On the formation of circulating patterns of excitation in anisotropic excitable media

We present a model of excitable media with the feature that it has a vulnerable phase during which a premature current stimulus will result in the formation of a reentrant selfsustained wave of excitation. The model exploits anisotropic coupling of identical cells, and is therefore useful as a model for the myocardium. We give rigorous verification that there is a vulnerable phase, and demonstrate numerically that permanently rotating waves are formed. Finally, it is shown that the direction of fastest propagation in myocardium is not necessarily the direction of highest safety factor, contrary to commonly accepted opinion.     

The emergence of wave emitting centres in an excitable medium

The response of an excitable biological medium to a double local stimulus is considered within the context of a mathematical model for a layer of starving cells of Dictyostelium discoideum, with both spatially one- and two-dimensional (1D and 2D) system being investigated. In contrast to the response usually seen in excitable media, whereby each superthreshold stimulus delivered to the relaxed medium results in the initiation of just one travelling wave, a source emitting a sequence of waves can develop in the present excitable medium after the second stimulus. In a 1D system, only transient wave sources forming a limited number of waves are found. In 2D systems, a permanent wave sources consisting in a pair of spirals are observed as well as the transient wave sources forming circular wave patterns. The general features of the medium dynamics that underlie the observed responses to the double stimulus are discussed.     

Electrical instability in cardiac muscle: phase singularities and rotors

A dynamical system is "excitable" at some stage in its behavior (e.g. at a rest state or while it is nearly at rest prior to a spontaneous event) if a small, but not too small, stimulus of the right kind elicits an immediate big reaction that eventually leads back to the original state. During this return to excitability a typical system is not excitable. An excitable system need not have an attracting rest state; a spontaneous oscillator can be excitable, too, as is common in biological and in chemical excitable kinetics. In a medium characterized by such excitable dynamics at every point, the excitation can propagate as a travelling pulse. Undamaged cardiac muscle shares with other excitable media certain features of such pulse propagation in two and three dimensions. Among the new electrophysiological phenomena thus anticipated are paired mirror-image vortices ("rotors") organized around phase singularities. These should arise in the myocardium near the intersection of a moving critical contour of phase in the normal cycle of excitation and recovery with a momentary critical contour of local stimulus strength. Such intersections, and the corresponding aftermath of paired rotors, should only occur following certain combinations of stimulus size and stimulus timing. Plotting those combinations on a "vulnerability diagram", one delineates a domain for creation of rotors (corresponding to tachycardia) surrounded on all sides by a halo of combinations at which just a few repetitive responses follow stimulation. The experiments called for to check these implications have now been carried out in the special case of electrically-induced tachycardia in healthy canine ventricle. They support the two-dimensional theory, so a new experiment is suggested to demonstrate wholly intramural three-dimensional vortex filaments.     

Real-time computer simulations of excitable media: JAVA as a scientific language and as a wrapper for C and FORTRAN programs.

We describe a useful setting for interactive, real-time study of mathematical models of cardiac electrical activity, using implicit and explicit integration schemes implemented in JAVA. These programs are intended as a teaching aid for the study and understanding of general excitable media. Particularly for cardiac cell models and the ionic currents underlying their basic electrical dynamics. Within the programs, excitable media properties such as thresholds and refractoriness and their dependence on parameter values can be analyzed. In addition, the cardiac model applets allow the study of reentrant tachyarrhythmias using premature stimuli and conduction blocks to induce or to terminate reentrant waves of electrical activation in one and two dimensions. The role of some physiological parameters in the transition from tachycardia to fibrillation also can be analyzed by varying the maximum conductances of ion channels associated with a given model in real time during the simulations. These applets are available for download at http://arrhythmia.hofstra.edu or its mirror site http://stardec.ascc.neu.edu/~fenton.     

Vulnerability in an excitable medium: analytical and numerical studies of initiating unidirectional propagation.

Cardiac tissue can display unusual responses to certain stimulation protocols. In the wake of a conditioning wave of excitation, spiral waves can be initiated by applying stimuli timed to occur during a period of vulnerability (VP). Although vulnerability is well known in cardiac and chemical media, the determinants of the VP and its boundaries have received little theoretical and analytical study. From numerical and analytical studies of reaction-diffusion equations, we have found that 1) vulnerability is an inherent property of Beeler-Reuter and FitzHugh-Nagumo models of excitable media; 2) the duration of the vulnerable window (VW) the one-dimensional analog of the VP, is sensitive to the medium properties and the size of the stimulus field; and 3) the amplitudes of the excitatory and recovery processes modulate the duration of the VW. The analytical results reveal macroscopic behavior (vulnerability) derived from the diffusion of excitation that is not observable at the level of isolated cells or single reaction units.     

Mechanistic inquiry into decrease in probability of defibrillation success with increase in complexity of preshock reentrant activity

Energy requirements for successful antiarrhythmia shocks are arrhythmia specific. However, it remains unclear why the probability of shock success decreases with increasing arrhythmia complexity. The goal of this research was to determine whether a diminished probability of shock success results from an increased number of functional reentrant circuits in the myocardium, and if so, to identify the responsible mechanisms. To achieve this goal, we assessed shock efficacy in a bidomain defibrillation model of a 4-mm-thick slice of canine ventricles. Shocks were applied between a right ventricular cathode and a distant anode to terminate either a single scroll wave (SSW) or multiple scroll waves (MSWs). From the 160 simulations conducted, dose-response curves were constructed for shocks given to SSWs and MSWs. The shock strength that yielded a 50% probability of success (ED(50)) for SSWs was found to be 13% less than that for MSWs, which indicates that a larger number of functional reentries results in an increased defibrillation threshold. The results also demonstrate that an isoelectric window exists after both failed and successful shocks; however, shocks of strength near the ED(50) value that were given to SSWs resulted in 16.3% longer isoelectric window durations than the same shocks delivered to MSWs. Mechanistic inquiry into these findings reveals that the two main factors underlying the observed relationships are 1) smaller virtual electrode polarizations in the tissue depth, and 2) differences in preshock tissue state. As a result of these factors, intramural excitable pathways leading to delayed breakthrough on the surface were formed earlier after shocks given to MSWs compared with SSWs and thus resulted in a lower defibrillation threshold for shocks given to SSWs.     

A condition for setting off ectopic waves in computational models of excitable cells

The purpose of this paper is to study the stability of steady state solutions of the Monodomain model equipped with Luo-Rudy I kinetics. It is well established that re-entrant arrhythmias can be created in computational models of excitable cells. Such arrhythmias can be initiated by applying an external stimulus that interacts with a partially refractory region, and spawn breaking waves that can eventually generate extremely complex wave patterns commonly referred to as fibrillation. An ectopic wave is one possible stimulus that may initiate fibrillation. Physiologically, it is well known that ectopic waves exist, but the mechanism for initiating ectopic waves in a large collection of cells is poorly understood. In the present paper we consider computational models of collections of excitable cells in one and two spatial dimensions. The cells are modeled by Luo-Rudy I kinetics, and we assume that the spatial dynamics is governed by the Monodomain model. The mathematical analysis is carried out for a reduced model that is known to provide good approximations of the initial phase of solutions of the Luo-Rudy I model. A further simplification is also introduced to motivate and explain the results for the more complicated models. In the analysis the cells are divided into two regions; one region (N) consists of normal cells as model by the standard Luo-Rudy I model, and another region (A) where the cells are automatic in the sense that they would act as pacemaker cells if they where isolated from their surroundings. We let delta denote the spatial diffusion and a denote a characteristic length of the automatic region. It has previously been shown that reducing diffusion or increasing the automatic region enhances ectopic activity. Here we derive a condition for the transition from stable resting state to ectopic wave spread. Under suitable assumptions on the model we provide mathematical and computational arguments indicating that there is a constant eta such that a steady state solution of this system is stable whenever delta approximately > etaa(2), and unstable whenever delta approximately < etaa(2).     

An electromechanical model of cardiac tissue: constitutive issues and electrophysiological effects

We present an electromechanical model of myocardium tissue coupling a modified FitzHugh-Nagumo type system, describing the electrical activity of the excitable media, with finite elasticity, endowed with the capability of describing muscle contractions. The high degree of deformability of the medium makes it mandatory to set the diffusion process in a moving domain, thereby producing a direct influence of the deformation on the electrical activity. Various mechano-electric effects concerning the propagation of cylindrical waves, the rotating spiral waves, and the spiral breakups are discussed.     

Success and failure of biphasic shocks: results of bidomain simulations.

The mechanisms behind the superiority of optimal biphasic defibrillation shocks over monophasic are not fully understood. This simulation study examines how the shock polarity and second-phase magnitude of biphasic shocks influence the virtual electrode polarization (VEP) pattern, and thus the outcome of the shock in a bidomain model representation of ventricular myocardium. A single spiral wave is initiated in a two-dimensional sheet of myocardium that measures 2 x 2 cm(2). The model incorporates non-uniform fiber curvature, membrane kinetics suitable for high strength shocks, and electroporation. Line electrodes deliver a spatially uniform extracellular field. The shocks are biphasic, each phase lasting 10 ms. Two different polarities of biphasic shocks are examined as the first-phase configuration is held constant and the second-phase magnitude is varied between 1 and 10 V/cm. The results show that for each polarity, varying the second-phase magnitude reverses the VEP induced by the first phase in an asymmetric fashion. Further, the size of the post-shock excitable gap is dependent upon the second-phase magnitude and is a factor in determining the success or failure of the shock. The maximum size of a post-shock excitable gap that results in defibrillation success depends on the polarity of the shock, indicating that the refractoriness of the tissue surrounding the gap also contributes to the outcome of the shock.     

Reentry in heterogeneous cardiac tissue described by the Luo-Rudy ventricular action potential model

Heterogeneity of cardiac tissue is an important factor determining the initiation and dynamics of cardiac arrhythmias. In this paper, we studied the effects of gradients of electrophysiological heterogeneity on reentrant excitation patterns using computer simulations. We investigated the dynamics of spiral waves in a two-dimensional sheet of cardiac tissue described by the Luo-Rudy phase 1 (LR1) ventricular action potential model. A gradient of action potential duration (APD) was imposed by gradually varying the local current density of K(+) current or inward rectifying K(+) current along one axis of the tissue sheet. We show that a gradient of APD resulted in spiral wave drift. This drift consisted of two components. The longitudinal (along the gradient) component was always directed toward regions of longer spiral wave period. The transverse (perpendicular to the gradient) component had a direction dependent on the direction of rotation of the spiral wave. We estimated the velocity of the drift as a function of the magnitude of the gradient and discuss its implications.     

A model for human ventricular tissue

The experimental and clinical possibilities for studying cardiac arrhythmias in human ventricular myocardium are very limited. Therefore, the use of alternative methods such as computer simulations is of great importance. In this article we introduce a mathematical model of the action potential of human ventricular cells that, while including a high level of electrophysiological detail, is computationally cost-effective enough to be applied in large-scale spatial simulations for the study of reentrant arrhythmias. The model is based on recent experimental data on most of the major ionic currents: the fast sodium, L-type calcium, transient outward, rapid and slow delayed rectifier, and inward rectifier currents. The model includes a basic calcium dynamics, allowing for the realistic modeling of calcium transients, calcium current inactivation, and the contraction staircase. We are able to reproduce human epicardial, endocardial, and M cell action potentials and show that differences can be explained by differences in the transient outward and slow delayed rectifier currents. Our model reproduces the experimentally observed data on action potential duration restitution, which is an important characteristic for reentrant arrhythmias. The conduction velocity restitution of our model is broader than in other models and agrees better with available data. Finally, we model the dynamics of spiral wave rotation in a two-dimensional sheet of human ventricular tissue and show that the spiral wave follows a complex meandering pattern and has a period of 265 ms. We conclude that the proposed model reproduces a variety of electrophysiological behaviors and provides a basis for studies of reentrant arrhythmias in human ventricular tissue.     

Image processing techniques applied to excitation waves in the chicken retina

Image processing techniques are described in detail that are used to gain information about the dynamics of wave propagation in excitable media. We focus on a phenomenon called spreading depression (SD) observed in the chicken retina, but the techniques described here concern a large variety of excitable systems. Despite the impressive progress both in SD research of the past 50 years and, during nearly the same period, in the theory of self-organization of wave patterns, there is still little mutual overlap. However, the increasing demands for understanding complex systems, like neuronal tissue, require such theoretical concepts. Arguments are given why the chicken retina is a nearly perfect experimental system for assessing and further developing these concepts.     

Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias

We introduce the concept of a contracting excitable medium that is capable of conducting non-linear waves of excitation that in turn initiate contraction. Furthermore, these kinematic deformations have a feedback effect on the excitation properties of the medium. Electrical characteristics resemble basic models of cardiac excitation that have been used to successfully study mechanisms of reentrant cardiac arrhythmias in electrophysiology. We present a computational framework that employs electromechanical and mechanoelectric feedback to couple a three-variable FitzHugh–Nagumo-type excitation-tension model to the non-linear stress equilibrium equations, which govern large deformation hyperelasticity. Numerically, the coupled electromechanical model combines a finite difference method approach to integrate the excitation equations, with a Galerkin finite element method to solve the equations governing tissue mechanics. We present example computations demonstrating various effects of contraction on stationary rotating spiral waves and spiral wave break. We show that tissue mechanics significantly contributes to the dynamics of electrical propagation, and that a coupled electromechanical approach should be pursued in future electrophysiological modelling studies.     

Microgravity dependence of excitable biological and physicochemical media

Neuronal tissue and especially the central nervous system (CNS) is an excitable medium. Self-organisation, pattern formation, and propagating excitation waves as typical characteristics in excitable media consequently have been found in neuronal tissue. The properties of such phenomena in excitable media do critically depend on the parameters (i.e., electromagnetic fields, temperature, chemical drugs) of the system and on small external forces to which gravity belongs. The spreading depression, a propagating excitation depression wave of neuronal activity, is one of the best described of the those wave phenomena in the CNS. Especially in the retina as a true part of the CNS it can be easily observed with optical techniques due to the high intrinsic optical signal of this tissue. Another of such waves in neuronal tissue is the propagating action potential in nerve fibres. In this paper, data from our laboratories concerning the influence of gravity on the velocity of propagating waves in excitable media are summarized mainly in terms of the retinal spreading depression and propagating action potentials. Additionally, we have used waves in gels of the Belousov-Zhabotinsky reaction as the physicochemical model system of biological activity as the properties of these waves follow the same theories as the spreading depression and action potentials and they have some striking similarities in wave behavior. Thus propagating Belousov-Zhabotinsky waves are described by their gravity dependence.     

A model for feature linking via collective oscillations in the primary visual cortex

A neural network model for explaining experimentally observed neuronal responses in cat primary visual cortex is proposed. In our model, the basic functional unit is an orientation column which is represented by a large homogeneous population of neurons modeled as integrate-and-fire type excitable elements. The orientation column exhibits spontaneous collective oscillations in activity in response to suitable visual stimuli. Such oscillations are caused by mutual synchronization among the neurons within the column. Numerical simulation for various stimulus patterns shows that as a result of activity correlations between different columns, the amplitude and the phase of the oscillation in each column depend strongly on the global feature of the stimulus pattern. These results satisfactorily account for experimental observations.     

Excitability with multiple thresholds. A new mode of dynamic behavior analyzed in a regulated biochemical system

We analyze in a biochemical model the phenomenon of excitability in which suprathreshold perturbations of a stable steady state are amplified in a pulsatory manner. The two-variable model is that of an autocatalytic enzyme reaction with recycling of product into the substrate. This model was previously studied for the coexistence between two stable periodic regimes (birhythmicity). We show that the multiplicity of dynamic behavioral modes extends to the phenomenon of excitability. Whereas excitable behavior is generally characterized by a single threshold for excitation, two distinct thresholds may coexist in this model. Moreover, in these conditions, two different plateaux are obtained for the response amplitude when the stimulus is gradually increased. By means of phase plane analysis we explain the origin of multiple thresholds for excitability and predict the conditions for their occurrence. Implications of the phenomenon for excitable cells, in particular for neurons, are discussed.     

Noise-induced spatiotemporal patterns in Hodgkin–Huxley neuronal network

The effect of noise on the pattern selection in a regular network of Hodgkin–Huxley neurons is investigated, and the transition of pattern in the network is measured from subexcitable to excitable media. Extensive numerical results confirm that kinds of travelling wave such as spiral wave, circle wave and target wave could be developed and kept alive in the subexcitable network due to the noise. In the case of excitable media under noise, the developed spiral wave and target wave could coexist and new target-like wave is induced near to the border of media. The averaged membrane potentials over all neurons in the network are calculated to detect the periodicity of the time series and the generated traveling wave. Furthermore, the firing probabilities of neurons in networks are also calculated to analyze the collective behavior of networks.     

The mechanisms for compression and reflection of cortical waves

Waves are common in cortical networks and may be important for carrying information about a stimulus from one local circuit to another. In a recent study of visually evoked waves in rat cortex, compression and reflection of waves are observed as the activation passes from visual areas V1 to V2. The authors of this study apply bicuculline (BMI) and demonstrate that the reflection disappears. They conclude that inhibition plays a major role in compression and reflection. We present several models for propagating waves in heterogeneous media and show that the velocity and thus compression depends weakly on inhibition. We propose that the main site of action of BMI with respect to wave propagation is on the threshold for firing which we suggest is related to action on potassium channels. We combine numerical and analytic methods to explore both compression and reflection in an excitable system with synaptic coupling.     

Virtual electrodes in cardiac tissue: a common mechanism for anodal and cathodal stimulation.

Traditional cable analyses cannot explain complex patterns of excitation in cardiac tissue with unipolar, extracellular anodal, or cathodal stimuli. Epifluorescence imaging of the transmembrane potential during and after stimulation of both refractory and excitable tissue shows distinctive regions of simultaneous depolarization and hyperpolarization during stimulation that act as virtual cathodes and anodes. The results confirm bidomain model predictions that the onset (make) of a stimulus induces propagation from the virtual cathode, whereas stimulus termination (break) induces it from the virtual anode. In make stimulation, the virtual anode can delay activation of the underlying tissue, whereas in break stimulation this occurs under the virtual cathode. Thus make and break stimulations in cardiac tissue have a common mechanism that is the result of differences in the electrical anisotropy of the intracellular and extracellular spaces and provides clear proof of the validity of the bidomain model.