The topology of defibrillation

We describe how Art Winfree's ideas about phase singularities can be used to understand the response of cardiac tissue with a random preexisting pattern of reentrant waves (fibrillation) to a large brief current stimulus. This discussion is organized around spatial dimension, beginning with a discussion of reentry on a periodic ring, followed by reentry in a two-dimensional planar domain (spiral waves), and ending with consideration of three-dimensional reentrant patterns (scroll waves). In all cases, we show how reentrant activity is changed by the application of a shock, describing conditions under which defibrillation is successful or not. Using topological arguments we draw the general conclusion that with a generic placement of stimulating electrodes, large-scale virtual electrodes do not give an adequate explanation for the mechanism of defibrillation.     

A physical approach to remove anatomical reentries: a bidomain study

Controlling cardiac chaos is often achieved by applying a large damaging electric shock-defibrillation. It removes all waves, without differentiating reentries and normal waves, anatomical and functional reentries. Anatomical reentries can be removed by anti-tachycardia pacing (ATP) as well. But ATP requires the knowledge of the position of the reentry, and an access to it with an invasive stimulating electrode. We show that the physics of electric field distribution between cardiac cells permits one to deliver an electric pulse exactly to the core of an anatomical reentry, without knowing its position and even to locations where access with a stimulating electrode is not possible. The energy needed is two orders of magnitude less than defibrillation energy. The results are insensitive to both a detailed ionic model and to the geometry of the fibers.     

Temporal oscillations in neuronal nets

Summary A model for the interactions of cortical neurons is derived and analyzed. It is shown that small amplitude spatially inhomogeneous standing oscillations can bifurcate from the rest state. In a periodic domain, traveling wave trains exist. Stability of these patterns is discussed in terms of biological parameters. Homoclinic and heteroclinic orbits are demonstrated for the space-clamped system.The research reported in this paper was supported in part by NIH GM2037     

Neural fields with fast learning dynamic kernel

We introduce a modified-firing-rate model based on Hebbian-type changing synaptic connections. The existence and stability of solutions such as rest state, bumps, and traveling waves are shown for this type of model. Three types of kernels, namely exponential, Mexican hat, and periodic synaptic connections, are considered. In the former two cases, the existence of a rest state solution is proved and the conditions for their stability are found. Bump solutions are shown for two kinds of synaptic kernels, and their stability is investigated by constructing a corresponding Evans function that holds for a specific range of values of the kernel coefficient strength (KCS). Applying a similar method, we consider exponential synaptic connections, where traveling wave solutions are shown to exist. Simulation and numerical analysis are presented for all these cases to illustrate the resulting solutions and their stability.     

On boundary stimulation and optimal boundary control of the bidomain equations

The bidomain equations with Neumann boundary stimulation and optimal control of these stimuli are investigated. First an analytical framework for boundary control is provided. Then a parallel finite element based algorithm is devised and its efficiency is demonstrated not only for the direct problem but also for the optimal control problem. The computations realize a model configuration corresponding to optimal boundary defibrillation of a reentry phenomenon by applying current density stimuli.     

Standing waves and traveling waves distinguish two circuits in visual cortex

The visual cortex represents stimuli through the activity of neuronal populations. We measured the evolution of this activity in space and time by imaging voltage-sensitive dyes in cat area V1. Contrast-reversing stimuli elicit responses that oscillate at twice the stimulus frequency, indicating that signals originate mostly in complex cells. These responses stand clear of the noise, whose amplitude decreases as 1/frequency, and yield high-resolution maps of orientation preference and retinotopy. We first show how these maps are combined to yield the responses to focal, oriented stimuli. We then study the evolution of the oscillating activity in space and time. In the orientation domain, it is a standing wave. In the spatial domain, it is a traveling wave propagating at 0.2-0.5 m/s. These different dynamics indicate a fundamental distinction in the circuits underlying selectivity for position and orientation, two key stimulus attributes.     

The effects of the size and shape of landscape features on the formation of traveling waves in cyclic populations

Recent field data indicate that in a number of cyclic populations, the cycles are organized spatially with the form of a periodic traveling wave. One way in which this type of wave is generated is when dispersing individuals encounter landscape features that impede movement in certain directions. In this article, we investigate the dependence of such periodic waves on ecological parameters and on the form of the landscape feature. Using a standard predator-prey model as a prototype for a cyclic population, we calculate the speed and amplitude of waves generated by a large landscape feature. This enables us to determine parameters for which the waves are stable; in other cases, they evolve into irregular oscillations. We then undertake for the first time a detailed study of the effects of the size and shape of a landscape feature on the waves that it generates. We show that size rather than shape is the key wave-forming property, with smaller obstacles generating waves with longer wavelength and waves from larger landscape features dominating those from smaller ones. Our results suggest that periodic traveling waves may be much more common than has previously been assumed in real ecological systems, and they enable quantitative predictions on the properties of these waves for particular cases.     

Dynamics of action potential head-tail interaction during reentry in cardiac tissue: ionic mechanisms

In a sufficiently short reentry pathway, the excitation wave front (head) propagates into tissue that is partially refractory (tail) from the previous action potential (AP). We incorporate a detailed mathematical model of the ventricular myocyte into a one-dimensional closed pathway to investigate the effects of head-tail interaction and ion accumulation on the dynamics of reentry. The results were the following: 1) a high degree of head-tail interaction produces oscillations in several AP properties; 2) Ca(2+)-transient oscillations are in phase with AP duration oscillations and are often of greater magnitude; 3) as the wave front propagates around the pathway, AP properties undergo periodic spatial oscillations that produce complicated temporal oscillations at a single site; 4) depending on the degree of head-tail interaction, intracellular [Na(+)] accumulation during reentry either stabilizes or destabilizes reentry; and 5) elevated extracellular [K(+)] destabilizes reentry by prolonging the tail of postrepolarization refractoriness.     

Bifurcation of periodic solutions of Hodgkin-Huxley model for the squid giant axon

The Hodgkin-Huxley model of the space-clamped squid giant axon is shown to admit unstable periodic solutions for current stimuli less than the stimulus at which the rest state becomes linearly unstable. The periodic solutions are demonstrated both by bifurcation theory and by numerical integration. The presence of subcritical unstable oscillations explains the discontinuous behaviour of the amplitude of the repetitive response as a function of current stimulus     

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.     

Assessment of spatial inhomogeneities in intima media thickness along an arterial segment using its dynamic behavior

To assess locally deviating structural and mechanical properties of arterial walls, the spatial variance in end-diastolic intima media thickness (IMT) and the change in IMT during the cardiac cycle (DeltaIMT) were determined along a short segment of the common carotid artery (15.86 mm), at 16 positions simultaneously. Intrasubject spatial inhomogeneities along the artery were revealed by a spatial variance significantly larger than the temporal variance over several beats. If differences between positions were confirmed, the extent of the inhomogeneity was obtained by comparison of IMT and DeltaIMT at each position with their spatial medians +/- the least-significant difference. Because no intersubject comparisons were necessary, a single session of several measurements was sufficient to assess inhomogeneities in the arterial wall properties of a subject, making the method independent of biological variability between subjects. The method was evaluated on 47 presumed healthy subjects (age range 21-75 yr). In 22 subjects, spatial inhomogeneities in DeltaIMT occurred (P < 0.05). In young subjects, DeltaIMT was locally decreased, i.e., in systole inhomogeneities were less compressed than their surrounding tissue. In older subjects, DeltaIMT was locally increased, i.e., the inhomogeneity was locally more compressed than its surrounding wall tissue.     

The spatial origin of a perceptual transition in binocular rivalry

When the left and the right eye are simultaneously presented with incompatible images at overlapping retinal locations, an observer typically reports perceiving only one of the two images at a time. This phenomenon is called binocular rivalry. Perception during binocular rivalry is not stable; one of the images is perceptually dominant for a certain duration (typically in the order of a few seconds) after which perception switches towards the other image. This alternation between perceptual dominance and suppression will continue for as long the images are presented. A characteristic of binocular rivalry is that a perceptual transition from one image to the other generally occurs in a gradual manner: the image that was temporarily suppressed will regain perceptual dominance at isolated locations within the perceived image, after which its visibility spreads throughout the whole image. These gradual transitions from perceptual suppression to perceptual dominance have been labeled as traveling waves of perceptual dominance. In this study we investigate whether stimulus parameters affect the location at which a traveling wave starts. We varied the contrast, spatial frequency or motion speed in one of the rivaling images, while keeping the same parameter constant in the other image. We used a flash-suppression paradigm to force one of the rival images into perceptual suppression. Observers waited until the suppressed image became perceptually dominant again, and indicated the position at which this breakthrough from suppression occurred. Our results show that the starting point of a traveling wave during binocular rivalry is highly dependent on local stimulus parameters. More specifically, a traveling wave most likely started at the location where the contrast of the suppressed image was higher than that of the dominant one, the spatial frequency of the suppressed image was lower than that of the dominant one, and the motion speed of the suppressed image was higher than that of the dominant one. We suggest that a breakthrough from suppression to dominance occurs at the location where salience (the degree to which a stimulus element stands out relative to neighboring elements) of the suppressed image is higher than that of the dominant one. Our results further show that stimulus parameters affecting the temporal dynamics during continuous viewing of rival images described in other studies, also affect the spatial origin of traveling waves during binocular rivalry.     

The pinwheel experiment revisited: effects of cellular electrophysiological properties on vulnerability to cardiac reentry

In normal heart, ventricular fibrillation can be induced by a single properly timed strong electrical or mechanical stimulus. A mechanism first proposed by Winfree and coined the "pinwheel experiment" emphasizes the timing and strength of the stimulus in inducing figure-of-eight reentry. However, the effects of cellular electrophysiological properties on vulnerability to reentry in the pinwheel scenario have not been investigated. In this study, we extend Winfree's pinwheel experiment to show how the vulnerability to reentry is affected by the graded action potential responses induced by a strong premature stimulus, action potential duration (APD), and APD restitution in simulated monodomain homogeneous two-dimensional tissue. We find that a larger graded response, longer APD, or steeper APD restitution slope reduces the vulnerable window of reentry. Strong graded responses and long APD promote tip-tip interactions at long coupling intervals, causing the two initiated spiral wave tips to annihilate. Steep APD restitution promotes wave front-wave back interaction, causing conduction block in the central common pathway of figure-of-eight reentry. We derive an analytical treatment that shows good agreement with numerical simulation results.     

Refinement and Pattern Formation in Neural Circuits by the Interaction of Traveling Waves with Spike-Timing Dependent Plasticity

Traveling waves in the developing brain are a prominent source of highly correlated spiking activity that may instruct the refinement of neural circuits. A candidate mechanism for mediating such refinement is spike-timing dependent plasticity (STDP), which translates correlated activity patterns into changes in synaptic strength. To assess the potential of these phenomena to build useful structure in developing neural circuits, we examined the interaction of wave activity with STDP rules in simple, biologically plausible models of spiking neurons. We derive an expression for the synaptic strength dynamics showing that, by mapping the time dependence of STDP into spatial interactions, traveling waves can build periodic synaptic connectivity patterns into feedforward circuits with a broad class of experimentally observed STDP rules. The spatial scale of the connectivity patterns increases with wave speed and STDP time constants. We verify these results with simulations and demonstrate their robustness to likely sources of noise. We show how this pattern formation ability, which is analogous to solutions of reaction-diffusion systems that have been widely applied to biological pattern formation, can be harnessed to instruct the refinement of postsynaptic receptive fields. Our results hold for rich, complex wave patterns in two dimensions and over several orders of magnitude in wave speeds and STDP time constants, and they provide predictions that can be tested under existing experimental paradigms. Our model generalizes across brain areas and STDP rules, allowing broad application to the ubiquitous occurrence of traveling waves and to wave-like activity patterns induced by moving stimuli.     

Theoretical analysis of the magnetocardiographic pattern for reentry wave propagation in a three-dimensional human heart model

We present a computational study of reentry wave propagation using electrophysiological models of human cardiac cells and the associated magnetic field map of a human heart. We examined the details of magnetic field variation and related physiological parameters for reentry waves in two-dimensional (2-D) human atrial tissue and a three-dimensional (3-D) human ventricle model. A 3-D mesh system representing the human ventricle was reconstructed from the surface geometry of a human heart. We used existing human cardiac cell models to simulate action potential (AP) propagation in atrial tissue and 3-D ventricular geometry, and a finite element method and the Galerkin approximation to discretize the 3-D domain spatially. The reentry wave was generated using an S1-S2 protocol. The calculations of the magnetic field pattern assumed a horizontally layered conductor for reentry wave propagation in the 3-D ventricle. We also compared the AP and magnetocardiograph (MCG) magnitudes during reentry wave propagation to those during normal wave propagation. The temporal changes in the reentry wave motion and magnetic field map patterns were also analyzed using two well-known MCG parameters: the current dipole direction and strength. The current vector in a reentry wave forms a rotating spiral. We delineated the magnetic field using the changes in the vector angle during a reentry wave, demonstrating that the MCG pattern can be helpful for theoretical analysis of reentry waves.     

Thrombin Activity Propagates in Space During Blood Coagulation as an Excitation Wave

Injury-induced bleeding is stopped by a hemostatic plug formation that is controlled by a complex nonlinear and spatially heterogeneous biochemical network of proteolytic enzymes called blood coagulation. We studied spatial dynamics of thrombin, the central enzyme of this network, by developing a fluorogenic substrate-based method for time- and space-resolved imaging of thrombin enzymatic activity. Clotting stimulation by immobilized tissue factor induced localized thrombin activity impulse that propagated in space and possessed all characteristic traits of a traveling excitation wave: constant spatial velocity, constant amplitude, and insensitivity to the initial stimulation once it exceeded activation threshold. The parameters of this traveling wave were controlled by the availability of phospholipids or platelets, and the wave did not form in plasmas from hemophilia A or C patients who lack factors VIII and XI, which are mediators of the two principal positive feedbacks of coagulation. Stimulation of the negative feedback of the protein C pathway with thrombomodulin produced nonstationary patterns of wave formation followed by deceleration and annihilation. This indicates that blood can function as an excitable medium that conducts traveling waves of coagulation.     

Evolution of the optimum bidirectional (+/- biphasic) wave for defibrillation

Introduction of the asymmetric bidirectional (+/- biphasic) current waveform has made it possible to achieve ventricular defibrillation with less energy and current than are needed with a unidirectional (monophasic) waveform. The symmetrical bidirectional (sinusoidal) waveform was used for the first human-heart defibrillation. Subsequent studies employed the underdamped and overdamped sine waves, then the trapezoidal (monophasic) wave. Studies were then undertaken to investigate the benefit of adding a second identical and inverted wave; little success rewarded these efforts until it was discovered that the second inverted wave needed to be much less in amplitude to lower the threshold for defibrillation. However, there is no physiologic theory that explains the mechanism of action of the bidirectional wave, nor does any theory predict the optimum amplitude and time dimensions for the second inverted wave. The authors analyze the research that shows that the threshold defibrillation energy is lowest when the charge in the second, inverted phase is slightly more than a third of that in the first phase. An ion-flux, spatial-K+ summation hypothesis is presented that shows the effect on myocardial cells of adding the second inverted current pulse.     

Devolution of inhomogeneities in bone structure--predictions of adaptive elasticity theory

In this paper we consider an initially inhomogeneous adaptive elastic body subjected to a steady homogeneous stress state. The adaptive elastic body, which is a model for living bone tissue, is inhomogeneous in both its anisotropic elastic properties and its density. The principal result of the paper is the determination of the devolution of the initially inhomogeneous body to a homogeneous body under the influence of the steady homogeneous stress state. A cylindrical body that is inhomogeneous along the axis of the cylinder, but homogeneous in each transverse plane of the cylinder, is used as an example. This cylindrical body is loaded by a steady uniform stress directed along the cylindrical axis. The temporal devolution of an inhomogeneity in the initial shape of a sine wave is illustrated. As time progresses the amplitude of the sine wave decreases, rapidly at first and then more slowly. As time becomes very large the sine wave becomes a straight line signifying that the cylinder has become homogeneous.     

Art Winfree and the bidomain model of cardiac tissue

This paper reviews Art Winfree's contributions to the bidomain model of cardiac tissue. Specifically, he first predicted quatrefoil reentry, he showed that an S1 refractory gradient is not required for an S2 stimulus to induce reentry, and his work on spiral wave meandering led to studies on how the path of the tip of a spiral wave is influenced by tissue anisotropy.     

Stimulus motion propels traveling waves in binocular rivalry

State transitions in the nervous system often take shape as traveling waves, whereby one neural state is replaced by another across space in a wave-like manner. In visual perception, transitions between the two mutually exclusive percepts that alternate when the two eyes view conflicting stimuli (binocular rivalry) may also take shape as traveling waves. The properties of these waves point to a neural substrate of binocular rivalry alternations that have the hallmark signs of lower cortical areas. In a series of experiments, we show a potent interaction between traveling waves in binocular rivalry and stimulus motion. The course of the traveling wave is biased in the motion direction of the suppressed stimulus that gains dominance by means of the wave-like transition. Thus, stimulus motion may propel the traveling wave across the stimulus to the extent that the stimulus motion dictates the traveling wave's direction completely. Using a computational model, we show that a speed-dependent asymmetry in lateral inhibitory connections between retinotopically organized and motion-sensitive neurons can explain our results. We argue that such a change in suppressive connections may play a vital role in the resolution of dynamic occlusion situations.