A Model for the Propagation of Action Potentials in Non-Uniformly Excitable Media

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.

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.

Analytical and simulation results for the stochastic spatial Fitzhugh-Nagumo model neuron

For the Fitzhugh-Nagumo system with space-time white noise, we use numerical methods to consider the generation of action potentials and the reliability of transmission in the presence of noise. The accuracy of simulated solutions is verified by comparison with known exact analytical results. Noise of small amplitude may prevent transmission directly, whereas larger-amplitude noise may also interfere by producing secondary nonlocal responses. The probability of transmission as a function of noise amplitude is found for both uniform noise and noise restricted to a patch. For certain parameter ranges, the recovery variable may be neglected to give a single-component nonlinear diffusion with space-time white noise. In this case, analytical results are obtained for small perturbations and noise, which agree well with simulation results. For the voltage variable, expressions are given for the mean, covariance, and variance and their steady-state forms. The spectral density of the voltage is also obtained. Numerical examples are given of the difference between the properties of nonlinear and linear cables, and the validity of the expressions obtained for the statistical properties is investigated as a function of noise amplitude. For given parameters, analytical results are in good agreement with simulation until a certain critical noise amplitude is reached, which can be estimated. The role of trigger zones in increasing the reliability of transmission is discussed.

Dynamics of spiral pairs induced by unidirectional propagating pulses

The dynamics of unidirectionally propagating pulses in a two-dimensional uniform excitable reaction-diffusion medium is investigated. It is shown that under weak diffusion coupling between medium points such a pulse can evolve into a pair of counter-rotating spirals (spiral pair). We analyze the drift of such a pair and examine the collisions between several drifting pairs. It is demonstrated that collisions can result in a special type of reflection or, alternatively, in new types of complex stationary spiral structures. A possible application of these findings for the diagnosis of cardiac arrhythmias is suggested.

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

Electrical heterogeneities play a role in the initiation of cardiac arrhythmias. In certain pathological conditions such as ischaemia, current sinks can develop in the diseased cardiac tissue. In this study, we investigate the effects of changing the amount of heterogeneity and intercellular coupling on wavefront stability in a cardiac cell culture system and a mathematical model of excitable media. In both systems, we observe three types of behaviour: plane wave propagation without breakup, plane wave breakup into spiral waves and plane wave block. In the theoretical model, we observe a linear decrease in propagation velocity as the number of heterogeneities is increased, followed by a rapid, nonlinear decrease to zero. The linear decrease results from the heterogeneities acting independently on the wavefront. A general scaling argument that considers the degree of system heterogeneity and the properties of the excitable medium is used to derive a dimensionless parameter that describes the interaction of the wavefront with the heterogeneities.

Stimulation of unidirectional pulses in excitable systems

Using a judicious spatial shape of input current pulses (and electrodes), responses of an excitable system (FitzHugh-Nagumo) appear as unidirectional pulses (UDP's) instead of bidirectional ones (in one dimension) or circular ones (in two dimensions). The importance of the UDP's for a possible mechanism for pinpointing the reentry cycle position and for a possible use in tachycardia suppression is discussed.

Abnormal frequency locking and the function of the cardiac pacemaker

A heterogeneous reaction-diffusion medium consisting of two adjoining uniform regions is analyzed. The first region is a purely oscillatory one, while the second is bistable (oscillatory/excitable). We show that such a construction allows an abnormal domination of the low natural frequency of the oscillatory regime over the whole medium (abnormal frequency locking). Bifurcations leading to the appearance of the bistable regime are discussed as well as the specific dynamics of the bistable oscillations. The abnormal frequency-locking phenomenon could explain some dynamical properties of the cardiac pacemaker.

Pseudoreflection from interface between two oscillatory media: extended driver

The dynamics of a reaction-diffusion medium composed of two uniform self-oscillating regions is considered. We analyze the phenomenon of pseudoreflection of waves at the region's interface. The reflected waves show an unusual change of wavelength, amplitude, and period. In contrast to our previous results, here this behavior can be perceived as an action of a spatially extended higher-frequency "driver." Observed also are the interesting phenomena of the appearance of narrow transient zones near the interface and of diffusion-induced bifurcations. Furthermore, the pseudoreflection is shown to be a possible mechanism of spiral and "target" waves generation. The relevance of the obtained results to the dynamics of the cardiac sinus node is discussed.

Reaction-diffusion dynamics in an oscillatory medium of finite size: pseudoreflection of waves

Wave propagation in an oscillatory reaction-diffusion, one-dimensional domain of finite size with Dirichlet boundary conditions is analyzed. For sizes below a certain threshold length, the medium cannot sustain wave motion. Above this threshold we find that for a relatively small domain extent, a strong correlation exists between the dynamics of the system and its size. This correlation gradually disappears with increasing domain size. For still larger sizes, we observe an effect of wave pseudo reflection near the boundary. It is shown both numerically and analytically that pseudoreflected waves are periodically generated inside the medium by a fast, self-generated "source" near the boundary.

The sinus node as a nonlinear dynamic system

The Sinus Node (SN) of the heart is the natural pacemaker driving electrical impulses into the atria, an ordinary excitable medium. In order to avoid wave interference, the SN operation must be of such a nature that no reflections from the atria should occur. It is shown that such a behavior is achieved when the SN operates in a nonlinear dynamical regime situated away from relaxation.

Inwards propagating waves in a limit cycle medium

The existence of a novel inwards propagating wave motion is demonstrated in a limit-cycle medium both for the FitzHugh-Nagumo and for modified Chernyak-Starobin-Cohen reaction-diffusion systems. The waves (pulses) are seen to be moving "backwards," that is, towards the point where the triggering pulse was initiated, instead of the regular propagation away from the origin. The feasibility of the phenomenon and some of its features are analyzed.