The role of heterogeneities and intercellular coupling in wave propagation in cardiac tissue

Modeling and Analysis of Cardiac Hybrid Cellular Automata via GPU-Accelerated Monte Carlo Simulation

The heart consists of a complex network of billions of cells. Under physiological conditions, cardiac cells propagate electrical signals in space, generating the heartbeat in a synchronous and coordinated manner. When such a synchronization fails, life-threatening events can arise. The inherent complexity of the underlying nonlinear dynamics and the large number of biological components involved make the modeling and the analysis of electrophysiological properties in cardiac tissue still an open challenge. We consider here a Hybrid Cellular Automata (HCA) approach modeling the cardiac cell-cell membrane resistance with a free variable. We show that the modeling approach can reproduce important and complex spatiotemporal properties paving the ground for promising future applications. We show how GPU-based technology can considerably accelerate the simulation and the analysis. Furthermore, we study the cardiac behavior within a unidimensional domain considering inhomogeneous resistance and we perform a Monte Carlo analysis to evaluate our approach.

Statistics of correlated percolation in a bacterial community

Signal propagation over long distances is a ubiquitous feature of multicellular communities, but cell-to-cell variability can cause propagation to be highly heterogeneous. Simple models of signal propagation in heterogenous media, such as percolation theory, can potentially provide a quantitative understanding of these processes, but it is unclear whether these simple models properly capture the complexities of multicellular systems. We recently discovered that in biofilms of the bacterium Bacillus subtilis, the propagation of an electrical signal is statistically consistent with percolation theory, and yet it is reasonable to suspect that key features of this system go beyond the simple assumptions of basic percolation theory. Indeed, we find here that the probability for a cell to signal is not independent from other cells as assumed in percolation theory, but instead is correlated with its nearby neighbors. We develop a mechanistic model, in which correlated signaling emerges from cell division, phenotypic inheritance, and cell displacement, that reproduces the experimentally observed correlations. We find that the correlations do not significantly affect the spatial statistics, which we rationalize using a renormalization argument. Moreover, the fraction of signaling cells is not constant in space, as assumed in percolation theory, but instead varies within and across biofilms. We find that this feature lowers the fraction of signaling cells at which one observes the characteristic power-law statistics of cluster sizes, consistent with our experimental results. We validate the model using a mutant biofilm whose signaling probability decays along the propagation direction. Our results reveal key statistical features of a correlated signaling process in a multicellular community. More broadly, our results identify extensions to percolation theory that do or do not alter its predictions and may be more appropriate for biological systems.

Hypoglycemic Effects of Intestinal Electrical Stimulation by Enhancing Nutrient-Stimulated Secretion of GLP-1 in Rats

PURPOSE

To find out the best location for intestinal electrical stimulation (IES) to decrease hyperglycemia, and mechanisms involving intraluminal nutrients and plasma glucagon-like peptide-1 (GLP-1) MATERIALS AND METHODS: Eight rats had electrodes implanted at the duodenum and ileums for IES. The oral glucose tolerance test (OGTT) was performed with IES and sham-IES and with/without GLP-1 antagonist, exendin. To study the role of intraluminal nutrients, the experiment was repeated using intraperitoneal glucose tolerance test (IPGTT). Glucagon was administrated in the OGTT/IPGTT to induce temporary hyperglycemia.

RESULTS

(1) In the OGTT, IES at the duodenum reduced blood glucose from 30 to 120 min after oral glucose (P < 0.05, vs. sham-IES) and the hypoglycemic effect was more potent than IES at the ileum. (2) The hypoglycemic effect of IES was absent in IPGTT experiment, suggesting the important role of intraluminal nutrients. (3) An increase in GLP-1 was noted in the OGTT with IES at the duodenum in comparison with sham-IES. Moreover, the blocking effect of exendin suggested the role of GLP-1 in the hypoglycemic effect of IES.

CONCLUSIONS

The best stimulation location for IES to decrease hyperglycemia is in the duodenum. The hypoglycemic effect of IES is attributed to the enhancement in nutrient-stimulated release of GLP-1.

Signal Percolation within a Bacterial Community

Signal transmission among cells enables long-range coordination in biological systems. However, the scarcity of quantitative measurements hinders the development of theories that relate signal propagation to cellular heterogeneity and spatial organization. We address this problem in a bacterial community that employs electrochemical cell-to-cell communication. We developed a model based on percolation theory, which describes how signals propagate through a heterogeneous medium. Our model predicts that signal transmission becomes possible when the community is organized near a critical phase transition between a disconnected and a fully connected conduit of signaling cells. By measuring population-level signal transmission with single-cell resolution in wild-type and genetically modified communities, we confirm that the spatial distribution of signaling cells is organized at the predicted phase transition. Our findings suggest that at this critical point, the population-level benefit of signal transmission outweighs the single-cell level cost. The bacterial community thus appears to be organized according to a theoretically predicted spatial heterogeneity that promotes efficient signal transmission.

In silico study of multicellular automaticity of heterogeneous cardiac cell monolayers: Effects of automaticity strength and structural linear anisotropy

The biological pacemaker approach is an alternative to cardiac electronic pacemakers. Its main objective is to create pacemaking activity from added or modified distribution of spontaneous cells in the myocardium. This paper aims to assess how automaticity strength of pacemaker cells (i.e. their ability to maintain robust spontaneous activity with fast rate and to drive neighboring quiescent cells) and structural linear anisotropy, combined with density and spatial distribution of pacemaker cells, may affect the macroscopic behavior of the biological pacemaker. A stochastic algorithm was used to randomly distribute pacemaker cells, with various densities and spatial distributions, in a semi-continuous mathematical model. Simulations of the model showed that stronger automaticity allows onset of spontaneous activity for lower densities and more homogeneous spatial distributions, displayed more central foci, less variability in cycle lengths and synchronization of electrical activation for similar spatial patterns, but more variability in those same variables for dissimilar spatial patterns. Compared to their isotropic counterparts, in silico anisotropic monolayers had less central foci and displayed more variability in cycle lengths and synchronization of electrical activation for both similar and dissimilar spatial patterns. The present study established a link between microscopic structure and macroscopic behavior of the biological pacemaker, and may provide crucial information for optimized biological pacemaker therapies.

Implantation of electronic pacemakers is a standard treatment to pathologically slow heart rhythm. Despite improving quality of life, those devices display many shortcomings. Bioengineered tissue pacemakers may be a therapeutic alternative, but associated design methods usually lack control of the way cells with spontaneous activity are scattered throughout the tissue. Our study is the first to use a mathematical model to rigorously define and thoroughly characterize how pacemaker cells scattering at the microscopic level may affect macroscopic behaviors of the bioengineered tissue pacemaker. Automaticity strength (ability of pacemaker cell to drive its non-pacemaker neighbors) and anisotropy (preferential orientation of cell shape) are also implemented and give unparalleled insights on how effects of uncontrollable scattered pacemaker cells may be modulated by available experimental techniques. Our model is a powerful tool to aid in optimized bioengineered pacemaker therapies.

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

How Does the Xenopus laevis Embryonic Cell Cycle Avoid Spatial Chaos?

Theoretical studies have shown that a deterministic biochemical oscillator can become chaotic when operating over a sufficiently large volume and have suggested that the Xenopus laevis cell cycle oscillator operates close to such a chaotic regime. To experimentally test this hypothesis, we decreased the speed of the post-fertilization calcium wave, which had been predicted to generate chaos. However, cell divisions were found to develop normally, and eggs developed into normal tadpoles. Motivated by these experiments, we carried out modeling studies to understand the prerequisites for the predicted spatial chaos. We showed that this type of spatial chaos requires oscillatory reaction dynamics with short pulse duration and postulated that the mitotic exit in Xenopus laevis is likely slow enough to avoid chaos. In systems with shorter pulses, chaos may be an important hazard, as in cardiac arrhythmias, or a useful feature, as in the pigmentation of certain mollusk shells.

Hyperglycemia-induced small intestinal dysrhythmias attributed to sympathovagal imbalance in normal and diabetic rats

BACKGROUND

Hyperglycemia is known to induce dysrhythmias in the stomach; however, it is unknown whether they are also induced in the small intestine. Autonomic dysfunction is commonly noted in diabetes but the role it plays in hyperglycemia-induced dysrhythmias remains unknown. This study aimed to explore the effects of hyperglycemia on intestinal myoelectrical activity and the role of autonomic functions in hyperglycemia.

METHODS

Small intestinal myoelectrical activity (slow wave and spike activity) and autonomic functions (assessed by the spectral analysis of heart rate variability) were measured in Goto-Kakizaki diabetic rats and control rats treated with acute glucagon. Blood glucose was measured and its correlation with intestinal slow waves was determined.

KEY RESULTS

(1) The diabetic rats showed reduced regularity in intestinal slow waves in fasting and fed states (p < 0.001 for both), and increased sympathovagal balance (p < 0.05) in comparison with the control rats. The regularity in intestinal slow waves was negatively correlated with the HbA1c level in all rats (r = -0.663, p = 0.000). (2) Glucagon injection in the control rats induced transient hyperglycemia, intestinal slow wave dysrhythmias and impaired autonomic functions, similar to those observed in the diabetic rats. The increase in blood glucose was correlated with the decrease in the regularity of intestinal slow waves (r = -0.739, p = 0.015).

CONCLUSIONS & INFERENCES

Both spontaneous and glucagon-induced hyperglycemia results in slow wave dysrhythmias in the small intestine. Impairment in autonomic functions (increased sympathovagal balance) may play a role in hyperglycemia-induced dysrhythmias.

Form and function remixed: developmental physiology in the evolution of vertebrate body plans

The most widely accepted model of evolutionary change, the Modern Evolutionary Synthesis, is based on the gradualism of Darwin and Wallace. They, in turn, developed their ideas in the context of 19th century concepts of how matter, including the tissues of animals and plants, could be reshaped and repatterned. A new physics of condensed, chemically, electrically and mechanically excitable materials formulated in the 20th century was, however, readily taken up by physiologists, who applied it to the understanding of dynamical, external condition-dependent and homeostatic properties of individual organisms. Nerve conduction, vascular and airway dynamics, and propagation of electrical excitations in heart and brain tissue all benefited from theories of biochemical oscillation, fluid dynamics, reaction-diffusion-based pattern instability and allied dissipative processes. When, in the late 20th century, the development of body and organ form was increasingly seen to involve dynamical, frequently non-linear processes similar to those that had become standard in physiology, a strong challenge to the evolutionary synthesis emerged. In particular, large-scale changes in organismal form now had a scientific basis other than gradualistic natural selection based on adaptive advantage. Moreover, heritable morphological changes were seen to be capable of occurring abruptly with little or no genetic change, with involvement of the external environment, and in preferred directions. This paper discusses three examples of morphological motifs of vertebrate bodies and organs, the somites, the skeletons of the paired limbs, and musculoskeletal novelties distinctive to birds, for which evolutionary origination and transformation can be understood on the basis of the physiological and biophysical determinants of their development.

A microstructural model of reentry arising from focal breakthrough at sites of source-load mismatch in a central region of slow conduction

Regions of cardiac tissue that have a combination of focal activity and poor, heterogeneous gap junction coupling are often considered to be arrhythmogenic; however, the relationship between the properties of the cardiac microstructure and patterns of abnormal propagation is not well understood. The objective of this study was to investigate the effect of microstructure on the initiation of reentry from focal stimulation inside a poorly coupled region embedded in more well-coupled tissue. Two-dimensional discrete computer models of ventricular monolayers (1 × 1 cm) were randomly generated to represent heterogeneity in the cardiac microstructure. A small, central poorly coupled patch (0.40 × 0.40 cm) was introduced to represent the site of focal activity. Simulated unipolar electrogram recordings were computed at various points in the tissue. As the gap conductance of the patch decreased, conduction slowed and became increasingly complex, marked by fractionated electrograms with reduced amplitude. Near the limit of conduction block, isolated breakthrough sites occurred at single cells along the patch boundary and were marked by long cell-to-cell delays and negative deflections on electrogram recordings. The strongest determinant of the site of wavefront breakthrough was the connectivity of the brick wall architecture, which enabled current flow through small regions of overlapping cells to drive propagation into the well-coupled zone. In conclusion, breakthroughs at the size scale of a single cell can occur at the boundary of source-load mismatch allowing focal activations from slow conducting regions to produce reentry. These breakthrough regions, identifiable by distinct asymmetric, reduced amplitude electrograms, are sensitive to tissue architecture and may be targets for ablation.

Spatial heterogeneity enhances and modulates excitability in a mathematical model of the myometrium

The muscular layer of the uterus (myometrium) undergoes profound changes in global excitability prior to parturition. Here, a mathematical model of the myocyte network is developed to investigate the hypothesis that spatial heterogeneity is essential to the transition from local to global excitation which the myometrium undergoes just prior to birth. Each myometrial smooth muscle cell is represented by an element with FitzHugh-Nagumo dynamics. The cells are coupled through resistors that represent gap junctions. Spatial heterogeneity is introduced by means of stochastic variation in coupling strengths, with parameters derived from physiological data. Numerical simulations indicate that even modest increases in the heterogeneity of the system can amplify the ability of locally applied stimuli to elicit global excitation. Moreover, in networks driven by a pacemaker cell, global oscillations of excitation are impeded in fully connected and strongly coupled networks. The ability of a locally stimulated cell or pacemaker cell to excite the network is shown to be strongly dependent on the local spatial correlation structure of the couplings. In summary, spatial heterogeneity is a key factor in enhancing and modulating global excitability.

Microscopic variations in interstitial and intracellular structure modulate the distribution of conduction delays and block in cardiac tissue with source-load mismatch

AIMS

Reentrant activity in the heart is often correlated with heterogeneity in both the intracellular structure and the interstitial structure surrounding cells; however, the combined effect of cardiac microstructure and interstitial resistivity in regions of source-load mismatch is largely unknown. The aim of this study was to investigate how microstructural variations in cell arrangement and increased interstitial resistivity influence the spatial distribution of conduction delays and block in poorly coupled regions of tissue.

METHODS AND RESULTS

Two-dimensional 0.6 cm × 0.6 cm computer models with idealized and realistic cellular structure were used to represent a monolayer of ventricular myocytes. Gap junction connections were distributed around the periphery of each cell at 10 μm intervals. Regions of source-load mismatch were added to the models by increasing the gap junction and interstitial resistivity in one-half of the tissue. Heterogeneity in cell shape and cell arrangement along the boundary between well-coupled and poorly coupled tissue increased variability in longitudinal conduction delays to as much as 10 ms before the onset of conduction block, resulting in wavefront breakthroughs with pronounced curvature at distinct points along the boundary. Increasing the effective interstitial resistivity reduced source-load mismatch at the transition boundary, which caused a decrease in longitudinal conduction delay and an increase in the number of wavefront breakthroughs.

CONCLUSION

Microstructural variations in cardiac tissue facilitate the formation of isolated sites of wavefront breakthrough that may enable abnormal electrical activity in small regions of diseased tissue to develop into more widespread reentrant activity.

Reentry near the percolation threshold in a heterogeneous discrete model for cardiac tissue

Arrhythmias in cardiac tissue are related to irregular electrical wave propagation in the heart. Cardiac tissue is formed by a discrete cell network, which is often heterogeneous. A localized region with a fraction of nonconducting links surrounded by homogeneous conducting tissue can become a source of reentry and ectopic beats. Extensive simulations in a discrete model of cardiac tissue show that a wave crossing a heterogeneous region of cardiac tissue can disintegrate into irregular patterns, provided the fraction of nonconducting links is close to the percolation threshold of the cell network. The dependence of the reentry probability on this fraction, the system size, and the degree of excitability can be inferred from the size distribution of nonconducting clusters near the percolation threshold.

Pacemaker interactions induce reentrant wave dynamics in engineered cardiac culture

Pacemaker interactions can lead to complex wave dynamics seen in certain types of cardiac arrhythmias. We use experimental and mathematical models of pacemakers in heterogeneous excitable media to investigate how pacemaker interactions can be a mechanism for wave break and reentrant wave dynamics. Embryonic chick ventricular cells are cultured in vitro so as to create a dominant central pacemaker site that entrains other pacemakers in the medium. Exposure of those cultures to a potassium channel blocker, E-4031, leads to emergence of peripheral pacemakers that compete with each other and with the central pacemaker. Waves emitted by faster pacemakers break up over the slower pacemaker to form reentrant waves. Similar dynamics are observed in a modified FitzHugh-Nagumo model of heterogeneous excitable media with two distinct sites of pacemaking. These findings elucidate a mechanism of pacemaker-induced reentry in excitable media.

Electrophysiological and contractile function of cardiomyocytes derived from human embryonic stem cells

Human embryonic stem cells have emerged as the prototypical source from which cardiomyocytes can be derived for use in drug discovery and cell therapy. However, such applications require that these cardiomyocytes (hESC-CMs) faithfully recapitulate the physiology of adult cells, especially in relation to their electrophysiological and contractile function. We review what is known about the electrophysiology of hESC-CMs in terms of beating rate, action potential characteristics, ionic currents, and cellular coupling as well as their contractility in terms of calcium cycling and contraction. We also discuss the heterogeneity in cellular phenotypes that arises from variability in cardiac differentiation, maturation, and culture conditions, and summarize present strategies that have been implemented to reduce this heterogeneity. Finally, we present original electrophysiological data from optical maps of hESC-CM clusters.

Control of turbulence in heterogeneous excitable media

Control of turbulence in two kinds of typical heterogeneous excitable media by applying a combined method is investigated. It is found that local-low-amplitude and high-frequency pacing (LHP) is effective to suppress turbulence if the deviation of the heterogeneity is minor. However, LHP is invalid when the deviation is large. Studies show that an additional radial electric field can greatly increase the efficiency of LHP. The underlying mechanisms of successful control in the two kinds of cases are different and are discussed separately. Since the developed strategy of combining LHP with a radial electric field can terminate turbulence in excitable media with a high degree of inhomogeneity, it has the potential contribution to promote the practical low-amplitude defibrillation approach.

Scaling properties of conduction velocity in heterogeneous excitable media

Waves of excitation through excitable media, such as cardiac tissue, can propagate as plane waves or break up to form reentrant spiral waves. In diseased hearts reentrant waves can be associated with fatal cardiac arrhythmias. In this paper we investigate the conditions that lead to wave break, reentry, and propagation failure in mathematical models of heterogeneous excitable media. Two types of heterogeneities are considered: sinks are regions in space in which the voltage is fixed at its rest value, and breaks are nonconducting regions with no-flux boundary conditions. We find that randomly distributed heterogeneities in the medium have a decremental effect on the velocity, and above a critical density of such heterogeneities the conduction fails. Using numerical and analytical methods we derive the general relationship among the conduction velocity, density of heterogeneities, diffusion coefficient, and the rise time of the excitation in both two and three dimensions. This work helps us understand the factors leading to reduced propagation velocity and the formation of spiral waves in heterogeneous excitable media.

Slow wave propagation and plasticity of interstitial cells of Cajal in the small intestine of diabetic rats

The number of myenteric interstitial cells of Cajal (ICC-MY), responsible for the generation and propagation of the slow wave in the small intestine, has been shown to decrease in diabetes, suggesting impairment of slow-wave (SW) propagation and related motility. To date, however, this expected decrease in SW propagation has neither been recorded nor analysed. Eleven rats were treated with streptozotocin and housed in pairs with 11 age-matched control animals. After 3 or 7 months, segments of duodenum, jejunum and ileum were isolated and divided into two parts. One part was processed for immediate freezing, cryosectioning and immunoprobing using anti-c-Kit antibody to quantify ICC-MY. The second part was superfused in a tissue bath, and SW propagation was recorded with 121 extracellular electrodes. In addition, a cellular automaton was developed to study the effects of increasing the number of inactive cells on overall propagation. The number of ICC-MY was significantly reduced after 3 months of diabetes, but rebounded to control levels after 7 months of diabetes. Slow-wave frequencies, velocities and extracellular amplitudes were unchanged at any stage of diabetes. The cellular automaton showed that SW velocity was not linearly related to the number of inactive cells. The depletion of ICC-MY is not as severe as is often assumed and in fact may rebound after some time. In addition, at least in the streptozotocin model, the initial reduction in ICC-MY is not enough to affect SW propagation. Diabetic intestinal dysfunction may therefore be more affected by impairments of other systems, such as the enteric system or the muscle cells.

Target-wave to spiral-wave pattern transition in a discrete Belousov-Zhabotinsky reaction driven by inactive resin beads

Wave pattern formation and transition in chemical and biochemical reaction systems can reveal the system properties. We investigate the pattern transition from target waves to spiral waves upon the increment of inactive beads in a discrete system model, where ion-exchange resin loaded with Belousov-Zhabotinsky catalyst corresponds to the active beads. The results show that inactive beads slow the propagation speed of target waves and increase the wave frequency. As the population of inactive beads increases, clusters are formed, which then break waves into segments where bounded spiral pairs are generated and separated into individual spirals. From this observation, we conclude that the population of inactive resin beads acts as the bifurcation parameter controlling the wave pattern transition from targets to spirals.

Spiral wave breakup in excitable media with an inhomogeneity of conduction anisotropy

Many conditions remodel the heart muscle such that it results in a perturbation of cells coupling. The effect of this perturbation on the stability of the spiral waves of electrochemical activity is not clear. We used the FitzHugh-Nagumo model of an excitable medium to model the conduction of the activation waves in a two-dimensional system with inhomogeneous anisotropy level. Inhomogeneity of the anisotropy level was modeled by adding Gaussian noise to diffusion coefficients corresponding with lateral coupling of the cells. Low noise levels resulted in a stable propagation of the spiral wave. For large noise level conduction was not possible due to insufficient coupling in direction perpendicular to fibers. For intermediate noise intensities, the initial wave broke up into several independent spiral waves or waves circulating around conduction obstacles. At an optimal noise intensity, the number of wavelets was maximized-a form of anti-coherent resonance was obtained. Our results suggest that the inhomogeneity of conduction anisotropy may promote wave breakup and hence play an important role in the initiation and perpetuation of the cardiac arrhythmias.

Effective medium theory for reaction rates and diffusion coefficients of heterogeneous systems

A simple effective medium theory is derived for spatially heterogeneous nonlinear reaction-diffusion media. Its validity is tested through comparisons with simulations of front and pulse propagation in systems with spatially varying diffusion coefficients and reaction rates. The theory is able to predict wave speeds if the characteristic front width is much larger than the length scale of the heterogeneities. This condition is violated in media with isolated or weakly connected sites. However, the theory nevertheless provides good results in cases where it correctly predicts the percolation threshold of the medium.

Suppression of spirals and turbulence in inhomogeneous excitable media

Suppression of spiral and turbulence in inhomogeneous media due to local heterogeneity with higher excitability is investigated numerically. When the inhomogeneity is small, control tactics by boundary periodic forcing (BPF) is effective against the existing spiral and turbulence. When the inhomogeneity of excitability is large, a rotating electric field (REF) is utilized to "smooth" regional heterogeneity based on driven synchronization. Consequently, a control approach combining BPF with REF is proposed to suppress the spiral and turbulence. The underlying mechanism of successful suppression is discussed in terms of dispersion relation.

Electrotonic myofibroblast-to-myocyte coupling increases propensity to reentrant arrhythmias in two-dimensional cardiac monolayers

In pathological conditions such as ischemic cardiomyopathy and heart failure, differentiation of fibroblasts into myofibroblasts may result in myocyte-fibroblast electrical coupling via gap junctions. We hypothesized that myofibroblast proliferation and increased heterocellular coupling significantly alter two-dimensional cardiac wave propagation and reentry dynamics. Co-cultures of myocytes and myofibroblasts from neonatal rat ventricles were optically mapped using a voltage-sensitive dye during pacing and sustained reentry. The myofibroblast/myocyte ratio was changed systematically, and junctional coupling of the myofibroblasts was reduced or increased using silencing RNAi or adenoviral overexpression of Cx43, respectively. Numerical simulations in two-dimensional models were used to quantify the effects of heterocellular coupling on conduction velocity (CV) and reentry dynamics. In both simulations and experiments, reentry frequency and CV diminished with larger myofibroblast/myocyte area ratios; complexity of propagation increased, resulting in wave fractionation and reentry multiplication. The relationship between CV and coupling was biphasic: an initial decrease in CV was followed by an increase as heterocellular coupling increased. Low heterocellular coupling resulted in fragmented and wavy wavefronts; at high coupling wavefronts became smoother. Heterocellular coupling alters conduction velocity, reentry stability, and complexity of wave propagation. The results provide novel insight into the mechanisms whereby electrical myocyte-myofibroblast interactions modify wave propagation and the propensity to reentrant arrhythmias.

Spiral wave dynamics in excitable media with spherical geometries

We describe the spatial and temporal organization of spiral and scroll waves in spherical shells of different sizes and solid spheres. We present simulation results for the evolution of the dynamics and clustering of spiral waves as a function of the excitability of the medium. The excitability, topology, and size of the domain places restrictions on how single and multiarmed spiral waves are organized in space. The results in spherical geometries are compared with those in planar two-dimensional media. These studies are relevant to the dynamics of spiral waves in a variety of media including the heart, and chemical reactions on spherical surfaces.

Initiation of re-entry in an excitable medium: structural investigation of cardiac tissue using a genetic algorithm

The detailed mechanisms by which re-entry and ventricular fibrillation are initiated in the heart remain poorly understood because they are difficult to investigate experimentally. We have used a simplified excitable media computational model of action potential propagation to systematically study how re-entry can be produced by diffuse regions of inexcitable tissue. Patterns of excitable and inexcitable tissue were generated using a genetic algorithm. The inexcitable tissue was modeled in two ways: (i) diffusive, electrically connected but inexcitable tissue, or (ii) zero-flux, areas of tissue electrically disconnected in the same way as zero-flux boundary conditions. We were able to evolve patterns of diffuse inexcitable tissue that favored re-entry, but no single structure or pattern emerged. Diffusive inexcitable regions were inherently less arrhythmogenic than zero-flux inexcitable ones.

Asymptotic properties of mathematical models of excitability

We analyse small parameters in selected models of biological excitability, including Hodgkin-Huxley (Hodgkin & Huxley 1952 J. Physiol.117, 500-544) model of nerve axon, Noble (Noble 1962 J. Physiol.160, 317-352) model of heart Purkinje fibres and Courtemanche et al. (Courtemanche et al. 1998 Am. J. Physiol.275, H301-H321) model of human atrial cells. Some of the small parameters are responsible for differences in the characteristic time-scales of dynamic variables, as in the traditional singular perturbation approaches. Others appear in a way which makes the standard approaches inapplicable. We apply this analysis to study the behaviour of fronts of excitation waves in spatially extended cardiac models. Suppressing the excitability of the tissue leads to a decrease in the propagation speed, but only to a certain limit; further suppression blocks active propagation and leads to a passive diffusive spread of voltage. Such a dissipation may happen if a front propagates into a tissue recovering after a previous wave, e.g. re-entry. A dissipated front does not recover even when the excitability restores. This has no analogy in FitzHugh-Nagumo model and its variants, where fronts can stop and then start again. In two spatial dimensions, dissipation accounts for breakups and self-termination of re-entrant waves in excitable media with Courtemanche et al. kinetics.

Mathematical models in physiology

Computational modelling of biological processes and systems has witnessed a remarkable development in recent years. The search-term (modelling OR modeling) yields over 58000 entries in PubMed, with more than 34000 since the year 2000: thus, almost two-thirds of papers appeared in the last 5-6 years, compared to only about one-third in the preceding 5-6 decades. The development is fuelled both by the continuously improving tools and techniques available for bio-mathematical modelling and by the increasing demand in quantitative assessment of element inter-relations in complex biological systems. This has given rise to a worldwide public domain effort to build a computational framework that provides a comprehensive theoretical representation of integrated biological function-the Physiome. The current and next issues of this journal are devoted to a small sub-set of this initiative and address biocomputation and modelling in physiology, illustrating the breadth and depth of experimental data-based model development in biological research from sub-cellular events to whole organ simulations.

A mathematical model of pacemaker activity recorded from mouse small intestine

The pacemaker activity of interstitial cells of Cajal (ICCs) has been known to initiate the propagation of slow waves along the whole gastrointestinal tract through spontaneous and repetitive generation of action potentials. We studied the mechanism of the pacemaker activity of ICCs in the mouse small intestine and tested it using a mathematical model. The model includes ion channels, exchanger, pumps and intracellular machinery for Ca2+ regulation. The model also incorporates inositol 1,4,5-triphosphate (IP3) production and IP3-mediated Ca2+ release activities. Most of the parameters were obtained from the literature and were modified to fit the experimental results of ICCs from mouse small intestine. We were then able to compose a mathematical model that simulates the pacemaker activity of ICCs. The model generates pacemaker potentials regularly and repetitively as long as the simulation continues. The frequency was set at 20 min(-1) and the duration at 50% repolarization was 639 ms. The resting and overshoot potentials were -78 and +1.2 mV, respectively. The reconstructed pacemaker potentials closely matched those obtained from animal experiments. The model supports the idea that cyclic changes in [Ca2+]i and [IP3] play key roles in the generation of ICC pacemaker activity in the mouse small intestine.