My main interest within Applied Mathematics is pattern formation. In the interface between mathematics, biology and the physical sciences, there are many yet unexplained spatial structures far from equilibria that are not well understood yet. I like to investigate symmetric and time-symmetric structures computationally and to explain the phenomena I see semi-analytically and from a dynamical-systems viewpoint.
Spiral waves occur in several different contexts of biology and chemistry: Certain slime mold amoebae (Dictyostelium discoideum) create spiral patterns in the early part of aggregation to from a multi-cellular slug, Spreading Depression occurs in most grey matter of the central nervous system and is linked to migraine and epilepsy, early stages of heart fibrillation exhibit spiral patterns of electrical activity. In chemistry, the Belousov -- Zhabotinsky reaction is a very good and easily studied model of spiral waves. Understanding and influencing the movement of spiral waves is an important step in trying to avoid or cure heart attacks and possibly migraine attacks.
I have been working on periodic forcing of spiral waves and three-dimensional scroll waves, both with numerical PDE models and in the case of spiral waves via an analysis of an ODE model given by the spiral symmetries. Periodic forcing at a resonance frequency causes the spiral to drift, i.e. move in a constant direction for a long time (in a direction depending on the phase of the forcing). This enables control of the spiral location and, via drift to the boundary, the extinction of spirals on finite domains. Having realized that in 2-D, there is no qualitative difference between a periodically forced rigidly rotating spiral and a meandering spiral (only the origin of the secondary rotation is different ), I decided to investigate the phenomenon of meander in three-dimensional scroll waves via periodic forcing.
In the simplest case, the rotationally symmetric scroll ring, the filament drifts along the rotational axis and shrinks without breaking the symmetry under periodic forcing. Adding to this a flower pattern of a few times the core size via forcing ( [IMAGE ], T1being the natural period) gives the expected overlay pattern. However, resonance drift ( [IMAGE ]) does not occur because the periodic forcing appears to amplify some of the natural fluctuations on the period, stopping the resonance drift after a while and leading to the collapse of the scroll ring. It is not clear whether this is a numerical or a genuine phenomenon.
For a twisted scroll ring, which has the same symmetry as a 2-D spiral wave, periodic forcing has the same qualitative effects perpendicular to the symmetry axis as on a spiral wave, plus a translation along the symmetry axis[3,4].
More recently, I have been looking at the collision and robustness of scroll waves. With resonant periodic forcing, it is possible to move a scroll wave locally in a direction normal to its filament. This can force scroll waves to collide. Viewing filaments as level contours of concentration fields or as kernels of continuous maps, the filaments generically only pass through each other in a certain way: anti-parallel filaments touch and re-connect in a common plane, changing the topological structure of the solution.
I have also started to investigate the bifurcations of spiral waves from meander to hyper-meander. While the Hopf-bifurcation from rigidly rotating spirals to meandering spirals does not change the underlying one-dimensional periodic wave-train structure and hence is clearly a two-dimensional effect, first numerical investigations of the bifurcation from meander to hyper-meander suggest that this bifurcation is accompanied by a change of the underlying periodic wave-train structure into a quasi-periodic wave train. It is not clear yet whether this one-dimensional effect causes the bifurcation to hyper-meander or is caused by the bifurcation.
There are several directions in which I aim to extend my research: Having studied the behavior of a twisted scroll ring under periodic forcing, I am interested in the behavior of less symmetric structures. Prime examples are several linked scroll rings and the trefoil knot. Winfree [6,7] reports loss of symmetry in these structures in the meandering regime of parameter space and some general squirming of the core regions. Now it is numerically possible to study the near-resonant behavior of these structures under periodic forcing, replacing Winfree's question `can these structures persist' with `can these structures be easily destroyed by near resonant forcing', and `what happens when some of these stable organizing centers collide'.
Another question I would like to look at is how easy it is to create initial conditions resulting in persistent scroll wave structures. In two dimensions, a wave front passing through a gap results in a persistent spiral-- antispiral pair. In three dimensions, the same setup would result in a scroll ring that persists only temporarily. Can persistent scroll wave structures be generated in a similarly simple fashion?
I am interested in participating in the 1998-99 program ``Mathematics in Biology'' because my work on spiral waves has important biological applications and because the 1997-98 program ``Applications of Dynamical Systems'' has awakened my interest in exploring the interface between Mathematics and Biology further. Especially the Spring quarter on dynamic models of ecosystems and epidemics promises to bring up interesting questions concerning pattern formation in systems with several space scales wich has many similarities with spiral waves where several time scales and space scales are involved in pattern formation.
More generally, I am interested in understanding mechanisms of pattern formation in excitable media and in modelling of similar physical and biological phenomena.