Nonlinear Dynamics of wave propagation in Heterogeneous Excitable Media[UNM Th 88]

Abstract

Excitable media is a generic term for a wide range of physical, chemical and biological systems that exhibit spontaneous formation of spatial patterns. Examples of such patterns include spiral waves in a two-dimensional medium and their generalization in a three-dimensional system, scroll waves. Under certain conditions, these waves may become unstable and break up, giving rise to spatiotemporal chaos. Controlling these patterns using low-amplitude external perturbations is of fundamental importance as these patterns are known to have critical functional consequences for vital biological systems, such as the heart. Specifically, spatial patterns of electrical excitation have been implicated in many life-threatening disturbances to the natural rhythm of the heart. Hence understanding the dynamics of these patterns is critical for developing safe and efficient clinical treatment for these disturbances. In this thesis we explore different aspects of the dynamics of spiral and scroll waves using both simple and realistic models of excitable media. Specifically, we study the dynamical evolution of these patterns upon their interaction with different kinds of heterogeneities in the medium. We also propose several low-amplitude control schemes to eliminate such patterns from an excitable medium. This thesis begins with a brief overview of various features of excitable systems in Chapter 1. In the first few sections, important concepts, terms and models that are used throughout the thesis are defined. This is followed by a brief discussion of the role of heterogeneities on spiral and scroll wave dynamics. Following this is a section with a detailed review of various lowamplitude chaos control schemes for spatially extended chaos in excitable media. In Chapter 2, we study the drift dynamics of spiral waves in the presence of different gradients using simple models of excitable media. The model parameters for which the spiral drifts to regions of lower and higher excitability are determined. Drift of a spiral wave to a region where it rotates faster is of special relevance as it suggests a possible mechanism for the onset of “mother-rotor” fibrillation. We discuss the possible mechanism underlying such anomalous drift. In Chapter 3, we discuss the conditions under which a pinned spiral can be unpinned using a high frequency wave-train in a simple model of excitable media. We then derive a relation between pacing period and the size of the obstacle. We also show that unpinning the spiral from an inexcitable obstacle becomes easier with the decrease of medium excitability. In Chapter 4 we study the breakup of an otherwise stable scroll wave in the presence of an inexcitable obstacle which does not extend throughout the medium. The scroll wave breaks up at the edge of the obstacle, where a transition from a quasi-two-dimensional propagation front to a fully three-dimensional spherical wave front occurs. In Chapter 5 we propose a non-global spatially extended low-amplitude chaos control scheme, using an array of control points. A travelling wave of control is simulated as the spatially separated array points are excited in a sequence. We find that, depending on wave velocity and spacing of the control points, the chaotic activity can be eliminated completely. Moreover our scheme is robust in the presence of heterogeneities. In Chapter 6 we apply sub-threshold stimuli, whose effect on a solitary wave propagating in an extended medium is negligible, on a system with spatiotemporally heterogeneous activity. Surprisingly, the signal which is not sufficient to excite a resting medium, fundamentally alters the recovery dynamics and terminates all activity in the medium. We determine model-independent generic conditions under which this effect can be observed. Finally we conclude in Chapter 7 with a summary of our results on the role of heterogeneities in the dynamics of excitable media, and how control of spatiotemporal patterns in these systems need to take into account the presence of such features.