SOCIAL AND ECOLOGICAL FACTORS INFLUENCING COLLECTIVE BEHAVIOR IN GIANT DANIO

A fundamental problem in biology is understanding how and why things group together. Collective behavior is observed on all organismic levels - from cells and slime molds, to swarms of insects, flocks of birds, and schooling fish, and in mammals, including humans. The long-term goal of this research is to understand the functions and mechanisms underlying collective behavior in groups. This dissertation focuses on shoaling (aggregating) fish. Shoaling behaviors in fish confer foraging and anti-predator benefits through social cues from other individuals in the group. However, it is not fully understood what information individuals receive from one another or how this information is propagated throughout a group. It is also not fully understood how the environmental conditions and perturbations affect group behaviors. The specific research objective of this dissertation is to gain a better understanding of how certain social and environmental factors affect group behaviors in fish. I focus on two ecologically relevant decision-making behaviors: (i) rheotaxis, or orientation with respect to a flow, and (ii) startle response, a rapid response to a perceived threat. By integrating behavioral and engineering paradigms, I detail specifics of behavior in giant danio Devario aequipinnatus (McClelland 1893), and numerically analyze mathematical models that may be extended to group behavior for fish in general, and potentially other groups of animals as well. These models that predict behavior data, as well as generate additional, testable hypotheses. One of the primary goals of neuroethology is to study an organism's behavior in the context of evolution and ecology. Here, I focus on studying ecologically relevant behaviors in giant danio in order to better understand collective behavior in fish. The experiments in this dissertation provide contributions to fish ecology, collective behavior, and biologically-inspired robotics.

Spectral graph theory with applications to quantum adiabatic optimization

In this dissertation I draw a connection between quantum adiabatic optimization, spectral graph theory, heat-diffusion, and sub-stochastic processes through the operators that govern these processes and their associated spectra. In particular, we study Hamiltonians which have recently become known as ``stoquastic'' or, equivalently, the generators of sub-stochastic processes. The operators corresponding to these Hamiltonians are of interest in all of the settings mentioned above. I predominantly explore the connection between the spectral gap of an operator, or the difference between the two lowest energies of that operator, and certain equilibrium behavior. In the context of adiabatic optimization, this corresponds to the likelihood of solving the optimization problem of interest. I will provide an instance of an optimization problem that is easy to solve classically, but leaves open the possibility to being difficult adiabatically. Aside from this concrete example, the work in this dissertation is predominantly mathematical and we focus on bounding the spectral gap. Our primary tool for doing this is spectral graph theory, which provides the most natural approach to this task by simply considering Dirichlet eigenvalues of subgraphs of host graphs. I will derive tight bounds for the gap of one-dimensional, hypercube, and general convex subgraphs. The techniques used will also adapt methods recently used by Andrews and Clutterbuck to prove the long-standing ``Fundamental Gap Conjecture''.