Networks of simple spiking neurons provide abstract models for studying the dynamics of biological neural tissue. At the expense of cellular-level complexity, they are a frame-work in which we can gain a clearer understanding of network-level dynamics. Substantial insight can be gained analytically, using methods from stochastic calculus and dynamical systems theory. This can be complemented by data generated from computational simulations of these models, most of which benefit easily from parallelisation. One cubic millimetre of mammalian cortical tissue can contain between fifty and one-hundred thousand neurons and display considerable homogeneity. Mammalian cortical tissue (or grey matter") also displays several distinct firing patterns which are widely and regularly observed in several species. One such state is the "input-free" state of low-rate, stochastic firing. A key objective over the past two decades of modelling spiking-neuron networks has been to replicate this background activity state using "biologically plausible" parameters. Several models have produced dynamically and statistically reasonable activity (to varying degrees) but almost all of these have relied on some driving component in the network, such as endogenous cells (i.e. cells which spontaneously fire) or wide-spread, randomised external input (put down to background noise from other brain regions). Perhaps it would be preferable to have a model where the system itself is capable of maintaining such a background state? This a functionally important question as it may help us understand how neural activity is generated internally and how memory works. There has also been some contention as to whether driven" models produce statistically realistic results. Recent numerical results show that there are connectivity regimes in which Self-Sustained, Asynchronous, Irregular (SSAI) firing activity can be achieved. In this thesis, I discuss the history and analysis of the key spiking-network models proposed in the progression toward addressing this problem. I also discuss the underlying constructions and mathematical theory from measure theory and the theory of Markov processes which are used in the analysis of these models. I then present a small adjustment to a well known model and provide some original work in analysing the resultant dynamics. I compare this analysis to data generated by simulations. I also discuss how this analysis can be improved and what the broader future is for this line of research.

In the approximation and solution of both ordinary and partial differential equations by finite difference equations, it is well-known that for different ratios of the time interval to the spatial intervals widely differing solutions are obtained. This problem was first attacked by John von Neumann using Fourier analysis. It has also been studied in the context of the theory of semi-groups of operators. It seemed that the problem could be studied with profit if set in a more abstract structure. The concepts of the stability of a linear operator on a (complex) Banach space and the stability of a Banach sub-algebra of operators were formed in an attempt to generalize the matrix 2 theorems of H.O. Kreiss as applied to the L² stability problem. Chapter 1 deals with the stability and strict stability of linear operators. The equivalence of stability and convergence is discussed in Chapter 2 and special cases of the Equivalence Theorem are considered in Chapters 3 and 4. In Chapter 5 a brief account of the theory of discretizations is given and used to predict instability in non-linear algorithms.

Includes bibliographical references. / We present a systematic development and application of Geometric Algebra, an extended vector calculus. The entire algebraic structure, which is a graded Clifford algebra, is developed. To illustrate the derived results, examples are given for two and three dimensions. Here it becomes clear, how rotations and Lorentz boosts can be formulated in the Geometric Algebra. Further we realize that the Geometric Algebra contains elements, which can be used as representations of the complex unit. Having derived the necessary tools, we turn our attention to physics. We give applications to classical mechanics, quantum mechanics, ï¬ eld theory, curved manifolds, electromagnetism, and gravity as a gauge theory.

Bibliography: pages 179-184. / Singular perturbation methods are used to obtain amplitude equations for the parametrically driven damped linear and nonlinear oscillator, the linear and nonlinear Klein-Gordon equations in the small-amplitude limit in various frequency regimes. In the case of the parametrically driven linear oscillator, we apply the Lindstedt-Poincare method and the multiple-scales technique to obtain the amplitude equation for the driving frequencies Wdr ~ 2ω₀,ω₀, (2/3)ω₀ and (1/2)ω₀. The Lindstedt-Poincare method is modified to cater for solutions with slowly varying amplitudes; its predictions coincide with those obtained by the multiple-scales technique. The scaling exponent for the damping coefficient and the correct time scale for the parametric resonance are obtained. We further employ the multiple-scales technique to derive the amplitude equation for the parametrically driven pendulum for the driving frequencies Wdr ~ 2ω₀, ω₀, (2/3)ω₀, (1/2)ω₀ and 4ω₀. We obtain the correct scaling exponent for the amplitude of the solution in each of these frequency regimes.

Bibliography: pages 82-83. / In most implementations of the finite element method for the solution of contact problems the model of friction used is the classic Amontons-Coulomb. This dissertation is an attempt to rectify the current situation by considering four more advanced friction models, and coding them in FORTRAN for use with the finite element program ABAQUS. The new models are: a quasi-steady-state sliding model proposed by Zhang, Moslehy and Rice; a nonlinear pressure-dependent model proposed by Wriggers, vu Van and Stein; and a model that includes a film of lubricant proposed by Wilson, Hsu and Huang. The friction models are described in detail, including the algorithmic implementation. The contact problem is then formulated in the Total Lagrangian and Updated Lagrangian formulations for contact between an elastic-plastic (Mises plasticity) body and a rigid tool. The variational (weak) form of the formulation is given and this is then discretised by the finite element method. To test and compare the models three common metal forming processes are simulated: hemispherical punching of a disk, two-dimensional plane strain and three-dimensional cold rolling of a strip, and axisymmetric cup deep-drawing. The results are presented in the form of contour plots of the second invariant of stress (Mises), and the plastic yield and maximum stress. Also graphs for the thickness strain are given. These results are presented for each combination of friction model and process to allow easy comparison of frictional behaviour.

Bibliography: pages 93-101. / The main aim of this thesis is to analyse two types of general finite element approximations to the solution of a time-dependent variational inequality. The two types of approximations considered are the following: 1. Semi-discrete approximations, in which only the spatial domain is discretised by finite elements; 2. fully discrete approximations, in which the spatial domain is again discretised by finite elements and, in addition, the time domain is discretised and the time-derivatives appearing in the variational inequality are approximated by backward differences. Estimates of the error inherent in the above two types of approximations, in suitable Sobolev norms, are obtained; in particular, these estimates express the rate of convergence of successive finite element approximations to the solution of the variational inequality in terms of element size h and, where appropriate, in terms of the time step size k. In addition, the above analysis is preceded by related results concerning the existence and uniqueness of the solution to the variational inequality and is followed by an application in elastoplasticity theory.

This dissertation consists of two parts. They are separate ideas, but both fall into the context of General Relativity using dynamical systems. Part one is titled Cyclic Universes. It is shown that a Friedmann model with positive spatial sections and a decaying dark energy term admits cyclic solutions which is shown graphically by the use of phase planes. Coupling the modified Friedmann model to a scalar field model with cross-sectional terms in order to model the reheating phase in the early universe, it is found that there is a violation of the energy condition, i.e. when the universe is in the contracting phase and re-collapses again. We suspect that the cause for this violation is due to the asymmetry of the solution of w together with the cross-sectional terms at the bounce preceding slow-roll inflation. Part two is titled Thought Experiment to Directly Detect Cosmic Expansion by Holonomy. Two thought experiments are proposed to directly measure the expansion of the universe by the parallel transfer of a vector around a closed loop in a curved spacetime. Generally, expansion would cause a measurable deficit angle between the vector’s initial and final positions. Using the McVittie spacetime (which describes a spherically symmetric object in an expanding universe) as a backdrop to perform these experiments it is shown that the expansion of the universe can be directly detected by measuring changes in the components of a gyroscopic spin axis. We find these changes to be small but large enough (∆S ∼ 10−7 ) to be measured if the McVittie spacetime were a representation of our universe.

In the first part of the work, we show a general relation between the spatially disjoint product of probability density functions and the sum of their Fisher information metric tensors. We then utilise this result to give a method for constructing the probability density functions for an arbitrary Riemannian Fisher information metric tensor given its associated Nash embedding. We note further that this construction is extremely unconstrained, depending only on certain continuity properties of the probability density functions and a select symmetry of their domains. In the second part of the work, with the aim of understanding the global, algebrao-topological nature of information manifolds, we exhibit some of the necessary category theory required to effect a discussion of homological algebra in a general setting.

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