Neural mean-field models attempt to describe the electrical signals of networks of neurons at length scales beyond the extent of a given neuron. We investigate periodic solutions to a particular mean-field model of the cortex (Liley et. al, Network-Comp Neural (2002) 13, 67-113) with certain spatial symmetries. The solutions we study originate from Turing-Hopf bifurcations, where spatially homogeneous equilibria destabilize into periodic solutions with some spatial dependence. The detection of these bifurcations can be done simply from analysis of the (low dimensional) spatially homogeneous reduction of the equations, but studying the spatiotemporal solutions that develop must be done with the full PDE model. We will present our method, and its implementation using PETSc, for finding and continuing these solutions using distributed finite difference discretization of the model. With submillimetre discretization, the centimetre length scales we are interested in result in millions of unknowns to be solved.
DAESA, Differential-Algebraic Equations Structural Analyzer, is a Matlab tool for structural analysis of DAEs. It allows convenient translation of a DAE system into Matlab and provides a small set of easy-to-use functions. DAESA can analyze systems that are fully nonlinear, high-index, and of any order. It determines structural index, degrees of freedom, constraints, variables to be initialized, and suggests a solution scheme.
This tool constructs a block-triangular structure of the problem and prescribes a solution scheme exploiting this structure. DAESA also provides functions for displaying the original structure of the DAE and functions for displaying its coarse and fine block-triangularizations.
Joint work with Gary Tan (McMaster University) and John Pryce (Cardiff University).
As with ordinary differential equations (ODEs), delay differential equations (DDEs) generally have to be solved numerically. The standard approach to do this is to look at existing ODE methods and extend them to accomodate the delay terms. However there is a lot to consider when making this extension, especially when the delay is state-dependent. In this talk we will consider some of the issues that arise when numerically integrating delay equations. We will also look at an application to an age-structured population model consisting of the McKendrick partial differential equation and a state-dependent delay arising from a threshold condition.
This talk will present the under appreciated ideas of rigorous numerics as a means of solving problems in PDE analysis. The core concepts such as interval arithmetic, automatic differentiation and radii polynomial will be discussed and recent extensions using mesh adaptivity will be presented. A fully automated curve following code for a class of elliptic problems will be explained.