A mathematical framework for critical transitions: normal forms, variance and applications

Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find early-warning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast subsystem bifurcations up to codimension two. The theory is applied to several models: the Stommel-Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator-inhibitor switch from systems biology, a predator-prey system from ecology and to the Euler buckling problem from classical mechanics. For the Stommel-Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator-inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator-prey model explosive population growth near a codimension two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio

Novel outcome measures for clinical trials in cystic fibrosis

Cystic fibrosis (CF) is a common inherited condition caused by mutations in the gene encoding the CF transmembrane regulator protein. With increased understanding of the molecular mechanisms underlying CF and the development of new therapies there comes the need to develop new outcome measures to assess the disease, its progression and response to treatment. As there are limitations to the current endpoints accepted for regulatory purposes, a workshop to discuss novel endpoints for clinical trials in CF was held in Anaheim, California in November 2011. The pros and cons of novel outcome measures with potential utility for evaluation of novel treatments in CF were critically evaluated. The highlights of the 2011 workshop and subsequent advances in technologies and techniques that could be used to inform the development of clinical trial endpoints are summarized in this review