Jon Borresen

FURTHER INVESTIGATION OF HYSTERESIS IN CHUA'S CIRCUIT

For a system to display bistable behavior (or hysteresis), it is well known that there needs to be a nonlinear component and a feedback mechanism. In the Chua circuit, nonlinearity is supplied by the Chua diode (nonlinear resistor) and in the physical medium, feedback would be inherently present, however, with standard computer models this feedback is omitted. Using Poincare first return maps, bifurcations for a varying parameter in the Chua circuit equations are investigated for both increasing and decreasing parameter values. Evidence for the existence of a small bistable region is shown and numerical methods are applied to determine the behavior of the solutions within this bistable region.

Oscillatory Threshold Logic

In the 1940s, the first generation of modern computers used vacuum tube oscillators as their principle components, however, with the development of the transistor, such oscillator based computers quickly became obsolete. As the demand for faster and lower power computers continues, transistors are themselves approaching their theoretical limit and emerging technologies must eventually supersede them. With the development of optical oscillators and Josephson junction technology, we are again presented with the possibility of using oscillators as the basic components of computers, and it is possible that the next generation of computers will be composed almost entirely of oscillatory devices. Here, we demonstrate how coupled threshold oscillators may be used to perform binary logic in a manner entirely consistent with modern computer architectures. We describe a variety of computational circuitry and demonstrate working oscillator models of both computation and memory.

Heteroclinic Switching in Coupled Oscillator Networks: Dynamics on Odd Graphs

We review some examples of dynamics displaying sequential switching for systems of coupled phase oscillators. As an illustration we discuss a simple family of coupled phase oscillators for which one can find robust heteroclinic networks between unstable cluster states. For N = 2 k+1 oscillators we show that there can be open regions in parameter space where the heteroclinic networks have the structure of an odd graph of order k; a class of graphs known from permutation theory. These networks lead to slow sequential switching between cluster states that is driven by noise and/or imperfections in the system. The dynamics observed is of relevance to modelling the emergent complex dynamical behaviour of coupled oscillator systems, e.g. for coupled chemical oscillators and neural networks.

Oscillations, Feedback and Bifurcations in Mathematical Models of Angiogenesis and Haematopoiesis

Angiogenesis and haematopoiesis are physiological processes which include the dynamics of tumour growth from a dormant to a malignant state. These dynamics involve many interacting oscillatory processes that operate on several scales of space and time. It is important to understand such processes as anti-angiogenic drugs can have both inhibitory and stimulatory effects on tumour growth. In this chapter, we consider ordinary and delay differential equations which model angiogenesis and haematopoiesis. By introducing a feedback mechanism, bifurcations involving critical points and limit cycles are investigated as parameter values are altered in the system. It is shown that the dynamics are history dependent and hysteresis is possible. MathematicaTM program files have also been listed so that the reader can reproduce the results listed in this chapter. The reader can also download working Mathematica programs from the web.

Community Structure and Multi-Modal Oscillations in Complex Networks

In many types of network, the relationship between structure and function is of great significance. We are particularly interested in community structures, which arise in a wide variety of domains. We apply a simple oscillator model to networks with community structures and show that waves of regular oscillation are caused by synchronised clusters of nodes. Moreover, we show that such global oscillations may arise as a direct result of network topology. We also observe that additional modes of oscillation (as detected through frequency analysis) occur in networks with additional levels of topological hierarchy and that such modes may be directly related to network structure. We apply the method in two specific domains (metabolic networks and metropolitan transport) demonstrating the robustness of our results when applied to real world systems. We conclude that (where the distribution of oscillator frequencies and the interactions between them are known to be unimodal) our observations may be applicable to the detection of underlying community structure in networks, shedding further light on the general relationship between structure and function in complex systems.

Josephson junction binary oscillator computing

The first modern computers were built at Manchester University in the late 1940s and amongst the principal components used were vacuum tube oscillators. Following the development of the transistor, such oscillator-based computers quickly became obsolete. Here, a novel application of superconducting Josephson Junction (JJ) technology based on neural dynamics is proposed. Neuron-type oscillators can be constructed from JJs and they can be connected together to form computer circuits in a manner entirely consistent with modern architectures. SIMetrix models of both computation and memory are presented and metrics are discussed. It has been estimated that JJs switch 100 times faster than their transistor counterparts and use 300 times less power (including the super-cooling). Utilizing the circuits described in this paper, it is also estimated that computers could be constructed using at least 10 times fewer components. Thus, theoretically these JJ-based computers could be up to 300,000 times more efficient than their CMOS counterparts.

Cdp-Choline Neuroprotection and Vascular Remodelling Via Irs-1 Mechanisms in Vascular Dementia

Evidence of reduced blood-brain barrier (BBB) integrity preceding other Alzheimers disease (AD) pathology provides a strong link between cerebrovascular pathology and AD. In animals models, amyloid-β peptide injected animals exhibited a commonality in perturbations of microvessels compared with those evident in AD brain (22) . It was suggested that amyloidogenesis promotes extensive neoangiogenesis leading to increased vascular permeability and subsequent hypervascularization in AD. In human patients hypervascularity was corroborated in a comparison of postmortem brain tissues from AD. Brain microvessels derived from patients with AD expressed numerous factors implicated in vascular activation and angiogenesis. Signaling cascades associated with vascular activation and angiogenesis were upregulated in AD-derived brain microvessels (21) . However, these newly formed blood vessels may be nonfunctional. All above provides a new paradigm for integrating vascular remodeling with the pathophysiology observed in AD (20) . Therefore, vascular activation hypothesis could be a novel, unexplored therapeutic target in AD. Background of our own recent investigations: In our recent study, we have, for the first time, demonstrated both a vascular protective, and proangiogenic effect of citicoline using in vivo and in vitro models (19) . Our data suggests a strong protective effect against the damaging process of excitotoxicity and hypoxia, similar to that experienced after acute ischaemic stroke. In regard to the possible mechanism our protein studies demonstrated that citicoline induced pERK1/2 expression, a key mitogenic signalling protein known to be involved in angiogenesis and generally stimulated by growth factors through interaction with their receptors (23) .