Electrical refractory period restitution and spiral wave reentry in simulated cardiac tissue

Theoretical and experimental studies have shown that restitution of the cardiac action potential (AP) duration (APD) plays a major role in predisposing ventricular tachycardia to degenerate to ventricular fibrillation, whereas its role in atrial fibrillation is unclear. We used the Courtemanche human atrial cell model and the Luo-Rudy guinea pig ventricular model to compare the roles of electrical restitution in destabilizing spiral wave reentry in simulated two-dimensional homogeneous atrial and ventricular tissue. Because atrial AP morphology is complex, we also validated the usefulness of effective refractory period (ERP) restitution. ERP restitution correlated best with APD restitution at transmembrane potentials greater than or equal to -62 mV, and its steepness was a reliable predictor of spiral wave phenotype (stable, meandering, hypermeandering, and breakup) in both atrial and ventricular tissue. Spiral breakup or single hypermeandering spirals occurred when the slope of ERP restitution exceeded 1 at short diastolic intervals. Thus ERP restitution, which is easier to measure clinically than APD restitution, is a reliable determinant of spiral wave stability in simulated atrial and ventricular tissue.     

Selection of Multiarmed Spiral Waves in a Regular Network of Neurons

Formation and selection of multiarmed spiral wave due to spontaneous symmetry breaking are investigated in a regular network of Hodgkin-Huxley neuron by changing the excitability and imposing spatial forcing currents on the neurons in the network. The arm number of the multiarmed spiral wave is dependent on the distribution of spatial forcing currents and excitability diversity in the network, and the selection criterion for supporting multiarmed spiral waves is discussed. A broken spiral segment is measured by a short polygonal line connected by three adjacent points (controlled nodes), and a double-spiral wave can be developed from the spiral segment. Multiarmed spiral wave is formed when a group of double-spiral waves rotate in the same direction in the network. In the numerical studies, a group of controlled nodes are selected and spatial forcing currents are imposed on these nodes, and our results show that l-arm stable spiral wave (l = 2, 3, 4,...8) can be induced to occupy the network completely. It is also confirmed that low excitability is critical to induce multiarmed spiral waves while high excitability is important to propagate the multiarmed spiral wave outside so that distinct multiarmed spiral wave can occupy the network completely. Our results confirm that symmetry breaking of target wave in the media accounts for emergence of multiarmed spiral wave, which can be developed from a group of spiral waves with single arm under appropriate condition, thus the potential formation mechanism of multiarmed spiral wave in the media is explained.     

Reentry as an Origin for Rotors

The aim of the study is to understand in depth the meaning of “reentry”, and to decipher if and how it can lead to malfunctions of the heart and possibly of the brain. A simple model is used to reveal the mechanism by which a single pulse of action potential rotating around a ring of excitable medium, the latter simulating a reentry circuit, can generate spirals (single and/or double) when the pulse can emerge from and develop outside the ring. Two mechanisms of spiral generation are demonstrated: (1) a mechanism in which a source of single spirals is created at the contact with the core soon after the pulse freeing action, their chirality being due to the sense of the preceding pulse rotation. Interestingly, these spirals, adhering to the core, become “double-spiral patterns” while leaving behind the seeds of the new single spirals. (2) A second possible mechanism, similar to the known “arms encountering methods”, in which a double spiral (a figure of eight) is repeatedly created on the other side of the core. Similar procedures are assumed to occur in the heart, leading to tachycardia and fibrillation and possibly in the brain leading to epilepsy. The exact processes of the hitherto assumed spiral generations by reentry were established. The novel deep understanding of the mechanisms involved in these processes can lead to new methods of treating heart fibrillation (e.g., by judicial ablation).     

Spiral waves of spreading depression in the isolated chicken retina

Existence of the theoretically predicted spiral waves of excitation in intact two-dimensional networks of excitable elements has been experimentally confirmed in the isolated chicken retina. The preparation supports the waves of Leão's spreading depression (SD) the concentric propagation of which from the point of origin can be directly observed as a change of the optical properties of the retinal tissue. The propagation rate of 3.7 mm/min (35°C) decreased to 1.5 mm/min for SD waves elicited during relative refractory period. When a several-mm long segment of the SD wave had been blocked by anodal polarization, the laterally opened ends of the wavefront started to spread after termination of polarization into the previously blocked tissue, gradually turning around and penetrating into the region recovering from the original SD. One or two simultaneously generated spiral waves of SD continued to rotate for several cycles. Spiral SD could also be elicited by punctiform cathodal polarization (1 mA) applied to the SD wave-rear. Since the new SD wave could only spread into the recovering tissue it formed a laterally open wavefront, the free ends of which eventually turned around and started spiral SD. With continued reverberation the nucleus of the spiral SD wave gradually migrated across the retina until it approached an obstacle (e.g., pecten) which stopped further spiral propagation. Spiral SD waves were elicited in 31 retinal preparations and lasted for 4.5 cycles on the average. Average cycle duration was 4.7 min. Spontaneous spiral SD waves were observed in preparations incubated in Mg²⁺-free media. The spiral SD waves in retina are compared with mathematical models of analogous phenomena. It is argued that spiral SD waves probably exist in the cerebral cortex of rats and account for generation of repetitive SD waves sometimes elicited by overlapping stimulation of two cortical regions.     

A computationally efficient electrophysiological model of human ventricular cells

Recent experimental and theoretical results have stressed the importance of modeling studies of reentrant arrhythmias in cardiac tissue and at the whole heart level. We introduce a six-variable model obtained by a reformulation of the Priebe-Beuckelmann model of a single human ventricular cell. The reformulated model is 4.9 times faster for numerical computations and it is more stable than the original model. It retains the action potential shape at various frequencies, restitution of action potential duration, and restitution of conduction velocity. We were able to reproduce the main properties of epicardial, endocardial, and M cells by modifying selected ionic currents. We performed a simulation study of spiral wave behavior in a two-dimensional sheet of human ventricular tissue and showed that spiral waves have a frequency of 3.3 Hz and a linear core of approximately 50-mm diameter that rotates with an average frequency of 0.62 rad/s. Simulation results agreed with experimental data. In conclusion, the proposed model is suitable for efficient and accurate studies of reentrant phenomena in human ventricular tissue.     

Transition from an excitable to an oscillatory state in dictyostelium discoideum

Under conditions of starvation, populations of the amoebae Dictyostelium discoideum aggregate are mediated by chemical excitation waves of cAMP. Two types of waves can be observed, either spiral or circular-shaped ones. We investigate transitions from rotating spirals to circular shaped waves (target patterns). Two different experiments demonstrating this phenomenon are presented. In the first case a continuous transition from the spiral type pattern to target waves was observed at the later stages of aggregation. In the second case the transition was induced by annihilation of waves by a spatially homogeneous cAMP pulse. Instead of the originally present spiral waves, oscillating spots bearing target patterns emerged. On the basis of a model for Dictyostelium aggregation, we provide a theoretical explanation for such transitions. It is shown that cell density can be an effective bifurcation parameter. Under certain conditions, the system is shifted from the excitable to the oscillatory state while the frequency of oscillations is proportional to the square root of the cell density. Thus, the regions with the highest cell density during the early stages of the spatial rearrangement of the cells become pacemakers and produce target patterns. The analytic results were confirmed in numerical simulations of the model.     

Evolution of spiral and scroll waves of excitation in a mathematical model of ischaemic border zone

Abnormal electrical activity from the boundaries of ischemic cardiac tissue is recognized as one of the major causes in generation of ischemia-reperfusion arrhythmias. Here we present theoretical analysis of the waves of electrical activity that can rise on the boundary of cardiac cell network upon its recovery from ischaemia-like conditions. The main factors included in our analysis are macroscopic gradients of the cell-to-cell coupling and cell excitability and microscopic heterogeneity of individual cells. The interplay between these factors allows one to explain how spirals form, drift together with the moving boundary, get transiently pinned to local inhomogeneities, and finally penetrate into the bulk of the well-coupled tissue where they reach macroscopic scale. The asymptotic theory of the drift of spiral and scroll waves based on response functions provides explanation of the drifts involved in this mechanism, with the exception of effects due to the discreteness of cardiac tissue. In particular, this asymptotic theory allows an extrapolation of 2D events into 3D, which has shown that cells within the border zone can give rise to 3D analogues of spirals, the scroll waves. When and if such scroll waves escape into a better coupled tissue, they are likely to collapse due to the positive filament tension. However, our simulations have shown that such collapse of newly generated scrolls is not inevitable and that under certain conditions filament tension becomes negative, leading to scroll filaments to expand and multiply leading to a fibrillation-like state within small areas of cardiac tissue.     

A Quantitative Comparison of the Behavior of Human Ventricular Cardiac Electrophysiology Models in Tissue

Numerical integration of mathematical models of heart cell electrophysiology provides an important computational tool for studying cardiac arrhythmias, but the abundance of available models complicates selecting an appropriate model. We study the behavior of two recently published models of human ventricular action potentials, the Grandi-Pasqualini-Bers (GPB) and the O''Hara-Virág-Varró-Rudy (OVVR) models, and compare the results with four previously published models and with available experimental and clinical data. We find the shapes and durations of action potentials and calcium transients differ between the GPB and OVVR models, as do the magnitudes and rate-dependent properties of transmembrane currents and the calcium transient. Differences also occur in the steady-state and S1–S2 action potential duration and conduction velocity restitution curves, including a maximum conduction velocity for the OVVR model roughly half that of the GPB model and well below clinical values. Between single cells and tissue, both models exhibit differences in properties, including maximum upstroke velocity, action potential amplitude, and minimum diastolic interval. Compared to experimental data, action potential durations for the GPB and OVVR models agree fairly well (although OVVR epicardial action potentials are shorter), but maximum slopes of steady-state restitution curves are smaller. Although studies show alternans in normal hearts, it occurs only in the OVVR model, and only for a narrow range of cycle lengths. We find initiated spiral waves do not progress to sustained breakup for either model. The dominant spiral wave period of the GPB model falls within clinically relevant values for ventricular tachycardia (VT), but for the OVVR model, the dominant period is longer than periods associated with VT. Our results should facilitate choosing a model to match properties of interest in human cardiac tissue and to replicate arrhythmia behavior more closely. Furthermore, by indicating areas where existing models disagree, our findings suggest avenues for further experimental work.     

Structure of bound water and topological rearrangement waves

The evolutionary conception of bound water and the structure of liquid water has been briefly described. The function of the radial distribution of water molecules has been analyzed. The intermediate maxima at 0.35 nm of the function is shown to be described in the frame of the model of continuous hydrogen bond net without invoking the mixed and two-state models of liquid water. Using a simple extensive parametric structure, spiral 30/11, a model of topological rearrangement waves including bifurcate bonds is suggested. These waves can transmit the information on conformational transitions between cellular elements through the virtual spirals of bound water.     

Interaction between spiral and paced waves in cardiac tissue

For prevention of lethal arrhythmias, patients at risk receive implantable cardioverter-defibrillators, which use high-frequency antitachycardia pacing (ATP) to convert tachycardias to a normal rhythm. One of the suggested ATP mechanisms involves paced-induced drift of rotating waves followed by their collision with the boundary of excitable tissue. This study provides direct experimental evidence of this mechanism. In monolayers of neonatal rat cardiomyocytes in which rotating waves of activity were initiated by premature stimuli, we used the Ca(2+)-sensitive indicator fluo 4 to observe propagating wave patterns. The interaction of the spiral tip with a paced wave was then monitored at a high spatial resolution. In the course of the experiments, we observed spiral wave pinning to local heterogeneities within the myocyte layer. High-frequency pacing led, in a majority of cases, to successful termination of spiral activity. Our data show that 1) stable spiral waves in cardiac monolayers tend to be pinned to local heterogeneities or areas of altered conduction, 2) overdrive pacing can shift a rotating wave from its original site, and 3) the wave break, formed as a result of interaction between the spiral tip and a paced wave front, moves by a paced-induced drift mechanism to an area where it may become unstable or collide with a boundary. The data were complemented by numerical simulations, which was used to further analyze experimentally observed behavior.     

Scroll-wave dynamics in human cardiac tissue: lessons from a mathematical model with inhomogeneities and fiber architecture

Cardiac arrhythmias, such as ventricular tachycardia (VT) and ventricular fibrillation (VF), are among the leading causes of death in the industrialized world. These are associated with the formation of spiral and scroll waves of electrical activation in cardiac tissue; single spiral and scroll waves are believed to be associated with VT whereas their turbulent analogs are associated with VF. Thus, the study of these waves is an important biophysical problem. We present a systematic study of the combined effects of muscle-fiber rotation and inhomogeneities on scroll-wave dynamics in the TNNP (ten Tusscher Noble Noble Panfilov) model for human cardiac tissue. In particular, we use the three-dimensional TNNP model with fiber rotation and consider both conduction and ionic inhomogeneities. We find that, in addition to displaying a sensitive dependence on the positions, sizes, and types of inhomogeneities, scroll-wave dynamics also depends delicately upon the degree of fiber rotation. We find that the tendency of scroll waves to anchor to cylindrical conduction inhomogeneities increases with the radius of the inhomogeneity. Furthermore, the filament of the scroll wave can exhibit drift or meandering, transmural bending, twisting, and break-up. If the scroll-wave filament exhibits weak meandering, then there is a fine balance between the anchoring of this wave at the inhomogeneity and a disruption of wave-pinning by fiber rotation. If this filament displays strong meandering, then again the anchoring is suppressed by fiber rotation; also, the scroll wave can be eliminated from most of the layers only to be regenerated by a seed wave. Ionic inhomogeneities can also lead to an anchoring of the scroll wave; scroll waves can now enter the region inside an ionic inhomogeneity and can display a coexistence of spatiotemporal chaos and quasi-periodic behavior in different parts of the simulation domain. We discuss the experimental implications of our study.     

Mechanisms of subcellular cytosolic Ca2+ signaling evoked by stimulation of the vasopressin V1a receptor.

Receptor activation may result in distinct subcellular patterns of Ca2+ release. To define the subcellular distribution of Ca2+i signals induced by stimulation of the vasopressin V1a receptor, we expressed the cloned receptor in Xenopus oocytes. Oocytes were then loaded with fluo-3 and observed using confocal microscopy. Vasopressin induced a single concentric wave of increased Ca2+ that radiated inward from the plasma membrane. With submaximal stimulation, however, regions of the Ca2+ wave spontaneously reorganized into repetitive (oscillatory) waves. Focal stimulation of a small part of the plasma membrane resulted in a Ca2+ wave which began at the point of stimulation, radiated toward the center of the cell, then reorganized into multiple foci of repetitive, colliding waves and spirals of increased Ca2+i. The pattern of Ca2+ signaling induced by focal or global stimulation was not altered in Ca(2+)-free medium, although signals did not propagate as fast. Finally, subcellular Ca2+ signaling patterns induced by vasopressin were inhibited by caffeine, while neither vasopressin nor microinjection of inositol trisphosphate blocked caffeine-induced increases in cytosolic Ca2+. Thus, stimulation of the V1a receptor in this cell system induces a complex pattern of Ca2+ signaling which is influenced by (1) the magnitude of the stimulus, (2) the distribution of the surface receptors that are stimulated, and (3) mobilization of Ca2+ from the extracellular space as well as from two distinct endogenous Ca2+ pools. The manner in which a single type of receptor is activated may represent an important potential mechanism for subcellular Ca2+i signaling.     

Increased vulnerability of human ventricle to re-entrant excitation in hERG-linked variant 1 short QT syndrome

The short QT syndrome (SQTS) is a genetically heterogeneous condition characterized by abbreviated QT intervals and an increased susceptibility to arrhythmia and sudden death. This simulation study identifies arrhythmogenic mechanisms in the rapid-delayed rectifier K(+) current (I(Kr))-linked SQT1 variant of the SQTS. Markov chain (MC) models were found to be superior to Hodgkin-Huxley (HH) models in reproducing experimental data regarding effects of the N588K mutation on KCNH2-encoded hERG. These ionic channel models were then incorporated into human ventricular action potential (AP) models and into 1D and 2D idealised and realistic transmural ventricular tissue simulations and into a 3D anatomical model. In single cell models, the N588K mutation abbreviated ventricular cell AP duration at 90% repolarization (APD(90)) and decreased the maximal transmural voltage heterogeneity (δV) during APs. This resulted in decreased transmural heterogeneity of APD(90) and of the effective refractory period (ERP): effects that are anticipated to be anti-arrhythmic rather than pro-arrhythmic. However, with consideration of transmural heterogeneity of I(Kr) density in the intact tissue model based on the ten Tusscher-Noble-Noble-Panfilov ventricular model, not only did the N588K mutation lead to QT-shortening and increases in T-wave amplitude, but δV was found to be augmented in some local regions of ventricle tissue, resulting in increased tissue vulnerability for uni-directional conduction block and predisposing to formation of re-entrant excitation waves. In 2D and 3D tissue models, the N588K mutation facilitated and maintained re-entrant excitation waves due to the reduced substrate size necessary for sustaining re-entry. Thus, in SQT1 the N588K-hERG mutation facilitates initiation and maintenance of ventricular re-entry, increasing the lifespan of re-entrant spiral waves and the stability of scroll waves in 3D tissue.     

Properties of two human atrial cell models in tissue: restitution, memory, propagation, and reentry

To date, two detailed ionic models of human atrial cell electrophysiology have been developed, the Nygren et al. model (NM) and the Courtemanche et al. model (CM). Although both models draw from similar experimental data, they have vastly different properties. This paper provides the first systematic analysis and comparison of the dynamics of these models in spatially extended systems including one-dimensional cables and rings, two-dimensional sheets, and a realistic three-dimensional human atrial geometry. We observe that, as in single cells, the CM adapts to rate changes primarily by changes in action potential duration (APD) and morphology, while for the NM rate changes affect resting membrane potential (RMP) more than APD. The models also exhibit different memory properties as assessed through S1-S2 APD and conduction velocity (CV) restitution curves with different S1 cycle lengths. Reentrant wave dynamics also differ, with the NM exhibiting stable, non-breaking spirals and the CM exhibiting frequent transient wave breaks. The realistic atrial geometry modifies dynamics in some cases through drift, transient pinning, and breakup. Previously proposed modifications to represent atrial fibrillation-remodeled electrophysiology produce altered dynamics, including reduced rate adaptation and memory for both models and conversion to stable reentry for the CM. Furthermore, proposed variations to the NM to reproduce action potentials more closely resembling those of the CM do not substantially alter the underlying dynamics of the model, so that tissue simulations using these modifications still behave more like the unmodified NM. Finally, interchanging the transmembrane current formulations of the two models suggests that currents contribute more strongly to RMP and CV, intracellular calcium dynamics primarily determine reentrant wave dynamics, and both are important in APD restitution and memory in these models. This finding implies that the formulation of intracellular calcium processes is as important to producing realistic models as transmembrane currents.     

A mathematical model for spreading cortical depression.

A mathematical model is derived from physiological considerations for slow potential waves (called spreading depression) in cortical neuronal structures. The variables taken into account are the intra- and extracellular concentrations of Na+, Cl-, K+, and Ca++, together with excitatory and inhibitor transmitter substances. The general model includes conductance changes for these various ions, which may occur at nonsynaptic and synaptic membrane together with active transport mechanisms (pumps). A detailed consideration of only the conductance changes due to transmitter release leads to a system of nonlinear diffusion equations coupled with a system or ordinary differential equations. We obtain numerical solutions of a set of simplified model equations involving only K+ and Ca++ concentrations. The solutions agree qualitatively with experimentally obtained time-courses of these two ionic concentrations during spreading depression. The numerical solutions exhibit the observed phenomena of solitary waves and annihilation of colliding waves.     

Impact of mitochondrial Ca2+ cycling on pattern formation and stability.

Energization of mitochondria significantly alters the pattern of Ca2+ wave activity mediated by activation of the inositol (1,4,5) trisphosphate (IP3) receptor (IP3R) in Xenopus oocytes. The number of pulsatile foci is reduced and spiral Ca2+ waves are no longer observed. Rather, target patterns of Ca2+ release predominate, and when fragmented, fail to form spirals. Ca2+ wave velocity, amplitude, decay time, and periodicity are also increased. We have simulated these experimental findings by supplementing an existing mathematical model with a differential equation for mitochondrial Ca2+ uptake and release. Our calculations show that mitochondrial Ca2+ efflux plays a critical role in pattern formation by prolonging the recovery time of IP3Rs from a refractory state. We also show that under conditions of high energization of mitochondria, the Ca2+ dynamics can become bistable with a second stable stationary state of high resting Ca2+ concentration.     

The influence of saturation on the predator-prey relations

The well-known Lotka-Volterra differential equations are modified in such way that the predators are supposed to be able to consume only a limited amount of preys in a unit of time. This saturation causes the appearance of nonperiodic solutions while the periodic ones are partly preserved. The paths in the phase plane which correspond to the nonperiodic solutions are expanding spirals of two different shapes. For a particular system of equations, all of the spirals either rotate for ever around the equilibrium point or straighten up at a certain point and head to infinity. The latter alternative occurs if the voracity of the predators is not too great. The biological significance of this result is in the possibility of a simultaneous progressive development of both populations.     

Microscopic spiral waves reveal positive feedback in subcellular calcium signaling.

The regenerative Ca(2+)-induced Ca2+ release mechanism is an important amplifier of signal transduction in diverse cells. In heart muscle cells, this mechanism contributes to the Ca2+ transient activating the mechanical contraction, but it is also believed to drive Ca2+ waves propagating within the cytosol. We investigated the subcellular Ca2+ distribution in heart muscle cells during spontaneous Ca2+ release using laser scanning confocal microscopy with a ratiometric fluorescent indicator technique. Besides planar Ca2+ waves with linear propagation, sequences of confocal optical sections also revealed spiral Ca2+ waves spinning around a subcellular core at approximately 1 Hz. Although the Ca2+ spirals were continuous processes they frequently exhibited an apparently oscillatory output function into the elongated cell body. These oscillatory waves emanating from the spiral at regular intervals were formally considered to be short outer segments of the spiral but could not be distinguished from planar Ca2+ waves propagating along the longitudinal cell axis. The complex spatiotemporal pattern of spiral Ca2+ waves implies the participation of an active process exhibiting a large degree of positive feedback, most likely the Ca(2+)-induced Ca2+ release mechanism.     

From retinal waves to activity-dependent retinogeniculate map development

A neural model is described of how spontaneous retinal waves are formed in infant mammals, and how these waves organize activity-dependent development of a topographic map in the lateral geniculate nucleus, with connections from each eye segregated into separate anatomical layers. The model simulates the spontaneous behavior of starburst amacrine cells and retinal ganglion cells during the production of retinal waves during the first few weeks of mammalian postnatal development. It proposes how excitatory and inhibitory mechanisms within individual cells, such as Ca(2+)-activated K(+) channels, and cAMP currents and signaling cascades, can modulate the spatiotemporal dynamics of waves, notably by controlling the after-hyperpolarization currents of starburst amacrine cells. Given the critical role of the geniculate map in the development of visual cortex, these results provide a foundation for analyzing the temporal dynamics whereby the visual cortex itself develops.     

A single spiral artefact in Arthropod cuticle

Spirals are often seen in sections transverse to the axes of bumped structures in arthropod cuticle. (Sections through arthropod cornea or exocones yield excellent examples.) As arthropod cuticle has a helicoidal architecture (Bouligand, 1965), it might be expected that the spirals are a simple consequence of that structure. According to a symmetry argument, the spirals thus predicted must be double spirals. In contrast, the observed spirals are usually single. We propose that the single spirals result from an interaction between the microtome knife and the cuticle architecture. The direction of knife travel defines an orientation within the cuticle, subverting the symmetry arguments that require double spirals. Bouligand (1972) presented a model for the interaction of the knife with the cuticle. However, we offer arguments and observations which show that Bouligand's model is incorrect. We argue from detailed observations of the single spiral that it is indeed a knifing artifact and that its explanation probably lies within a certain class of models. Two related models based on relative movements of cuticle components are examined via computer techniques.