Dimension Reduction for Heterogeneous Populations of Oscillators

Abstract

This dissertation discusses coarse-graining methods and applications for simulations of large heterogeneous populations of neurons. These simulations are structured as large coupled sets of ordinary differential equations describing the state evolution for many qualitatively similar, but quantitatively distinct individual units. In full generality, the direct coupling between these units is not all-to-all, but is mediated through a directed network. With sufficiently strong coupling and weak heterogeneity across the population, a common outcome for such simulations is synchronization. Here, the states of individual units, while not identical, can be neatly approximated by a smooth function of some latent independent variable. It is this smooth structure that we seek to exploit in this dissertation for both didactic and computational purposes. Briefly put, the polynomial chaos expansion (PCE) methods used in this dissertation are reminiscent of Fourier expansions, recast in a setting where the spatial domain is a parameter space rather than a physical space, and, as such, has an associated probability density.

After describing methods and introducing a common terminology in chapter 1, We examine several applications in the remaining chapters, each illustrating different computational benefits and concerns for PCE. The applications range in complexity from the simple Kuramoto coupled phase oscillator model in chapter 3 to a biophysically realistic simulation of circadian rhythms with up to 23 dynamic quantities per unit. For each, some coarse computational technique(s) are demonstrated, including the computation of fixed points and limit cycles, continuation of branches of these invariant features, and the delineation of resonance horns for forced oscillations.

It is our belief that such a coarse-graining approach is the correct way to summarize the state of coupled heterogeneous populations, and so must play a part in any scheme for efficiently simulating large populations.