The geometry and motion of reaction-diffusion waves on closed two-dimensional manifolds

Chemical or biological systems modelled by reaction diffusion (R.D.) equations which support simple one-dimensional travelling waves (oscillatory or otherwise) may be expected to produce intricate two or three-dimensional spatial patterns, either stationary or subject to certain motion. Such structures have been observed experimentally. Asymptotic considerations applied to a general class of such systems lead to fundamental restrictions on the existence and geometrical form of possible structures. As a consequence of the geometrical setting, it is a straightforward matter to consider the propagation of waves on closed two-dimensional manifolds. We derive a fundamental equation for R.D. wave propagation on surfaces and discuss its significance. We consider the existence and propagation of rotationally symmetric and double spiral waves on the sphere and on the torus. On leave of absence from: Department of Mathematics, Glasgow College of Technology, Cowcaddens Road, Glasgow G4 0BA, Scotland, UK     

Existence and stability of periodic travelling wave solutions to Nagumo's nerve equation

Summary Nagumo's nerve conduction equation has a one-parameter family of spatially periodic travelling wave solutions. First, we prove the existence of these solutions by using a topological method. (There are some exceptional cases in which this method cannot be applied in showing the existence.) A periodic travelling wave solution corresponds to a closed orbit of a third-order dynamical system. The Poincaré index of the closed orbit is determined as a direct consequence of the proof of the existence. Second, we prove that the periodic travelling wave solution is unstable if the Poincaré index of the corresponding closed orbit is + 1. By using this result, together with the result of the author's previous paper, it is concluded that the slow periodic travelling wave solutions are always unstable. Third, we consider the stability of the fast periodic travelling wave solutions. We denote by L(c) the spatial period of the travelling wave solution with the propagation speed c. It is shown that the fast solution is unstable if its period is close to L_min, the minimum of L(c).     

Toroidal elastic supercoiling of DNA

Covalently closed circular DNA can exist in different configurations known as circular, toroidal, and interwound. Changes among these forms can be made in several ways, including the insertion of dye molecules between adjacent base pairs, which tends to untwist the double-helical structure. The aim of this paper is to discuss these configurations, and the changes among them, in the context of classical elastomechanics. The concepts of twisting, linkage and writhing are explained. Simple experiments on a twisted linear-elastic rod are described, and it is shown that although the circular and interwound forms may be modeled in this way, the toroidal form does not occur, being mechanically unstable. Theoretical energy calculations by Levitt on bent and twisted DNA show that DNA exhibits a particular kind of nonlinear elasticity in which there is an unusual coupling between bending and twisting. The aim of the paper is to show qualitatively that this special kind of elasticity can stabilize the toroidal form of closed circular DNA.     

Dynamics of a transmural scroll wave in ground squirrel myocardium

A study was made of tachyarrhythmia evoked by a premature stimulus (4 ms at 4–5 diastolic thresholds) following a train of rectangular pulses (4 ms at 2 diastolic thresholds, repetition rate of 0.5 or 2 s^?1). The spatiotemporal distribution of the potential over the endo- and epicardial surfaces of a thin (～1 mm) specimen of ground squirrel ventricular myocardium was monitored with two arrays of 32 unipolar electrodes each. The electrographic data were processed into isochrone maps reflecting the spread of activation over the surfaces. These maps were further analyzed to infer the 3D structure and dynamics of the vortex (scroll) wave. During the evolution of a transmural scroll, (i) the filament could be normal to the myocardial surfaces as well as oblique at varying angles (down to 12°); (ii) the scroll could drift as a whole, whereby the filament remained self-parallel or changed its inclination; in other cases, one (endocardial) core was anchored while the other changed its position (precession of the filament); (iii) the vortex cores on both or only one surface changed in size and shape; (iv) the filament could be repeatedly twisted through various angles and untwisted. Scroll rotation was attended with excitation breakthroughs that might have originated from filament bending as well as from focal sources.     

Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

 In this paper we study the existence of one-dimensional travelling wave solutions u(x, t)=φ(x−ct) for the non-linear degenerate (at u=0) reaction-diffusion equation u _t =[D(u)u _x ] _x +g(u) where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c=0, 2. The existence of a unique value c ^*>0 of c for which φ(x−c ^* t) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for c≠c ^*. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation. Received 15 December 1995; received in revised form 14 May 1996     

Travelling wave solutions of diffusive Lotka-Volterra equations

We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speed c there is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c^* dividing the waves into two types, waves of speed c < c^* being one type, waves of speed c ? c^* being of the other type. We present numerical evidence that for this system the wave of speed c^* is stable, and that c^* is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves.     

Thermal stability of carbon [n,5] prismanes (n = 2–4): a molecular dynamics study

Tight-binding molecular dynamics simulations are carried out to analyse the thermal stability of the carbon [n,5] prismanes with n = 2–4 over a wide temperature range. The results obtained demonstrate that the isomerisation activation energy as well as the frequency factor in the Arrhenius equation of these metastable nanostructures rapidly decreases with an increase of n. Therefore, the increase in the effective length of [n,5] prismane leads to the decrease in its lifetime up to the moment of its isomerisation. Nevertheless, the stability of [n,5] prismanes is confirmed to be sufficient for their existence at the liquid-nitrogen temperature. The main identified mechanism of [n,5] prismanes isomerisation is the interlayer C–C bond breaking leading to their transformation to the hypostrophene-based molecular systems.     

Are buffers boring? Uniqueness and asymptotical stability of traveling wave fronts in the buffered bistable system

Traveling waves of calcium are widely observed under the condition that the free cytosolic calcium is buffered. Thus it is of physiological interest to determine how buffers affect the properties of calcium waves. Here we summarise and extend previous results on the existence, uniqueness and stability of traveling wave solutions of the buffered bistable equation, which is the simplest possible model of the upstroke of a calcium wave. Taken together, the results show that immobile buffers do not change the existence, uniqueness or stability of the traveling wave, while mobile buffers can eliminate a traveling wave. However, if a wave exists in the latter case, it remains unique and stable.      

Criterion for stable reentry in a ring of cardiac tissue

We model electrical wave propagation in a ring of cardiac tissue using an mth-order difference equation, where m denotes the number of cells in the ring. Under physiologically reasonable assumptions, the difference equation has a unique equilibrium solution. Applying Jury’s stability test, we prove a theorem concerning the local asymptotic stability of this equilibrium solution. Our results yield conditions for sustained reentrant tachycardia, a type of cardiac arrhythmia.      

Spatiotemporal dynamics of two generic predator–prey models

We present the analysis of two reaction–diffusion systems modelling predator–prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L ^∞-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L ^p -estimate, uniform in time, for all p≥1, implies L ^∞-uniform bounds, given any nonnegative L ^∞-initial data. The applicability of the L ^∞-estimate to general reaction–diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be ‘trapped’ in an invariant region of phase space.     

Stability of periodic travelling wave solutions of a nerve conduction equation

Summary Nagumo's nerve conduction equation has travelling wave solutions of pulse type and periodic wave type. We consider the stability of the latter ones. We denote byL(c) the minimum spatial period of a periodic travelling wave solution whose propagation speed isc. It is shown that this travelling wave solution is unstable ifL′(c)<0.     

Dynamics of a vortex with the U-shaped filament (ends “anchored”) in the heart of the ground squirrel

The dynamics of an electrical scroll wave with the U-shaped filament with both ends of the filament being “anchored” on the endocardial surface and the dependence of the structure of pseudoECG on the dynamics of the vortex during the development of polymorphic tachysystolia have been studied by applying premature stimuli to the “target phase” with subsequent registration of the spatial and temporal distribution of electrical potential throughout the surface (endocardial and epicardial) of a thin (≈1 mm) preparation. It was found that (1) the pseudoECG of the polymorphic form during the tachysystolia attack can be observed in the case that the position of the filament ends on the surfaces of the preparation does not practically change from turn to turn (filament ends are “anchored”); (2) the thread of a scroll wave during this attack can twist and untwin (twisted filament), just as it was the case for scroll waves with a straight filament; (3) in the case of pseudoECG of polymorphic form, the twisting and untwining of the filament were stronger (the angle of maximal twisting was 120 degrees and more), and the angle of twisting changed by a substantially greater value from turn to turn as compared with the pseudoECG of monomorphic form; (4) in the case of pseudoECG of polymorphic form, the time interval between the appearance of waves on the surfaces of the preparation (T _epi-endo) was substantially greater and changed to a greater extent from turn to turn of the vortex; and (5) simultaneously with the appearance of pseudoECG of polymorphic form and the onset of changes in the twisting of the scroll and the T _epi-endo interval indicated in (2–4), significant changes in the patterns of coverage of the surface by excitation occurred. Based on the results obtained, an explanation of the reasons for the appearance of excitation breakdown patterns on the surface of the myocardium was proposed, which differs from the traditional viewpoint. These patterns may be the result of reflection on myocardial surfaces of the activity of not different simultaneously occurring sources of initiation of excitation but of a single three-dimensional vortex whose filament twists when passing through the thickness of the myocardium and can closely approach one or the other surface.     

Traveling wave solutions of a nonlinear reaction–advection equation

 We establish the existence of traveling wave solutions for a nonlinear partial differential equation that models a logistically growing population whose movement is governed by an advective process. Conditions are presented for which traveling wave solutions exist and for which they are stable to small semi-finite domain perturbations. The wave is of mathematical interest because its behavior is determined by a singular differential equation and those with small speed of propagation steepen into a shock-like solutions. Finally, we indicate that the smoothing presence of diffusion allows wave persistence when an advective slow moving wave may collapse. Received: 24 November 1997 / Revised version: 13 July 1998     

The Geographic Spread of “El mal de las caderas” in Capybaras (Hydrochaeris hydrochaeris)

The aim of this work is to describe an epidemiological model for a capybara (Hydrochaeris hydrochaeris) population. The model considers a tabanid (“mutuca”) population (Diptera: tabanidae), as a vector for the disease called “mal de las caderas” in Estero del Ibera, Corrientes, Argentina. The study of this problem has ecological and economical importance since the meat and the hide of the capybara can be an exploitation resource. At first, a threshold value is determined as a function of the model parameters, obtaining a critical carrying capacity which determines the disease propagation or eradication. Then as the carrying capacity condition for the disease existence is satisfied, the existence of traveling wave solution is studied. Independent speeds are considered for the susceptible capybaras, the noninfected insect, and the disease. The speed of propagation for this model is obtained as function of model parameters followed by a discussion of strategies for controlling the spread of the disease. N.A. Maidana is a fellowship Fapesp and partially supported by Grant Fapesp (temático).     

Equilibrium solutions of age-specific population dynamics of several species

Summary A mathematical model describing the dynamics of a population consisting of several species is studied. The interactions in the population are assumed to be age-specific. Using an evolution equation approach, sufficient conditions for well-posedness in L ¹ of the dynamics and for existence as well as for stability of equilibrium solutions are given.     

Traveling waves in the discrete fast buffered bistable system

We study the existence and uniqueness of traveling wave solutions of the discrete buffered bistable equation. Buffered excitable systems are used to model, among other things, the propagation of waves of increased calcium concentration, and discrete models are often used to describe the propagation of such waves across multiple cells. We derive necessary conditions for the existence of waves, and, under some restrictive technical assumptions, we derive sufficient conditions. When the wave exists it is unique and stable.      

Geometry and mechanics of DNA superhelicity

This paper analyzes the elastic equilibrium conformations of duplex DNA constrained by the constancy of its molecular linking number, Lk. The DNA is regarded as having the mechanical properties of a homogeneous, linearly elastic substance with symmetric cross section. Integral representations of the writhing number Wr and of Lk are developed, in terms of which the equilibria are given as solutions to an isoperimetric problem. It is shown that the Euler angles defining equilibrium conformations must obey equations identical to those governing unconstrained equilibria. A scaling law is developed stating that molecules supercoiled the same amount ΔLk will have geometrically similar elastic equilibria regardless of their length. Thus, comparisons among molecules of properties related to their large-scale tertiary structure should be referred to differences in ΔLk rather than to their superhelix densities. Specific conditions on the elastic equilibrium conformations are developed that are necessary for ring closure. The equilibrium superhelical conformations accessible to closed-ring molecules are shown to approximate toroidal helices. Questions relating to the stability and nonuniqueness of equilibria are treated briefly. A comparison is made between these toroidal conformations and interwound configurations, which are shown to be stable, although they are not equilibria in the present sense. It is suggested that entropic factors are responsible for favouring the toroidal conformation in solution.     

Effects of Mechano-Electric Feedback on Scroll Wave Stability in Human Ventricular Fibrillation

Recruitment of stretch-activated channels, one of the mechanisms of mechano-electric feedback, has been shown to influence the stability of scroll waves, the waves that underlie reentrant arrhythmias. However, a comprehensive study to examine the effects of recruitment of stretch-activated channels with different reversal potentials and conductances on scroll wave stability has not been undertaken; the mechanisms by which stretch-activated channel opening alters scroll wave stability are also not well understood. The goals of this study were to test the hypothesis that recruitment of stretch-activated channels affects scroll wave stability differently depending on stretch-activated channel reversal potential and channel conductance, and to uncover the relevant mechanisms underlying the observed behaviors. We developed a strongly-coupled model of human ventricular electromechanics that incorporated human ventricular geometry and fiber and sheet orientation reconstructed from MR and diffusion tensor MR images. Since a wide variety of reversal potentials and channel conductances have been reported for stretch-activated channels, two reversal potentials, −60 mV and −10 mV, and a range of channel conductances (0 to 0.07 mS/µF) were implemented. Opening of stretch-activated channels with a reversal potential of −60 mV diminished scroll wave breakup for all values of conductances by flattening heterogeneously the action potential duration restitution curve. Opening of stretch-activated channels with a reversal potential of −10 mV inhibited partially scroll wave breakup at low conductance values (from 0.02 to 0.04 mS/µF) by flattening heterogeneously the conduction velocity restitution relation. For large conductance values (>0.05 mS/µF), recruitment of stretch-activated channels with a reversal potential of −10 mV did not reduce the likelihood of scroll wave breakup because Na channel inactivation in regions of large stretch led to conduction block, which counteracted the increased scroll wave stability due to an overall flatter conduction velocity restitution.     

Dynamics of Scroll Wave in a Three-Dimensional System with Changing Gradient

The dynamics of a scroll wave in an excitable medium with gradient excitability is studied in detail. Three parameter regimes can be distinguished by the degree of gradient. For a small gradient, the system reaches a simple rotating synchronization. In this regime, the rigid rotating velocity of spiral waves is maximal in the layers with the highest filament twist. As the excitability gradient increases, the scroll wave evolutes into a meandering synchronous state. This transition is accompanied by a variation in twisting rate. Filament twisting may prevent the breakup of spiral waves in the bottom layers with a low excitability with which a spiral breaks in a 2D medium. When the gradient is large enough, the twisted filament breaks up, which results in a semi-turbulent state where the lower part is turbulent while the upper part contains a scroll wave with a low twisting filament.     

Pressure profiles of a plasma confined by the magnetic field of a ring current

Pressure profiles p(ψ) marginal with respect to convective instability in a toroidal tubular plasma confined by the magnetic field of an internal levitated ring current and external ring currents are studied as functions of the shape of the magnetic separatrix. Configurations are found in which the maximum plasma pressure in a finite-width layer near the plasma boundary decreases by two orders of magnitude at the expense of artificially raising the effective length (characterized by the integral ∮dl/B) of the magnetic field lines near the separatrix surface. It is shown that, in the case of a straight cylindrical tubular plasma, which is the limiting case of a toroidal configuration with an arbitrarily large aspect ratio, the sufficient condition for the plasma to be MHD stable against both convective and kink perturbations is satisfied for local values β≤0.4. __________ Translated from Fizika Plazmy, Vol. 26, No. 6, 2000, pp. 519–528. Original Russian Text Copyright ￠ 2000 by Popovich, Shafranov.