ResearchI am a member of the Biomathematics and Biostatistics Working Group (BioM&S) in the Department of Mathematics and Statistics at the University of Guelph.
Mathematically my research is centred around dynamical systems models of biological or other processes. It is weighted toward the applied end of the mathematics spectrum, often concerned with implementation details and issues of making the mathematics accessible to biologists and engineers, issues which I feel could receive more attention from the applied mathematics community. My research spans a relatively wide set of application areas, from ion channels to robot path planning, and a relatively wide set of disciplines, including algorithm development, numerical analysis, dynamic programming, and bifurcation theory, but dynamics is a unifying theme throughout. I enjoy working closely with experimentalists helping solve the problems in which they are interested, but am also interested in the more mathematical issues which often surface from this research.
Here are descriptions of some of the projects on which I am or have recently been working.
Parameter range reduction for ODE modelsThe problem of identifying appropriate parameter values for nonlinear ODE models of some observed phenomenon is quite difficult. Rather than attempting to find optimal values of the parameters which minimize some measure of the difference between the noisy data and the model output, I have attacked this problem from the standpoint of trying to find ranges of the parameter values which give model output that is consistent with the data. I have introduced a novel class of linear multi-step methods called Cumulative Backward Differentiation Formulas which are used to discretize the system. This class of formulas has the advantage of preserving any monotonicity properties of the vector field with respect to the parameters. This idea is then used to reduce a priori ranges of the parameter values by removing extremal portions of the ranges which yield model output that is inconsistent with the data.
Bifurcation with SymmetryTogether with William Langford and Petko Kitanov, we have been studying the effects of symmetry, particularly the symmetry of identical oscillators, on the normal forms of bifurcations in such systems. The work is motivated by the problem of Huygen's clocks where two identical clocks, coupled weakly through vibrations in a wall, exhibit entrained, anti-phase motion after a short while.
Robot Path PlanningThe determination of appropriate safe paths for the movement of autonomous robots is a problem that has been studied for some time. But not much work has been done for fully dynamic environments where the location of targets and obstacles changes in time in unpredictable ways. Together with Dr. Simon Yang of the School of Engineering, we have developed a dynamic programming based algorithm for robot path planning in such an environment, and are continuing to extend and improve it.
Parameter estimation for Hodgkin-Huxley modelsHodgkin-Huxley models have been used extensively in the past 50 years to describe the behaviour of ion currents in neurons. The model has continued to be extremely useful when analyzing currents in whole cells, and especially useful as a succinct way of characterizing channel properties. Unfortunately, the parameter estimation methods employed for these models have not changed substantially since Hodgkin and Huxley's time. I have been using modern numerical optimization techniques to obtain better parameter estimates for these models from standard voltage-clamp data. In order for neurobiologists to take advantage of these methods, we have implemented them in a usable software package, NEUROFIT.
Multi-state ion channel modelsMulti-state models of ion channel gating are a generalization of the Hodgkin-Huxley model. Here the channel is assumed to have several closed states and one open state which allows the passage of ions. These models are quite general and can reproduce nearly any data. The difficult questions are how to choose an appropriate model from this class that is minimally complex to sufficiently describe the data at hand, and how to accurately fit such a model to the data. A fundamental understanding of how these models behave relative to Hodgkin-Huxley models or other models is helpful. We have recently submitted a paper which identifies some characteristics of these models.
Horizontal gene transferHorizontal gene transfer (HGT) is the movement of genes between different organisms (even across species) as opposed to the vertical transmission which takes place from parent to offspring. It is believed that HGT is an important factor in the rapid spread of antibiotic resistance among pathogenic bacteria. One of my recent joint papers models the ability of antibiotic resistance genes, which typically live on plasmids, to transfer horizontally to other cells that may be arrested in their growth due to the presence of antibiotics. Even if this transfer does not immediately make the recipient cell resistant to the antibiotic, in situations where the antibiotic is alternately present and absent, it can allow both an increase in overall plasmid abundance and the persistence of plasmid presence in situations where it otherwise would die out from the population.