Evans functions for integral neural field equations with Heaviside firing rate function

In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model and a limiting case is shown to recover recent results of L. Zhang [On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16, (2003), pp.513-536.]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speed. Such fronts may be connected and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses.

'Danger Theory: The Link between AIS and IDS?'

Abstract We present ideas about creating a next generation Intrusion Detection System (IDS) based on the latest immunological theories. The central challenge with computer security is determining the difference between normal and potentially harmful activity. For half a century, developers have protected their systems by coding rules that identify and block specific events. However, the nature of current and future threats in conjunction with ever larger IT systems urgently requires the development of automated and adaptive defensive tools. A promising solution is emerging in the form of Artificial Immune Systems (AIS): The Human Immune System (HIS) can detect and defend against harmful and previously unseen invaders, so can we not build a similar Intrusion Detection System (IDS) for our computers? Presumably, those systems would then have the same beneficial properties as HIS like error tolerance, adaptation and self-monitoring. Current AIS have been successful on test systems, but the algorithms rely on self-nonself discrimination, as stipulated in classical immunology. However, immunologist are increasingly finding fault with traditional self-nonself thinking and a new 'Danger Theory' (DT) is emerging. This new theory suggests that the immune system reacts to threats based on the correlation of various (danger) signals and it provides a method of 'grounding' the immune response, i.e. linking it directly to the attacker. Little is currently understood of the precise nature and correlation of these signals and the theory is a topic of hot debate. It is the aim of this research to investigate this correlation and to translate the DT into the realms of computer security, thereby creating AIS that are no longer limited by self-nonself discrimination. It should be noted that we do not intend to defend this controversial theory per se, although as a deliverable this project will add to the body of knowledge in this area. Rather we are interested in its merits for scaling up AIS applications by overcoming self-nonself discrimination problems.