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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.
We present systematic evidence for the origins of 1/f -type temporal scaling in human heart rate. The heart rate is regulated by the activity of two branches of the autonomic nervous system: the parasympathetic (PNS) and the sympathetic (SNS) nervous systems. We examine alterations in the scaling property when the balance between PNS and SNS activity is modified, and find that the relative PNS suppression by congestive heart failure results in a substantial increase in the Hurst exponent H towards random-walk scaling 1/ f(2) and a similar breakdown is observed with relative SNS suppression by primary autonomic failure. These results suggest that 1/f scaling in heart rate requires the intricate balance between the antagonistic activity of PNS and SNS.
We consider an extension of Granger causality to nonlinear bivariate time series. In this frame, if the prediction error of the first time series is reduced by including measurements from the second time series, then the second time series is said to have a causal influence on the first one. Not all the nonlinear prediction schemes are suitable to evaluate causality; indeed, not all of them allow one to quantify how much knowledge of the other time series counts to improve prediction error. We present an approach with bivariate time series modeled by a generalization of radial basis functions and show its application to a pair of unidirectionally coupled chaotic maps and to physiological examples.
Action potential duration (APD) restitution, which relates APD to the preceding diastolic interval (DI), is a useful tool for predicting the onset of abnormal cardiac rhythms. However, it is known that different pacing protocols lead to different APD restitution curves (RCs). This phenomenon, known as APD rate dependence, is a consequence of memory in the tissue. In addition to APD restitution, conduction velocity restitution also plays an important role in the spatiotemporal dynamics of cardiac tissue. We present results concerning rate-dependent restitution in the velocity of propagating action potentials in a one-dimensional fiber. Our numerical simulations show that, independent of the amount of memory in the tissue, the wave-back velocity exhibits pronounced rate dependence and the wave-front velocity does not. Moreover, the discrepancy between wave-back velocity RCs is most significant for a small DI. We provide an analytical explanation of these results, using a system of coupled maps to relate the wave-front and wave-back velocities. Our calculations show that rate-dependent wave-back velocity can be present even if neither APD nor wave-front velocity exhibits rate dependence.
The cardiac muscle is well known to conduct electric impulses anisotropically, showing a larger conduction velocity along than across fibers. Fiber orientation is not uniform within the cardiac wall, but rotates by as much as 180 degrees throughout the wall thickness. Numerical simulations and experiments have indicated that this rotational anisotropy considerably affects the spread of excitation in cardiac tissue: the wave front shows a complex intramural shape with trailing cusps. The cusps can travel across layers and reach the epicardial and endocardial surfaces where they cause apparent accelerations of propagation. In the present study we provide an analytical description of the asymptotic wave front, as well as of cusp waves. We investigate the motion of cusp waves, based on the assumption that they occur at the intersection of asymptotic solutions, and we show that our theoretical analysis is in close agreement with numerical simulations. The asymptotic solutions are found to be determined purely by the fiber organization within the cardiac wall, independent of the excitable properties of cardiac tissue.
The bidomain model can be used for calculating the electrical potential in the heart during defibrillation. However, this model consists of a coupled system of two partial differential equations that are, in general, difficult and time consuming to solve. In this paper, we present an approximate, iterative method of solving the bidomain equations. After working out the general method, we apply it to four problems: (i) a cylindrical strand in a uniform electric field, (ii) a nonuniform electric field applied to tissue with straight fibers, (iii) a spherical heart in a uniform electric field, and (iv) a two-dimensional sheet of cardiac tissue with curving fibers. Finally, we analyze the general case of three dimensions.
We study numerically the dynamics of conduction blocks using a detailed electrophysiological model. We find that this dynamics depends critically on the size of the paced region. Small pacing regions lead to stationary conduction blocks while larger pacing regions can lead to conduction blocks that travel periodically towards the pacing region. We show that this size-dependence dynamics can lead to a novel arrhythmogenic mechanism. Furthermore, we show that the essential phenomena can be captured in a much simpler coupled-map model.
We study numerically the dynamics of a scroll wave in a three-dimensional (3D) excitable medium in the presence of substantial meandering of the corresponding 2D spiral wave in the Aliev-Panfilov model. We identify three types of dynamics of the scroll wave filament--quasi-2D, periodic, and aperiodic meandering--and we study their dependence on parameter settings and thickness of the medium.
Motivated by the fact that many empirical time series--including changes of heartbeat intervals, physical activity levels, intertrade times in finance, and river flux values--exhibit power-law anticorrelations in the variables and power-law correlations in their magnitudes, we propose a simple stochastic process that can account for both types of correlations. The process depends on only two parameters, where one controls the correlations in the variables and the other controls the correlations in their magnitudes. We apply the process to time series of heartbeat interval changes and air temperature changes and find that the statistical properties of the modeled time series are in agreement with those observed in the data.
Spiral wave formation and disorganized activity in excitable media require the existence of broken waves and are related to partial wave block. The determinants of wave block in excitable systems are incompletely understood, especially for cardiac excitable tissue. Previous work in one-dimensional cardiac models has suggested that wave break of a premature excitation (PE) requires critical timing and that the conditions for broken waves are improbable. We analyzed the mechanism of unidirectional wave block that occurs when two consecutive PEs interact with a normal plane wave in a generic one-dimensional spatial excitable medium. A nondimensional coupled-map model built from mesoscopic characteristics of the substrate (the velocity and action potential duration restitution functions) shows that block can occur over a large interval of timing between the two PEs and leads to wave break in two-dimensional media. This mechanism may be an important determinant of spiral wave formation by the response to premature excitations.
We analyze the effect of blood flow through large arteries of peripheral circulation on heat transfer in living tissue. Blood flow in such arteries gives rise to fast heat propagation over large scales, which is described in terms of heat superdiffusion. The corresponding bioheat heat equation is derived. In particular, we show that under local strong heating of a small tissue domain the temperature distribution inside the surrounding tissue is affected substantially by heat superdiffusion.
The dynamics of reentry in a model of a two-dimensional annulus of homogeneous cardiac tissue, with a Beeler-Reuter type formulation of the membrane ionic currents, is examined. The bifurcation structure of the sustained propagated solutions is described as a function of Rin and Rout, the inner and outer radii of the annulus. The transition from periodic to quasiperiodic reentry occurs at a critical Rin, which first diminishes and then saturates as Rout is increased. The reduction of the critical Rin is a consequence of the increase of the wave-front curvature. There is a range of Rin below the critical radius in which two distinct quasiperiodic solutions coexist. Each of these solutions disappears at its own specific value of Rin, and their annihilation is preceded by a new type of bifurcation leading to a regime of propagation with transient successive detachments of the wave front from the inner border of the annulus.
In this work, we reanalyze the heart rate variability (HRV) data from the 2002 Computers in Cardiology (CiC) Challenge using the concept of large-scale dimension densities and additionally apply this technique to data of healthy persons and of patients with cardiac diseases. The large-scale dimension density (LASDID) is estimated from the time series using a normalized Grassberger-Procaccia algorithm, which leads to a suitable correction of systematic errors produced by boundary effects in the rather large scales of a system. This way, it is possible to analyze rather short, nonstationary, and unfiltered data, such as HRV. Moreover, this method allows us to analyze short parts of the data and to look for differences between day and night. The circadian changes in the dimension density enable us to distinguish almost completely between real data and computer-generated data from the CiC 2002 challenge using only one parameter. In the second part we analyzed the data of 15 patients with atrial fibrillation (AF), 15 patients with congestive heart failure (CHF), 15 elderly healthy subjects (EH), as well as 18 young and healthy persons (YH). With our method we are able to separate completely the AF (rho (mu/ls) = 0.97 +/- 0.02) group from the others and, especially during daytime, the CHF patients show significant differences from the young and elderly healthy volunteers (CHF, 0.65 +/- 0.13; EH, 0.54 +/- 0.05; YH, 0.57 +/- 0.05; p < 0.05 for both comparisons). Moreover, for the CHF patients we find no circadian changes in rho (mu/ls) (day, 0.65 +/- 0.13; night, 0.66 +/- 0.12; n.s.) in contrast to healthy controls (day, 0.54 +/- 0.05; night, 0.61 +/- 0.05; p=0.002). Correlation analysis showed no statistical significant relation between standard HRV and circadian LASDID, demonstrating a possibly independent application of our method for clinical risk stratification.
Scroll waves in an excitable medium rotate about tubelike filaments, whose ends, when they exist, can lie on the external boundary of the medium or be pinned to an inclusion. We derive a topological rule that governs such pinning. It implies that some configurations cannot occur although they might otherwise have been expected. Heart tissue provides an application of these concepts. Computational illustrations based on a FitzHugh-Nagumo model are given.
Pinning of vortices by defects plays an important role in various physical (superconductivity, superfluidity, etc.) or biological (propagation in cardiac muscle) situations. Which defects act as pinning centers? We propose a way to study this general problem by using an advection field to quantify the attraction between an obstacle and a vortex. A full solution is obtained for the real Ginzburg-Landau equation (RGLE). Two pinning mechanisms are found in excitable media. Our results suggest strong analogies with the RGLE when the heterogeneity is excitable. Unpinning from an unexcitable obstacle is qualitatively harder, resulting in a stronger pinning force. We discuss the implications of our results to control vortices and propose experiments in a chemical active medium and in cardiac tissue.
Entrainment by a pacemaker, representing an element with a higher frequency, is numerically investigated for several classes of random networks which consist of identical phase oscillators. We find that the entrainment frequency window of a network decreases exponentially with its depth, defined as the mean forward distance of the elements from the pacemaker. Effectively, only shallow networks can thus exhibit frequency locking to the pacemaker. The exponential dependence is also derived analytically as an approximation for large random asymmetric networks.
A simulation algorithm for elastic membrane sheets is presented. Overdamped stochastic dynamics including hydrodynamic coupling to surrounding solvent and arbitrary external forces are generated by employing Fourier modes of the sheet as the primary dynamic variables. Simulations over the micron length scale and second time scale are easily achieved. The dynamics of a lipid bilayer attached to an underlying network of cytoskeletal filaments is used to estimate the diffusion constant of membrane-bound proteins on the surface of the red blood cell.
An oscillatory heterogeneous excitable medium undergoes a transition from periodic target patterns to a bursting rhythm driven by the spontaneous initiation and termination of spiral waves as coupling or density is reduced. We illustrate these phenomena in monolayers of chick embryonic heart cells using calcium-sensitive fluorescent dyes. These results are modeled in a heterogeneous cellular automaton in which the neighborhood of interaction and cell density is modified. Parameters that give rise to bursting rhythms are organized in distinct zones in parameter space, leading to a global organization that should be applicable to the dynamics in a large class of excitable media.