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Active media (AM) with chemical reactions and diffusion and their analogs play an important role in various branches of biology, chemistry and physics [l–7]. A simple example of such media is furnished by a bistable AM [1, 3, 5, 7] described by the equations $${\partial _t}u + div\,\vec j = f\left( u \right),{\kern 1pt} \quad \quad \vec j = - D\,\;grad\;u\quad \quad \left( {1a} \right)$$ (1a) or, similarty, by $${\partial _t}u = D\Delta u + D \bullet grad\;u + f\left( u \right)\quad \quad \left( {1b} \right)$$ (1b) .

The formation and maintenance of spiral structure in galaxies is discussed in terms of spiral modes with a general perception consistent with the hypothesis of quasi-stationary spiral structure. The latter is explained in some detail to give it the original proper perspective and to contrast it with recent studies of non-stationary spiral structures. We again emphasize the possible coexistence of fast-evolving spiral features and slow-evolving spiral grand designs. Spiral modes are described in terms of three categories of propagating waves. The reason is given why isolated non-barred galaxies with spiral grand design are found to be mostly two-armed. Some suggestions are made for future research.

“the term ‘mathematical model’ … will be used for any complete and consistent set of mathematical equations which is thought to correspond to some other entity, its prototype. The prototype may be a physical, biological, social, psychological or conceptual entity, perhaps even another mathematical model …” This quote, from Aris’ [1978] interesting book on mathematical modelling, gives a basic description of a mathematical model. As Aris points out, mathematical modelling largely remains an art and it is difficult to communicate modelling skills to the uninitiated.

It is documented that the evolution of spatial-temporal organization in catalytic oxidation processes on metal surfaces through the synchronization of oscillators can be simulated in an at least qualitatively satisfying way by means of an “infection” model expressed in form of a cellular automaton.

Recently, the development of the equilibrium and stability theory of flat gravitating systems has occurred, mainly with the aim of understanding the galactic spiral structure. From the vast material accumulated here we have selected only some problems1 which are, in our opinion, most closely related to the subject of the book.

The system of narrow elliptical Uranian rings is one of the most mysterious astronomical objects. At present they may be considered as attractors. But the question of their forming in the circumplanetary Uranian disk at its protostage is still open.

A new chemical reaction with either excitable or periodic dynamics appears every month in the theoretical journals (in 1978). But only one has been widely studied experimentally in ways that reveal wave-like organization in space. There are already two entire books about it: Zhabotinsky (1974, in Russian) and Tyson (1976).

The behaviour of a reverberator in a bounded medium of interacting nonlinear oscillators is considered. The expressions obtained for the reverberator drift velocity and the shift of its frequency may be interpreted in terms of “field-particle”.

Rotating spiral waves (SW) represent one of the most fascinating examples of the self-organization phenomena. Along with the leading centers and vortices they represent the basic types of autonomous wave generators in active media. Experimental data indicate [1, 2] that all SW have the same rotation frequency which is determined only by the properties of the medium and not by the initial conditions. Hence the analytical estimation of the SW frequency is an important task of any theoretical treatment.

The properties of disks accreting by global spiral shocks are reviewed. The discussion includes the formation of these waves, the way they transport angular momentum, a selfsimilar model for spiral shocked accretion and recent numerical simulations. These simulations suggest that at least in disks produced by mass overflow in close binaries the efficiency of accretion (in terms of an effective α-value) can be substantial.

The study of spatiotemporal organization in reactive media due to the coupling of chemical reactions with transport processes has become, in recent years, a subject of intensive research in the field of non-equilibrium dynamics in chemical systems. The attention given to the evolution of spatial inhomogeneities on a macroscopic scale has been rapidly growing, ever since the remarkable phenorusnon of chemical wave propagation in the Belousov-Zhabotinskii reaction was discovered almost 20 years ago [1,2]. However, as pointed out by TYSON and KAGAN in their historical review in this Volume, one should remember that numerous observations of chemical patterns in heterogeneous systems were reported before systematic research on temporal and spatial periodicities in homogeneous liquid-phase reactions began.

My contribution to this Joint Discussion can hardly be termed very novel or original. Rather, I just want to call your attention here to two headaches — one perhaps only a hangover, but the other a real migraine. Sooner or later these two headaches beset anyone who persists in asking how the random motions of stars in the only vicinity where we tolerably observe them can possibly jibe with the sensible presumption that not only our neighborhood but this entire Galaxy should by now be reasonably stable.

The beautiful spiral waves of oxidation in the Belousov-Zhabotinskii reaction [20] are the source of many interesting questions about periodic structures in excitable media. Because they are easy to produce and photograph, pictures of these spirals have appeared in a number of popular science oriented magazines. The study of spirals takes on a more personal interest when one realizes that fibrillation and sudden cardiac arrest from heart attacks may also be due to the appearance of rotating spiral waves of electrical activity on the ventricular myocardium [1], [14]. Immediately questions like “How do spirals form?” and “Can Spirals be prevented?” or “Can one predict if a heart attack will be fatal?” spring to mind. Going beyond questions of self preservation, we may also ask about the properties of spirals, such as their wavelength and frequency, or the conditions necessary to. sustain spiral activity in a given medium.

Several algorithms are tested to characterize order and disorder in digitised images of spatial patterns. As source of images, the BelousovZhabotinskii reaction coupled to convective motion in a thin liquid layer is used.

The autowaves [1–3] form a type of waves which are characteristic of strongly nonlinear active media. To here belong self-sustained signals inducing local release of stored energy in an active medium, which is spent to trigger the same process in adjacent regions. The examples are provided by waves of combustion, of phase transitions, concentrational waves in chemical reactions, and also by many biological autowave processes (propagation of nerve impulses, excitation waves in heart muscle, epidemic waves in ecological communities, spreading waves in the cerebral cortex). These examples stress the importance of autowave phenomena.

Spatial structures are actually observed in an evergrowing class of chemically active media and their possible relation to numerous important biological phenomena is often emphasized [l, 2]. Hence the problem of their origin and selection is of particular importance and we review here the main results of the theory of spatial chemical patterns based on a stochastic reaction-diffusion description of nonlinear chemical networks.

Complexity in space and time is frequently observed in the dynamical evolution of chemical reactions kept far from thermodynamic equilibrium. First observations were made more than 100 years ago, but systematic investigations on such phenomena both in theory and experiment have been carried out only during about the last two decades. The most striking features are chemical oscillations and the formation of remarkably regular spatial patterns (for recent reviews see [1–3]). Historically, the first periodic phenomena were found in electrochemical reactions [4], while liquid phase oscillations, as already reported by Bray in 1921 [5], remained unnoticed by the scientific community for over 40 years. The discovery of the Belousov-Zhabotinskii (BZ) reaction [6,7] and simultaneous observations in the area of biochemical oscillations [8] finally led, in the mid-sixties, to an appreciation of oscillatory phenomena which so far did not fit into the framework of classical thermodynamics.

Whereas cellular patterns in driven hydrodynamic systems have been known for nearly a century /1/, the observation of organized behavior (periodic oscillations, waves,…) in complex chemical systems is much more recent. It goes back to the work of Belousov and Zhabotinsky and came as a surprise to most chemists. They were indeed at first considered as rather exotic phenomena (although they had been shown, by Glansdorff and Prigogine /2/, not to be in contradiction with basic thermodynamic principles), but it is now apparent that an increasing number of complex chemical networks may exhibit this type of behavior when functioning sufficiently far from thermal equilibrium /3/. Much effort has lately gone into the precise determination of the characteristics of the various kind of waves (fronts /4/, target patterns /5/ and spiral waves /6,7/) in order to understand the mechanism of their formation.

The dynamics of networks is closely connected with the development of the modern science of non-linear irreversible processes [1–6].

In a two-dimensional nonlinear system with a spatially uniform state unstable against finite-wavenumber oscillatory perturbations, stability of near-onset travelling-rectangle and oscillatory triangle patterns against long-wave modulations are investigated within the framework of coupled amplitude equations of the Newell-Whitehead-Segel type.

In this short communication, I intend to comment on the studies of the structure of the Milky Way in view of current progress made in the density wave theory of spiral modes. I shall first make a brief report of these developments.

A model developed previously for signal relay and adaptation in the cellular slime mold Dictyostelium discoideum is applied in the context of wave propagation during aggregation. The ability of the sensory system in D. discoideum to adapt to constant background stimulus levels underlies wave propagation, and a brief general analysis of adaptation is given first. We then present the model for D. discoideum and show how the requirement of adaptation places constraints on the parameters in the model. Even with these constraints the model exhibits excitability or periodic oscillations for ranges of the parameters. Lastly we discuss some numerical results on wave propagation. It is found that the computed amplitude and speed of the axisymmetric cyclic AMP waves (or target patterns) that exist in the model agree well with the experimentally-observed values.

The wave patterns in the distributed Belousov-Zhabotinsky reaction are generated by two different types of sources. Those are spiral waves [1, 2] and leading centers [1, 3] producing the so called “target patterns”. The nature of leading centers remains unclarified up to now. Target patterns may be generated either by some small auto-oscillatory regions of the medium or by a special type of sources (echo-sources) arising in a medium with non-oscillatory (excitable) kinetics [4].

Galactic dynamics has been influenced considerably in recent years by plasma dynamics. For instance, Lin’s gravitational theory of spiral structure is using methods that were first developed for a plasma. The same is true for many studies of collisionless and collisional relaxation of stellar systems, problems of stability, etc.

A geometrical-kinematical approach is used to describe spiral wave behaviour. The dynamic behaviour depends only on the motion of the tip as it follows a pre-determinated trajectory. This tipis considered to be a source for the points making up the spiral wave front. Circular and meandering tip motions are studied in our model. Super-spiral waves are generated when highly meandering tip motion occurs.

Aim: Computer models enabled the study of the fundamental mechanisms responsible for arrhythmias and have the potential of optimizing the clinical procedure for an individual patients pathology. The model complexity and the computational costs affecting computer models hamper their application on a routinely performed procedure. In this work, we aim to design a computer model suitable for clinical time scales. Methods: We adopt a (multi-front) eikonal model that adapts the conduction velocity to the underlying electrophysiology; we describe the diffusion current using a parametrised form, fitted to reproduce the monodomain profile. Results: We simulated spiral waves on a 3D tissue slab and bi-atrial anatomy. We compared the numerical results obtained with a monodomain formulation with those obtained with the new method. Both models provided the same pattern of the spiral waves. While the monodomain model presented slower propagation fronts, the eikonal model captured the correct value of the conduction velocity CV even using a coarse resolution. Conclusion: The eikonal model has the potential of enabling computer-guided procedures when adapts the conduction velocity to the underlying electrophysiology and characterises the diffusion current with a parametrised form.

Calcium is critically important for a vast array of cellular functions, as discussed in detail in Chap. 1. There are a number of Ca2+ control mechanisms operating on different levels, all designed to ensure that Ca2+ is present in sufficient quantity to perform its necessary functions, but not in too great a quantity in the wrong places. Prolonged high concentrations of Ca2+ are toxic. For instance, it is known that cellular Ca2+ overload can trigger apoptotic cell death, a process in which the cell kills itself. Indeed, control of Ca2+ homeostasis is so crucial that even just disruptions in the normal Ca2+ fluxes can lead to initiation of active cell death. There are many reviews of Ca2+ physiology in the literature: in 2003 an entire issue of Nature Reviews was devoted to the subject and contains reviews of Ca2+ homeostasis (Berridge et al., 2003), extracellular Ca2+ sensing (Hofer and Brown, 2003), Ca2+ signaling during embryogenesis (Webb and Miller, 2003), the Ca2+-apoptosis link (Orrenius et al., 2003), and the regulation of cardiac contractility by Ca2+ (MacLennan and Kranias, 2003). Other useful reviews are Berridge, (1997) and Carafoli (2002).

During macropinocytosis, cells remodel their morphologies for the uptake of extracellular matter. This endocytotic mechanism relies on the collapse and closure of precursory structures, which are propagating actin-based, ring-shaped vertical undulations at the dorsal (top) cell membrane, a.k.a. circular dorsal ruffles (CDRs). As such, CDRs are essential to a range of vital and pathogenic processes alike. Here we show, based on both experimental data and theoretical analysis, that CDRs are propagating fronts of actin polymerization in a bistable system. The theory relies on a novel mass-conserving reaction–diffusion model, which associates the expansion and contraction of waves to distinct counter-propagating front solutions. Moreover, the model predicts that under a change in parameters (for example, biochemical conditions) CDRs may be pinned and fluctuate near the cell boundary or exhibit complex spiral wave dynamics due to a wave instability. We observe both phenomena also in our experiments indicating the conditions for which macropinocytosis is suppressed.

Spiral waves in cardiac tissue can pin to tissue heterogeneities and form stable pinned waves. These waves can be unpinned by electric stimuli applied close to the pinning center during the vulnerable window of the spiral. Using a phase transition curve (PTC), we quantify the response of a pinned wave in a cardiac monolayer to secondary excitations generated electric field pulses. The PTC can be used to construct a one-dimensional map that faithfully predicts the pinned wave's response to periodic field stimuli. Based on this 1D map, we predict that pacing at a frequency greater than the spiral frequency, over drive pacing, leads to phase locking of the spiral to the stimulus, which hinders unpinning. In contrast, under drive pacing can lead to scanning of the phase window of the spiral, which facilitates unpinning. The predicted mechanisms of phase scanning and phase locking are experimentally tested and confirmed in the same monolayers that were used to obtain the PTC. Our results have the potential to help choose optimal parameters for low energy antifibrillation pacing schemes.

The interaction of spiral waves of excitation with atrial anatomy remains unclear. This simulation study isolates the role of atrial anatomical structures on spiral wave spontaneous drift in the human atrium. We implemented realistic and idealised 3D human atria models to investigate the functional impact of anatomical structures on the long-term (∼40 s) behaviour of spiral waves. The drift of a spiral wave was quantified by tracing its tip trajectory, which was correlated to atrial anatomical features. The interaction of spiral waves with the following idealised geometries was investigated: (a) a wedge-like structure with a continuously varying atrial wall thickness; (b) a ridge-like structure with a sudden change in atrial wall thickness; (c) multiple bridge-like structures consisting of a bridge connected to the atrial wall. Spiral waves drifted from thicker to thinner regions and along ridge-like structures. Breakthrough patterns caused by pectinate muscles (PM) bridges were also observed, albeit infrequently. Apparent anchoring close to PM-atrial wall junctions was observed. These observations were similar in both the realistic and the idealised models. We conclude that spatially altering atrial wall thickness is a significant cause of drift of spiral waves. PM bridges cause breakthrough patterns and induce transient anchoring of spiral waves.

Identifying the atrial tissue that is capable of supporting sustained re-entrant spiral wave activation patterns offers a potential ablation target for atrial arrhythmias. Current strategies for identifying this substrate require the patient to be in atrial fibrillation and require a large specialized catheter or an inverse ECG vest. We propose a novel method to personalize biophysical ionic models from standard multi-electrode catheter measurements and to predict spiral wave stability using computer simulations of a tissue region. The developed method was applied to 5 clinical cases; the spiral wave stability was analyzed on a 5×5 cm2 homogeneous tissue slab and stable (2/5) and unstable self-terminating (3/5) rotors were identified.

The ray methods of geometrical optics are used to calculate both steady and unsteady patterns of waves produced by localized sources in galactic disks. Caustics or ray envelopes are shown to form in a number of instances, and are important because they mark the most prominent features of wave patterns. Unsteady spiral wave patterns in a stellar disk can have caustics that propagate both toward and away from the galactic center. The study of these waves in a specific model shows that one-armed spiral waves have the widest range of propagation, can propagate into the galactic center and be reflected there, and allow the development of leading as well as trailing spiral wave patterns.

62.2±4.5 mV, p

Background Functional reentrant excitation in atrial tissue (i.e., reentry with no central anatomical obstacle) was proposed to be caused by the “leading circle” mechanisms (Allessie's model). According to this concept, there is no excitable gap and the central core of functional conduction block (FCB) results from continuous centripetal invasion of wavefronts that prevent the cells in the core to recover. Hypothesis The mechanism of functional reentry in atrial tissue is caused by a spiral wave with an excitable gap and a central core of FCB caused by the steep curvature of the spiral wave tip. Methods Blocks of 3.0 by 4.0 cm of canine left and right atrial tissues (N = 3) were mounted in a tissue bath with endocardial surface up and reentrant activity initiated by premature stimulation and/or by rapid pacing and acetylcholine (10 -6 to 10 -5 M). Isochronal activation maps of the induced reentrant excitation were constructed using 512 bipolar electrodes with 1.6 mm interelectrode distance and were also displayed dynamically on the computer screen to visualize the pattern of activation. Results Ten episodes of both nonsustained and sustained atrial reentry was induced and mapped. The induced reentrant activity was clearly a spiral wave with a rotation period of 184 ± 74 (110 to 350) msec and rotating in a counterclockwise direction in all. The area of FCB in the central core of the spiral wave was caused by the steep curvature of the tip of the spiral wave that was excitable during regular pacing but not excited during the spiral wave activity. Spiral wave activity could be terminated by electrical stimulation indicating the presence of an excitable gap. Conclusion Functional reentry in normal atrial tissue is caused by a spiral wave of excitation with an excitable gap.

Recent advances in Kalman filtering to estimate system state and parameters in nonlinear systems have offered the potential to apply such approaches to spatiotemporal nonlinear systems. We here adapt the nonlinear method of unscented Kalman filtering to observe the state and estimate parameters in a computational spatiotemporal excitable system that serves as a model for cerebral cortex. We demonstrate the ability to track spiral wave dynamics, and to use an observer system to calculate control signals delivered through applied electrical fields. We demonstrate how this strategy can control the frequency of such a system, or quench the wave patterns, while minimizing the energy required for such results. These findings are readily testable in experimental applications, and have the potential to be applied to the treatment of human disease.

It is well known that alternans can lead to cardiac arrhythmias. Alternans typically occurs in single cells due to voltage or calcium cycling instabilities. We showed in a previous study that alternans can also result from an instability caused by supernormal excitability (negative slope of the conduction velocity (CV) restitution curve), in the absence of voltage or calcium cycling instabilities.In this study, we used computer simulations of cables and two-dimensional domains of Luo-Rudy phase 1 model cells in which supernormal conduction was induced by decreasing the extracellular potassium concentration ([K+]o, 2.0 mmol/L) to explore arrhythmogenesis in the presence of supernormal excitability.Conduction was stable in a homogeneous cable (10 cm long) paced at one end at a basic cycle length (BCL) of 290 ms. However, when a slightly premature stimulus was applied and pacing at BCL was resumed, alternans developed gradually towards the end of the cable. This alternans was caused by supernormal CV restitution. The amplitude of alternans progressively increased until conduction block occurred at approximately 2/3 of the length of the cable.Conduction was also stable in a homogeneous two-dimensional domain (10x10 cm) paced on the left border at BCL=280 ms. However, when a premature pulse was applied at the lower half of the left border and pacing at BCL was resumed, alternans developed in the lower half of the domain via the same mechanism as in the 1D cable, while conduction in the upper half remained stable. This eventually led to conduction block and wavebreak formation in the lower half of the domain, which initiated spiral wave reentry.In conclusion, supernormal excitability represents a novel mechanism of alternans, which can lead to conduction block, wavebreak formation and reentry in cardiac tissue.