Case Western Reserve University
Department of Mathematics, Applied Mathematics and Statistics (primary)
Department of Biology; Department of Cognitive Science
Department of Electrical Engineering and Computer Science (secondary)
Mathematical Modeling and Systems Biology: Mechanisms, Stochastic Phenomena, and Dimension Reduction
ABSTRACT: The discipline of systems biology arose through the application of large-scale computational and analytical methods, originating in systems engineering, to quantitative biology. In the talk I will discuss examples from my own work in which mathematical modeling contributes to understand biological systems. 1. In collaboration with neurophysiologist C. Wilson, we showed that the dynamical mechanism underlying robust generation of the respiratory rhythm by central pattern generator circuits in the brainstem may be fundamentally different in the open-loop case (corresponding to classic ex vivo studies) than in the closed-loop case (i.e. in vivo) [Diekman et al, 2017]. 2. In collaboration with theoretical biophysicist B. Lindner, we have shown how to adapt phase reduction methods for biological oscillators to the case of stochastic oscillators systems, such as arise in hybrid Markov models of randomly gating ion channels in electrically active membranes in nerve cells [Thomas and Lindner, 2014, Anderson et al 2015]. 3. In a different type of dimension reduction, neuroscientist R. Galan and I are analyzing stochastic shielding approximation methods for accurately simplifying stochastic network models. By identifying optimal projections within the sample space we find reduced complexity models of neural systems with minimal error along sample paths [Schmandt and Galan 2012, Schmidt and Thomas 2014, Schmidt et al 2018]. 4. Accurate complexity reduction methods for stochastic processes have broad potential application within systems biology, for instance in understanding signal transduction pathways. With information theorist A. Eckford, I pioneered the development of exactly solvable communications channel models specific to signal transduction [Thomas and Eckford 2016].
Anderson, David F., Bard Ermentrout, and Peter J. Thomas. "Stochastic
representations of ion channel kinetics and exact stochastic simulation
of neuronal dynamics." Journal of computational neuroscience 38.1
Diekman, Casey O., Peter J. Thomas, and Christopher G. Wilson. "Eupnea,
tachypnea, and autoresuscitation in a closed-loop respiratory control
model." Journal of Neurophysiology 118.4 (2017): 2194-2215.
Schmandt, Nicolaus T., and Roberto F. Galán. "Stochastic-shielding
approximation of Markov chains and its application to efficiently
simulate random ion-channel gating." Physical review letters 109.11
Schmidt, Deena R., and Peter J. Thomas. "Measuring edge importance: a
quantitative analysis of the stochastic shielding approximation for
random processes on graphs." The Journal of Mathematical Neuroscience
4.1 (2014): 6.
Schmidt, Deena R., Roberto F. Galán, and Peter J. Thomas. "Stochastic
Shielding and Edge Importance for Markov Chains with Timescale
Separation." in revision, 2018.
Thomas, Peter J., and Benjamin Lindner. "Asymptotic phase for stochastic
oscillators." Physical review letters 113.25 (2014): 254101.
Thomas, Peter J., and Andrew W. Eckford. "Capacity of a simple
intercellular signal transduction channel." IEEE Transactions on
information Theory 62.12 (2016): 7358-7382.
Thomas, Peter J., and Andrew W. Eckford. "Shannon capacity of signal
transduction for multiple independent receptors." Information Theory
(ISIT), 2016 IEEE International Symposium on. IEEE, 2016.
Joint work with
C. Diekman (New Jersey Institute of Technology)
A. Eckford (York University, Toronto, Canada)
R. Galan (Case Western Reserve University)
B. Lindner (Humboldt University, Berlin, Germany)
D. Schmidt (University of Nevada, Reno)
C. Wilson (Loma Linda University)