The dynamics of neuronal networks are often modeled using Integrate-and-Fire (IF) neurons, consisting of an ODE to evolve the membrane potential and a spiking/reset mechanism.   Perhaps the simplest of these is a current based IF model, where each neuron behaves as an RC circuit.  When coupled in a network, these neurons synchronize their firing times within a large range of model parameters.  This model provides an ideal situation to study the underlying mechanism responsible for synchrony in the presence of noisy input.  For my thesis, I characterized the competition between the desynchronizing noisy input to each neuronal voltage and the synchronizing instantaneous pulse-coupling between the neurons in the network.

A random packed state of macroscopic particles can be described by its local statistical quantities (i.e. neighbors, contacts and cell volumes) as well as by its global thermodynamic-like quantities (i.e. entropy and compactivity).  We use positional disorder in a `granocentric’ model to efficiently generate local cells by Monte Carlo simulation that accurately reproduce the statistical distributions of the local quantities from experimental packings of spheres.  Generating accurate local cell statistics allows global quantities like entropy and compactivity to be calculated directly, without assumptions for the statistical distributions or for reference values. 

I am interested in transition times between metastable states in stochastic systems.  It is understood that  in over damped gradient systems the mean transition time varies exponentially with the inverse temperature and energy barrier height.  We have found similar scalings for non-gradient systems in the absence of an energy to serve as a Lyapunov function as well as in deterministic systems where "dynamical bottlenecks" causes metastability.  

Micromagnetic systems under the influence of spin-transfer torques cannot be written as a gradient system, however they do operate in an underdamped regime where the energy changes on a long time scale as compared to the period of nearly Hamiltonian orbits.  We showed that an averaged equation describing the diffusion of energy on a graph captures the low-damping dynamics of these systems. From this equation we obtained the bifurcation diagram of the magnets, including the critical currents to induce stable precessional states and magnetization switching, as well as give a theoretical basis for a N\'eel-Brown-type formula with an effective energy barrier for the thermally assisted magnetization reversal times.

Spatially extended micromagnetic systems can be modeled by a Hamiltonian partial differential equation system.  In this deterministic system, metastability is observed due to dynamical bottlenecks; the system spends a long time exploring a large region of phase-space before finding its way through a narrow passage way to another large region.  Although the evolution in time is deterministic, weak solutions can be arbitrarily rough as a function of the spatial position.  A large amount of energy is stored in rough functions, causing the energy to behave as an extensive variable that scales with N, the size of the discretized system.   By defining a temperature parameter to quantify the roughness of an initial condition, we find the mean time between transitions can be written in terms of the exponential of the inverse of this temperature parameter and the energy barrier height.