Peter J. Thomas

Assistant Professor in the Department of Mathematics
with secondary appointments in the Department of Biology
and the Department of Cognitive Science.

Office: Yost 212A (216-368-3623)

Lab: Yost 212B (216-368-8710)

E-mail: pjthomas( AT )case.edu

   
B.A., Yale University, 1990 (Physics and Philosophy)
M.S., The University of Chicago, 1994 (Mathematics)
M.A., The University of Chicago, 2000 (Conceptual Foundations of Science)
Ph.D., The University of Chicago, 2000 (Mathematics)
 

Research: Mathematical Biology, Theoretical Neuroscience, Computational Cell Biology.
Google Scholar Profile
Curriculum vitae
Research, Teaching, and Service Statements

Links:

Teaching: (Fall 2006) MATH 224: Differential Equations
(Spring 2007) MATH 319: Applied Probability and Stochastic Processes for Biology (cross-list: BIOL 319/419, EECS 319)
(Spring 2007) MATH 342: Preparation for Research in Mathematical Biology (cross-list: BIOL 309)
(Spring 2008) MATH 223: Calculus for Science and Engineering III
(Spring 2008) MATH 378/478: Computational Neuroscience (cross-list: BIOL 378/478, COGS 378, NEUR 478, EBME 478, EECS 478)
(Fall 2008) MATH 223: Calculus for Science and Engineering III
(Fall 2008) MATH 319: Applied Probability and Stochastic Processes for Biology (cross-list: BIOL 319/419, EECS 319, PHOL 419, EBME 419)
(Fall 2009) MATH 223: Calculus for Science and Engineering III
(Spring 2010) MATH 378/478: Computational Neuroscience (cross-list: BIOL 378/478, COGS 378, NEUR 478, EBME 478, EECS 478)
(Spring 2010) MATH 435: Ordinary Differential Equations
(Fall 2010) MATH 223: Calculus for Science and Engineering III. Meets MWF 2-2:50 p.m. in Bingham 305.
(Fall 2010) MATH 321/421: Fundamentals of Analysis I. Meets WF 12:30-1:45 p.m. in Bingham 305.
(Spring 2011) MATH 319: Applied Probability and Stochastic Processes for Biology (cross-list: BIOL 319/419, EECS 319, PHOL 419, EBME 419) Meets MWF 2:00-2:50, location TBA.
(Spring 2011) MATH 322/422: Fundamentals of Analysis II. Meets WF 12:30-1:45 p.m. in Yost 101.
(Fall 2011) Pretenure teaching release.
(Spring 2012) MATH 224: Differential Equations. Meets MWF 11:30-12:20 in Bingham 304. Here is the google docs page for the course.
(Spring 2012) MATH 378/478: Computational Neuroscience (cross-list: BIOL 378/478, COGS 378, NEUR 478, EBME 478, EECS 478). Meetings MWF 2:00-2:50 in Bingham 204. Here is the google docs page for the course.
(Fall 2012) MATH 223: Calculus for Science and Engineering III. Meets MWF 10:30-11:20 in Bingham 304.
(Fall 2012) MATH 441: Mathematical Modeling. Meets MWF 2:00-2:50 in Bingham 304.
(Spring 2013) MATH 224: Differential Equations. Meets MWF 11:30-12:20 in Clapp 305.
(Spring 2013) MATH 378/478: Computational Neuroscience (cross-list: BIOL 378/478, COGS 378, NEUR 478, EBME 478, EECS 478). Meetings MW 2:00-2:50 in Clapp 305 and F 2:00-2:50 in Clapp 304.

Office Hours: (Spring 2013) Tuesdays 11:00-noon. Wednesdays 10:15-11:15. Fridays 12:45-1:45. Or by appointment.

Research Interests & Publications

* denotes undergraduate coauthors

Stochastic Phenomena, Spike Time Patterns, and Control in Neural Dynamics



Effects of phase offset on response to a two-frequency current stimulus in a cortical cell recorded in vitro. Top: stimulus. Bottom: spike train responses (time vs phase). From Thomas et al 2003.

Ion channel fluctuations, irregular synaptic barrages and other sources of ``noise'' limit the precision and reliability with which nerve cells produce action potentials. But highly precise and reliable patterns of spike times have been observed experimentally both in vitro and in vivo. What is the origin and functional significance of precise temporal patterns in the ``neural code''? Problems of current interest include (1) Relation of noise spectrum and intensity and input shape and amplitude to spike time precision in single cell models (integrate-and-fire, conductance based models). (2) Genericity of spike time convergence in simple deterministic neural oscillator models. (3) Construction of biophysical models for single cell and network activity including the effects of noise in comparison with experimental data from the sea hare Aplysia (collaboration with the Chiel laboratory). (4) Development of novel statistical techniques for identifying patterns in multiunit recordings in both Aplysia and in mammalian hippocampus.

Gradient Sensing, Signal Transduction, and Information Theory



MCell simulation of a cell in a field of signaling molecules. Top: uniform background distribution. Bottom: distribution after imposing flux conditions.

Signal transduction networks are the biochemical systems by which living cells sense their environments, make and act on decisions -- all without the benefit of a nervous system. How do cells use networks of chemical reactions to process information? We are combining mathematical ideas from the theory of stochastic point processes and Brownian motion with information theory to develop a framework for understanding information processing in biochemical systems. As a case study we are studying gradient sensing in neutrophils (white blood cells) and the social amoebae (Dictyostelium) from the points of view of information theory and statistical signal detection theory. Projects range from highly theoretical (devising information measures for time varying continuous time Markov processes) to highly computational (simulation of gradient sensing networks using explicit Monte Carlo techniques such as MCell. Gradient sensing work is being pursued in collaboration with the Baskaran laboratory.

  • Edward K. Agarwala*, Hillel J. Chiel, Peter J. Thomas, ``Pursuit of Food versus Pursuit of Information in Markov Chain Models of a Perception-Action Loop''. Journal of Theoretical Biology, in press 2012.
  • Peter J. Thomas, ``Cell Signaling: Every Bit Counts'', Science, 334(6054), 21 October, 2011, pp:321-322. DOI: 10.1126/science.1213834. Links courtesy of Science refer service: Summary, Reprint, Full Text.
  • J.M. Kimmel*, R. M. Salter, P.J. Thomas, ``An Information Theoretic Framework for Eukaryotic Gradient Sensing", Advances in Neural Information Processing Systems 19, MIT Press, pp 705-712, 2007. PDF and Supplementary Materials.
  • P.J. Thomas, D.J. Spencer, S.K. Hampton*, P. Park* and J. Zurkus, ``The Diffusion-Limited Biochemical Signal-Relay Channel'', Advances in Neural Information Processing Systems 16, MIT Press, 2004. PDF.
  • W.J. Rappel, P.J. Thomas, H. Levine and W.F. Loomis, ``Establishing Direction during Chemotaxis in Eukaryotic Cells'', Biophysical Journal 83, 1361-1367, September 2002. PDF.

Pattern Formation in the Visual Cortex


Monte Carlo sampling of the Heisenberg XY Model with (+) center (-) surround lateral interaction. Color represents preferred orientation angle (Thomas 2000, thesis).

Bifurcation planform corresponding to a predicted spontaneous hallucination pattern (Bressloff et al 2001A).

The pathway from the eyes to the visual cortex organizes spontaneously during development using a combination of intrinsic chemical markers and correlation-based, activity-dependent (``Hebbian") mechanisms. The resulting cortical architecture shows fascinating quasiregular patterns with elements including pinwheel and other phase singularity lattices in the cortical maps representing orientation, ocular dominance, retinotopic position and other features of the visual world. Using methods from equivariant bifurcation theory -- the study of branching solutions in the presence of symmetry -- an elegant theory has been developed that accounts for many aspects of the structure of cortical maps. The same mathematical structure underlies the forms of geometric visual hallucinations reported by subjects experiencing sensory deprivation or treatment with mescal, cannabis and other hallucinogens.

  • Peter J. Thomas, Jack D.Cowan, ``Generalized Spin Models for Coupled Cortical Feature Maps Obtained by Coarse Graining Correlation Based Synaptic Learning Rules". Journal of Mathematical Biology, in press (2011). PDF.
  • P.J.Thomas, J.D. Cowan, ``Simultaneous constraints on pre- and post-synaptic cells couple cortical feature maps in a 2D geometric model of orientation preference'', Mathematical Medicine and Biology, 23 (2):119-138, June 2006. PDF.
  • P.J. Thomas, J.D. Cowan, ``Symmetry induced coupling of cortical feature maps'', Physical Review Letters, 92 (18):188101, May 7, 2004. PDF.
  • P.C. Bressloff, J.D. Cowan, M. Golubitsky, P.J. Thomas and M.C. Wiener, ``What geometric visual hallucinations tell us about the visual cortex'', Neural Computation 14, 473-491, 2002. PDF.
  • P.C. Bressloff, J.D. Cowan, M. Golubitsky, P.J. Thomas and M.C. Wiener, ``Geometric visual hallucinations, Euclidean symmetry, and the functional architecture of striate cortex'', Phil. Trans. R. Soc. Lond. B 356, 299-330, 2001A. PDF.
  • P.C. Bressloff, J.D. Cowan, M. Golubitsky and P.J. Thomas, ``Scalar and pseudoscalar bifurcations motivated by pattern formation on the visual cortex'', Nonlinearity. 14, 739-775, 2001B. PDF.
  • P.J. Thomas ``Order and Disorder in Visual Cortex: Spontaneous Symmetry-Breaking and Statistical Mechanics of Pattern Formation in Vector Models of Cortical Development'', Dissertation, University of Chicago Department of Mathematics, August 2000.

Bioinformatics, Data Mining, and Malaria


Anopheles mosquito, capable of transmitting malaria.

Analysis of molecular diagnostic data. X-axis: fluorescence signal associated with drug sensitive allele. Y-axis: signal for drug resistant allele. Sloped lines: diagnostic thresholds.

Molecular diagnostic technology pioneered by collaborator P. Zimmerman in CWRU's Center for Global Health and Diseases promise faster, more accurate means for tracking and combatting the spread of drug resistance in endemic malaria populations worldwide. In order to achieve their potential the new methods require novel analysis tools. Mathematical ideas as simple as coordinate transformation and histogram segmentation have proven effective in boosting the accuracy of molecular genotyping techniques for discriminating drug-sensitive from drug-resistant infections.

Applications of Graph Theory to Biological Networks


For more information, please contact Dr. Thomas.

Updated: August 3, 2010