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
Research, Teaching, and Service Statements
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.
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
(4) Development of novel statistical techniques for identifying patterns in
multiunit recordings in both Aplysia and in mammalian hippocampus.
Kendrick M. Shaw, Young-Min Park*, Hillel J. Chiel and Peter J. Thomas, ``Phase Resetting in an Asymptotically Phaseless System: On the Phase Response of Limit Cycles Verging on a Heteroclinic Orbit.'' SIAM Journal on Applied Dynamical Systems 11(1):350-391, 2012.
See the movies profiled on DS Web!
Hillel J. Chiel and Peter J. Thomas, ``Applied Neurodynamics: From Neural Dynamics to Neural Engineering''. Journal of Neural Engineering in press 2011.
D. Michael Ackermann, Niloy Bhadra, Meana Gerges and Peter J. Thomas, ``Dynamics and Sensitivity Analysis of High Frequency Conduction Block". Journal of Neural Engineering in press 2011.
Peter J. Thomas, ``A Lower Bound for the First Passage Time Density of the Suprathreshold
Ornstein-Uhlenbeck Process". Journal of Applied Probability 48(2):420-434, June 2011.
- J. Vincent Toups, Jean-Marc Fellous, Peter J. Thomas,
Terrence J. Sejnowski, Paul H. Tiesinga, ``Finding the event structure of neuronal spike trains''.
Neural Computation, 2011 Sep;23(9):2169-208. Epub 2011 Jun 14.
- K.M. Stiefel, J.M. Fellous, P.J. Thomas, T.J. Sejnowski,
``Intrinsic subthreshold oscillations extend the influence of inhibitory synaptic inputs on cortical pyramidal neurons". European Journal of Neuroscience.
31(6):1019-26, March 2010. (Epub Mar 8, 2010).
- P.B. Kruskal*, J.J. Stanis, B.L. McNaughton, P.J. Thomas,
``A binless correlation measure reduces the variability of memory
reactivation estimates", Statistics in Medicine, 26(21):3997-4008,
Sep 20, 2007 (Epub June 26, 2007).
- J.M. Fellous, P.H.E. Tiesinga, P.J. Thomas and T.J. Sejnowski, ``Discovering
Spike Patterns in Neuronal Responses'', Journal of Neuroscience, 24 (12), 2989-3001,
March 24, 2004.
- P.J. Thomas, P.H. Tiesinga, J.M. Fellous and T.J.
Sejnowski, ``Reliability and Bifurcation in Neurons Driven by Multiple Sinusoids'',
Neurocomputing 52-54, 955-961, 2003.
- J.D. Hunter, J.G. Milton, P.J. Thomas and J.D. Cowan, ``A Resonance Effect
for Neural Spike Time Reliability'', J. Neurophysiol. 80, 1427-1438, 1998.
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
- 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:
- 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.
- 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.
- 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.
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).
- 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.
- P.J. Thomas, J.D. Cowan, ``Symmetry induced coupling of cortical feature
maps'', Physical Review Letters, 92 (18):188101, May 7, 2004.
- 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.
- 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.
- 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.
- 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.