From patterns to processes: Phase and density dependencies in the Canadian lynx cycle

Across the boreal forest of North America, lynx populations undergo 10-year cycles. Analysis of 21 time series from 1821 to the present demonstrates that these fluctuations are generated by nonlinear processes with regulatory delays. Trophic interactions between lynx and hares cause delayed density-dependent regulation of lynx population growth. The nonlinearity, in contrast, appears to arise from phase dependencies in hunting success by lynx through the cycle. Using a combined approach of empirical, statistical, and mathematical modeling, we highlight how shifts in trophic interactions between the lynx and the hare generate the nonlinear process primarily by shifting functional response curves during the increase and the decrease phases.

From cell death to embryo arrest: Mathematical models of human preimplantation embryo development

Human preimplantation embryos exhibit high levels of apoptotic cells and high rates of developmental arrest during the first week in vitro. The relation between the two is unclear and difficult to determine by conventional experimental approaches, partly because of limited numbers of embryos. We apply a mixture of experiment and mathematical modeling to show that observed levels of cell death can be reconciled with the high levels of embryo arrest seen in the human only if the developmental competence of embryos is already established at the zygote stage, and environmental factors merely modulate this. This suggests that research on improving in vitro fertilization success rates should move from its current concentration on optimizing culture media to focus more on the generation of a healthy zygote and on understanding the mechanisms that cause chromosomal and other abnormalities during early cleavage stages.

Neural codes: Firing rates and beyond

Computational neuroscience has contributed significantly to our understanding of higher brain function by combining experimental neurobiology, psychophysics, modeling, and mathematical analysis. This article reviews recent advances in a key area: neural coding and information processing. It is shown that synapses are capable of supporting computations based on highly structured temporal codes. Such codes could provide a substrate for unambiguous representations of complex stimuli and be used to solve difficult cognitive tasks, such as the binding problem. Unsupervised learning rules could generate the circuitry required for precise temporal codes. Together, these results indicate that neural systems perform a rich repertoire of computations based on action potential timing.

The operator algebra approach to quantum groups

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.