Peter J. Thomas, Assistant Professor

Department of Mathematics
Department of Biology (secondary appointment)
Department of Cognitive Science (secondary appointment)

Co-Director, Undergraduate Program in
Research at the Interface of the Biological and Mathematical Sciences (RIBMS)

Yost 210
216-368-3623
peter( D0T )j( D0T )thomas( 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.

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)

Office Hours: Monday & Wednesday 2:00-3:30, and by appointment.

Research Interests & Opportunities

Noise and Reliability in Neural Spike Time Patterns



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.

  • J.V. Toups, J.M. Fellous, P.J. Thomas, P.H. Tiesinga, T.J. Sejnowski, ``Spike pattern stability under changes in stimulus strength" (submitted).
  • 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). Pubmed. PDF.
  • 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 Baskaran laboratory.

  • 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.

  • 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.

For more information, please contact Dr. Thomas.

Updated: August 29, 2008