by Jürgen Parisi and Stefan C. Müller

In many examples of physical, chemical and biological systems, excitation states occur that propagate in waves as spiralling, rotating structures. These phenomena are important in biomedicine, among other areas. As part of a priority programme of the German Research Foundation, in which the Department of Energy and Semiconductor Research in Oldenburg is involved, numerous projects on such spatio-temporal structure formations in so-called dissipative continuous systems are being funded, which originate from the fields of hydrodynamics, physical chemistry and semiconductor physics. Due to their interdisciplinary significance, the funded projects have an impact on neighbouring areas of scientific research.

Many examples of physical, chemical and biological systems show states of excitement which are propagated as spiral waves. This phenomenon is, among other fields, of importance in biomedicine. Numerous projects of these spatio-temporal developments of structure are supported by a cooperative programme of the German Research Foundation. Focus of research are projects from hydrodynamics, physical chemistry and semiconductor physics. The Department of Energy and Semiconductor Research at the University of Oldenburg takes part in these investigations. The interdisciplinary relevance of these projects make them important for adjacent fields of natural sciences.

In the life cycle of the slime mould Dictyostelium discoideum, individual amoeba cells grow together to form a multicellular structure when food is scarce, by means of chemotactic movements. This type of movement is based on the ability of the cells to recognise the spatial concentration gradient of a signalling substance by means of specific receptors and to use it as a directional guideline for independent locomotion. The resulting structure then differentiates into a slime mould consisting of stalk and fruiting body, whose spores later initiate a new cycle of amoeba growth. In the early stages of cell aggregation, circular and spiral patterns can be visualised in the dark-field microscope due to differences in the light scattering behaviour of resting and moving cells.

The decisive factor is the spatial distribution of the signalling substance cyclic adenosine monophosphate (cAMP), which is produced by the individual cells in an oscillatory manner and released into the environment. The resulting concentration gradients cause the cells to perform a pulsating chemotactic movement in the direction of the future aggregation centre.

Such a cAMP spiral causes a push of amoeboid cells towards the centre per revolution, and a vortex-like movement structure initially develops there. Later, the accumulation of cells in this area leads to the growth of a fungus-like pseudoplasmodium out of the plane and upwards. However, the precise mechanism that transforms the initial cell movement into the later pseudoplasmodium is still unknown.

The dynamic behaviour of the slime mould during aggregation can be treated excellently within the framework of so-called reaction-diffusion models. Thus, this behaviour is one of the exceptions to the rule that biological structures often fit well into the framework of dissipative continuous systems but, with a few exceptions, usually only allow qualitative statements to be made when comparing experiment and theory.

The slime mould Dictyostelium discoideum is an example of an excitable system. Probably the best known example of this is the nerve fibre, which is in an excitable initial state - mediated by a resting potential. A brief opening of ion channels causes an excitation pulse, which runs along the nerve fibre as a potential change (spike), followed by a recovery or refractory phase. During this phase, charge differences between membranes are restored before another excitation pulse can be triggered. The excitation process that characterises nerve conduction is a non-linear dynamic phenomenon that also manifests itself in numerous other spatially extended media. Excitability can lead to the propagation of sharply defined fronts through the effect of short-range coupling mechanisms such as diffusion, which carry an active, excited state at a certain speed through a spatial volume. The active state can be the increased concentration of a rapidly produced chemical or biochemical substance; in the Belousov-Zhabotinsky (BZ) reaction, a chemical reaction that oscillates periodically depending on the choice of initial concentrations and exhibits the typical characteristics of an excitable system, this is, for example, hypobromous acid as a chemical intermediate. The geometric shapes that form such fronts are circular arrays arranged concentrically in thin layers, so-called "target disc patterns", or, as in the slime mould Dictyostelium discoideum, the even more remarkable spirals, the tips of which circle undauntedly around a central core area, with one front being emitted into the surroundings in rhythmic succession per revolution. The first illustration (p. 23) contains some snapshots of spiral waves from chemistry and biology. Figure A shows a spiral system in the FC reaction as a classic example. Figure B shows the spiral distributions of the signalling messenger in the amoeba colonies of Dictyostelium as one of numerous examples from biology. The third image C leads to biomedically relevant questions regarding the significance of excitation waves in the so-called "spreading depression" in neuronal tissue. This dynamic phenomenon in the central nervous system, in which a wave of greatly reduced electrical activity of the nerve cells spreads at a speed of only 3 mm/min after mechanical or chemical stimulation, is demonstrated here in the chicken retina. Spreading depression is discussed in connection with the occurrence of migraine and focal epilepsy.

Structure formation in systems that are not in equilibrium, such as the systems mentioned above, has become an important interdisciplinary field of research in recent years, which has a stimulating effect on many branches of the natural sciences and mathematics. The German Research Foundation (DFG) has therefore been supporting around 30 research groups working on "Structure formation in dissipative continuous systems" in one of its priority programmes for several years, which are based at numerous scientific institutions in Germany. In contrast to the non-linear dynamics of systems with few degrees of freedom, which is an active field of work in many places and has much in common with the dynamics of continuous systems, this DFG Priority Programme emphasises those aspects for whose description the continuum properties and non-linear transport equations are important.

The development of precise measuring methods and new theoretical concepts as well as the greatly increased capacity of computers have made it possible to follow the non-linear development of regular and chaotic structures in continuous media in detail and to gain a better understanding of them. Numerous examples can be found in hydrodynamics and in systems in which non-linear chemical or biochemical reactions are coupled with a transport process, in particular with molecular diffusion. Here the addition "Experiment and theory in quantitative comparison", which follows the title of the DFG Priority Programme, becomes essential in order to emphasise (which should actually be common practice) that individual sub-questions on structure formation should be worked on with comparable intensity in the laboratory as well as in theory, modelling and simulation.

In terms of content, this focus was originally conceived as a "core with a shell" under the coordination of Prof. Dr. Friedrich H. Busse (Bayreuth) and in close organisational cooperation with the authors: At the centre of the essential questions are experimentally accessible structures from the field of hydrodynamics and the aforementioned reaction-diffusion systems in physical chemistry and biophysics. This core area includes quantitative numerical studies of the individual systems and general theoretical methods and concepts, which are intended to establish an overarching universal context. Beyond these basic pillars, this core area should radiate out into a surrounding "mantle area", which concerns a fairly large number of neighbouring research areas - such as structure formation in optically active media, biological structures, physiological phenomena or numerical studies in these areas through to meteorology and astrophysics. Dynamic structuring phenomena in solids, especially semiconductors, have always been of particular interest. Their treatment was therefore explicitly included in the priority programme, and these research projects have since established themselves as permanent sub-areas of the core area.

How can an excitable medium be characterised in a compact way so that classical biological processes such as nerve conduction, electrophysiological phenomena on the contracting heart muscle and the aforementioned phenomena are also included? Essentially, the three states already mentioned for nerve fibres are sufficient to describe typical excitation kinetics: either the system remains in an excitable resting state; or it is in an active, excited state, which can be generated by a local disturbance, for example by immersing a thin, hot wire in the reactive chemical solution layer; or it is refractory, i.e. it cannot be temporarily re-excited during the recovery phase, the return from the excited to the resting state

A simple mathematical model of such kinetic behaviour is based on the assumption that a fast activator variable (in the FC reaction the autocatalytically generated hypobromous acid, with changes on a time scale of milliseconds) develops in interaction with a much slower inhibitor variable (in the FC reaction the catalyst ferroin, with changes on a time scale of seconds). The temporal dynamics are generally described mathematically by non-linear functions. Suitable functions have been introduced for a number of excitable systems and have been successfully used for qualitative or semi-quantitative comparisons with experimental data. For example, the FC reaction is described using the so-called "Oregonator" model with the variables specified above; comparable models exist for amoeba aggregation in Dictyostelium and for electrophysiological systems.

The essential features of the dynamics of the system can then be derived from the properties of these model functions. For example, the excitable resting state is stable against small disturbances, but a rapid jump to an excited state takes place when the activator is increased above the threshold. If the inhibitor rises to its maximum value in the excitation phase, the system falls into the third state, the recovery or refractory phase. A new excitation pulse can then only be triggered after some time.

How does front propagation occur in spatially extended systems? The basic mechanism is the coupling of reaction and diffusion events: Starting from an excitation nucleus generated by an external stimulation, such as in the FC reaction by the hot wire or by injection of the activator, molecular diffusion takes over to increase the activator concentration. Neighbouring areas are thus "infected" to produce the activator. For reasons of symmetry, this results in a circular wave front of the active state, followed by a refractory zone that specifies a certain minimum distance at which the next circular wave can occur. Mathematically, this coupling is reflected in the addition of diffusion terms to the functions describing the reaction.

Due to the very different time scales of the activator and inhibitor variables, special methods can be used for the analytical treatment of the resulting systems of equations. Two relationships characteristic of wave solutions can be derived from this: Wave propagation obeys a dispersion relation, i.e. the speed of plane wave fronts decreases when the frequency of wave generation increases and thus the distance of successive fronts decreases. Secondly, there is a dependence of the wave velocity on the local radius of curvature of a wave front. The latter relationship predicts the existence of a minimum radius for circular waves, below which there is no outward propagation. Furthermore, the relationship ensures a stabilisation of the front geometry.

The initial situation for the formation of spirals is the existence of an open wave end, which is obtained in an aqueous solution layer of the BZ reaction by breaking up a closed front by means of an air blast from a pipette. In biological systems such as amoeba aggregation, open ends often arise spontaneously at an inhomogeneity that acts as an obstacle to the travelling front. If the excitability of the system is high enough, the end winds up over time to form a regular, almost Archimedean spiral. Such a fully developed spiral structure rotates uniformly around a core area. Special conditions prevail in the core area of the spiral. This area is exempt from any excitation and forms a centre at rest.

It should be possible to extend the current understanding of excitation waves to other areas of biology. For example, current work on isolated heart muscle tissue underlines the fact that electrophysiological activity in the heart can take the form of spiral patterns. Spiral peaks are trapped and anchored by natural inhomogeneities in tissue such as arteries, as is also observed in FC solutions with artificial obstacles. They remain in a stable rotation around this centre and are difficult to detach from this anchor. This could initiate the life-threatening process of cardiac fibrillation. Consequently, it is a medically urgent concern to understand how a rotating electrophysiological excitation can be reduced within the shortest possible deadline. A current research focus is therefore on how such internally controlled dynamics can be influenced from the outside by suitable means in order to subject the spiral behaviour to targeted control.

The biological examples briefly outlined here underline the importance of research into spatio-temporal self-organisation on the basis of reaction-diffusion coupling. Physico-chemical model systems are particularly suitable for controlled experiments in the laboratory.

Prof Dr Jürgen Parisi (48), Head of the Department of Energy and Semiconductor Research, was appointed to Oldenburg in 1995. He began his studies at the University of Stuttgart and completed them at the University of Tübingen, where he also gained his doctorate in 1982 and habilitated five years later. In 1990, following a guest professorship at the University of Zurich, he was appointed to the University of Bayreuth. Numerous research visits have taken him abroad, including to Enschede, Cardiff, Grenoble, Beijing and Sao Paulo. The work presented above is a joint product with the Department of Physics at the University of Magdeburg. Co-author Prof Dr Stefan C. Müller is a member of the Institute of Physics, Department of Biophysics.