Coming together individuals at different scales working for a common goal

Abstract

Patterns in biology, chemistry, physics and mathematics can occur from self-organization and the interaction of constituents. In this thesis defense, I will explore patterns in two very different systems: (i) “chimera states” in a biologically-relevant model of excitable tissue, namely a modified version of the FitzHugh-Nagumo model, and (ii) collective motion of living many-agent systems such as swarms of brine shrimp. The FitzHugh-Nagumo model is a simple dynamical system that adequately describes many phenomena in excitable biological systems, such as firing neurons. The excitability is modeled via cubic terms added to the otherwise linear differential equations that describe the time evolution of two dependent variables that characterize the state of a cell. When many of these cells are then coupled in space, the model results in either a stable fixed point or a stable limit cycle which describes synchronized oscillating cells. However, chimera states in which stable fixed-point and limit-cycle regions coexist are not described within this model, even though they are observed in the heart and the brain. By adding a 5th order term in the membrane potential to this 3rd order system, we can recover chimeras, dependent on only initial conditions of the cells. Chimeras have previously been shown in systems with non-local coupling. Interestingly, however, they appear in this new system with purely local coupling. We study the dynamics of these chimeras in a few situations: in 1-dimensional cables and rings with two different simultaneous dynamics and in 2-dimensional grids representing tissues. Switching gears, I then discuss the patterns that occur in swarming, a self-organization phenomenon exhibited in many biological systems such as flocks of bird and insect, schools of fish, and collections of bacteria. This sort of behavior emerges spontaneously, arising without any sort of centralized control or leadership. Many crustaceans such as brine shrimp produce swarms, in which individuals cluster together rather than spread out uniformly in their environment. The size and distribution of these swarms are governed by local interactions between individuals. We will discuss the three-dimensional patterns that can be observed in brine shrimp swarms, specifically of the Great Salt Lake strain of Artemia franciscana, at high concentration. These patterns can be easily observed with simple tabletop experiments. We experimentally test the effects of certain environmental conditions on the dynamics of the individuals and on the development of these swarms.