|
Organizers |
Topological Approaches to the Analysis of Spatiotemporal Organization in Chemical and Biological Systems
by
Leon Glass
Department of Physiology, McGill University, Montreal, Quebec, Canada
Chemical and biological systems often display complex spatiotemporal patterns that can be affected by stimuli delivered directly to the system. In this talk, I will discuss the use of phase maps to help analyze and predict unexpected dynamic behavior in chemistry and biology. In a phase map, each point in a manifold is labelled by a phase, i.e. a point on the unit circle. I assume that the phase maps are continuous except at a finite number of singular points (or perhaps singular lines, planes or hyperplanes). I first consider phase maps on two dimensional, compact, oriented manifolds (perhaps containing holes) that are continuous except at a finite number of singular points. I define the indices of the holes and the singular points. The sum of the indices is zero [1]. This result is applicable to excitable media (such as the Belousov-Zhabotinsky chemical reaction, or cardiac tissue), and helps us to understand the geometry of spiral waves propagating in inhomogeneous spherical shells [2].
As originally emphasized by Art Winfree, phase maps can also be useful to think about the resetting of limit cycle oscillations. A state X, in the basin of attraction of a limit cycle oscillation and a state Y, on the limit cycle oscillation have the same asymptotic phase if the distance between trajectories starting at X and Y at t=0, approach each other in the limit as t goes to infinity. If a perturbation of fixed magnitude to a limit cycle oscillation always leaves one in the basin of attraction of the limit cycle independent of the phase of delivery of the perturbation, then from the continuity of the phase maps, the resulting resetting curves will be continuous. Hence, numerical and wet-lab experimental observations of discontinuities in resetting experiments pose problems for interpretation. I discuss discontiuities in phase resetting curves that arise from perturbations given excitable waves circulating on a one-dimensional ring [3, 4] and from perturbations delivered to spontaneous cardiac oscillations [5].
Although they might seem abstract, these results pertain to practical matters, such as the initiation and termination of potentially fatal cardiac arrhythmias. By emphasizing qualitative topological aspects, I focus on results that should be testable for broad classes of systmes independent of the detailed equations of motion.
[1] L. Glass. Patterns of supernumerary limb regeneration. Science 198, 321-322 (1977).
[2] J. Davidsen, L. Glass, R. Kapral. Topological constraints on spiral wave dynamics in spherical shells with inhomogeneous excitability. Physical Review E (2004, In Press).
[3] L. Glass, M.E. Josephson. Resetting and annihilation of reentrant abnormally rapid heartbeat. Physical Review Letters 75, 2059-2063 (1995).
[4] T. Gedeon, L. Glass. Continuity of resetting curves for FitzHugh-Nagumo equations on the circle. In: Fields Institute Communications: Differential Equations with Applications to Biology, 225-236 (1998).
[5] T. Krogh-Madsen, L. Glass, E. Doedel, M.R. Guevara. Apparent discontinuities in the phase-resetting response of cardiac pacemakers. Journal of Theoretical Biology (2004, In Press).
Date received: September 15, 2004
Copyright © 2004 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # capf-02.