a cellular automaton model of excitable media iii fitting the belousov zhabotinskiic reaction abstract we obtain a set of parameter values for our cellular automaton model that gives a reasonably good description of the belousov zhabotinskii reaction as determined by comparison with experimental results and with calculations on the oregonator a differential equation model of the chemical system

kinematics of rigidly rotating spiral waves spiral waves rigidly rotating in excitable media are studied by use of a free boundary approach this study reveals the selection principle which determines the shape and the rotation frequency of spiral waves in an unbounded medium with a given excitability it is shown that a rigidly rotating spiral in a medium with a strongly reduced refractoriness is supported within a range of the medium excitability restricted by two universal limits at the low excitability limit the spiral core radius diverges while at the high excitability limit it vanishes the simulations performed for the medium excitability higher than the high excitability limit reveal nonstationary rotating waves which considerably differ from well studied meandering spiral waves it is shown how the proposed free boundary approach can be extended to the case of an arbitrary refractoriness the predictions of the free boundary approach are in good agreement with the results from numerical simulations of the underlying reaction diffusion model and with asymptotics derived earlier for highly and weakly excitable media

classification of steady solutions of the full kinematic model abstract the geometric type of kinematic model governs any moving plane curve which propagates in the normal direction in the medium there have appeared some results in the literature via this model however these results are all based upon some simplification on the model or prior phenomenological assumptions on the solutions in this paper we use really full kinematic model to classify steady solutions i e curvature is independent of time with positive or negative curvature various type of solutions have been obtained such as rotating and translating waves especially for the rotating spiral waves spiral waves arise in many biological chemical and physiological systems the kinematical model can be used to describe the motion of the spiral arms approximated as curves in the plane in fact using our results theorem 3 8 we can answer the following questions is there any steadily rotating wave for a given weakly excitable medium if yes what kind of information we can know about these rotating waves e g the tip s curvature the tip s tangential velocity and the rotating frequency comparing our results with previous ones in the literature there are some differences between them the only evolving plane curves with positive or negative curvature must have monotonous curvatures via simplified model but full model admits solutions with any given oscillating number of the curvatures

the simplified fitzhugh nagumo model with action potential duration restitution effects on 2d wave propagation abstract a modification of the simplified fitzhugh nagumo fn equations is proposed for introducing a residual component of the slow variable which determines the restitution of action potential duration apd also known as the interval excitation duration relationship the three step wise approximation of e which is widely used in current publications is replaced in a new model by a four step approximation this change is used for studying by computer simulation the effects of apd restitution properties independently of the apd and refractory period on 2d wave propagation in an isotropic matrix made by 128 128 nodes the method for fitting the model to the given experimental restitution data obtained from myocardial cells is presented the computer simulations implemented on a massively parallel computer connection machine showed at least three important qualitative distinctions in behavior which demonstrate the effect of apd restitution changes in the speed and wavelength of propagated waves with the period of stimulation non stationary propagation of spiral waves and site specific induction of spiral waves with premature stimulation not on the tail of the previous wave quantitative effects of differing restitution properties are expressed in the size and location of a window of vulnerability in 2d excitable media these windows are characterized by the appearance of single and double spiral waves in response to premature stimulation applied inside the window thus the apd restitution incorporated in the fn model produces a significant effect on the formation and propagation of spiral waves

dynamic behavior of a paced cardiac fiber consider a typical experimental protocol in which one end of a one dimensional fiber of cardiac tissue is periodically stimulated or paced resulting in a train of propagating action potentials there is evidence that a sudden change in the pacing period can initiate abnormal cardiac rhythms in this paper we analyze how the fiber responds to such a change in a regime without arrhythmias in particular given a fiber length l and a tolerance we estimate the number of beats n n l required for the fiber to achieve approximate steady state in the sense that spatial variation in the diastolic interval di is bounded by we track spatial di variation using an infinite sequence of linear integral equations which we derive from a standard kinematic model of wave propagation the integral equations can be solved in terms of generalized laguerre polynomials we then estimate n by applying an asymptotic estimate for generalized laguerre polynomials we find that for fiber lengths characteristic of cardiac tissue it is often the case that n effectively exhibits no dependence on l more exactly i there is a critical fiber length l such that if l l the convergence to steady state is slowest at the pacing site and ii often l is substantially larger than the diameter of the whole heart

curvature dependence of propagating velocity for a simplified calcium model it is known that the relation between curvature and wave speed plays a key role in the propagation of two dimensional waves in an excitable model for typical excitable models e g the fitzhugh nagumo fhn model such a relation is believed to obey the linear eikonal equation which states that the relation between the normal velocity and the local curvature is approximately linear in this paper we show that for a caricature model of intracellular calcium dynamics although its temporal dynamics can be investigated by analogy with the fhn model the curvature relation does not obey the linear eikonal equation even in the limiting case hence this caricature calcium model may be an unexpected excitable system whose wave propagation properties cannot be always understood by analogy with the fhn model

spiral waves in the belousov zhabotinskii reaction abstract the beautiful spiral waves of oxidation in the belousov zhabotinskii reaction are the source of many interesting and important questions about periods structures in excitable media it has long been known that these spirals are similar to involutes of circles at least some distance from the center but until now no way has been known to determine the correct wavelength and frequency in this paper we show that the parameters of a spiral wave can be viwed s eigenvalues of a problem with unique solution the critical ingredients of the theory are the effects of curvature on the propagation of wavefronts in two dimensional media and the dispersion of plane waves our analytical results are shown to be in good agreement with experimental data for the belousov zhabotinskii reagent

two dimensional spectrophotometry of spiral wave propagation in the belousov zhabotinskii reaction i experiments and digital data representation abstract the spiral shaped wave of chemical activity propagating in a 1 mm layer of an excitable solution of the belousov zhabotinskii reaction is analysed quantitatively for a 5 min time interval in a 4 5 4 5 mm area the spatial distribution of the reaction catalyst and indicator ferroin is measured by light absorption at 490 nm with a video and computer based two dimensional spectrophotometer digital images with 10 m spatial and 256 digital units intensity resolution are acquired at a frequency of 20 per minute initially the spiral has a pitch of 1 18 mm a revolution time of 17 2 s and the wavefronts propagate outwards with a velocity of 69 m s after 5 min these quantities have changed by up to 5 three dimensional graphic procedures and image analysis reveal that the wave profiles are symmetric close to the spiral center and become highly asymmetric during their outward displacement the core of the spiral has a radius of 0 35 mm its center is a singular site diameter

convolution generated motion as a link between cellular automata and continuum pattern dynamics cellular automata have been used to model the formation and dynamics of patterns in a variety of chemical biological and ecological systems however for patterns in which sharp interfaces form and propagate automata simulations can exhibit undesirable properties including spurious anisotropy and poor representation of interface curvature effects these simulations are also prohibitively slow when high accuracy is required even in two dimensions also the highly discrete nature of automata models makes theoretical analysis difficult in this paper we present a method for generating interface motions that is similar to the threshold dynamics type cellular automata but based on continuous convolutions rather than discrete sums these convolution generated motions naturally achieve the fine grid limit of the corresponding automata and they are also well suited to numerical and theoretical analysis because of this the desired pattern dynamics can be computed accurately and efficiently using adaptive resolution and fast fourier transform techniques and for a large class of convolutions the limiting interface motion laws can be derived analytically thus convolution generated motion provides a numerically and analytically tractable link between cellular automata models and the smooth features of pattern dynamics this is useful both as a means of describing the continuum limits of automata and as an independent foundation for expressing models for pattern dynamics in this latter role it also has a number of benefits over the traditional reaction diffusion ginzburg landau continuum pde models of pattern formation which yield true moving interfaces only as singular limits we illustrate the power of this approach with convolution generated motion models for pattern dynamics in developmental biology and excitable media

singular filaments organize chemical waves in three dimensions iii knotted waves abstract this is the third of a series of papers on the anatomy of three dimensional organizing centers in excitable media we here ask whether all self consistent waves in excitable media are topologically equivalent to the experimentally verified scroll ring whose axis lies in a plane as a test case we examine a scroll ring whose axis contains a knot it proves to be incompatible with the requirements of physical chemistry unless simultaneously twisted by an amount equal to the writhing number of its axis which is zero for planar closed curves appropriate initial conditions are suggested for experimentally creating a wave whose source is a scroll ring knotted and twisted in this way

a new upper bound for the minimum of an integral lattice of determinant 1 let be an n dimensional integral lattice of determinant 1 we show that for all sufficiently large n the minimal nonzero squared length in does not exceed n 6 10 this bound is a consequence of some new conditions on the theta series of these lattices these conditions also enable us to find the greatest possible minimal squared length in all dimensions n 33 in particular we settle the no roots problem there is a determinant 1 lattice containing no vectors of squared length 1 or 2 precisely when n 23 n 25 there are also analogues of all these results for codes

formation of space time structure in a forest fire model we present a general stochastic forest fire model which shows a variety of different structures depending on the parameter values the model contains three possible states per site tree burning tree empty site and three parameters tree growth probability p lightning probability f and immunity g we review analytic and computer simulation results for a quasideterministic state with spiral shaped fire fronts for a percolation like phase transition and a self organized critical state possible applications to excitable systems are discussed

helical and circular scroll wave filaments abstract we present a formal theory of the motion of circular and helical scroll wave filaments with uniform twist some implications of the theory are derived and compared to numerical computations of scroll wave evolution

existence and uniqueness of stabilized propagating wave segments in wave front interaction model abstract recent experimental studies of photosensitive belousov zhabotinskii reaction have revealed the existence of propagating wave segments the propagating wave segments are unstable but can be stabilized by using a feedback control to continually adjust the excitability of the medium experimental studies also indicate that the locus of the size of a stabilized wave segment as a function of the excitability of the medium gives the excitability boundary for the existence of 2d wave patterns with free ends in excitable media to study the properties of this boundary curve we use the wave front interaction model proposed by zykov and showalter this is equivalent to study a first order system of three ordinary differential equations which includes a singular nonlinearity using two different reduced first order systems of two ordinary differential equations we first show the existence of wave segments for any given propagating velocity then the wave profiles can be classified into two types namely convex and non convex types more precisely when the normalized propagating velocity is small we show that the wave profile is of convex type while the wave profile is of non convex type when the normalized velocity is close to 1

topology optimization using a reaction diffusion equation abstract this paper presents a structural topology optimization method based on a reaction diffusion equation in our approach the design sensitivity for the topology optimization is directly employed as the reaction term of the reaction diffusion equation the distribution of material properties in the design domain is interpolated as the density field which is the solution of the reaction diffusion equation so free generation of new holes is allowed without the use of the topological gradient method our proposed method is intuitive and its implementation is simple compared with optimization methods using the level set method or phase field model the evolution of the density field is based on the implicit finite element method as numerical examples compliance minimization problems of cantilever beams and force maximization problems of magnetic actuators are presented to demonstrate the method s effectiveness and utility

interconnected turing patterns in three dimensions we study numerically the turing pattern in three dimensions in a fitzhugh nagumo type reaction diffusion system we have found that interconnected periodic domain structures such as a gyroid fddd and perforated lamellar structures appear in three dimensions which never exist in lower dimensions the stability analysis of these structures is also performed by means of a mode expansion

symmetric spirals in media with relaxation kinetics and two diffusing species abstract we find an approximate solution for the shape and rotation frequency of spirals in excitable media with anti symmetric kinetics and diffusion in both species using singular perturbation theory we approximate the governing diffusion reaction system with a free boundary problem and then map this free boundary problem onto a fixed domain problem the resulting fixed domain problem is viewed as a nonlinear eigenvalue problem which can be solved iteratively using a numerical shooting algorithm solutions of the approximate free boundary problem are compared with numerical solutions of the full system of diffusion reaction equations and shown to be in good agreement

shadow bounds for self dual codes conway and sloane 1990 have previously given an upper bound on the minimum distance of a singly even self dual binary code using the concept of the shadow of a self dual code we improve their bound finding that the minimum distance of a self dual binary code of length n is at most 4 n 24 4 except when n mod 24 22 when the bound is 4 n 24 6 we also show that a code of length a multiple of 24 meeting the bound cannot be singly even the same technique gives similar results for additive codes over gf 4 relevant to quantum coding theory

robin type boundary conditions in transition from reaction diffusion equations in 3d domains to equations in 2d domains abstract we consider a singular limit of diffusion equations in 3d domains of thickness converging to zero in the 2d limit the resulting reaction diffusion equation has a source term resulting from the robin type boundary conditions imposed on boundaries of the original 3d domain the proposed approach can be applied to constructing approximate solutions of diffusion problems in thin planar cylindrical or spherical layers between two membranes as an example we refer to the problem of activation of b lymphocytes which typically have large nuclei and a thin cytoplasmic layer which can be considered as a spherical shell for this example assuming additionally axial symmetry we provide a rigorous convergence theorem in the language of semigroups of operators

the eikonal equation stability of reaction diffusion waves on a sphere abstract the eikonal approach to excitable reaction diffusion systems provides an analytical hold on the phenomenon of wave propagation on curved surfaces and the evolution of toroidal scroll structures in the neighbourhood of organising filaments the extent to which the eikonal equation contains information on the stability of these wave solutions is important from a theoretical and experimental point of view we initiated this study in 1987 by demonstrating the stability of an expanding spherical wave recent experiments on wave propagation on a sphere encouraged us further to investigate stability of waves on surfaces we analyse in this paper the geometrical stability of symmetric counter rotating spiral waves propagating on the unit sphere we demonstrate that these solutions are stable under small perturbations normal to the wave fron lying on the unit sphere

singular perturbation theory of traveling waves in excitable media a review abstract waves of chemical or electrical activity traveling through space have been observed in several contexts chemical reaction mixtures in non convecting liquid phase cell suspensions nerve axons and neuromuscular tissues typically wave supporting preparations are excitable that is they respond and sensitivity to perturbations are rapidly damped out but suprathreshold disturbances trigger an abrupt and substantial response the abruptness of the response can be exploited by the methods of singular perturbation theory to obtain a mathematical description of wave propagation in spatially distributed excitable media singular perturbation analysis of propagating waves in one spatial dimension is straightforward and uncontentious but the analysis of propagating waves in one spatial dimension is straightforward and several fundamentally different ways we compare and contrast the approaches taken by greenberg zykov fife krinskii and his collaborators and ourselves with particular emphasis on the case of rotating spiral waves our intention is to bring some order to the important but difficult theory of propagating waves in two dimensional excitable media in conclusion we discuss briefly some possible extensions of the singular perturbation approach to propagating wave surfaces in three dimensional space

alternative stable rotors in an excitable medium abstract a given excitable medium continuous uniform isotropic simply connected two or three dimensional may support alternative types of vortex like solution radiating from a rotor these have distinct periods wavelengths behaviors near interfaces and in rotationally symmetric vortex rings different contraction or expansion rates and drift rates initial conditions select among the discrete alternatives rotors of different types may coexist within one medium until the interface between their domains encroaches upon the rotor of longer period examples are given using the fitzhugh nagumo model they are interpreted in terms of the measured dispersion curve for one dimensional propagation and a hypothetical curvature relation with one undetermined parameter

a one dimensional variational problem with continuous lagrangian and singular minimizer we construct a continuous lagrangian strictly convex and superlinear in the third variable such that the associated variational problem has a lipschitz minimizer which is non differentiable on a dense set more precisely the upper and lower dini derivatives of the minimizer differ by a constant on a dense hence second category set in particular we show that mere continuity is an insufficient smoothness assumption for tonelli s partial regularity theorem

a cellular automaton model of excitable media ii curvature dispersion rotating waves and meandering waves abstract a cellular automaton model of excitable media recently introduced by the authors is examined in more detail here we show that i the model is consistent with a linear dependence of wave speed on wave front curvature in two dimensional space ii the model exhibits a stepwise increasing relation between wave speed and temporal period the dispersion relation whose precise shape can be controlled by parameter settings iii the model generates rigidly rotating spiral waves whose frequency speed and core size are consistent with theoretical expectations and iv spiral waves rotate non rigidly showing short term flowerlet meandering which seems to arise by bifurcation from the rigidly rotating solution and long term irregular wandering over large distances these observations are consistent with many known features of wave propagation in two dimensional continuous excitable media governed by reaction diffusion equations suggesting that our cellular automaton model is not only a fast but also a reliable method for investigating the behavior of excitable media

pattern selection and self organization induced by random boundary initial values in a neuronal network abstract regular spatial patterns could be observed in spatiotemporal systems far from equilibrium states artificial networks with different topologies are often designed to reproduce the collective behaviors of nodes or neurons which the local kinetics of node is described by kinds of oscillator models it is believed that the self organization of network much depends on the bifurcation parameters and topology connection type indeed the boundary effect is every important on the pattern formation of network in this paper a regular network of hindmarsh rose neurons is designed in a two dimensional square array with nearest neighbor connection type the neurons on the boundary are excited with random stimulus it is found that spiral waves even a pair of spiral waves could be developed in the network under appropriate coupling intensity otherwise the spatial distribution of network shows irregular states a statistical variable is defined to detect the collective behavior by using mean field theory it is confirmed that regular pattern could be developed when the synchronization degree is low the potential mechanism could be that random perturbation on the boundary could induce coherence resonance like behavior thus spiral wave could be developed in the network

forest fire model with immune trees abstract we present a generalization of the forest fire model of p bak et al by including the immunity g which is the probability that a tree is not ignited although one of its neighbors is burning when g reaches a critical value gc p which depends on the tree growth rate p the fire cannot survive any more i e a continuous phase transition takes place from a steady state with fire to a steady state without fire we present results of computer simulations and explain them by analytic calculations the fire spreading at the phase transition represents a new type of percolation which is called fluctuating site percolation

global analysis of pattern selection and bifurcations in monostable reaction diffusion systems we study a piecewise linear version of a one component monostable reaction diffusion model in a bounded domain subjected to partially reflecting boundary conditions albedo b c we analyze the local and the global stability of the merging patterns and detect a bifurcation of the uniform solution induced by changes in the reflectivity of the boundaries

one and two dimensional wave fronts in diffusive systems with discrete sets of nonlinear sources abstract we study the dynamics of one and two dimensional diffusion systems with sets of discrete nonlinear sources we show that wave fronts propagating in such systems are pinned if the diffusion constant is below a critical value which corresponds to a saddle node bifurcation of the dynamics in two dimensions we find that the dissipation is enhanced and moving plain and circular fronts are stable with respect to any perturbations

stability of vortex rotation in an excitable cellular medium abstract stability of a vortex was studied in a modified cellular automation model the model includes dependence of wave velocity of excitation interval and front curvature two kinds of vortex instability are observed as found also in models based on partial differential equations meandering of the vortex and vortex drift in a heterogeneous medium

the role of curvature and wavefront interactions in spiral wave dynamics abstract a theory of rotating spiral waves in excitable media is presented that allows the study of dynamical aspects such as nonsteady rotation an approximate spiral wave solution is proposed in the form of a superposition of curved solitary wavefronts parallel to each other this form is used by means of singular perturbation theory to derive an evolution equation for the spiral arm numerical solutions of that equation are found that describe one and two frequency rotations the transition to two frequency dynamics tip meandering is attributed to a destabilizing curvature effect

curvature dependence of a model for calcium wave propagation abstract the propagation of a two dimensional wave front in an excitable medium is dependent on the curvature of the front current theories of excitable reaction diffusion models predict that when reaction is much faster than diffusion the normal wave speed n is approximately related to the curvature of the wave front the plane wave speed c and the diffusion coefficient of the propagator variable d by the eikonal equation n c d we show that a simple model for intracellular calcium ca2 wave propagation does not obey the eikonal equation and postulate an alternative curvature equation that is dependent on the parameter values used in the model this new curvature relation is confirmed by numerical simulations we raise the possibility that different models for ca2 wave propagation will have qualitatively different spiral wave patterns providing a new way of distinguishing between proposed models the theory developed here also necessitates a reconsideration of methods previously used to measure the intracellular diffusion coefficient of ca2

turing patterns in three dimensions we investigate three dimensional turing patterns in two component reaction diffusion systems the fitzhugh nagumo equation the brusselator and the gray scott model are solved numerically in three dimensions several interconnected structures of domains as well as lamellar hexagonal and spherical domains are obtained as stable motionless equilibrium patterns the relative stability of these structures is studied analytically based on the reduction approximation the relation with the microphase separated structures in block copolymers is also discussed

spiral wave solutions for reaction diffusion equations in a fast reaction slow diffusion limit abstract a two species reaction diffusion equation is studied in a distinguished limit of fast reaction and slow diffusion asymptotic matching yields a spiral solution whose rotation frequency is a unique function of the physical parameters and the constant pitch of the spiral using an idea of fife the problem is first reduced to a free boundary problem the solution is obtained by matching the core where diffusion balances the curvature of the free boundary to the far field where the effects of diffusion are small the equations of motion are averaged over the rotation frequency to yield steady equations in the two regions helically symmetric solutions to the averaged equation reduce to solving a fourth order ode in the core region matched to a second order ode in the far field speculations on the dynamics of the system and coupling between the core and far field are presented

instabilities of front patterns in reaction diffusion systems abstract recent experiments in chemical reaction diffusion systems with externally imposed concentration gradients may provide access to a host of spatio temporal pattern formation phenomena these systems tend to form steep reaction fronts in response to the external gradient we use singular perturbation techniques normal form calculations and numerical simulations to investigate the existence and the stability of such sustained fronts in one dimensional systems the theoretical predictions are found in quantitative agreement with direct simulations of the hopf bifurcation from steady to periodically oscillating front structures observed in the couette flow reactor also conditions are found under which oscillations of the spatial structure become chaotic in two dimensional systems we address the issue of realizing an experimental situation hitherto unattained a one dimensional chain of coupled oscillators at the onset of the hopf destabilization of the front structure we point out the intimate relationship between the frequency of oscillation of the homogeneous front pattern and the characteristic wavelength of the turing pattern that can develop along the front 2 d 1 2 we comment on subsequent bifurcations that may result from the nonlinear interaction between hopf and turing instabilities as the precursors to spatio temporal chaos in conclusion we emphasize the possibility of probing the transition to mediated defect turbulence in thin film gel reactors

diffusive effects on dispersion in excitable media perturbation and numerical methods are used to study traveling and standing wave solutions for a system of reaction diffusion equations modeled after excitable media and the effect of variation of the diffusion coefficients it is shown that traveling waves and standing waves can coexist and that as a certain diffusion coefficient is varied a branch of traveling waves can bifurcate from a branch of standing waves providing a transition between only traveling solutions in one limit and only standing waves in the other limit key words traveling waves standing waves bifurcation dispersion reaction diffusion equations singular perturbation

an electrothermal instability the influence of albedo boundary conditions on the stationary states of an exactly solvable model abstract in order to study the effect of partially reflective albedo boundary conditions on pattern formation and stability in reaction diffusion systems we have analyzed an exactly soluble model of an electrothermal instability the ballast resistor the present results allow a continuous interpolation between the results corresponding to neumann and dirichlet boundary conditions and they allow qualitative changes in the patterns to be followed

self organized pulse generator abstract numerical simulations of a reaction diffusion system of the bonhoeffer van der pol type will be presented the system admits a coesistence of a stable stationary solution and a stable limit cycle which leads to the generation of a self organized pulse generator pattern dynamics of the system are explored in one and two dimensional spaces

wave front interaction in steadily rotating spirals abstract we develop a self consistent model for excitable media the so called wave front interaction model wfm in which two waves the front and the back interact with a relaxational field c 2 we study numerically the solutions corresponding to spirals rotating steadily with an angular velocity with a tip rotating around an unexcited region of radius r 0 it is found that these two quantities are determined uniquely as a function of e the ratio between fast reaction time and refractory time and the excitability corresponding shapes for spirals are determined and in particular the shape and the size of the core a semi analytical solution for the spiral shape is found in the large r 0 limit

singularities in a modified kuramoto sivashinsky equation describing interface motion for phase transition abstract phase transitions can be modeled by the motion of an interface between two locally stable phases a modified kuramoto sivashinsky equation h t 2 h 4 h 1 h 2 2 h 2 h xx h yy h xy 2 describes near planar interfaces which are marginally long wave unstable we study the question of finite time singularity formation in this equation in one and two space dimensions on a periodic domain such singularity formation does not occur in the kuramoto sivashinsky equation 0 for all 1 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite time blow up in which a second derivative of h becomes unbounded using a bifurcation theory analysis we show a parallel between the stability of steady periodic solutions and the question of finite time blow up in one dimension finally we consider the local structure of the blow up in the one dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self similar grid the simulations resolve the singularity to over 25 decades in h xx l and indicate that the singularities are all locally described by a unique self similar profile in h xx we discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics

spirals in excitable media the free boundary limit with diffusion abstract we solve numerically for the steady state spiral in the thin interface limit including the effects of diffusion of the slow field the calculation is performed using a generalization of the hybrid scheme of keener in this method the diffusion equation is solved on a suitable mapped lattice while the eikonal equation relating the field on the interface to the interfacial velocity and curvature is solved independently we present results for the selected frequency and tip radius as a function of the various parameters we note that a stability analysis based on these results may be performed

dynamics of wave fronts in excitable media abstract the dynamics of wave fronts in two dimensional excitable media is analytically studied by means of the asymptotic perturbation method for wave fronts in reaction diffusion systems we derive a space derivative burgers equation which governs the shape of boundary layers and their propagation application of lie group analysis of differential equations yields group invariant solutions of this equation and their classification some classes of the group invariant solutions determine the remarkable patterns expanding circle waves rotating spiral waves initial value problems for the burgers equation are also studied and a spiral wave is shown to be asymptotically realized under some proper boundary conditions

the leech lattice new proofs of several known results about the leech lattice are given in particular i prove its existence and uniqueness and prove that its covering radius is the square root of 2 i also give a uniform proof that the 23 holy constructions of the leech lattice all work

dispersion of traveling waves in the belousov zhabotinskii reaction abstract we use perturbation and numerical methods to calculate the dispersion relation for periodic wave train solutions of diffusion reaction equations modeling the belousov zhabotinskii reaction these results are compared with experimental data for the belousov zhabotinskii reagent and shown to be in good agreement

theory of the spiral core in excitable media abstract a rigorous asymptotic spiral solution to an excitable reaction diffusion system is found by separating space into two scaling regions an outer region having the same spatial scaling as the overall spiral structure which exhibits a singularity at the spiral tip and a core region around the spiral tip where space is scaled so as to resolve the tip singularity the stability of the spiral structure is investigated for both the outer region which is found to possess no intrinsic instability and the core region which is found to be unstable both the zero and small diffusion systems are found to exhibit qualitatively similar behaviour the implication of these results for both experiment and simulation in particular within the context of the observed meandering instability are addressed

knotted scroll wave filaments in excitable media abstract it is shown that certain general systems of diffusion reaction equations can have invariant knotted scroll wave solutions the theory is based on the assumption that the system of equations supports permanently rotating spiral solutions and from this assumptions equations governing the motion of the scroll wave filament can be derived then bifurcatilon theory is used to show that the equations of motion have torus knots as invariant solutions analytical methods are used to determine qualitative and quantitative features of these knotted solutions

third generation cellular automaton for modeling excitable media abstract this paper introduces a new cellular automation model of excitable media with improved treatments of 1 diffusion and wave propagation and 2 slow dynamics of the recovery variable the automaton is both computationally efficient and faithful to the underlying partial differential equations

theory of reentrant excitation in a ring of cardiac tissue a cardiac pathology reentrant tachycardia is caused by the sustained circulation of activation along a reentrant path in cardiac tissue a new theoretical model for wave propagation in discrete excitable media employing coupled maps with continuous time is used to derive analytical stability criteria for the uni directional circulation of the excitation pulse on the ring of cardiac tissue numerical simulations of the model that incorporate experimentally measurable characteristics of the cardiac tissue reproduce the experimental data quantitatively the complex oscillations in the cycle length of unstable tachycardia observed experimentally are quasiperiodic oscillations that arise via the multiple hopf bifurcation in high dimensional maps

stable particle like solutions to the nonlinear wave equations of three dimensional excitable media so called excitable media support nonlinear waves that propagate as pulses e g the action potential in nerve fibers and heart muscle and the oxidation pulse in certain chemical reactions in two dimensional media the right stimulus will also create rotating vortices which radiate such pulses periodically in every direction their investigation in diverse excitable media has dramatically expanded in the usa and ussr since their discovery in both countries in 1970 in three dimensional media the pivot of the rotating structure is not a point but a curved line generically a closed ring rings linked together and knotted constitute topologically distinctive organizing centers for periodic activity throughout the medium it is of interest to understand how stimuli create organizing centers and how organizing centers persist or decay the numerically discovered stable organizing centers exhibit enough symmetries to encourage mathematical attention this paper illustrates organizing centers by computer g

theory of spike spiral waves in a reaction diffusion system we discovered a type of spiral wave solutions in reaction diffusion systems spike spiral wave which significantly differs from the spiral waves observed in the models of fitzhugh nagumo type we present an asymptotic theory of these waves in the gray scott model chem sci eng 38 29 1983 we derive the kinematic relations describing the shape of this spiral and find the dependence of its main parameters on the control parameters the theory does not rely on the specific features of the gray scott model and thus is expected to be applicable to a broad range of reaction diffusion systems

reflection and attachment of spirals at obstacles for the fitzhugh nagumo and beeler reuter models in this paper the fitzhugh nagumo fhn equations and a modified fhn mfhn are considered for the modified version the recovery variable v has three different time scales by considering different parameters in the local dynamics of the mfhn equations it is observed that the phenomenon of reflection and annihilation at an impermeable boundary is observed just as in the beeler reuter model the interaction of spirals obtained with the fhn mfhn and beeler reuter model and an obstacle is also considered the phenomenon of reflection of the spiral wave at a boundary changes when the boundary becomes an obstacle four properties for attachment of a spiral wave to an obstacle are presented in this work

existence and uniqueness of travelling waves for a neural network a one dimensional scalar neural network with two stable steady states is analysed it is shown that there exists a unique monotone travelling wave front which joins the two stable states some additional properties of the wave such as the direction of its velocity are discussed

on the stability of steady rotating waves in the free boundary formulation abstract we perform a linear stability analysis of the steadily rotating solutions found from the free boundary problem of wave propagation in single diffusive excitable media two kind of modes can be considered a tangential mode which moves the spiral tip tangentially to the circle limiting the unexcited region and a radial mode which moves the spiral tip outwards the same circle in the radial direction the study of the second kind of modes is complicated by a technical difficulty associated to the divergence of the refractory field perturbations at the spiral tip square root difficulty the tangential modes are found to be stable whatever the angular velocity of the steady solution

spirals in excitable media ii meandering transition in the diffusive free boundary limit abstract in this paper we present a numerical stability calculation for steadily rotating spirals in an excitable medium while experiments as well as numerical simulations of two field reaction diffusion models have shown the existence of a hopf bifurcation from steady rotations to a meandering state all the analytical approaches so far have failed to predict this transition this mismatch between analysis and simulations raises the question whether meandering critically depends on the finite diffusivity of the interface separating between the excited and the refractory phases our calculations show that this is not the case the meandering transition takes place even in the limit of an infinitely sharp interface the boundaries of the meandering transition as function of the model parameters are traced we discuss possible explanations for the failure of previous analytical approaches

a numerical study of singularly perturbed generalized burgers huxley equation using three step taylor galerkin method in the present work a numerical study has been carried out for the singularly perturbed generalized burgers huxley equation using a three step taylor galerkin finite element method a burgers huxley equation represents the traveling wave phenomena in singular perturbed problems a very small positive parameter e called the singular perturbation parameter is multiplied with the highest order derivative term as this parameter tends towards zero the problem exhibits boundary layers the traditional methods fail to capture the boundary layers when e becomes very small in this paper a three step taylor galerkin finite element method is used to capture the boundary layers the method is third order accurate and has inbuilt upwinding stability analysis has been carried out and the numerical results show that the method is efficient in capturing the boundary layers

numerical bifurcation and stability analysis of solitary pulses in an excitable reaction diffusion medium abstract we present a systematic computer assisted study of the bifurcations and instabilities of solitary pulses in an excitable medium capable of displaying both stable pulse propagation and spatiotemporally chaotic dynamics over intervals of parameter space the reaction diffusion model used is of the activator inhibitor type only the activator diffuses in this medium the control parameters are the ratio of time scales of the activator and inhibitor dynamics and the excitation threshold this study focuses on travelling pulses their domain of existence and the bifurcations that render them unstable these pulses are approximated as a homoclinic orbits in a travelling wave ode frame and b as solutions of the full partial differential equation pde with periodic boundary conditions in large domains a variety of bifurcations in the travelling wave ode frame are observed including heteroclinic loops so called t points a r champneys and y a kuznetsov numerical detection and continuation of codimension 2 homoclinic bifurcations int j bif chaos 4 1994 785 h kokobu homoclinic and heteroclinic bifurcations of vectorfields japan j appl math 5 1988 455 instabilities in the full pde frame include both hopf bifurcations to modulated travelling waves involving the discrete pulse spectrum as well as transitions involving the continuous spectrum such as the so called backfiring transition m bar m hildebrand m eiswirth m falcke h engel and m neufeld chemical turbulence and standing waves in a surface reaction model the influence of global coupling and wave instabilities chaos 4 1994 499 the stability of modulated pulses is computed through numerical floquet analysis and a cascade of period doubling bifurcations is observed as well as certain global bifurcations these results corroborated by observations from direct numerical integration provide a skeleton around which many features of the overall complex spatiotemporal dynamics of the pde are organized

singular filaments organize chemical waves in three dimensions iv wave taxonomy abstract this is the last of four papers on the anatomy of three dimensional organizing centers in excitable media these organizing centers are made of scroll ring waves with twists and knots in them diversely linked together each such aggregate of linked rings is characterized by a handful of topological quantum numbers which partition the family of organizing centers into discrete classes or atoms we derive the rules which govern nuclear structure in this periodic table as well as the selection rules for transmutation

self similar solutions for the kinematic model equation of spiral waves we present a class of self similar solutions of the kinematic model equation introduced by v a davydov a s mikhailov and v s zykov this equation is designed to describe the dynamics of spiral waves in excitable media in this model the sharply located spiral fronts are regarded as planar curves if the tip neither grows nor retracts in the tangential direction and if their normal velocity with the eikonal approximation is assumed to possess no driving force then the kinematic equation admits self similar solutions with nonzero curvature we show the global structure of both forward and backward self similar solutions which implies mathematically the existence of various types of spiral waves

initiation of propagation in a one dimensional excitable medium abstract this study examines the initiation of propagation in a one dimensional fiber by local stimulation with a small electrode the membrane dynamics is based on the generic fitzhugh nagumo model reduced in a singular limit to a nonlinear heat equation a steady state solution of this nonlinear heat equation defines the critical nucleus a time independent distribution of potential that acts as a threshold for propagating wavefronts the criterion for initiation of propagation from the initial conditions on potential is obtained by re writing the nonlinear heat equation as a gradient flow of an energy and projecting this gradient flow onto an approximate solution space assuming that the evolving potential has a shape of a gaussian pulse the solution space consists of the amplitude of the pulse a and the inverse of its width k the evolution of the potential is visualized on the a k phase plane in which the rest state is a stable node and the critical nucleus solution is a saddle point the criterion for initiating propagation takes the form of a pair of separatrices that bisect all possible pulse widths for a specific pulse width the separatrices determine the minimum amplitude necessary to start propagation infinitely broad pulses space clamped fiber require amplitude equal to the membrane excitation threshold as the width of the pulse decreases the requirement on the amplitude grows in a limit of very narrow pulses the pulse width and the amplitude are related by a linear relationship corresponding to a constant charge delivered by the pulse

growth of domains under global constraints abstract we develop growth rates for domains in bistable systems far from equilibrium the governing equation is a bistable reaction diffusion equation including a nonlocal inhibiting term the latter one will be responsible for the existence of stable localized domains the growth of the size of different domains shows quite surprising similarities with coalescence phenomena of droplets in van der waals gases eventually we discuss the possibility of breathing of the domains induced by a time delayed nonlocal interaction

a model for fast computer simulation of waves in excitable media abstract starting from a two variable system of reaction diffusion equations an algorithm is devised for efficient simulation of waves in excitable media the spatio temporal resolution of the simulation can be varied continuously for fine resolutions the algorithm provides accurate solution of the underlying reaction diffusion equations for coarse resolutions the algorithm provides qualitative simulations at small computational cost

the dynamics of three dimensional scroll waves in excitable media abstract we investigate the motion of the central filament of a scroll wave in excitable medium we assume that two dimensional spiral solutions are known and use these solutions to produce slowly varying scroll waves in three dimensional space using standard ideas from perturbation theory multiscale techniques we find equations governing the motion of the central filament of the scroll and the twist of the scroll wave these equations are specialized to the case of a circular filament with uniformly distributed twist to determine the collapse or expansion and vertical drift of the spiral filament and to a helical filament with drift the predictions of these equations are in good qualitative agreement with the numerical experiments

simplifying the complexity of pipe flow transitional pipe flow is modeled as a one dimensional excitable and bistable medium models are presented in two variables turbulence intensity and mean shear that evolve according to established properties of transitional turbulence a continuous model captures the essence of the puff slug transition as a change from excitability to bistability a discrete model which additionally incorporates turbulence locally as a chaotic repeller reproduces almost all large scale features of transitional pipe flow in particular it captures metastable localized puffs puff splitting slugs localized edge states a continuous transition to sustained turbulence via spatiotemporal intermittency directed percolation and a subsequent increase in turbulence fraction toward uniform featureless turbulence

diffusion and wave propagation in cellular automaton models of excitable media in this paper the authors 1 examine general masks as discrete approximations to the diffusion equation showing how to calculate the diffusion coefficient from the elements of the mask 2 combine the mask with a thresholding operation to simulate the propagation of waves in excitable media showing that for well chosen masks the waves obey a linear speed curvature relation with slope given by the predicted diffusion coefficient and 3 assess the utility of different masks in terms of computational efficiency and adherence to a linear speed curvature relation

existence and stability of traveling waves in buffered systems we study wave propagation in the buffered bistable equation i e the bistable equation where the diffusing species reacts with immobile buffers that restrict its diffusion such a model describes wave front propagation in excitable systems where the diffusing species is buffered in particular the study of the propagation of waves of increased calcium concentration in a variety of cell types depends directly upon the analysis of such buffered excitability however despite the biological importance of these types of equations there have been almost no analytical studies of their properties here we study the question of whether or not the inclusion of multiple buffers can eliminate propagated waves first we prove that a unique up to translation traveling wave front exists moreover the wave speed is also unique then we prove that this traveling wave front is stable i e that any initial condition which vaguely resembles a traveling wave front in a way we make precise evolves to the unique wave front we thus prove that multiple stationary buffers cannot prevent the existence of a traveling wave front in the buffered bistable equation and may not eliminate stable wave fronts this suggests although we do not prove that the same result is true for more complex and realistic models of calcium wave propagation a result of direct physiological relevance

two types of performance of an isotropic cellular automaton stationary turing patterns and spiral waves abstract an isotropic cellular automaton containing only the crudest caricature of activator inhibitor systems is shown to be capable of producing stationary spatial structures turing patterns in qualitative agreement with a variety of experiments in chemistry due to the isotropy of our automaton we can obtain spots arranged in regular hexagons a pattern that cannot be obtained with a classical square lattice labyrinthine stripes like those observed in magnetic domains in thin films are also obtained spiral waves are obtained upon increasing the radius of action of the activator relative to that of the inhibitor as predicted by theory

fronts pulses sources and sinks in generalized complex ginzberg landau equations an important clement in the long time dynamics of pattern forming systems is a class of solutions we will call coherent structures these are states that are either themselves localized or that consist of domains of regular patterns connected by localized defects or interfaces this paper summarizes and extends recent work on such coherent structures in the one dimensional complex ginzburg landau equation and its generalizations for which rather complete information can be obtained on the existence and competition of fronts pulses sources and sinks for the special subclass of uniformly translating structures the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three dimensional phase space fixed points of the flow correspond to the two basic building blocks of coherent structures uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero a study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures the dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts and is expressed in terms of a set of conjectures for front propagation that generalize the marginal stability and pinch point approaches of earlier authors these rules together with an exact front solution whose dynamics plays an important role in the selection of patterns yield an analytic expression for the upper limit of the range of existence of pulse solutions as well as a determination of the regions of parameter space where uniformly translating fron t solutions can exist extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses in the parameter ranges where no uniformly translating fronts can exist examples are shown of irregularly spreading fronts that generate strongly chaotic regions as well as nonuniformly translating fronts that lead to uniform amplitude states recent perturbative treatments based on expansions about the nonlinear schr6dinger equation are generalized to perturbations of the cubic quintic and derivative schr idinger equations for which both pulses and fronts exist in the unperturbed system comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions nevertheless those that are obtained are found to be consistent with the general conjectures and in particular they provide an analytic demonstration of front pulse competition while the discussion of the competition between fronts and pulses focusses on the complex ginzburg landau equation with quintic terms and a subcritical bifurcation a number of results are also presented for the cubic equation in particular the existence of a family of moving source solutions derived by bekki and nozaki for this equation contradicts the naive counting arguments we attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits

dynamics of one and two dimensional fronts in a bistable equation with time delayed global feedback propagation failure and control mechanisms we study the evolution of fronts in a bistable equation with time delayed global feedback in the fast reaction and slow diffusion regime this equation generalizes the hodgkin grafstein and allen cahn equations we derive a nonlinear equation governing the motion of fronts which includes a term with delay in the one dimensional case this equation is linear we study the motion of one and two dimensional fronts finding a much richer dynamics than for the previously studied cases without time delayed global feedback we explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a hopf bifurcation as the delay time increases we show that for certain delay times the prevailing phase is different from that corresponding to the system in the absence of global coupling numerical simulations of the partial differential equation are in agreement with the analytical predictions

spiral waves in reaction diffusion equations we consider the reaction diffusion system begin gathered r t nabla 2 r r left 1 r 2 vec nabla theta cdot vec nabla theta right hfill r theta t r nabla 2 theta 2 vec nabla r cdot vec nabla theta qr 3 hfill end gathered this system governs the solutions of reaction diffusion systems near a hopf bifurcation point in two spatial dimensions we use formal asymptotic expansions to contruct one armed and multi armed archimedean spiral waves for small values of the parameter q we then show that the one armed spiral waves are probably stable and the multi armed ones are unstable for q small next by numerical continuation methods we construct spiral waves for all q these calculations show that the one armed spiral waves are unstable when q 1 397 cdots we also find the explicit one dimensional analogues of spiral waves for all q and show that they are unstable for q small

a geometrical theory for spiral waves in excitable media in this paper we develop a geometrical theory for waves in excitable reacting media using singular perturbation arguments and dispersion of traveling plane wave trains we derive an approximate theory of wave front propagation which has strong resemblance to the geometrical diffraction theory of high frequency waves in hyperbolic systems governed by the eikonal equation using this theory we study the effect of curvature on waves in excitable media specifically rotating spiral patterns in planar regions from this theory we are able to determine the frequency and wavelength for spiral patterns in excitable nonoscillatory media

a geometric theory for scroll wave filaments in anisotropic excitable media scroll waves are an important example of self organisation in excitable media in cardiac tissue scroll waves of electrical activity underlie lethal ventricular arrhythmias and fibrillation they rotate around a topological line defect which has been termed the filament numerical investigation has shown that anisotropy can substantially affect the dynamics of scroll waves it has recently been hypothesised that stationary scroll wave filaments in cardiac tissue describe geodesics in a space whose metric is the inverse diffusion tensor several computational studies have validated this hypothesis but until now no quantitative theory has been provided to study the effects of anisotropy on scroll wave filaments here we review in detail the recently developed covariant formalism for scroll wave dynamics in general anisotropy and derive the equations of motion of filaments these equations are fully covariant under general spatial coordinate transformations and describe the motion of filaments in a curved space whose metric tensor is the inverse diffusion tensor our dynamic equations are valid for thin filaments and for general anisotropy and we show that stationary filaments obey the geodesic equation we extend previous work by allowing spatial variations in the determinant of the diffusion tensor and the reaction parameters leading to drift of the filament

on the performance of anisotropic mesh adaptation for scroll wave turbulence dynamics in reaction diffusion systems abstract nonlinear reaction diffusion systems are widely employed to study the spatio temporal chaotic behavior that occurs in excitable media such as cardiac tissue where sufficiently strong perturbations can excite nonlinear propagating waves which can form spiral waves in two dimensions or scroll waves in three dimensions the numerical simulation of these waves calls for grids that are extremely fine over the whole computational domain to accurately predict the trajectories and multiplication of wave fronts and therefore leads to huge computational challenges mesh adaptation methods can reduce the number of degrees of freedom required for a given accuracy but they also have a cost and it is not clear if they are competitive with respect to very fine uniform meshes previous mesh adaptation techniques applied to spatio temporal chaotic behavior have been mostly limited to the two dimensional case the purpose of this paper is to explore the efficiency of a three dimensional anisotropic finite element mesh adaptivity for simulating scroll wave turbulence the computational efficiency of the proposed method is assessed using reference solutions obtained on a uniform refined mesh with more than 44 millions degrees of freedom the proposed method reduces significantly the number of elements leading to huge saving in memory as well as in computational time examples of the dynamics of ventricular fibrillation in cardiac tissue will be presented illustrating the performance of the overall methodology

the dynamics of scroll waves in excitable media an overview is given of the attempts to understand the behavior of scroll waves in three dimensional excitable media using analytical methods and these results are compared with recent numerical and experimental results the behavior of untwisted planar scroll rings twisted rings and helices and knotted scroll wave filaments is discussed

evolution of reaction diffusion patterns in infinite and bounded domains abstract we introduce a semi analytical method to study the evolution of spatial structures in reaction diffusion systems it consists in writing an integral equation for the relevant densities from the propagator of the linear part of the evolution operator in order to test the method we perform an exhaustive study of a one dimensional reaction diffusion model associated to an electrical device the ballast resistor we consider the evolution of step and bubble shaped initial density profiles in free space as well as in a semi infinite domain with dirichlet and neumann boundary conditions the piecewise linear form of the reaction term which preserves the basic ingredients of more complex nonlinear models makes it possible to obtain exact wave front solutions in free space and stationary solutions in the bounded domain short and long time behaviour can also be analytically studied whereas the evolution at intermediate times is analyzed by numerical techniques we paid particular attention to the features introduced in the evolution by boundary conditions

rotating spiral waves in λ ω systems on circular domains abstract systems are reaction diffusion systems whose reaction kinetics admit a stable limit cycle it is known that systems can possess various types of solutions among them spiral waves are the most fascinating pattern however the effects of the diffusivity the sizes of the domains and the reaction kinetics on spiral waves are largely unknown in this paper we investigate how these quantities affect the properties of m armed spiral waves in a generalized class of system on a circular disk with no flux boundary condition first we derive a criterion for the existence of m armed spiral waves specifically we show that m armed spiral waves do not exist for d 0 r 2 j m 2 while for d 0 0 r 2 j m 2 there exists an m armed spiral wave if the twist parameter q is small here d is the diffusivity for the system r is the radius of the circular disk 0 is the value of the function a at a 0 and j m is the first positive zero of the first derivative of the bessel function of the first kind of order m we also show that the critical diffusivity d 0 r 2 j m 2 is a bifurcation point next we use the numerical simulation to show that for small twist parameter the rotational frequency increases with increasing domain size while for large twist parameter the dependence of the rotational frequency on the domain size is not monotonic moreover small circular domains may change the properties of spiral waves drastically these numerical results are in contrast to those in excitable media finally the stability of spiral waves is investigated numerically

solitary waves in excitable systems with cross diffusion we consider a fitzhughnagumo system of equations where the traditional diffusion terms are replaced with linear cross diffusion of components this system describes solitary waves that have unusual

the effect of boundaries on the asymptotic wavenumber of spiral wave solutions of the complex ginzburg landau equation abstract in this paper we consider an oscillatory medium whose dynamics are modeled by the complex ginzburg landau equation in particular we focus on n armed spiral wave solutions of the complex ginzburg landau equation in a disk of radius d with homogeneous neumann boundary conditions it is well known that such solutions exist for small enough values of the twist parameter q and large enough values of d we investigate the effect of boundaries on the rotational frequency of the spirals which is an unknown of the problem uniquely determined by the parameters d and q we show that there is a threshold in the parameter space where the effect of the boundary on the rotational frequency switches from being algebraic to exponentially weak we use the method of matched asymptotic expansions to obtain explicit expressions for the asymptotic wavenumber as a function of the twist parameter and the domain size for small values of q

a retarded differential equation model of wave propagation in a thin ring we propose a mathematical model for wave propagation in a narrow ring filled with an excitable medium the speed of the wave fronts is assumed to depend on the time that has passed since the last impulse from this assumption we derive a system of nonlinear functional differential equations we prove that it has a special solution for which the speed of the fronts is the same constant determined by the dispersion relation and the fronts are distributed uniformly any initial distribution of the fronts apart from certain exceptional cases tends to this distribution in this sense it is the asymptotic state that result is in agreement with chemical experimental observations namely that the long term distribution of the fronts is uniform in an annular reactor our functional differential equation is transformed into a system of delay differential equations after this transformation a global stability theorem is proved

stable three dimensional action potential circulation in the fitzhugh nagumo model abstract excitable media generally support vortex rings of self excitation depending on the exact nature of the medium such a ring may expand or contract possibly to a stable radius we describe one such case encountered during numerical experiments on a simple model of electrophysiological excitability in nerve and cardiac muscle membrane the ring s rate of shrinkage depends parabolically on curvature and on proximity to other rings the vortex period also depends on curvature so rings of different sizes compete for territory we associate the stability of the ring with repulsive forces which we show are present between two dimensional rotors the observed minimal distance for repulsion agrees with the stable radius of the vortex ring

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen