One of the successfully tested methods to design genetically engineered cardiac pacemaker cells consists in transfecting a human mesenchymal stem cell (hMSC) with a HCN2 gene and connecting it to a myocyte. We develop and study a mathematical model, describing a myocyte connected to a hMSC transfected with a HCN2 gene. The cardiac action potential is described both with the simple Beeler–Reuter model, as well as with the elaborate dynamic Luo–Rudy model. The HCN2 channel is described by fitting electrophysiological records, in the spirit of Hodgkin–Huxley. The model shows that oscillations can occur in a pair myocyte-stem cell, that was not observed in the experiments yet. The model predicted that: (1) HCN pacemaker channels can induce oscillations only if the number of expressed IK1 channels is low enough. At too high an expression level of IK1 channels, oscillations cannot be induced, no matter how many pacemaker channels are expressed. (2) At low expression levels of IK1 channels, a large domain of values in the parameter space (n, N) exists, where oscillations should be observed. We denote N the number of expressed pacemaker channels in the stem cell, and n the number of gap junction channels coupling the stem cell and the myocyte. (3) The expression levels of IK1 channels observed in ventricular myocytes, both in the Beeler–Reuter and in the dynamic Luo–Rudy models are too high to allow to observe oscillations. With expression levels below ∼1/4 of the original value, oscillations can be observed. The main consequence of this work is that in order to obtain oscillations in an experiment with a myocyte-stem cell pair, increasing the values of n, N is unlikely to be helpful, unless the expression level of IK1 has been reduced enough. The model also allows us to explore levels of gene expression not yet achieved in experiments, and could be useful to plan new experiments, aimed at improving the robustness of the oscillations.

The effectiveness of chaos control in large systems increases with the number of control sites. We find that electric field induced wave emission from heterogeneities (WEH) in the heart gives a unique opportunity to have as many control sites as needed. The number of pacing sites grows with the amplitude of the electric field. We demonstrate that WEH has important advantages over methods used in clinics, and opens a new way to manipulate vortices in experiments, and potentially to radically improve the clinical methods of chaos control in the heart.

Many arrhythmias are believed to be triggered by ectopic sources arising from the border of the ischemic tissue. However, the development of ectopic activity from individual sources to a larger mass of cardiac tissue remains poorly understood. To address this critical issue, we used monolayers of neonatal rat cardiomyocytes to create conditions that promoted progression of ectopic activity from single cells to the network that consisted of hundreds of cells. To explain complex spatiotemporal patterns observed in these experiments we introduced a new theoretical framework. The framework's main feature is a parameter space diagram, which uses cell automaticity and coupling as two coordinates. The diagram allows one to depict network behavior, quantitatively address the heterogeneity factor, and evaluate transitions between different regimes. The well-organized wave trains were observed at moderate and high cell coupling values and network heterogeneity was found to be qualitatively unimportant for these regimes. In contrast, at lower values of coupling, spontaneous ectopic activity led to the appearance of fragmented ectopic waves. For these regimes, network heterogeneity played an essential role. The ectopic waves occasionally gave rise to spiral activity in two different regions within the parameter space via two distinct mechanisms. Together, our results suggest that localized ectopic waves represent an essential step in the progression of ectopic activity. These studies add to the understanding of initiation and progression of arrhythmias and can be applied to other phenomena that deal with assemblies of coupled oscillators.

Controlling cardiac chaos is often achieved by applying a large damaging electric shock-defibrillation. It removes all waves, without differentiating reentries and normal waves, anatomical and functional reentries. Anatomical reentries can be removed by anti-tachycardia pacing (ATP) as well. But ATP requires the knowledge of the position of the reentry, and an access to it with an invasive stimulating electrode. We show that the physics of electric field distribution between cardiac cells permits one to deliver an electric pulse exactly to the core of an anatomical reentry, without knowing its position and even to locations where access with a stimulating electrode is not possible. The energy needed is two orders of magnitude less than defibrillation energy. The results are insensitive to both a detailed ionic model and to the geometry of the fibers.

Rotating waves in cardiac muscle may be pinned to a heterogeneity, as it happens in superconductors or in superfluids. We show that the physics of electric field distribution between cardiac cells permits one to deliver an electric pulse exactly to the core of a pinned wave, without knowing its position, and even to locations where a direct access is not possible. Thus, unpinning or removal of rotating waves can be achieved. The energy needed is 2 orders of magnitude less than defibrillation energy. This opens a way to new manipulations with pinned vortices both in experiments and in cardiac clinics.

Pinning of vortices by defects plays an important role in various physical (superconductivity, superfluidity, etc.) or biological (propagation in cardiac muscle) situations. Which defects act as pinning centers? We propose a way to study this general problem by using an advection field to quantify the attraction between an obstacle and a vortex. A full solution is obtained for the real Ginzburg-Landau equation (RGLE). Two pinning mechanisms are found in excitable media. Our results suggest strong analogies with the RGLE when the heterogeneity is excitable. Unpinning from an unexcitable obstacle is qualitatively harder, resulting in a stronger pinning force. We discuss the implications of our results to control vortices and propose experiments in a chemical active medium and in cardiac tissue.

By using both experimental and theoretical means, we have addressed the progression of ectopic activity from individual cardiac cells to a multicellular two-dimensional network. Experimental conditions that favor ectopic activity have been created by local perfusion of a small area of cardiomyocyte network (I-zone) with an isoproterenol-heptanol containing solution. The application of this solution initially slowed down and then fully blocked wave propagation inside the I-zone. After a brief lag period, ectopically active cells appeared in the I-zone, followed by evolution of the ectopic clusters into slowly propagating waves. The changing pattern of colliding and expanding ectopic waves confined to the I-zone persisted for as long as the isoproterenol-heptanol environment was present. On restoration of the control environment, the ectopic waves from the I-zone broke out into the surrounding network causing arrhythmias. The observed sequence of events was also modeled by FitzHugh-Nagumo equations and included a cell's arrangement of two adjacent square regions of 20 x 20 cells. The control zone consisted of well-connected, excitable cells, and the I-zone was made of weakly coupled cells (heptanol effect), which became spontaneously active as time evolved (isoproterenol effect). The dynamic events in the system have been studied numerically with the use of a finite difference method. Together, our experimental and computational data have revealed that the combination of low coupling, increased excitability, and spatial heterogeneity can lead to the development of ectopic waves confined to the injured network. This transient condition appears to serve as an essential step for the ectopic activity to "mature" before escaping into the surrounding control network.

Recent experiments [R. A. Gray et al., Phys. Rev. Lett. 87, 168104 ( 2001)] have revealed striking standing wave patterns in cardiac muscle. In excitable media, such as cardiac tissue where colliding waves annihilate, standing wave patterns result from a fully nonlinear mechanism. We present a possible physical mechanism explaining these patterns. The phenomenon does not depend on the precise excitable model chosen. Analogies are drawn with weak links in superconductors, and phase-slip solutions in the Ginzburg-Landau equations.

Pattern formation occurs spontaneously in endothelial cell cultures, leading to the formation of capillary networks, which eventually grow to form blood vessels. This phenomenon occurs on a time scale of a few days.We show here that patterns can also be induced on a much shorter time scale, by using the Faraday hydrodynamic instability, resulting from an oscillatory motion of the container. Close to the threshold of instability, the patterns observed are very sharp concentric rings or stripes. The patterns can be induced only inside a very narrow time window, ~ 5 min. Cells attachment then develops, and pattern formation can no longer be induced. The time window for pattern formation was diminished by favoring cell attachment, for instance by treating culture dishes with cationic macromolecules, such as poly-L-Lysine. It was increased by cooling the cells to 18°C, or by a prolonged exposure of the cells to trypsin, which is known to digest adhesion molecules.The Faraday instability leads to a method to characterize cell attachment. It also permits the production of heterogeneous cultures with several cell types, with a well controlled heterogeneity. This can be used to study heterotypic cell interactions in vitro.

We investigate the drift of a spiral wave core in a homogeneous excitable medium under the influence of a periodic stimulation by wave trains close to the core. Two important results were found. First, as opposed to existing theories of spiral wave drift, we observe drift induced by wave trains with periods larger than the period of the freely rotating spiral wave. Second, when investigating the drift of meandering spirals we found that the property of meandering of spirals is not robust against periodic stimulations. Simple phenomenological arguments are provided to explain these observations. (c) 2001 American Institute of Physics.

A pure platinum (Pt) catalyst and a new industrial catalyst, the silica fiber glass catalyst (SFGC) doped with only 0.2% Pt, were studied and compared. For propane–air catalytic combustion, the ignition of the reaction and its stability were investigated. The reaction, ignited by a laser beam, propagated as a wave front with constant amplitude and velocity along both catalysts, ∼10 mm/s for Pt, and 1–2 mm/s for SFGC. Laser ignition thresholds and the parameter space for front propagation were determined for both catalysts. Spatial instability (resulting in the destruction of the homogeneous regime of combustion and in the formation of hot spots) was found for Pt, but not for SFGC. The new catalyst (SFGC) containing only 10 −3 of Pt has the same catalytic activity as pure Pt and better spatial stability.

For the particular case of an excitable FitzHugh–Nagumo system with diffusion, we investigate the transition from annihilation to crossing of the waves in the head-on collision. The analysis exploits the similarity between the local and the global phase portraits of the system. We find that the transition has features typical of the nucleation theory of first-order phase transitions, and may be understood through purely geometrical arguments. In the case of periodic boundary conditions, the transition is an infinite-dimensional analog of the creation and the vanishing of limit cycles via a homoclinic Andronov bifurcation. Both before and after the transition, the behavior of a single cell continues to be typical for excitable systems: a stable equilibrium state, and a threshold above which an excitation pulse can be induced. The generality and qualitative character of our argument shows that the phenomenon described can be observed in excitable systems well beyond the particular case presented here.

An advective field induces drift of a vortex in excitable media. The component of the drift velocity C( perpendicular ) perpendicular to the field is known to change its sign with the chirality of the vortex. In an experiment with vortices in an electric field in a chemical excitable medium, we have found unexpectedly that C( perpendicular ) changes its sign also independently of chirality with changing composition of the medium. We did not succeed to explain this phenomenon by using existing mathematical models of chemical excitable media. The experiment described calls for more realistic models.(c) 1999 American Institute of Physics.

The possibility of terminating cardiac arrhythmias with electric fields of moderate intensity is a challenging problem from a fundamental point of view and an important issue for clinical applications. In an effort to understand how anatomical re-entries are affected by electric fields, we found that a weak shock, with an amplitude of an order of magnitude less than the defibrillating shock, may unpin the vortices rotating around the defects (obstacles). The unpinning results from a depolarization of the tissue near the obstacle, induced by an external electric field within a distance of order lambda approximately 1 mm. Unpinning was observed both in the FitzHugh model of excitable tissue, and in a specific Beeler-Reuter model of cardiac tissue. This theoretical observation suggests that anatomical re-entries can be transformed into functional re-entries, an effect that can be tested in experiments with cardiac muscle.

Excitation and deexcitation are fundamental phenomena in the electrophysiology of excitable cells. Both of them can be induced by stimulating a cell with intracellularly injected currents. With extracellular stimulation, deexcitation was never observed; only cell excitation was found. Why? A generic model with two variables (FitzHugh) predicts that an extracellular stimulus can both excite the cell and terminate the action potential (AP). Our experiments with single mouse myocytes have shown that short (2-5 ms) extracellular pulses never terminated the AP. This result agrees with our numerical experiments with the Beeler-Reuter model. To analyze the problem, we exploit the separation of time scales to derive simplified models with fewer equations. Our analysis has shown that the very specific form of the current-voltage (I-V) characteristics of the time-independent potassium current (almost no dependence on voltage for positive membrane potentials) is responsible here. When the shape of the I-V characteristics of potassium currents was modified to resemble that in ischemic tissues, or when the external potassium concentration (K-0) is increased, the AP was terminated by extracellular pulses. These results may be important for understanding the mechanisms of defibrillation.

The tunneling of chemical waves through the gaps of different geometrical shapes was studied using a new experimental model based on the polysulphone membranes with the fixed catalyst. Period doubling of the wave train, formation of the wave breaks on the split in polysulphone membrane, drift of the spiral wave along the widening split were observed. The effects were confirmed and studied in details using computer simulation.

Chemical waves of the Belousov-Zhabotinsky type are studied applying bathoferroin catalyst fixed on a polysulfone membrane. A new method is developed to create contacting high- (H) and low-amplitude (L) regions for chemical waves. The amplitude is high in zones (H) loaded with high catalyst concentrations, and it is low in zones (L) loaded with low catalyst concentrations. An asymmetric wave propagation is found: waves coming from region H can initiate waves in region L across the HL boundary with a higher frequency than vice versa. The ratio of the cross-recovery times R(L --> H) and R(H --> L) is 1.7 in the experiments reported here. To measure this ratio, rotating chemical waves were applied. The waves propagate in two concentric annular zones-the inner zone with low and the outer with high catalyst concentration-and the HL. boundary forms a circle. It was found that in such a reactor complex wave patterns (so-called chemical pinwheels) can rotate nearly independently in the H and L zones, interacting only weakly across the HL boundary.

Heterogeneities, such as gap junctions, defects in periodical cellular lattices, intercellular clefts and fiber curvature allow one to understand the effect of an electric field in cardiac tissue. They induce membrane potential variations even in the bulk of the myocardium, with a characteristic sawtooth shape. The sawtooth potential, induced by heterogeneities at large scales (tissue strands) can be more easily observed, and lead to stronger effects than the one induced at the cellular level. In the generic model of propagation in cardiac tissue (FitzHugh), 4 mechanisms of defibrillation were found, two mechanisms based on excitation (E(A),E(M)), and two-on de-excitation (D(A),D(M)). The lowest electric field is required by an E(M) mechanism. In the Beeler-Reuter ionic model, mechanism D(M) is impossible. We critically review the experimental basis of the theory and propose new experiments. (c) 1998 American Institute of Physics.

Rotating vortices are easily pinned to any defects in excitable media. Moderate advective fields cannot remove the pinned vortices. We have found and tested experimentally a mechanism for unpinning a vortex. To unpin a vortex, we created additional waves close to its core. A vortex can be unpinned when a new created wave is pinned to the same obstacle and has a topological defect. This mechanism works perfectly both in experiment and in numerical simulations. A hypothesis is put forward that similar mechanisms may underlay pacing for tachycardia control used in cardiac clinics.