Whether you're trying to hold together a social network on Facebook or looking for ways to treat a medical problem, mathematical modeling may be able to help find the best solution.
Assistant Professor Juan Restrepo teaches courses in mathematical modeling and numerical analysis while conducting research on the dynamics of complex networks. Among his projects are the modeling of heart and brain cell behavior, which constitute their own dynamic networks, and a study of complex networks aimed at improving strategies to predict their collapse.
Restrepo joined the Department of Applied Mathematics in fall 2008, after completing a post-doctoral research position at Northeastern University where he focused on cardiac dynamics. He created a mathematical model and computer simulation that attempted to explain electrical "alternans," a condition that shows up in a heart patient's electrocardiogram as an alternate-beat variation in the amplitude of the cardiac wave. Alternans is associated with higher risk for sudden cardiac death, so it is a significant area of study among biologists and medical researchers.
Applied mathematicians can play a helpful role through modeling techniques, Restrepo says, by reducing a situation to its core essentials so that the cause and consequences of a condition can more easily be discovered. His research on cardiac dynamics, for example, demonstrated that certain proteins that regulate the concentration of calcium and its stability are important factors in alternans. Graduate student Sebastian Skardal is continuing this work on the mathematical aspects of alternans in cardiac tissue at CU-Boulder.
While that project has specific parameters that make it difficult to generalize information about other complex systems, a separate project focusing on the brain cells of rats can be applied to other physical phenomena, Restrepo says. In this project, which is underway currently with graduate student Daniel Larremore, Restrepo is modeling a set of physical experiments being done at the National Institutes of Health, in which the researcher is measuring the electrical activity of brain circuits in vitro to see how they process information.
Restrepo and Larremore are trying to construct a theory that can describe the response of the neuron network to an external stimulus—or that of any complex network of coupled excitable systems—and the dynamic processes that take place on them.
Similarly, his research with graduate student Dane Taylor into the robustness of complex networks and what happens when certain nodes, or connections, are removed could be helpful in developing strategies to prevent the collapse of social or infrastructure networks.
Restrepo notes that the study of networks is an interdisciplinary field where applied mathematicians, physicists, computer and social scientists, and biologists can all make contributions. Two undergraduate students, Kyla Maletsky and Marshall Carpenter, are also working on related projects.