I am a professor 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 been working.
Simple models of Climate Change
Energy balance models are simple models used to study climate. We have used such models to simulate changes in the Arctic climate, both paleoclimates and modern ones. The primary feature of these models is that they exhibit a pair of saddle-node bifurcations with a region of bi-stability. The models predict that as CO2 levels rise, the Arctic climate will move from a stable cold state, through a region where a cold and warm state are both stable, to a region where only a warm state is possible. Further, the models exhibit hysteresis, so that once the single warm state is achieved, CO2 levels must be brought back down much farther before the cold state is again realized. Thus they predict that if we allow CO2 levels to rise too much, we will end up with an year-round ice-free Arctic which we will not be able to return to a cold state for at least hundreds of years.
Modelling in general. I like to work with other researchers helping them model and simulate things that they are studying. The topics can be quite diverse and unrelated. Some of the projects I have worked on include: drain heat-recovery devices, cat-bladder volume determination from ultrasound images, automatic detection of peaks in NMR data, E. coli contamination of ground beef in large processing plants, reconstructing fish exploitation amounts from fishers diary data, and influenza in a farrow-to-finish swine farm.
Parameter range reduction for ODE models
The 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 Symmetry
Together 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.
Parameter estimation for Hodgkin-Huxley models
Hodgkin-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.