but 1 for a spiral anchored around a fixed heterogeneity. We interpret this finding analytically using a simple theory where spiral wave unstable modes with different numbers of line defects correspond to quantized solutions of a Helmholtz equation. Furthermore, the slow inward rotation of spiral line defects is described in different regimes.

Sinklike spiral waves in oscillatory media with a disk-shaped inhomogeneity

Spiral wave propagation in oscillatory media with a disk-shaped inhomogeneity is examined. Depending on the properties of the medium as well as the inhomogeneity (different frequencies in two regions), distinct spiral waves including sinklike spirals and dense-sparse spirals, are able to emerge. We find that, unlike the previously found outward group velocity for spiral waves (normal spirals or antispirals), the direction of the velocity of the sinklike spiral wave points inward. A qualitative analysis of the possible mechanism underlying their formation is discussed, considering the inhomogeneity as a wave sink or source. Numerical simulations performed on other typical reaction-diffusion models confirm this analysis and suggest that our findings are robust and could be observed in experiments.

Single spiral wave induced by noise

Reaction systems far from thermodynamic equilibrium can form a fascinating wealth of spatio-temporal pattern. Noise is usually thought to spoil order. In our numerical simulation, the reaction diffusion system disturbed by environmental noise can constitute a stable well-ordered spatio-temporal pattern, chemical single spiral waves in two dimensions spatial domain. It is counterintuitive that noise plays a constructive role in the production and propagation of single spiral waves in this letter. When noise intensity exceeds a threshold, it will destroy the well-ordered spatio-temporal pattern and make system uniform. Too weak or too strong noise will make against the formation and propagation of spiral waves. In a certain noise level, spiral wave can be maintained in a medium, where such spiral cannot be observed in the absence of the noise.

SPIRAL WAVE MAINTAINED BY NOISE

The effect of Gaussian white noise on a chemical wavefront is studied in a modified FitzHugn–Nagumo model by applying numerical simulations. A rotating spiral waves can be formed if the medium is excitable enough and the fronts has a free end, when the reaction diffusion system is disturbed by a certain non-zero level noise. It is counterintuitive that noise plays a constructive role in the product and propagation of single spiral waves in this letter. Weak or strong noise will make against the product and propagation of spiral waves. In a certain noise level, spiral wave can be maintained in a medium, where such spiral waves cannot be observed in the absence of the noise.

Spiral wave break-up and planar front formation in two-dimensional reactive–diffusive media with straining

The propagation of spiral waves in two-dimensional reactive–diffusive media subject to a velocity field with straining is studied numerically. It is shown that, depending on the magnitude of the strain rate, the effects of straining on wave propagation can be classified in three regimes: the no break-up, the transitional and the break-up regimes. In the no break-up regime, the spiral wave preserves its integrity, whereas, in the transitional one, the spiral wave is deformed and transformed into a curved front which propagates almost along one of the principal directions of the strain rate tensor. In the break-up regime, spiral waves are broken up into many spiral waves, and planar fronts aligned with the principal direction of the strain rate tensor corresponding to elongation/stretching are formed. These fronts may be split into two new ones or are bands, the thickness of which decreases as the strain rate is increased.

Spiral waves in three-dimensional excitable media with light-sensitive reaction

The propagation of spiral waves in three-dimensional excitable media with and without forcing is studied numerically, and it is shown that, when localized forcing is imposed in a cube located in the middle of the computational domain, the spiral wave preserves its integrity, although it exhibits a dent along the cube’s periphery and is unable to penetrate into the cube. However, when forcing is applied in a parallelepiped which extends across the entire domain in the vertical direction and is located in the middle of the three-dimensional excitable medium, the spiral wave is destroyed by the strip forcing and two active regions are created on either side of the strip. These active regions are characterized by high concentration of the activator, move towards the boundaries parallel to the strip, and, upon reaching them, move rather quickly in a direction parallel to the strip. They also form complex spatial patterns, e.g., ear-shaped regions, and convoluted annular regions on both sides of the strip, where the activator’s concentration is small. These spatio-temporal patterns are periodic with a period and a width which are larger than those corresponding to the localized forcing. It is also shown that strip forcing not only destroys the spiral wave, it also increases the period and width of the breathing-type motions that result from such a destruction.

Spatio-temporal patterns in two-dimensional excitable media subject to Robin boundary conditions

A numerical study of spiral wave propagation in two-dimensional excitable media subject to homogeneous Robin boundary conditions is presented, and it is shown that, depending on the magnitude of the transfer coefficient, periodic, almost periodic and nonperiodic dynamics may be observed. For transfer coefficients equal to or smaller than one, spiral waves almost identical to those obtained with homogeneous Neumann boundary conditions are observed, whereas arms of spiral waves, layers along the boundaries of the domain, break-up, attachment to and detachment from boundaries, islands and complex spatio-temporal patterns result when the transfer coefficients for both the activator and inhibitor are larger than one along either the top and bottom boundaries or the left and right boundaries. When the transfer coefficients are larger than one along all the boundaries of the domain, it has been found that there is a layer of high activator’s concentration along the boundaries and an arm of spiral waves may appear and interact with the boundary layers, and the dynamics of the excitable medium is complex and nonperiodic.

Pattern formation in two-dimensional reactive-diffusive media with straining

The propagation of spiral waves in two-dimensional reactive-diffusive media subject to a velocity field with straining and no-flux boundary conditions is studied numerically. It is shown that, at low strain rates, the spiral wave preserves its integrity, whereas, at moderate ones, the spiral wave is deformed and transformed into a curved front which propagates almost along one of the principal directions of the strain rate tensor. At large strain rates, corner layers and elliptical fronts that become planar ones are observed.

Dynamics of spiral waves in excitable media with local time-periodic modulation

The propagation of spiral waves in excitable media with locally inhomogeneous time-periodic modulations is studied numerically, and it is shown that the size of the local inhomogeneity and the amplitude and frequency of the periodic forcing play a major role in determining the spiral wave dynamics, suppression and breakup. It is also shown that, in the absence of breakup, spiral waves flatten and thicken as the size of the inhomogeneity is increased, and that spiral waves may penetrate into the modulation region if the forcing frequency is sufficiently small. At high frequencies, however, it is shown that the inhomogeneity behaves as an obstacle. It is also reported that, for sufficiently large modulation regions, the spiral wave may be annihilated and become an almost cylindrical wave with initially high concentration of the activator in the center of the inhomogeneity.

Tile patterns in excitable media subject to non-solenoidal flow fields

The propagation of spiral waves in excitable media subject to a non-solenoidal advective field which satisfies the no-penetration condition on the boundaries of the domain is studied numerically, and it is shown that, depending on the amplitude and spatial frequencies of the velocity field, the spiral wave may be distorted highly, break up into a number of smaller spiral waves, or exhibit polygonal shapes or tile patterns. These patterns reflect the symmetry/asymmetry of the velocity field and are characterized by thick regions of high concentration at stagnation points where the velocity gradient is largest, and thin ones which are parallel to the velocity vector. It is also shown that the advective field distorts the spiral wave by decreasing its thickness where the velocity is largest due to the stretching of the wave, and by increasing it at the stagnation points where the curvature of the wave is largest.

Convection-induced anisotropy in excitable media subject to solenoidal advective flow fields

The effects of solenoidal velocity fields on the propagation of spiral waves in excitable media is studied numerically by means of a time-linearized method. It is shown that the advective field distorts the spiral wave at moderate frequencies, whereas, at large frequencies, the average shape of the spiral wave is nearly identical to that in the absence of convection, although its inner and outer parts exhibit spatial oscillations whose frequency increases as that of the velocity field is increased. At low frequencies and high amplitudes of the velocity field, the concentration of the activator and the wave propagation are controlled by the symmetry of the velocity and the number and location of the stagnation points, and the concentration of the activator may exhibit either counter-rotating regions or a layered structure.

Spatio-temporal patterns in excitable media with non-solenoidal flow straining

The propagation of spiral waves in excitable media with the Belousov–Zhabotinsky reactions in a non-solenoidal, time-independent velocity field is studied numerically as a function of the amplitude and frequency of the velocity. It is shown that the spiral wave is slightly distorted for small amplitudes and low frequencies, whereas it breaks-up into new spiral waves which merge and form periodic, cusped fronts at moderate amplitudes and small frequencies. For larger amplitudes but still small frequencies, the spiral wave undergoes a second transition to thick fronts characterized by small curvature, and the radius of curvature increases as the amplitude of the velocity field is increased. It is also shown that an increase in the frequency of the velocity field results in front distorsion and corrugations which are due to the increase in the number of stagnation points as the frequency is increased, straining of the front at stagnation points and the non-solenoidal velocity field employed in the paper. An explanation of these corrugations in terms of the straining, gradient of the transverse velocity along the normal to the front and compressibility is provided.

Propagation of spiral waves in anisotropic media: from waves to stripes

The effects of anisotropic diffusion on the propagation of spiral waves in excitable media are studied numerically by means of time-linearized methods in very refined meshes and with very small time steps. It is shown that the anisotropy of the inhibitor's diffusivity tensor does not play as important role on wave propagation as that of the activator. It is also shown that the off-diagonal components of the activator's diffusivity tensor cause stretching of the activator's concentration along the principal directions of this tensor, and that a large difference between the diffusion coefficients in the x- and y-directions may result in the spiral wave annihilation and stripe formation.

Three-dimensional Simulations Of Spiral Waves In Reacting AndDiffusing Media On DSM Computers

Parallel ADI and CG methods based on the time-linearization of the three-dimensional, reaction-diffusion equations which govern the propagation of spiral waves in excitable media, have been implemented on both shared- and distributedmemory computers. ADI has been implemented by means of a dynamic block cartesian decomposition and its efficiency tends to that of the CG technique as the mesh is refined. The largest efficiency of CG was obtained on an Origin-2000 with a message passing model, private memory and 16 processors, and the efficiency of CG is larger than that of ADI.

Controlling Spiral Waves in Confined Geometries by Global Feedback

The evolution of spiral waves on a circular domain and on a spherical surface is studied by numerical integration of a reaction-diffusion system with a global feedback. It is shown that depending on intensity, sign, and/or time delay in the feedback loop a global coupling can be effectively used either to stabilize the rigid rotation of a spiral wave or to completely destroy spiral waves and to suppress self-sustained activity in a confined domain of an excitable medium. An explanation of the numerically observed effects is produced by a kinematical model of spiral wave propagation.