Adaptive stochastic continuation with a modified lifting procedure applied to complex systems.

Many complex systems occurring in the natural or social sciences or economics are frequently described on a microscopic level, e.g., by lattice- or agent-based models. To analyze the states of such systems and their bifurcation structure on the level of macroscopic observables, one has to rely on equation-free methods like stochastic continuation. Here we investigate how to improve stochastic continuation techniques by adaptively choosing the parameters of the algorithm. This allows one to obtain bifurcation diagrams quite accurately, especially near bifurcation points. We introduce lifting techniques which generate microscopic states with a naturally grown structure, which can be crucial for a reliable evaluation of macroscopic quantities. We show how to calculate fixed points of fluctuating functions by employing suitable linear fits. This procedure offers a simple measure of the statistical error. We demonstrate these improvements by applying the approach in analyses of (i) the Ising model in two dimensions, (ii) an active Ising model, and (iii) a stochastic Swift-Hohenberg model. We conclude by discussing the abilities and remaining problems of the technique.

The effect of the G1-S transition checkpoint on an age structured cell cycle model.

Knowledge of how a population of cancerous cells progress through the cell cycle is vital if the population is to be treated effectively, as treatment outcome is dependent on the phase distributions of the population. Estimates on the phase distribution may be obtained experimentally however the errors present in these estimates may effect treatment efficacy and planning. If mathematical models are to be used to make accurate, quantitative predictions concerning treatments, whose efficacy is phase dependent, knowledge of the phase distribution is crucial. In this paper it is shown that two different transition rates at the G1-S checkpoint provide a good fit to a growth curve obtained experimentally. However, the different transition functions predict a different phase distribution for the population, but both lying within the bounds of experimental error. Since treatment outcome is effected by the phase distribution of the population this difference may be critical in treatment planning. Using an age-structured population balance approach the cell cycle is modelled with particular emphasis on the G1-S checkpoint. By considering the probability of cells transitioning at the G1-S checkpoint, different transition functions are obtained. A suitable finite difference scheme for the numerical simulation of the model is derived and shown to be stable. The model is then fitted using the different probability transition functions to experimental data and the effects of the different probability transition functions on the model's results are discussed.

Participatory development and analysis of a fuzzy cognitive map of the establishment of a bio-based economy in the Humber region.

Fuzzy Cognitive Mapping (FCM) is a widely used participatory modelling methodology in which stakeholders collaboratively develop a 'cognitive map' (a weighted, directed graph), representing the perceived causal structure of their system. This can be directly transformed by a workshop facilitator into simple mathematical models to be interrogated by participants by the end of the session. Such simple models provide thinking tools which can be used for discussion and exploration of complex issues, as well as sense checking the implications of suggested causal links. They increase stakeholder motivation and understanding of whole systems approaches, but cannot be separated from an intersubjective participatory context. Standard FCM methodologies make simplifying assumptions, which may strongly influence results, presenting particular challenges and opportunities. We report on a participatory process, involving local companies and organisations, focussing on the development of a bio-based economy in the Humber region. The initial cognitive map generated consisted of factors considered key for the development of the regional bio-based economy and their directional, weighted, causal interconnections. A verification and scenario generation procedure, to check the structure of the map and suggest modifications, was carried out with a second session. Participants agreed on updates to the original map and described two alternate potential causal structures. In a novel analysis all map structures were tested using two standard methodologies usually used independently: linear and sigmoidal FCMs, demonstrating some significantly different results alongside some broad similarities. We suggest a development of FCM methodology involving a sensitivity analysis with different mappings and discuss the use of this technique in the context of our case study. Using the results and analysis of our process, we discuss the limitations and benefits of the FCM methodology in this case and in general. We conclude by proposing an extended FCM methodology, including multiple functional mappings within one participant-constructed graph.

Modelling and detecting tumour oxygenation levels.

Tumours that are low in oxygen (hypoxic) tend to be more aggressive and respond less well to treatment. Knowing the spatial distribution of oxygen within a tumour could therefore play an important role in treatment planning, enabling treatment to be targeted in such a way that higher doses of radiation are given to the more radioresistant tissue. Mapping the spatial distribution of oxygen in vivo is difficult. Radioactive tracers that are sensitive to different levels of oxygen are under development and in the early stages of clinical use. The concentration of these tracer chemicals can be detected via positron emission tomography resulting in a time dependent concentration profile known as a tissue activity curve (TAC). Pharmaco-kinetic models have then been used to deduce oxygen concentration from TACs. Some such models have included the fact that the spatial distribution of oxygen is often highly inhomogeneous and some have not. We show that the oxygen distribution has little impact on the form of a TAC; it is only the mean oxygen concentration that matters. This has significant consequences both in terms of the computational power needed, and in the amount of information that can be deduced from TACs.

Spontaneous emergence of long-range shape symmetry.

Self-organization of matter is essential for natural pattern formation, chemical synthesis, as well as modern material science. Here we show that isovolumetric reactions of a single organometallic precursor allow symmetry breaking events from iron nuclei to the creation of different symmetric carbon structures: microspheres, nanotubes, and mirrored spiraling microcones. A mathematical model, based on mass conservation and chemical composition, quantitatively explains the shape growth. The genesis of such could have significant implications for material design.