Five projects in pattern formation, fluid dynamics and computational neuroscience

This dissertation is a compilation of five projects, which span the fields of pattern formation, fluid dynamics and computational neuroscience. The first two chapters deal with spatio-temporal patterns that may arise in various spatially extended systems driven far from equilibrium. In the first, the interaction of dispersionless traveling waves with a long-wave mode is studied, revealing two distinct instabilities, one of the phase and the other of the amplitude of the traveling waves. The amplitude-driven instability can lead to localized pulses in some cases. In the second the effect of anisotropy in rotating convection is investigated using a phenomenological Swift-Hohenberg equation. Stability results from an amplitude-equation approach are used to explain spirals and targets seen in experiment. In the third chapter the question of fluid slip at a solid interface is addressed at a molecular scale using methods from solid-state physics and nonlinear dynamics theory. We propose a modification of the well-known Frenkel-Kontorova model based on results from molecular dynamics simulations. The resulting dynamics explain the phenomenon of slip as the motion of defects in the fluid layer adjacent to the solid. The fourth and fifth chapters enter the realm of computational neuroscience. The fourth chapter discusses how CA1 pyramidal neurons in the hippocampus actively integrate their two principal excitatory inputs. Extensive numerical simulations using NEURON with realistic morphological data reveals that dendritic sodium spikes may forward-propagate to the soma causing a somatic action potential if the inputs are sufficiently synchronous. The final chapter introduces a model of excitable integrate-and-fire neurons in a small-world network, a network with regular, local coupling and a number randomly placed, long-range connections. The ensemble average of the network dynamics exhibits a transition from persistent activity to failure as a function of the density of global connections. Patterns of activity can be periodic or chaotic depending on parameters and the network is bistable in the persistent regime.