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In the fields of Mathematical Biology and Biomathematics, mathematicals models are designed and explored in order to better understand biological systems. My research interests involve mathematical models and computer simulations that describe the electrical behavior in nerve cells or neurons.

This has included, for example, analyzing models of nerve fibers or axons which are nonuniform in diameter, and models of dendrites with active spines nonuniformly distributed along the dendrites. These are modeled by systems of partial differential equations. The methods of study are analytical and computational. The behaviors discovered in such models include propagation speeds, propagation failure, and reversed propagation. Some results of my research were summarized in the articles: "Study of Propagation along Nonuniform Excitable Fibers" (with J. Bell) published in the Journal of Mathematical Biosciences, Vol.119: 169-203, (1994); and "Unique Wave Front for Dendritic Spines with the Nagumo Dynamics" also published in the Journal of Mathematical Biosciences, Vol.148: 205-225, (1998).

When collections of interacting nerve cells are considered, systems of nonlinear ordinary differential equations can be used to model the behavior of these neural networks. Similarly for patches of membranes of different types of neurons. The mathematical studies of these types of networks can be quantitative or qualitative. My work here uses dynamical systems and bifurcation theory and to study some qualitative properties of such networks. Sample papers are "Including a Second Inward Conductance in Morris and Lecar Dynamics" (with W. Gall) in the journal of Neurocomputing, Vol. 26-27: 131-136, (1999); "An Organizing Center for Planar Neural Excitability" (with W. Gall) in Neurocomputing, Vol. 32-33: 757-765, (2000); and "Networks Of Planar Neural Organizing Centers" (with W. Gall) also in the journal Neurocomputing, Vol. 44-46: 799-803, (2002).

In Miss K.M. Nabb's senior honor's project under my direction, a C

• computer program was used to simulate a network of thousands of neurons, modeled by a large system of differential equations, in order to investigate the formation of ocular dominance columns in the brain of many mammals. These ocular dominance columns organize patches of nerve cells in the visual cortex into "strips" that alternately react to input from only either the left or right eye.


• Issues and interesting problems that arise from my daily teaching are also of great interest to me. For example, students in my Math 181 Applied Basic Mathematics class and I found that some homework problems had more than one correct solution. Driven by curiosity, I studied the calculations involved, and was able to identify and rectify some confusion that results from the treatment of approximate numbers usually given in textbooks. If you are interested, see the article "A Closer Look at the Treatment of Approximate Numbers in Technical Mathematics - Do the Rules Really Work?" (with W. Gall) in PRIMUS: (Problems, Resources, and Issues in Undergraduate Mathematics), Vol. VIII, No. 3: 209-218, (1998). ​​​