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Biological processes occur naturally over a staggering range of temporal and spatial scales. Hearts beat several times a minute but cardiac myopathies take years to develop. Pancreatic endocrine cell calcium rises and falls on the order of seconds and several minutes, insulin (glucose) regulation, however, can take up to hours, and even varies with seasons. Neuronal membrane spikes fire in millisecond intervals but conditioned learning can take minutes or hours to develop, and remarkably, this memory can persist for years. To understand how heart tissue contracts and relaxes over several centimeters, we have to understand dynamics not only of the cell themselves but also of crucial subcellular - nanoscale - domains. I am interested in understanding the dynamical behavior of such biological processes through mathematical modeling. My research focuses on three prominent areas:
Glucose metabolism produces reactive oxygen species (ROS) as a natural by-product of respiration. As a corollary, glycemic stress produces oxidative stress (the un-neutralized, over production of ROS). However, OS is much more than simply a consequence of high glucose. In fact, OS is a central causal factor in the development of insulin resistance. Together with the observation that OS also impairs insulin secretion, this means that OS is not merely a consequence of diabetes. Rather, it is likely to be responsible for the very development of diabetes. We study various implications of this phenomenon. One of our hypotheses is the following. We have show in newly diagnosed diabetic patients that glucose control leads to recovery from OS. This ''dose-response'' relationship is nonlinear, with a well defined threshold. We have used this feature, the EC50 of the curve, to predict personalized targets of glucose control for a patient. We are now hoping to pursue the study of OS in diabetes, and more specifically, its use in the diagnostics of anti-diabetic treatment, in larger clinical trials. More generally, we believe that biomarkers of oxidative stress hold crucial information regarding diabetic management and therapy.
Beta-cells are the predominant functional units of the pancreas: insulin secreted from these cells regulates glucose metabolism. The release of insulin in beta-cells is a complex process that involves oscillatory electrical activity, strongly coupled to intracellular calcium and metabolic dynamics. Beta-cells are grouped together in irregular clusters in the pancreas, called the Islets of Langerhans. These islets are rich structures organized on several spatial scales: not only are the beta-cells capable of exchanging ions via gap junctions, intracellular sub-structures such as the endoplasmic reticuli are critical in shaping global behavior. I am currently developing multiscale models to understand how interaction across the ERs and gap junctions shapes the temporal behavior of membrane bursting. In related work, I am interested in the role that ion channels stochasticity plays in determining the excitability properties of beta-cells, and of islets.
For a detailed PDE model of myocyte calcium cycling that includes biophysically realistic models of pumps and exchangers see Higgins et al. (JTB, 2007). To account for the complexity of geometry of the sarcoplasmic reticulum, we derived a calcium bidomain model using homogenization theory (Goel et al., SIAM MMS, 2006).
Phase response curves (PRCs) and spatiotemporal properties of neuronal networks are studied Goel and Ermentrout (2002). Brain rhythms appear in Jensen et al.(2002) and Jensen et al. (2005). Input current observers feature in Goel and Robenack (2005) and Robenack and Goel (2007). A Limax network model of second-order conditioning is studied in Goel and Gelperin (2006).