Travelling waves of activity have been experimentally observed in many neural systems. The functional significance of such travelling waves is not always clear. Elucidating the mechanisms of wave initiation, propagation and bifurcation may therefore have a role to play in ascertaining the function of such waves. Previous treatments of travelling waves of neural activity have focussed on the mathematical analysis of travelling pulses and numerical studies of travelling waves. it is the aim of this thesis to provide insight into the propagation and bifurcation of travelling waveforms in biologically realistic systems. There is a great deal of experimental evidence which suggests that the response of a neuron is strongly dependent upon its previous activity. A simple model of this synaptic adaptation is incorporated into an existing theory of strongly coupled discrete integrate-and-fire (IF) networks. Stability boundaries for synchronous firing shift in parameter space according to the level of adaptation, but the qualitative nature of solutions is unaffected. The level of synaptic adaptation is found to cause a switch between bursting states and those which display temporal coherence.

No description available.

No description available.

We consider integrodifference equations (IDEs), which are of the form N_{t+1}(x) = \int K(x-y)F(N_t(y))dy, where K is a probability distribution and F is a growth function. It is already known that for monotone growth functions, solutions of the IDE will have spreading speeds and are sometimes in the form of travelling waves. We are interested in the case where F has a stable 2-point cycle, namely for the Ricker function and the logistic function [May, 1975]. It was claimed in [Kot, 1992] that the solution of this IDE alternates between two profiles, all the while moving with a certain speed. However, simulations revealed that not only do the profiles alternate, but the solution is a succession of two travelling objects with different speeds. Using the theory from [Weinberger, 1982], we can prove the existence of two speeds and establish their theoretical formulas. To explain the succession of travelling objects, we relate to the concept of dynamical stabilization [Malchow, 2002].

In this thesis, we study a class of multi-stable reaction-diffusion systems used to model competing species. Systems in this class possess uniform stable steady states representing semi-trivial solutions. We start by considering a bistable, interaction, where the interactions are of classic “Lotka-Volterra” type and we consider a particular problem with relevance to applications in population dynamics: essentially, we study under what conditions the interplay of relative motility (diffusion) and competitive strength can cause waves of invasion to be halted and reversed. By establishing rigorous results concerning related degenerate and near-degenerate systems,we build a picture of the dependence of the wave speed on system parameters. Our results lead us to conjecture that this class of competition model has three “zones of response” in which the wave direction is left-moving, reversible and right-moving, respectively and indeed that in all three zones, the wave speed is an increasing function of the relative motility. Moreover, we study the effects of domain size on planar and non-planar interfaces and show that curvature plays an important role in determining competitive outcomes. Finally, we study a 3-species Lotka-Volterra model, where the third species is treated as a bio-control agent or a bio-buffer and investigate under what conditions the third species can alter the existing competition interaction.

Travelling wave fault locators (TWFL) have the possibility to get rid of the limitation of typical locators based on the 50Hz impedance. Their principles were invented in the early 1900's, but only recently became economically viable. Some TWFL devices are now commercialized.Despite the recent commercialization of TW fault locators, actual field experience of TWFL is hard to acquire and rarely presented in the literature. Due to this, most studies are based on simplified simulation models.Practical experience in the form of TW records are important. It is the basis to understand the practical difficulties of applying TWFL algorithms. It is also necessary to illustrate the simulations limitations, and to test algorithms on real records.The work performed in this thesis was supported by Siemens with the hope to develop TWFL devices based on the results. The aim of the work was first to acquire experience in the practical side of TWs and their recording in substations. Based on this practical experience, the second objective was to study a TWFL that includes a new method for wave detection: the pattern recognition algorithm (PRA). The practical experience was acquired with a measurement campaign performed in the Belgian transmission network, and with laboratory tests that reproduce the measurements of currents inside a substation.Fault records suitable to TW studies were acquired during the measurement campaign, and are analysed in this report. The fault records and the laboratory tests highlighted and characterized the impact of the substation measurements on the waves recorded. Modelling those measurement systems is shown to improve the accuracy of the simulation tools.This report also presents a theoretical development of the PRA. The algorithm is adapted to take into account the practical difficulties previously analysed. An applicable version of the algorithm is proposed and tested. The algorithm proposal provides a precision better than 300m when applied to the simulation models. This precision varies with the set of parameters used, with the line topology, and with the fault location. On the field record acquired, the algorithm provides the fault location with a precision of 110m.A prototype has been developed by Siemens, and some devices have been installed at the end of this thesis. The TW records that will be acquired by those prototypes will provide a significant help in continuing the work presented in this report. / Doctorat en Sciences de l'ingénieur et technologie / info:eu-repo/semantics/nonPublished

No description available.

No description available.

In this thesis, we study a class of reaction-diffusion equations of the form $\frac{\partial u}{\partial t} = \mathcal{L}u + \phi u - \tfrac{1}{k} u^{k+1}$ where $\mathcal{L}$ is the stochastic generator of a Markov process, $\phi$ is a function of the space variables and $k\in \mathbb{R}\backslash\{0\}$. An important example, in the case when $k > 0$, is equations of the FKPP-type. We also give an example from the theory of utility maximisation problems when such equations arise and in this case $k < 0$. We introduce a new representation, for the solution of the equation, as the optimal value of an optimal control problem. We also give a second representation which can be seen as a dual problem to the first optimisation problem. We note that this is a new type of dual problem and we compare it to the standard Lagrangian dual formulation. By choosing controls in the optimisation problems we obtain upper and lower bounds on the solution to the PDE. We use these bounds to study the speed of the wave front of the PDE in the case when $\mathcal{L}$ is the generator of a suitable Lévy process.

Η παρούσα εργασία ασχολείται με μεθόδους εύρεσης κυματικών λύσεων καθώς και λύσεων οδευόντων κυμάτων επί σειράς πολύ γνωστών μερικών διαφορικών εξισώσεων καθώς και με θεωρήματα μελέτης της ύπαρξης και της μοναδικότητας, ευστάθειας, ασυμπτωτικής συμπεριφοράς και μονοτονίας των ανωτέρω λύσεων. Θα περιοριστούμε σε μερικές ansatze μεθόδους εύρεσης κυματικών λύσεων, καθώς και στην ύπαρξη και μοναδικότητα ειδικών κατηγοριών κυματικών λύσεων. / --