Mathematical Biosciences Institute
Ohio State University
381 Jennings Hall
1735 Neil Ave
Columbus, OH 43210
614-688-3493
mrempe AT mbi.ohio-state.edu
I am currently a post-doctoral fellow in the Mathematical Biosciences Institute (MBI) at Ohio State University. I am working on several projects under the direction of David Terman.
Previously, I was a graduate student in the Engineering Sciences and Applied Mathematics at Northwestern University under the direction of David Chopp.
While in graduate school I worked with an applied mathematician (David Chopp) and an experimental neurophysiologist (Nelson Spruston) to carry out research in computational neuroscience. My research focused on developing spatially adaptive numerical methods for simulating electrical activity in neurons. Many neurons have a tree-like shape, with several slender branches extending out from a cell body. The governing set of equations for this system consists of a reaction-diffusion equation for the voltage across the cell membrane plus kinetics equations for the motion of ions through membrane channels. The most common numerical methods that have been developed to solve this system on branched structures are implicit methods. These methods are desirable for their numerical stability properties and robustness. However, a trade-off of this approach is the fact that the voltage update for the entire cell is coupled, meaning that every location in the cell is updated at each time step regardless of the amount of local activity. This is a significant issue particularly in neural simulations because electrical activity in neurons is often very localized in space, meaning that much efficiency can be gained if computations are focused only on those parts of the cell that are active. The numerical algorithm I developed allows for de-coupling the computation on distinct regimes of the cell structure by splitting apart the branches using a predictor-corrector scheme.
Once the computation is reduced to the level of branches instead of cells, spatial adaptivity is straightforward: the active regions of the cell are detected and computational effort is focused there, while effort is saved in other regions of the cell that are at or near rest. I applied this novel numerical method to reproduce several computational studies of hippocampal pyramidal cells originally performed in the Spruston lab. In each case the simulations performed with adaptivity were more efficient than the original simulations. Additionally, I used the algorithm to perform some novel simulations that would be difficult or impossible with previous numerical methods. Overall, I found that when using the adaptive algorithm the computational cost of a simulation scales with the amount of activity present, rather than the physical size of the system being simulated. For certain problems spatial adaptivity reduces the computation time by up to 80%.
One of the main ways that I'm applying my dissertation research now is in the study of starburst amacrine cells in the retina. These cells respond to a bar of light moving in only one direction (for instance, right-to-left, but not left-to-right) even though the cells themselves are spatially symmetric. Called directionally-selective cells, their mechanism for responding to light motion in only one direction is still not fully understood even though much experimental work has been done in this area.
Experimentalists in the neuroscience department of OSU have determined several key factors in this phenomena, but they recognize that a model could be very useful in elucidating some of the mechanisms. Previously, only a relatively simple model of the cells has been used to investigate the hypotheses of the experimentalists. Using the algorithm I developed during my graduate training, I am constructing a detailed, morphologically-accurate model of a network of these cells to investigate the mechanism of the directional response. In this case the shape of the cells is important since each cell receives input from the light signal everywhere on each of its branches, but only produces output from the tips. Therefore, single compartment models are not appropriate for these cells as a model must include the cell's detailed morphology to account for the phenomena of interest.
The main project that I am working on as a postdoc in the Mathematical Biosciences Institute is a study of sleep in the human brain. This is a collaboration with David Terman and Janet Best, both of the mathematics department at OSU. While sleep is a daily process for most of us, compared to other physiologic processes, it is relatively poorly understood. Through both animal and human studies, several of the brain regions that are important for sleep have been identified, but it is still not entirely clear how they interact to bring about the separate stages of sleep and wakefulness.
An important conceptual model for understanding the sleep/wake cycle is called the ``flip/flop'' model. In this model the regions of the brain that cause wakefulness oppose those that cause sleepiness so the system is stable in either state, but does not spend much time in-between sleep and wakefulness. This is called a flip-flop because the system quickly ``flips'' from one state into another, instead of gradually changing from one to the other. The same researchers also proposed a second flip/flop switch for REM sleep. This model is helpful conceptually, but it does not have a mathematical foundation. In an effort to better understand sleep dynamics, and the ``flip/flop'' model, we use reduced Hodgkin-Huxley style oscillators to model the activity of four brain regions: one that is active during sleep, one that is active during wakefulness, one that is active during REM sleep, and one that is active during non-REM sleep. Taking this approach, we can analyze the dynamics of the system using phase plane analysis. This type of analysis helps us to understand the possible underlying mechanisms for the sleep-wake system, including the sleep disorder narcolepsy.
Our model does a good job of matching many of the important features of the human sleep-wake cycle including the timing of normal sleep, the main features of the sleep disorder narcolepsy, and the dynamics of REM sleep. To see the correct behavior out of the model, we needed to make an additional assumption about the connections present between cell groups. This serves as a hypothesis that can be investigated by experimentalists. The construction of the model was not simply done to reproduce the features of the sleep cycle, but to help us gain insight and make predictions about how features of the sleep cycle arise from the interaction of a few relatively simple oscillators.
Rempe, M.J., Best, J. and Terman, D.
A Neurobiological Model of the Human Sleep/Wake Cycle.
Submitted (available as MBI Technical Report No. 72)
Rempe, M.J., Spruston, N., Kath, W.L.,Chopp, D.L.
Compartmental Neural Simulations with Spatial Adaptivity.
J. Comp. Neuro. 25:3 465-480 (2008)
Rempe, M.J. & W.L.,Chopp, D.L.
A Predictor-Corrector Algorithm for Reaction-Diffusion Equations
Associated with Neural Activity on Branched Structures.
SIAM J. Sci. Comput. 28:6 2139-2161 (2006)
Staab, P.L. & Rempe, M.J. & Kassoy, D.R.
Acoustic-Rotational Internal Flow Caused by
Transient Sidewall Mass Addition..
SIAM J. Applied Math. 65 587-617 (2005)
I enjoy teaching and I hope to continue to improve at it. While at Northwestern I taught incoming freshmen as part of Northwestern's EXCEL program. At OSU I have taught MATH 153. I will be teaching again at OSU during the spring quarter 2009, but I don't know which course yet. A copy of my teaching philosophy can be found here.