Hysteresis phenomena between periodic and stationary solutions in a model of pacemaker and nonpacemaker coupled cardiac cells

Pacemaking function of two simplified cell models

Simplified nonlinear models of biological cells are widely used in computational electrophysiology. The models reproduce qualitatively many of the characteristics of various organs, such as the heart, brain, and intestine. In contrast to complex cellular ion-channel models, the simplified models usually contain a small number of variables and parameters, which facilitates nonlinear analysis and reduces computational load. In this paper, we consider pacemaking variants of the Aliev-Panfilov and Corrado two-variable excitable cell models. We conducted a numerical simulation study of these models and investigated the main nonlinear dynamic features of both isolated cells and 1D coupled pacemaker-excitable systems. Simulations of the 2D sinoatrial node and 3D intestine tissue as application examples of combined pacemaker-excitable systems demonstrated results similar to obtained previously. The uniform formulation for the conventional excitable cell models and proposed pacemaker models allows a convenient and easy implementation for the construction of personalized physiological models, inverse tissue modeling, and development of real-time simulation systems for various organs that contain both pacemaker and excitable cells.

Effects of pacing magnitudes and forms on bistability width in a modeled ventricular tissue

Bistability in periodically paced cardiac tissue is relevant to cardiac arrhythmias and its control. In the present paper, one-dimensional tissue of the phase I Luo-Rudy model is numerically investigated. The effects of various parameters of pacing signals on bistability width are studied. The following conclusions are obtained: (i) Pacing can be classified into two types: pulsatile and sinusoidal types. Pulsatile pacing reduces bistability width as its magnitude is increased. Sinusoidal pacing increases the width as its amplitude is increased. (ii) In a pacing period the hyperpolarizing part plays a more important role than the depolarizing part. Variations of the hyperpolarizing ratio in a period evidently change the width of bistability and its variation tendency. (iii) A dynamical mechanism is proposed to qualitatively explain the phenomena, which reveals the reason for the different effects of pulsatile and sinusoidal pacing on bistability. The methods for changing bistability width by external pacing may help control arrhythmias in cardiology.

Reduced models for unidirectional block conduction and their geometrical setting

This article revisits a reduced model of cardiac electro-physiology which was proposed to understand the genesis of unidirectional block pathology and of ectopic foci. We underline some specificities of the model from the viewpoint of dynamical systems and bifurcation theory. We point out that essentially the same properties are shared by a simpler system more accessible to analysis. With this simpler system, it becomes possible to give a new presentation of the phenomenon in a phase plane with time moving slow manifolds. This presentation can be of interest both for cardiac electro-physiologists and for mathematicians.

ROLE OF CELLULAR COUPLING AND DISPERSION OF REFRACTORINESS IN CARDIAC ARRHYTHMIAS: A SIMULATION STUDY

Computer simulation is applied to study the role of cellular coupling, dispersion of refractoriness as well as both of them, in the mechanisms underlying cardiac arrhythmias. We first assumed that local ischemia mainly induces cell to cell dispersion in the coupling resistance (case 1), refractory period (case 2) or both (case 3). Numerical experiments, based on the van Capelle and Durrer model, showed that vortices could not be induced in these conditions. In order to be more realistic about coronary circulation we simulated a patchy dispersion of cellular properties, each patch corresponding to the zone irrigated by a small coronary artery. In these conditions, a single activation wave could give rise to abnormal activities. Probabilities of reentry, estimated for the three cases cited above, showed that a severe alteration of the coupling resistance may be an important factor in the genesis of reentry. Moreover, use of isochronal maps revealed that vortices were both stable and sustained with an alteration of coupling alone or along with reductions of action potential duration. Conversely, simulations with reduction of the refractoriness alone induced only transient patterns.

Hysteresis modeling for calcium-mediated ciliary beat frequency in airway epithelial cells

A hysteresis model is proposed to describe calcium-mediated Ciliary beat frequency (CBF) in airway epithelial cells. In this dynamic model, the kinetics of coupling between calcium and CBF is posited as a two-step configuration. First, Ca²+ directly binds to or indirectly acts on the axonemal proteins to modulate the activity of axonemal proteins. This step can be modeled by a Hill function in biochemistry. In the second step, the activity of axonemal proteins interacts with the sliding velocity of axonemal microtubules, the equivalent to regulating the CBF. The well-known Bouc-Wen model for hysteresis in mechanical engineering, which can only generate the stable clockwise hysteresis loops, is modified to describe the counter clockwise hysteresis loops commonly observed in the biological experiments. Based upon this new hysteresis model, the dynamic behavior of calcium-regulated CBF in epithelial airway cells is investigated through simulation studies. The numerical results demonstrate that the CBF dynamics in airway epithelial cells predicted by the hysteresis model is more consistent with the experimental observations than that predicted by previous static model in the literature.

Hysteresis and bistability in periodically paced cardiac tissue

Hysteresis in periodically paced cardiac tissue is an important issue due to its relevance to cardiac arrhythmias. In the present paper, the mechanism of hysteresis formation and the related properties are interpreted by numerically investigating the phase I Luo-Rudy model. A formula calculating the width of hysteresis is proposed and well confirmed by numerical simulations. We also find that hysteresis in cardiac tissue shows several characteristics due to couplings among cardiac cells which are absent in a single cell. The influences of the physiological parameters are studied in detail. The model dependence of hysteresis is elucidated by considering a number of well-known models of excitable media. Moreover, the influence of bistability on controlling arrhythmias is revealed.

Spontaneous Initiation and Termination of Complex Rhythms in Cardiac Cell Culture

INTRODUCTION

Complex cardiac arrhythmias often start and stop spontaneously. These poorly understood behaviors frequently are associated with pathologic modification of the structural heterogeneity and functional connectivity of the myocardium. To evaluate underlying mechanisms, we modify heterogeneity by varying the confluence of embryonic chick monolayer cultures that display complex bursting behaviors. A simple mathematical model was developed that reproduces the experimental behaviors and reveals possible generic mechanisms for bursting dynamics in heterogeneous excitable systems.

METHODS AND RESULTS

Wave propagation was mapped in embryonic chick myocytes monolayers using calcium-sensitive dyes. Monolayer confluence was varied by plating cultures with different cell densities and by varying times in culture. At high plating densities, waves propagate without breaks, whereas monolayers plated at low densities display spirals with frequent breaks and irregular activation fronts. Monolayers at intermediate densities display bursting rhythms in which there is paroxysmal starting and stopping of spiral waves of activity. Similar spatiotemporal patterns of activity were also observed as a function of the time in culture; irregular activity dominates the first 30 hours, followed by repetitive bursting dynamics until 54 hours, after which periodic target patterns or stable spirals prevail. In some quiescent cultures derived from older embryos, it was possible to trigger pacemaker activity following a single activation. We are able to reproduce all of these behaviors by introducing spatial heterogeneity and varying neighborhood size, equivalent to cell connectivity, in a spontaneous cellular automaton model containing a rate-dependent fatigue term.

CONCLUSION

We observe transitions from irregular propagating waves, to spiral waves that spontaneously start and stop, to target waves originating from localized pacemakers in cell culture and a simple theoretical model of heterogeneous excitable media. The results show how physiologic properties of spontaneous activity, heterogeneity, and fatigue can give rise to a wide range of different complex dynamic behaviors similar to clinically observed cardiac arrhythmias.

Role of the dispersion of refractoriness on cardiac reentries

We used computer simulation to study the possible role of the dispersion of cellular coupling, refractoriness or both, in the mechanisms underlying cardiac arrhythmias. Local ischemia was first assumed to induce cell to cell dispersion of the coupling resistance (case 1), refractory period (case 2), or both (case 3). Our numerical experiments based on the van Capelle and Durrer model showed that vortices could not be induced. On the other hand, with cellular properties dispersed in a patchy way within the ischemic zone, a single activation wave could give rise to abnormal activities. This demonstrates the stability of the wave front under small inhomogeneities. Probabilities of reentry, estimated for the three cases cited above showed that a severe alteration of the coupling resistance may be an important factor in the genesis of reentry. Moreover, use of isochronal maps revealed that vortices were both stable and sustained with an alteration of the coupling alone or along with a reduction of the action potential duration. Conversely, simulations with reduction of the refractoriness alone, inducing only transient patterns, could exhibit functionally determined reentries.

Analytical modeling of the hysteresis phenomenon in guinea pig ventricular myocytes

In the present study, we have demonstrated hysteresis phenomena in the excitability of single, enzymatically dissociated guinea pig ventricular myocytes. Membrane potentials were recorded with patch pipettes in the whole-cell current clamp configuration. Repetitive stimulation with depolarizing current pulses of constant cycle length and duration but varying strength led to predictable excitation (1:1) and non-excitation (1:0) patterns depending on current strength. In addition, transition between patterns depended on the direction of current intensity change and stable hysteresis loops were obtained in stimulus:response pattern vs. current intensity plots in 14 cells. Increase of pulse duration and decrease of stimulation rate contributed to a reduction in hysteresis loop areas. Changes in amplitude and shape of the subthreshold responses during the transitions from one stable pattern to the other, suggested that activity led to an increase in membrane resistance, particularly in the voltage domain between resting potential, and threshold. Therefore, we modelled the dynamic behaviour of the single cells as a function of diastolic membrane resistance, using previously published analytical solutions. Numerical iteration of the analytical model equations closely reproduced the experimental hysteresis loops in both qualitative and quantitative ways. In particular, the effect of stimulation frequency on the model was similar to the experimental findings. The overall study suggests that the excitability pattern of guinea pig ventricular myocytes accounts for hysteresis and bistabilities when current intensity is allowed to fluctuate around threshold levels.

Hysteresis in the excitability of isolated guinea pig ventricular myocytes

Hysteresis phenomena were demonstrated in the excitability of single, enzymatically dissociated guinea pig ventricular myocytes. Membrane potentials were recorded with patch pipettes in the whole-cell current-clamp configuration. Repetitive stimulation with depolarizing current pulses of constant cycle length and duration but varying strength led to predictable excitation (1:1) and nonexcitation (1:0) patterns depending on current strength. However, transition between patterns depended on the direction of current strength change, and stable hysteresis loops were obtained in stimulus-response pattern versus current strength plots in 31 cells. Increase of pulse duration and decrease of stimulation rate contributed to a reduction in hysteresis loop areas. In addition, at the abrupt transitions from 1:0 to 1:1 patterns, a latency adaptation phenomenon was consistently observed. Bath application of tetrodotoxin (30 microM) produced no change of hysteresis, whereas hysteresis was substantially decreased in cobalt (2 mM) superfusion experiments. Analysis of the changes in amplitude and shape of the subthreshold responses during the transitions from one stable pattern to the other suggested that activity led to an increase in membrane resistance, particularly in the voltage domain between resting and threshold potentials. We therefore modeled the dynamic behavior of the single cells, using an analytical solution aimed at calculating the recovery of activation latency as a function of diastolic membrane resistance. Numerical iteration of the analytical model equations closely reproduced the experimental hysteresis loops in both qualitative and quantitative ways. The effect of stimulation frequency on the model was similar to the experimental findings. The overall study suggests that the excitability pattern of guinea pig ventricular myocytes is responsible for hysteresis and bistabilities when current intensity is allowed to fluctuate around threshold levels.

Bistabilities and annihilation phenomena in electrophysiological cardiac models

We have investigated the oscillatory behavior of cardiac cellular elements simulated by two electrophysiological models: the van Capelle and Durrer (VCD) model and the sinoatrial node cell model of Yanagihara, Noma, and Irisawa (YNI). The VCD model behavior was examined systematically by using continuation-bifurcation analysis. Bifurcation diagrams were constructed as a function of Qit1, an intrinsic parameter of the model, which sets both maximum diastolic potential and depolarization threshold of the cell. The existence of stable high amplitude oscillations was evidenced between two Hopf bifurcation points (HB). Near each HB, a zone of bistability was detected. Close to the HB that corresponded to high values of Qit1, a high amplitude periodic stable state coexisted with a stable steady state. Close to the other HB, in a narrow range of lower Qit1 values, a relatively high amplitude periodic stable state coexisted with a low amplitude periodic stable state. There was no stable steady state in the latter bistability zone. Through the use of phase-plane representations and the determination of separatrices between the different attractor basins, we could deduce the conditions of timing, polarity, and strength needed for a pulse perturbation to send the system from one state to another and vice versa. The YNI model was analyzed by numerical simulation, and the oscillatory behavior of the sinoatrial node cell was explored while applying a depolarizing bias current of various strengths. Results were similar to those obtained from the VCD model in that there were two bistability regions for two different ranges of applied bias current. Depending on current intensity, annihilation of pacemaker activity could be achieved in both zones. However, the coexistence of two oscillatory stable states was never observed in the YNI model. From the behavioral similarities of these different models, we can conclude that bistabilities and annihilation phenomena can be found in transitional zones between quiescence and rhythmic activity.