Arthur Sherman

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  Cross-currents between Biology and Mathematics on Models of Bursting
  Arthur Sherman
  I will trace the history of models for bursting, concentrating on square-wave bursters descended from the Chay-Keizer model for pancreatic beta cells. The model was originally developed on a biophysical and intutive basis but was put into a mathematical context by John Rinzel's fast-slow analysis. Rinzel also began the process of classifying bursting oscillations based on the bifurcations undergone by the fast subsystem, which led to important mathematical generalization by others. Further mathematical work, notably by Terman, Mosekilde and others, focused rather on bifurcations of the full bursting system, which showed a fundamental role for chaos in mediating transitions between bursting and spiking and between bursts with different numbers of spikes. The development of mathematical theory was in turn both a blessing and a curse for those interested in modeling the biological phenomena - having a template of what to expect made it easy to construct a plethora of models that were superficially different but mathematically redundant. This may also have steered modelers away from alternative ways of achieving bursting, but instructive examples exist in which unbiased adherence to the data led to discovery of new bursting patterns. Some of these had been anticipated by the general theory but not previously instantiated by Hodgkin-Huxley-based examples. A final level of generalization has been the addition of multiple slow variables. While often mathematically reducible to models with a one-variable slow subsystem, such models also exhibit novel resetting properties and enhanced dynamic range. Analysis of the dynamics of such models remains a current challenge for mathematicians.
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  Bistability of Beta Cell Mass in Diabetes
  Arthur Sherman

  Insulin is the master hormone that controls fuel usage by body tissues. After a meal, glucose is plentiful and stimulates insulin secretion, which allows muscle and fat to take up glucose. When blood glucose falls, insulin falls as well and tissues revert to using fat as a fuel. Obesity causes insulin resistance, meaning that more insulin is needed to produce a given amount of glucose uptake. If the number of insulin-secreting pancreatic beta cells, or secretion per cell, increases sufficiently, this excess demand for insulin can be met. If expansion of mass is inadequate, type 2 diabetes, a rise in glucose to levels that are harmful to cells, results. Diabetes leads to cardiovascular disease, blindness, kidney failure and premature death. We update the seminal model of Topp et al (J. Theor. Biol. 2001) for the regulation of beta-cell mass by glucose and present a comprehensive picture of how diabetes develops and may either be avoided or reversed. Although many details of the model are in doubt, we show that any successful model results in a bistable bifurcation structure, with normal and elevated glucose levels separated by a threshold. This simple picture unifies and explains a striking diversity of experimental data, including why prevention is much easier than cure and why bariatric surgery is able to reverse longstanding diabetes within a week.