Pattern Formation In Reaction Diffusion Systems

Many of the properties of macroscopic systems driven out of equilibrium by flows of matter and/or energy are not merely an additive effect. Instead, these properties emerge from cooperative interactions of their constituents, having no direct counterpart in the properties or dynamics of the microscopic constituents considered in isolation. While there is a general understanding of how such emergent properties arise in equilibrium — phase transitions, such as occur in the boiling or freezing of a liquid, and their description by renormalization group theory are one of the prime examples — this is not the case for systems far away from equilibrium.

A particularly interesting class of such self-organized structures are nonlinear waves. They occur in a multitude of systems including chemical and biological systems. For example, if chemical reagents are continuously supplied to and removed from a container where an oxidation reaction takes place on a catalytic surface, in many circumstances the chemical reaction does not occur homogeneously over the entire surface but instead proceeds by the propagation of chemical waves of oxidation that travel across the catalytic surface. The combination of nonlinear chemical kinetics and conditions that force the reaction to occur in far-from-equilibrium conditions is responsible for the existence of the evolving patterns of chemical waves seen on the surface of the catalyst (Davidsen et al. 2005). Similarly, in biological systems the nonlinear chemistry associated with biochemical networks, in combination with diffusion of chemical species, can lead to the formation of chemical waves which are often implicated in the mechanisms responsible for biological function.

Davidsen & Kapral (2002, 2003); Wei et al. (2006); Davidsen et al. (2003, 2004a,b, 2008a) have addressed fundamental questions of pattern formation related to spiral and scroll wave dynamics in nonlinear or “active” media. The motion of spiral cores or “vortices” is of particular relevance since it is related to some cardiac arrhythmias. We showed that the breaking of the rotation symmetry of spiral waves in a large class of media is accompanied by an intrinsic drift of the pattern giving rise, in multi-spiral regimes, to a novel “vortex liquid”. We also showed that spiral breakup in these media leads to defect-mediated turbulence where the spatiotemporal pattern is dominated by the rapid motion, nucleation, and annihilation of vortices. While many of the characteristics of this turbulent state are identical to those seen in diverse hydrodynamical and chemical experiments and simulations, the fluctuations in the number of vortices allow to distinguish between different media. Our theoretical results are currently tested by different experimental groups.

Defect-mediated turbulence appears in a wide range of processes including cardiac fibrillation — a disorganized electrical wave activity that destroys the coherent contraction of the ventricular muscle and its main pumping function leading to sudden cardiac death. At present, our understanding of the transition to fibrillation remains largely speculative: Only the surface of the heart is accessible to experimental observation, which prevents the study of the full dynamics. In a first step to overcome this barrier, we have shown that the statistical properties of defect-mediated turbulence on the surface of bounded three-dimensional excitable media can be used to distinguish between two major classes of mechanisms for turbulence (Davidsen et al., 2008a). This can be related to the different creation and annihilation processes of vortices and quantitatively captured by a theoretical model. As a next step, the analysis will be extended to the generic case of oscillatory media and more realistic heart models.