Bifurcation analysis of a normal form for excitable media: are stable dynamical alternans on a ring possible?

We present a bifurcation analysis of a normal form for traveling waves in one-dimensional excitable media. The normal form that has been recently proposed on phenomenological grounds is given in the form of a differential delay equation. The normal form exhibits a symmetry-preserving Hopf bifurcation that may coalesce with a saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We study here the Hopf bifurcation for the propagation of a single pulse in a ring by means of a center manifold reduction, and for a wave train by means of a multiscale analysis leading to a real Ginzburg-Landau equation as the corresponding amplitude equation. Both the center manifold reduction and the multiscale analysis show that the Hopf bifurcation is always subcritical independent of the parameters. This may have links to cardiac alternans, which have so far been believed to be stable oscillations emanating from a supercritical bifurcation. We discuss the implications for cardiac alternans and revisit the instability in some excitable media where the oscillations had been believed to be stable. In particular, we show that our condition for the onset of the Hopf bifurcation coincides with the well known restitution condition for cardiac alternans.     

Pulse trains propagating through excitable media subjected to external noise

We study the propagation of periodic pulse trains in excitable media exposed to external spatio-temporal noise using the light-sensitive Belousov-Zhabotinsky reaction with the underlying Oregonator model as representative example. In the weak noise approximation we find noise-induced transitions in the dispersion relation of pulse trains. We discuss noise-enhanced propagation of pulse trains within a certain wave-length range caused by external noise of moderate strength.     

Excitable optical waves in semiconductor microcavities

We demonstrate experimentally and theoretically the existence of excitable optical waves in semiconductor microcavities. Although similar to those observed in biological and chemical systems, these excitable optical waves are self-confined. This is due to a new dynamical scenario, where a stationary Turning pattern controls the propagation of waves in an excitable medium, thus bringing together the two paradigms of dynamical behavior (waves and patterns) in active media.     

Anomalous pulse interaction in dissipative media

We review a number of phenomena occurring in one-dimensional excitable media due to modified decay behind propagating pulses. Those phenomena can be grouped in two categories depending on whether the wake of a solitary pulse is oscillatory or not. Oscillatory decay leads to nonannihilative head-on collision of pulses and oscillatory dispersion relation of periodic pulse trains. Stronger wake oscillations can even result in a bistable dispersion relation. Those effects are illustrated with the help of the Oregonator and FitzHugh-Nagumo models for excitable media. For a monotonic wake, we show that it is possible to induce bound states of solitary pulses and anomalous dispersion of periodic pulse trains by introducing nonlocal spatial coupling to the excitable medium.     

A normal form for excitable media

We present a normal form for traveling waves in one-dimensional excitable media in the form of a differential delay equation. The normal form is built around the well-known saddle-node bifurcation generically present in excitable media. Finite wavelength effects are captured by a delay. The normal form describes the behavior of single pulses in a periodic domain and also the richer behavior of wave trains. The normal form exhibits a symmetry preserving Hopf bifurcation which may coalesce with the saddle node in a Bogdanov-Takens point, and a symmetry-breaking spatially inhomogeneous pitchfork bifurcation. We verify the existence of these bifurcations in numerical simulations. The parameters of the normal form are determined and its predictions are tested against numerical simulations of partial differential equation models of excitable media with good agreement.     

Conduction block in one-dimensional heart fibers

We present a nonlinear dynamical systems analysis of the transition to conduction block in one-dimensional cardiac fibers. We study a simple model of wave propagation in heart tissue that depends only on the recovery of action potential duration and conduction velocity. If the recovery function has slope >or=1 and the velocity recovery function is nonconstant, rapid activation causes dynamical heterogeneity and finally conduction block away from the activation site. This dynamical mechanism may play a role in the initiation and breakup of spiral waves in excitable media.     

Nonlocal control of pulse propagation in excitable media

We study the effects of nonlocal control of pulse propagation in excitable media. As ageneric example for an excitable medium the FitzHugh-Nagumo model with diffusion in theactivator variable is considered. Nonlocal coupling in form of an integral term with aspatial kernel is added. We find that the nonlocal coupling modifies the propagatingpulses of the reaction-diffusion system such that a variety of spatio-temporal patternsare generated including acceleration, deceleration, suppression, or generation of pulses,multiple pulses, and blinking pulse trains. It is shown that one can observe these effectsfor various choices of the integral kernel and the coupling scheme, provided that thecontrol strength and spatial extension of the integral kernel is appropriate. In addition,an analytical procedure is developed to describe the stability borders of the spatiallyhomogeneous steady state in control parameter space in dependence on the parameters of thenonlocal coupling.     

The blogosphere as an excitable social medium: Richter’s and Omori’s Law in media coverage

We study the dynamics of public media attention by monitoring the content of online blogs. Social and media events can be traced by the propagation of word frequencies of related keywords. Media events are classified as exogenous–where blogging activity is triggered by an external news item–or endogenous where word frequencies build up within a blogging community without external influences. We show that word occurrences exhibit statistical similarities to earthquakes. Moreover the size distribution of events scales with a similar exponent as found in the Gutenberg–Richter law. The dynamics of media events before and after the main event can be satisfactorily modeled as a type of process which has been used to understand fore–and aftershock rate distributions in earthquakes–the Omori law. We present empirical evidence that for media events of endogenous origin the overall public reception of the event is correlated with the behavior of word frequencies at the beginning of the event, and is to a certain degree predictable. These results imply that the process of opinion formation in a human society might be related to effects known from excitable media.     

Stimulus-induced response patterns of medium-embedded neurons

Neuronal ensembles in living organisms are often embedded in a media that provides additional interaction pathways and autoregulation. The underlying mechanisms include but are not limited to modulatory activity of some distantly propagated neuromediators like serotonin, variation of extracellular potassium concentration in brain tissue, and calcium waves propagation in networks of astrocytes. Interaction of these diverse processes can lead to formation of complex spatiotemporal patterns, both self-sustained or triggered by external signal. Besides network effects, many dynamical features of such systems originate from reciprocal interaction between single neuron and surrounding medium. In the present paper we study the response of such systems to the application of a single stimulus pulse. We use a minimal mathematical model representing a forced excitable unit that is embedded in a diffusive or (spatially inhomogeneous) excitable medium. We illustrate three different mechanisms for the formation of response patterns: (i) self-sustained depolarization, (ii) propagation of depolarization due to “nearest-neighbor” networks, and (iii) re-entrant waves.     

Exact solution to the problem of nonlinear pulse propagation through random layered media and its connection with number triangles

An energy pulse refers to a spatially compact energy bundle. In nonlinear pulse propagation, the nonlinearity of the relevant dynamical equations could lead to pulse propagation that is nondispersive or weakly dispersive in space and time. Nonlinear pulse propagation through layered media with widely varying pulse transmission properties is not wave-like and a problem of broad interest in many areas such as optics, geophysics, atmospheric physics and ocean sciences. We study nonlinear pulse propagation through a semi-infinite sequence of layers where the layers can have arbitrary energy transmission properties. By assuming that the layers are rigid, we are able to develop exact expressions for the backscattered energy received at the surface layer. The present study is likely to be relevant in the context of energy transport through soil and similar complex media. Our study reveals a surprising connection between the problem of pulse propagation and the number patterns in the well known Pascal’s and Catalan’s triangles and hence provides an analytic benchmark in a challenging problem of broad interest. We close with comments on the relationship between this study and the vast body of literature on the problem of wave localization in disordered systems.     

EXPLICIT REPRESENTATION OF PLANAR WAVE SPEED AND DISPERSIVE RELATION IN GENERAL EXCITABLE MEDIA

We transform the problem of seeking the planar wave speed in general excitable media into a cubically algbraic equation, thus we can find the explicit representation of the planar wave speed (in one-dimensional case, the traveling wave speed) and dispersive relation. Especially for a few typical excitable media, we give the formula for calculating the corresponding wave speed.     

Extended acoustic waves in diluted random systems

In this paper we study the propagation of acoustic waves in an one-dimensional diluted random media which is composed of two interpenetrating chains with pure and random elasticity. We considered a discrete one-dimensional version of the wave equation where the elasticity distribution appears as an effective spring constant. By using a matrix recursive reformulation we compute the localization length within the band of allowed frequencies. In addition, we apply a second-order finite difference method for both time and spatial variables, and study the nature of the waves that propagate in the chain. We numerically demonstrate that the diluted random elasticity distribution promotes extended acoustic modes at high-frequencies.     

Study of superluminal-pulse propagation in one-dimensional photonic crystals by the FDTD method

We study the pulse propagation in a one-dimensional photonic crystal using the finite-difference time-domain method. The wave propagation inside the crystal is the result of superposition of forward and backward waves. We observed the superluminal phenomena and negative values of the velocity of the energy-density maximum. The energy velocities within the crystal never exceed the speed of light in vacuum. We hope that our study contributes to a further understanding of the superluminal phenomena.     

Dynamics of electromagnetic pulses with wide spectra in semiconductor superlattices

We considered the Maxwell equations for electromagnetic-field propagation in a solid with a one-dimensional semiconductor superlattice of quantum dots in the case where the spectral width of the electromagnetic pulse is sufficient to excite transitions between different minizones. A phenomenological equation was obtained in the form of the classical one-dimensional sine-Gordon equation with the perturbation caused by quantum transitions between the minizones. Quantum behavior of electrons was considered using the microscopic Hamiltonian, in the assumption that the pulse duration is small enough for the phonon effects to be neglected. The equation obtained was analyzed numerically, and cases where the adiabatic perturbation theory for the sine-Gordon equation can be used were found. Numerical solutions were obtained, and the domain where transitions between the minizones play a significant role in the electromagnetic-pulse dynamics was found. Talk presented at the oral issue of J. Russ. Laser Res. dedicated to the memory of Professor Vladimir A. Isakov, Professor Alexander S. Shumovsky, and Professor Andrei V. Vinogradov held in Moscow February 21–22, 2008.     

Wave propagation along interacting fiber-like lattices

A fiber-like lattice with resistively coupled electronic elements mimicking a 1-D discrete reaction-diffusion system is considered. The chosen unit or element in the fiber is the paradigmatic Chua's circuit, capable of exhibiting bistable, excitable, oscillatory or chaotic behavior. Then the dynamics of a structure of two such interacting parallel active fibers is studied. Suitable conditions for the interaction to yield synchronization and other forms of collective behavior involving both fibers are obtained. They include wave front propagation, pulse reentry and pulse propagation failure, overcoming of propagation failure, and the appearance of a source of synchronized pulses. The possibility of designing controlled dynamic contacts by means of one or a few inter-fiber couplings is also discussed. Received 12 December 1998     

Projective synchronization of two coupled excitable spiral waves

Interaction of two identical excitable spiral waves in a bilayer system is studied. We find that the two spiral waves can be completely synchronized if the coupling strength is sufficiently large. Prior to the complete synchronization, we find a new type of weak synchronization between the two coupled systems, i.e., the spiral wave of the driven system has the same geometric shape as the spiral wave of the driving system but with a much lower amplitude. This general behavior, called projective synchronization of two spiral waves, is similar to projective synchronization of two coupled nonlinear oscillators, which has been extensively studied before. The underlying mechanism is uncovered by the study of pulse collision in one-dimensional systems.     

Negative filament tension in the Luo-Rudy model of cardiac tissue

Scroll waves are vortices that occur in three-dimensional excitable media. Scroll waves have been observed in a variety of systems including cardiac tissue, where they are associated with cardiac arrhythmias. The disorganization of scroll waves into chaotic behavior is thought to be the mechanism of ventricular fibrillation, which lethality is widely known. One of the possible mechanisms of scroll wave instability is negative filament tension, which was studied theoretically using low-dimensional models of excitable medium. In this article we perform a numerical study of negative filament tension using the Luo-Rudy phase 1 model, which is widely used in cardiac electrophysiology. We show that this instability exists in this model, study its manifestation and discuss its relation to cardiac arrhythmogenesis.     

Chirped optical pulse propagation in saturating nonlinear media

Using a variational method, we have investigated the propagation characteristics of a chirped optical pulse in anomalously dispersive media possessing saturating nonlinearity. For the special case of uniform loss less media, the dynamics of the temporal width of the pulse is shown to be equivalent to an oscillator of unit mass which is executing its motion under some effective potential well. The potential is examined and four different types of behavior of the pulse width are noticed. The role of saturation parameter and the initial chirp in determining the propagation characteristics have been examined. It is found that, both high value of chirp and saturation are detrimental to stable pulse propagation. Particularly, the effect of chirp becomes severe with the increase in the value of saturation. We have shown that incorporation of saturation in the nonlinearity leads to the existence of bistable soliton. For the case of a lossy medium, net broadening of width takes place over many cycles of oscillation. The net broadening decreases with the increase in the value of saturation.     

Oscillatory dispersion and coexisting stable pulse trains in an excitable medium

For planar wave trains in excitable media, we found a novel type of anomalous dispersion distinguished by bistable domains in the dependence of the propagation velocity on the wavelength. Within one medium alternative stable pulse trains can coexist having the same wavelength but different velocities. The phenomenon is related to oscillatory recovery of excitations, which causes small amplitude oscillations in the refractory tail of pulses. Crucial for the bistability is that the pulses in the trains are locked into one oscillation maximum in the tail of the preceding pulse in the train.     

Reaction-diffusion fronts in periodically layered media

We compute the effective wavefront speeds of reaction-diffusion equations in periodically layered media with coefficients that have small-amplitude oscillations around a uniform mean state. We compare them with the corresponding wavefront speeds in the uniform state. We analyze a one-dimensional model where wave propagation is along the layering direction of the medium and a two-dimensional shear flow model where wave propagation is othogonal to the layering direction. We find that the effective wave speed is smaller in the one-dimensional model and is larger in the two-dimensional model for both bistable cubic and quadratic nonlinearities of the Kolmogorov-Petrovskii-Piskunov form. We derive approximate expressions for the wave speeds in the bistable case.Dedicated to Jerry Percus on the occasion of his 65th birthday.