Curvature and Propagation Velocity of Chemical Waves

Curvature dependences of wave propagation in reaction–diffusion models

Reaction–diffusion waves in multiple spatial dimensions advance at a rate that strongly depends on the curvature of the wavefronts. These waves have important applications in many physical, ecological and biological systems. In this work, we analyse curvature dependences of travelling fronts in a single reaction–diffusion equation with general reaction term. We derive an exact, non-perturbative curvature dependence of the speed of travelling fronts that arises from transverse diffusion occurring parallel to the wavefront. Inward-propagating waves are characterized by three phases: an establishment phase dominated by initial and boundary conditions, a travelling-wave-like phase in which normal velocity matches standard results from singular perturbation theory and a dip-filling phase where the collision and interaction of fronts create additional curvature dependences to their progression rate. We analyse these behaviours and additional curvature dependences using a combination of asymptotic analyses and numerical simulations.

Traveling waves propagating through coupled microbeads in the Belousov-Zhabotinsky reaction

Spatio-temporal patterns, namely global oscillations (GO) and traveling waves (TW), were investigated in spherical microbeads loaded with a catalyst for the Belousov-Zhabotinsky (BZ) reaction onto the surface (2D-loaded) or the entire volume of the bead (3D-loaded). GO and TW selectively appeared in the 2D- and 3D-loaded beads, respectively, placed on a polyethylene terephthalate (PET) sheet in the catalyst-free BZ solution. We examined two types of coupling of the two beads: 2D-3D and 3D-3D couplings. In both cases, synchronization occurred when the minimum distance between the two beads, l, was shorter than the threshold. Herein, we reported not only temporal information, that is, phase difference, but also spatial information, that is, the directions of the TW propagating through the coupled BZ beads. In the synchronization for the 2D-3D coupling, TW in the 3D-loaded bead were initiated from the point near the 2D-loaded bead as a pacemaker and propagated in the opposite direction. By contrast, the directions of the TW in the 3D-loaded bead changed depending on l in the synchronization for the 3D-3D coupling. These experimental results can be quantitatively reproduced by numerical calculations based on the diffusion dynamics of an activator of the BZ reaction. Our results suggest that the features of spatio-temporal wave propagation are indicative of the configuration of the oscillators.

Geographic tongue as a reaction-diffusion system

Geographic tongue or benign migratory glossitis is a condition of an unknown cause characterized by chronic lesions that slowly migrate across the surface of the tongue. The condition's characteristic wavefronts suggest that it can be modeled as a reaction-diffusion system. Here, we present a model for geographic tongue pattern evolution using reaction-diffusion equations applied to portions of spheroids and paraboloids that approximate a tongue shape. We demonstrate that the observed patterns of geographic tongue lesions can be explained by propagating reaction-diffusion waves on these variably curved surfaces.

Excitable Wave Patterns in Temporal Systems with Two Long Delays and their Observation in a Semiconductor Laser Experiment

Excitable waves arise in many spatially extended systems of either a biological, chemical, or physical nature due to the interplay between local reaction and diffusion processes. Here we demonstrate that similar phenomena are encoded in the time dynamics of an excitable system with two, hierarchically long delays. The transition from 1D localized structures to curved wave segments is experimentally observed in an excitable semiconductor laser with two feedback loops and reproduced by numerical simulations of a prototypical model. While closely related to those found in 2D excitable media, wave patterns in delayed systems exhibit unobserved features originating from causality related constraints. An appropriate dynamical representation of the data uncovers these phenomena and permits us to interpret them as the result of an effective 2D advection-reaction-diffusion process.

Influence of reaction-induced convection on quorum sensing in enzyme-loaded agarose beads

In theory, groups of enzyme-loaded particles producing an acid or base may show complex behavior including dynamical quorum sensing, the appearance of synchronized oscillations above a critical number or density of particles. Here, experiments were performed with the enzyme urease loaded into mm-sized agarose beads and placed in a solution of urea, resulting in an increase in pH. This behavior was found to be dependent upon the number of beads present in the array; however, reaction-induced convection occurred and plumes of high pH developed that extended to the walls of the reactor. The convection resulted in the motion of the mm-sized particles and conversion of the solution to high pH. Simulations in a simple model of the beads demonstrated the suppression of dynamical quorum sensing in the presence of flow.

Front tracking velocimetry in advection-reaction-diffusion systems

In advection-reaction-diffusion systems, the spreading of a reactive scalar can be significantly influenced by the flow field in which it grows. In systems with sharp boundaries between reacted and unreacted regions, motion of the reaction fronts that lie at those boundaries can quantify spreading. Here, we present an algorithm for measuring the velocity of reaction fronts in the presence of flow, expanding previous work on tracking reaction fronts without flow. The algorithm provides localized measurements of front speed and can distinguish its two components: one from chemical dynamics and another from the underlying flow. We validate that the algorithm returns the expected front velocity components in two simulations and then show that in complex experimental flows, the measured front velocity maps fronts from one time step to the next self-consistently. Finally, we observe a variation of the chemical speed with flow speed in a variety of experiments with different time scales and length scales.

Front tracking for quantifying advection-reaction-diffusion

We present an algorithm for measuring the speed and thickness of reaction fronts, and from those quantities, the diffusivity and the reaction rate of the active chemical species. This front-tracking algorithm provides local measurements suitable for statistics and requires only a sequence of concentration fields. Though our eventual goal is front tracking in advection-reaction-diffusion, here we demonstrate the algorithm in reaction-diffusion. We test the algorithm with validation data in which front speed and thickness are prescribed, as well as simulation results in which diffusivity and reaction rate are prescribed. In all tests, measurements closely match true values. We apply the algorithm to laboratory experiments using the Belousov-Zhabotinsky reaction, producing speed, diffusivity, and reaction rate measurements that are statistically more robust than in prior studies. Finally, we use thickness measurements to quantify the concentration profile of chemical waves in the reaction.

Analyzing critical propagation in a reaction-diffusion-advection model using unstable slow waves

The effect of advection on the propagation and in particular on the critical minimal speed of traveling waves in a reaction-diffusion model is studied. Previous theoretical studies estimated this effect on the velocity of stable fast waves and predicted the existence of a critical advection strength below which propagating waves are not supported anymore. In this paper, an analytical expression for the advection-velocity relation of the unstable slow wave is derived. In addition, the critical advection strength is calculated taking into account the unstable slow wave solution. We also analyze a two-variable reaction-diffusion-advection model numerically in a wide parameter range. Due to the new control parameter (advection) we can find stable wave propagation in the otherwise non-excitable parameter regime, if the advection strength exceeds a critical value. Comparing theoretical predictions to numerical results, we find that they are in good agreement. Theory provides an explanation for the observed behaviour.

Tissue voltage discordance during tachycardia versus sinus rhythm: implications for catheter ablation

BACKGROUND

Electroanatomic mapping systems are an important tool to identify cardiac chamber voltage and assess channels of slow conduction.

OBJECTIVE

To assess the correlation between electroanatomic mapping voltage maps obtained during macroreentrant tachycardia compared to sinus rhythm (SR) with a contact mapping system.

METHODS

We retrospectively evaluated patients with atrial flutter (AFL) referred for radiofrequency ablation with electroanatomic voltage maps obtained during AFL and SR. The atrium was divided into predetermined segments. Overall atrial and segmental peak-to-peak bipolar voltages in AFL and SR were assessed. To directly compare a region within the same patient, tissue voltage differences during AFL and SR were assessed on the basis of mean voltage difference.

RESULTS

Sixteen patients (87% men) had available voltage maps. Eighty-one percent had typical cavotricuspid isthmus-dependent right AFL. A mean of 441.7±153.9 vs 398.1±125.4 total points (P = .22) were sampled during AFL and SR, with a mean of 99.5±58.9 vs 91.2±60.4 points (P = .45) sampled per region. Overall right atrial mean voltage was significantly higher during AFL than SR (0.554±0.092mV vs 0.473±0.079mV; P≤.001), with the lateral wall (0.707±0.120mV vs 0.573±0.097mV; P = .0004) and the cavotricuspid isthmus (0.559±0.100mV vs 0.356±0.066mV; P<.0001) also showing higher mean voltage during AFL. When compared within an individual patient, 19% (14 of 75) of the patient regions had a>0.5mV mean voltage difference and 40% (30 of 75) had a>0.25mV mean voltage difference.

CONCLUSIONS

These data suggest that voltage maps performed during macroreentrant atrial arrhythmias often vary significantly from maps obtained during SR.

Oscillatory kinetics and spatio-temporal self-organization in reactions at solid surfaces

Chemical reactions far from equilibrium on solid surfaces may exhibit typical phenomena of nonlinear dynamics, as exemplified by the catalytic oxidation of carbon monoxide on a platinum(110) single-crystal surface. Depending on the external parameters (temperature and partial pressures of the reactants), the temporal variation of the reaction rate may become oscillatory or even chaotic. In a parallel way, the concentration distributions of the adsorbed species on the surface form spatio-temporal patterns including propagating and standing waves, rotating spirals, as well as irregular and rapidly changing structures denoted "chemical turbulence."

Critical size and curvature of wave formation in an excitable chemical medium

The critical radius for the outward propagation of waves in an excitable solution of the Belousov-Zhabotinskii reaction was experimentally analyzed and found to be approximately 20 mum, being in a range predicted by theory. Thus, the wave initiation depends on the critical radius in an all-or-none fashion. For waves having high positive curvature of wave fronts, a linear relationship between the curvature and their normal velocity was established, allowing computation of a diffusion coefficient of 1.9 x 10(-5) cm(2)/s for the autocatalytic species, which agrees well with results previously obtained for negatively curved wave fronts. The analysis of the dispersion of wave velocity yielded the decrease of wave velocity for small initiation periods as predicted theoretically.

Micropatterning Chemical Oscillations: Waves, Autofocusing, and Symmetry Breaking

Arrays of chemical oscillators are micropatterned by Wet Stamping. The technique is used to demonstrate that chemical waves can be initiated and controlled in oscillatory systems and that such waves can give rise to phenomena not observed in excitable media. Interoscillator coupling and synchronization, kinetic autofocusing, and twist-symmetry breaking are a consequence of the dependence of the oscillation phase on the local concentrations of reagents and on systems' geometry. Conditions under which a generic oscillatory system would exhibit such behaviors are determined.

Reentry wave formation in excitable media with stochastically generated inhomogeneities

Clinical research shows that the frequency of arrhythmia events depends on the number and area of the border zones of infarct scars. We investigate the possibility that arrhythmia is initiated by reentry waves generated by the inhomogeneity of conduction velocity at the border zone. The interaction of a plane wave with a spatially extended inhomogeneity is simulated in the FitzHugh- Nagumo model. The inhomogeneity is introduced into the model by modifying the spatial dependence of the diffusion coefficient in a stochastic manner. This results in a rich variety of spatial distributions of conductivity. A plane wave propagating through such a system may break up on the regions with low conductivity and produce numerous spiral waves. The frequency of reentry wave formation is studied as a function of the parameters of the inhomogeneity generation algorithm. Three main scenarios of reentry wave formation were found: unidirectional block, main wave-wavelet collision, and wave break up during collision, on a region in which a conduction velocity gradient occurs. These scenarios are likely candidates for the mechanisms of arrhythmia initiation in a damaged tissue, e.g., the border zone of an infarct scar.

Front dynamics in an oscillatory bistable Belousov-Zhabotinsky chemical reaction

We observe breathing front dynamics which select three distinct types of bistable patterns in the 2:1 resonance regime of the periodically forced oscillatory Belousov-Zhabotinsky reaction. We measure the curvature-driven shrinking of a circular domain R approximately t(1/2) at forcing frequencies below a specific value, and show that the fast time scale front oscillations (breathing) drive this slow time scale shrinking. Above a specific frequency, we observe fronts of higher curvature grow instead of shrink and labyrinth patterns form. Just below the transition frequency is a relatively narrow range of frequencies where the curvature-driven coarsening is balanced by a competing front interaction, which leads to a pattern of localized structures. The length scale of the localized structure and labyrinth patterns is set by the front interactions.

Pattern selection in a cooperative biochemical in vitro amplification system: the role of parasites

Previously, numerical simulations have shown that evolving systems can be stabilized against emerging parasites by pattern formation in spatially extended flow reactors. Hence, it can be argued that pattern formation is a prerequisite for any experimental investigation of the biochemical evolution of cooperative function. Here, we study a model of an experimental biochemical system for the cooperative in vitro amplification of DNA strands and show that emerging parasites can induce a complex pattern formation even when no pattern formation occurs without parasites. In an adiabatic approximation where the cooperative amplification reaction is assumed to adapt fast to slowly emerging parasites, the parasite concentration itself acts as a Steuer parameter for the selection of various complex patterns. Without such an adiabatic approximation only transient patterns emerge. As any species can grow for very low concentrations, the parasite is able to infect the entire reactor and the system is finally diluted out. In the experimental biochemical system, however, the species are individual molecules and the growth of spatially separated, non-infected regions becomes feasible. Hence a cutoff threshold for the minimal concentration is applied. In these simulations the otherwise lethal infection by parasites induces the formation of spatiotemporal spirals, and this spatial structure help the host and parasitoid species to survive together. These theoretical results describe an inherent property of cooperative reactions and have an important impact on experimental investigations on the molecular evolution and complex function in spatially extended reactors. Since the formation of the complex pattern is restricted either to a rather large cutoff value or a special choice of the kinetic parameters, we, however, conclude that the persistence of evolving cooperative amplification is not possible in a simple reaction-diffusion reactor. Experimental set-ups with patchy environments, e.g. biomolecular amplification in coupled microstructured flow chambers or in microemulsion, are eligible candidates for the observation of such a self-organized pattern selection.

Propagation through heterogeneous substrates in simple excitable media models

The interaction of waves and obstacles is simulated by adding heterogeneities to a FitzHugh-Nagumo model and a cellular automata model. The cellular automata model is formulated to account for heterogeneities by modelling the interaction between current sources and current sinks. In both models, wave fronts propagate if the size of the heterogeneities is small, and block if the size of the heterogeneities is large. For intermediate values, wave fronts break up into numerous spiral waves. The theoretical models give insights concerning spiral wave formation in heterogeneous excitable media. (c) 2002 American Institute of Physics.

Calcium waves in agarose gel with cell organelles: implications of the velocity curvature relationship

Calcium oscillations and waves have been observed not only in several types of living cells but also in less complex systems of isolated cell organelles. Here we report the determination of apparent Ca2+ diffusion coefficients in a novel excitable medium of agarose gel with homogeneously distributed vesicles of skeletal sarcoplasmic reticulum. Spatiotemporal calcium patterns were visualized by confocal laser scanning fluorescence microscopy. To obtain characteristic parameters of the velocity curvature relationship, namely, apparent diffusion coefficient, velocity of plane calcium waves, and critical radius, positively and negatively curved wave fronts were analyzed. It is demonstrated that gel-immobilized cell organelles reveal features of an excitable medium. Apparent Ca2+ diffusion coefficients of the in vitro system, both in the absence or in the presence of mitochondria, were found to be higher than in cardiac myocytes and lower than in unbuffered agarose gel. Plane calcium waves propagated markedly slower in the in vitro system than in rat cardiac myocytes. Whereas mitochondria significantly reduced the apparent Ca2+ diffusion coefficient of the in vitro system, propagation velocity and critical size of calcium waves were found to be nearly unchanged. These results suggest that calcium wave propagation depends on the kinetics of calcium release rather than on diffusion.

Preferential depression of conduction around a pivot point in rabbit ventricular myocardium by potassium and flecainide

INTRODUCTION

During reentrant arrhythmias, the circulating wavefront often makes a sharp turn around a functional or anatomic barrier. We tested the hypothesis that lowering the safety factor for conduction by high K+ or flecainide preferentially depresses conduction of sharply turning wavefronts.

METHODS AND RESULTS

In 16 Langendorff-perfused rabbit hearts, a thin layer of anisotropic ventricular myocardium was made using a cryoprocedure. In this layer, a linear radiofrequency lesion was made parallel to the fiber orientation. The tip of the lesion was extended by a short incision. U-turning wavefronts were initiated by pacing at one side of the lesion. A mapping electrode (240 electrodes, resolution 350 to 700 microm) was used to measure conduction times and velocity of planar waves (longitudinal and transverse) and U-turning wavefronts. The safety factor for conduction was lowered by high potassium (8, 10, and 12 mmol/L) and flecainide (1 and 2 mg/L). On average, high potassium and flecainide increased the conduction times of U-turning wavefronts 1.6 times more than longitudinal or transverse planar wavefronts (P < 0.01). At a critical lowering of the excitatory current, functional conduction block occurred at the pivot point, which forced the wavefront to make a longer U-turn. In these cases, the total U-turn conduction time increased from 27+/-9 msec to 75+/-37 msec. About 40% of this delay was caused by a shift of the pivot point and consequent lengthening of the returning pathway.

CONCLUSION

Lowering the amount of excitatory current by potassium or flecainide preferentially impairs U-turn conduction. The occurrence of long delays and conduction block at pivot points may explain the mode of action of Class I drugs.

Factors determining clockwise and counterclockwise conduction patterns in atrial reentrant tachycardias: a rabbit model of atrial flutter

INTRODUCTION

In the development of atrial flutter due to reentry, the crista terminalis is supposed to pose a conduction barrier, but the role of its longitudinal conduction in determining the propagation pattern of the reentrant impulse is not known. In rabbit right atrial preparations, we induced reentrant atrial tachycardias and examined the effects of transverse section of the crista terminalis on the development and conduction patterns of arrhythmias.

METHODS AND RESULTS

Right atrial preparations from 12 albino rabbits were placed endocardial surface down in a chamber with an array of 48 bipolar electrodes to draw activation maps. A single premature stimulus was delivered to induce tachycardias at the free wall. In the control, five instances of tachycardia per preparation were induced and another five were induced after cutting the crista terminalis. In the control, the mean duration of tachycardia was 127.1+/-25.2 seconds. The tachycardia was counterclockwise in 39 of 60 instances, clockwise in 12, and undetermined in 4 defined as "atypical." After transverse section of the crista terminalis, the duration was prolonged to 372.6+/-30.4 seconds, but the conduction patterns were not changed. In the free wall, counterclockwise reentry had a broader wavefront and faster conduction than clockwise reentry.

CONCLUSION

Longitudinal conduction block at the crista terminalis contributed to maintenance of reentrant atrial tachycardias, but had no influence on their propagation patterns. Clockwise and counterclockwise rotation of impulses in reentrant tachycardias had different paths and velocities of the wavefront in the free wall of the right atrium.

Electrical turbulence as a result of the critical curvature for propagation in cardiac tissue

In cardiac tissue, the propagation of electrical excitation waves is dependent on the active properties of the cell membrane (ionic channels) and the passive electrical properties of cardiac tissue (passive membrane properties, distribution of gap junctions, and cell shapes). Initiation of cardiac arrhythmias is usually associated with heterogeneities in the active and/or passive properties of cardiac tissue. However, as a result of the effect of wave front geometry (curvature) on propagation of cardiac waves, inexcitable anatomical obstacles, like veins and arteries, may cause the formation of self-sustained vortices and uncontrolled high-frequency excitation in normal homogeneous myocardium. (c) 1998 American Institute of Physics.

Velocity-curvature relationship of colliding spherical calcium waves in rat cardiac myocytes

Colliding spherical calcium waves in enzymatically isolated rat cardiac myocytes develop new wavefronts propagating perpendicular to the original direction. When investigated by confocal laser scanning microscopy (CLSM), using the fluorescent Ca2+ indicator fluo-3 AM, "cusp"-like structures become visible that are favorably approximated by double parabolae. The time-dependent position of the vertices is used to determine propagation velocity and negative curvature of the wavefront in the region of collision. It is evident that negatively curved waves propagate faster than positively curved, single waves. Considering two perfectly equal expanding circular waves, we demonstrated that the collision of calcium waves is due to an autocatalytic process (calcium-induced calcium release), and not to a simple phenomenon of interference. Following the spatiotemporal organization in simpler chemical systems maintained under conditions far from the thermodynamic equilibrium (Belousov-Zhabotinskii reaction), the dependence of the normal velocity on the curvature of the spreading wavefront is given by a linear relation. The so-called velocity-curvature relationship makes clear that the velocity is enhanced by curvature toward the direction of forward propagation and decreased by curvature away from the direction of forward propagation (with an influence of the diffusion coefficient). Experimentally obtained velocity data of both negatively and positively curved calcium waves were approximated by orthogonal weighted regression. The negative slope of the straight line resulted in an effective diffusion coefficient of 1.2 x 10(-4) mm2/s. From the so-called critical radius, which must be exceeded to initiate a traveling calcium wave, a critical volume (with enhanced [Ca2+]i) of approximately 12 microm3 was calculated. This is almost identical to the volume that is occupied by a single calcium spark.

Vortex shedding as a precursor of turbulent electrical activity in cardiac muscle

In cardiac tissue, during partial blockade of the membrane sodium channels, or at high frequencies of excitation, inexcitable obstacles with sharp edges may destabilize the propagation of electrical excitation waves, causing the formation of self-sustained vortices and turbulent cardiac electrical activity. The formation of such vortices, which visually resembles vortex shedding in hydrodynamic turbulent flows, was observed in sheep epicardial tissue using voltage-sensitive dyes in combination with video-imaging techniques. Vortex shedding is a potential mechanism leading to the spontaneous initiation of uncontrolled high-frequency excitation of the heart.

The relation between atrial fibrillation wavefront characteristics and accessory pathway conduction

Although the source-sink relationship for impulse propagation in cardiac tissues has been demonstrated in vitro, there has been no verification of this hypothesis in humans. Accordingly, eight patients undergoing surgical division of their accessory pathways were studied. A 56-channel (7 x 8) bipolar plaque electrode array was placed over the atrioventricular groove on the accessory pathway and atrial fibrillation electrically induced. 10 episodes of QRS transition from consecutively preexcited to nonpreexcited complexes were analyzed. This showed that consecutively preexcited QRS complexes were always associated with uniform large atrial wavefronts. Immediately prior to QRS transition, four general types of changes were observed: (a) premature invasion by secondary wavefronts creating local conduction block (n = 5); (b) wavefront collision leading to wavefront curvature (n = 2); (c) transition from a uniform large atrial wavefront to multiple fractionated small wavefronts (n = 1); and (d) uniform atrial wavefronts "marching" into the accessory pathway refractory period (n = 2). We conclude that local atrial wavefront characteristics are important factors influencing impulse propagation through the accessory pathway. The findings that local wavefront collision, curvature, or fractionation often precede loss of accessory pathway conduction support the notion that source-sink relationship is an important determinant of the safety factor for impulse propagation in the human heart.

Effects of bipolar point and line stimulation in anisotropic rabbit epicardium: assessment of the critical radius of curvature for longitudinal block

Excitation front shape and velocity were studied in anisotropic perfused rabbit epicardium stained with potentiometric fluorescent dye. In the combined results from all experiments, convex excitation fronts produced by stimulation with a single electrode propagated longitudinally 13.3% slower than flat excitation fronts produced by stimulation with a line of electrodes. For transverse propagation, the two stimulation methods produced similar flat excitation fronts and velocities. The critical excitation front radius of curvature for longitudinal block (Rcr), calculated from excitable media theory, was 92 microns in control hearts. In hearts exposed to diacetyl monoxime (20 mmol/L), which decreases inward sodium current, Rcr was 175 microns. The slower longitudinal propagation velocity of convex fronts versus flat fronts and the theoretically predicted critical radius of curvature may be important for propagation and block of ectopic depolarizations in the heart.

Wave-front curvature as a cause of slow conduction and block in isolated cardiac muscle

We have investigated the role of wave-front curvature on propagation by following the wave front that was diffracted through a narrow isthmus created in a two-dimensional ionic model (Luo-Rudy) of ventricular muscle and in a thin (0.5-mm) sheet of sheep ventricular epicardial muscle. The electrical activity in the experimental preparations was imaged by using a high-resolution video camera that monitored the changes in fluorescence of the potentiometric dye di-4-ANEPPS on the surface of the tissue. Isthmuses were created both parallel and perpendicular to the fiber orientation. In both numerical and biological experiments, when a planar wave front reached the isthmus, it was diffracted to an elliptical wave front whose pronounced curvature was very similar to that of a wave front initiated by point stimulation. In addition, the velocity of propagation was reduced in relation to that of the original planar wave. Furthermore, as shown by the numerical results, wave-front curvature changed as a function of the distance from the isthmus. Such changes in local curvature were accompanied by corresponding changes in velocity of propagation. In the model, the critical isthmus width was 200 microns for longitudinal propagation and 600 microns for transverse propagation of a single planar wave initiated proximal to the isthmus. In the experiments, propagation depended on the width of the isthmus for a fixed stimulation frequency. Propagation through an isthmus of fixed width was rate dependent both along and across fibers. Thus, the critical isthmus width for propagation was estimated in both directions for different frequencies of stimulation. In the longitudinal direction, for cycle lengths between 200 and 500 milliseconds, the critical width was < 1 mm; for 150 milliseconds, it was estimated to be between 1.3 and 2 mm; and for the maximum frequency of stimulation (117 +/- 15 milliseconds), it was > 2.5 mm. In the transverse direction, critical width was between 1.78 and 2.32 mm for a basic cycle length of 200 milliseconds. It increased to values between 2.46 and 3.53 mm for a basic cycle length of 150 milliseconds. The overall results demonstrate that the curvature of the wave front plays an important role in propagation in two-dimensional cardiac muscle and that changes in curvature may cause slow conduction or block.

Three-dimensional waves of excitation during Dictyostelium morphogenesis

Cells in Dictyostelium slugs follow well-defined patterns of motion. We found that the chemotactic cell response is controlled by a scroll wave of messenger concentration in the highly excitable prestalk zone of the slug that decays in the less-excitable prespore region into planar wave fronts. This phenomenon is investigated by numerical solutions of partial differential equations that couple local nonlinear kinetics and diffusive transport of the chemotactic signal. In the interface of both regions a complex twisted scroll wave is formed that reduces the wave frequency in the prespore zone. The spatio-temporal dynamics of waves and filaments are followed over 33 periods of rotation. These results yield an explanation of collective self-organized cell motion in a multicellular organism.

Three-dimensional scroll waves organize Dictyostelium slugs

To test the hypothesis that periodic signals and chemotaxis direct later morphogenesis in Dictyostelium discoideum, we investigated cell behavior and cell movement in slugs. Trails of neutral red-stained prestalk and anterior-like cells were recorded by high-resolution digital image processing. Neutral red-stained anterior-like cells in the prespore zone of slugs move straight forward in the direction of slug migration and, furthermore, show coherent periodic cell movement. In contrast, cells in the prestalk zone move along completely different trajectories. Prestalk cells move perpendicular to the direction of slug migration; that is, they rotate around the tip. The cell movement data show that the chemotactic signal in the slug propagates as a three-dimensional scroll wave in the prestalk zone and as a planar wave in the prespore zone. The different behavior of prestalk and prespore cells is most likely caused by a difference in the oscillatory properties of the two cell types. We provide evidence that the slug stage of Dictyostelium behaves like an excitable system and that a (twisted) scroll wave organizes slug formation and migration.

Phototaxis of Spiral Waves

The drift of spiral waves toward regions of higher light intensity was observed experimentally in the ruthenium-catalyzed Belousov-Zhabotinsky reaction. A light gradient can thus be used to manipulate optical information in new computational systems based on photochemical media. The drift of a gradient that is rotationally invariant in space is three to four times as fast as that of a translationally invariant gradient. Simulations based on the use of a cellular automaton, which is made isotropic by a semirandom distribution of cells, are in agreement with the experimental results.

An eikonal-curvature equation for action potential propagation in myocardium

We derive an "eikonal-curvature" equation to describe the propagation of action potential wavefronts in myocardium. This equation is used to study the effects of fiber orientation on propagation in the myocardial wall. There are significant computational advantages to the use of an eikonal-curvature equation over a full ionic model of action potential spread. With this model, it is shown that the experimentally observed misalignment of spreading action potential "ellipses" from fiber orientation in level myocardial surfaces is adequately explained by the rotation of fiber orientation through the myocardial wall. Additionally, it is shown that apparently high propagation velocities on the epicardial and endocardial surfaces are the result of propagation into the midwall region and acceleration along midwall fibers before reemergence at an outer surface at a time preceding what could be accomplished with propagation along the surface alone.

Spiral calcium wave propagation and annihilation in Xenopus laevis oocytes

Intracellular calcium (Ca2+) is a ubiquitous second messenger. Information is encoded in the magnitude, frequency, and spatial organization of changes in the concentration of cytosolic free Ca2+. Regenerative spiral waves of release of free Ca2+ were observed by confocal microscopy in Xenopus laevis oocytes expressing muscarinic acetylcholine receptor subtypes. This pattern of Ca2+ activity is characteristic of an intracellular milieu that behaves as a regenerative excitable medium. The minimal critical radius for propagation of focal Ca2+ waves (10.4 micrometers) and the effective diffusion constant for the excitation signal (2.3 x 10(-6) square centimeters per second) were estimated from measurements of velocity and curvature of circular wavefronts expanding from foci. By modeling Ca2+ release with cellular automata, the absolute refractory period for Ca2+ stores (4.7 seconds) was determined. Other phenomena expected of an excitable medium, such as wave propagation of undiminished amplitude and annihilation of colliding wavefronts, were observed.