Peter J. Thomas

Co-Editor-in-Chief, Biological Cybernetics

Department of Mathematics, Applied Mathematics, and Statistics (primary)
Department of Computer and Data Sciences (secondary)
Department of Electrical, Computer, and Systems Engineering (secondary)
Department of Cognitive Science (secondary)
Department of Biology (secondary)

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

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

E-mail: pjthomas( AT )

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)
Postdoctoral, The Salk Institute for Biological Studies, 2000-2004 (Computational Neurobiology)

Research: Systems & Computational Neuroscience, Theoretical Biophysics, Mathematical Biology, Mathematical Neuroscience, Control & Information Theory in Biology.
Google Scholar Profile
Curriculum vitae
Research, Teaching, and Service Statements (2012)
Research, Teaching, and Service Statements (2017)

Simulation and Analysis Codes:


(Fall 2023) not teaching this semester
(Spring 2024) TBD
Previous courses.

Office Hours: (Fall 2023) by appointment.

Research Interests & Publications

* denotes undergraduate coauthors

Stochastic Phenomena, Spike Time Patterns, and Statistical Analysis of Neural Data

Pu and Thomas' article ``Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics" provided the cover art for the October 2020 issue of Neural Computation.
Semianalytic approximations of the power spectrum of a stochastic heteroclinic oscillator, see Giner-Baldo et al 2017.

Deterministic and stochastic isochrons of a planar conductance based model, see Thomas and Lindner, 2014.

Morris-Lecar model with discrete sodium and calcium channels, Anderson et al 2015.

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) Reexamination of the concept of ``asymptotic phase" and ``asymptotic isostable" coordinates for stochastic oscillators, in collaboration with Prof. Dr. Benjamin Lindner at the Bernstein Center for Computational Neuroscience, Humboldt University, Berlin. (2) 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). (3) Data assimilation methods for intracellular recordings of spontaneous spike data in isolated single neurons. (4) Genericity of spike time convergence in simple deterministic neural oscillator models.

  • Alberto Perez-Cervera, Boris Gutkin, Peter J. Thomas, and Benjamin Lindner. ``A Universal Description of Stochastic Oscillators.'' Proceedings of the National Academy of Sciences, 120 (29) e2303222120, July 11, 2023. Link to arXiv. Link to journal. DOI: 10.1073/pnas.2303222120.
  • Nicholas W. Barendregt, Peter J. Thomas. ``Heteroclinic cycling and extinction in May-Leonard models with demographic stochasticity." Journal of Mathematical Biology (2023). ArXiv link. Journal link.
  • Alberto Perez-Cervera, Benjamin Lindner, Peter J. Thomas. ``Quantitative comparison of the mean-return-time phase and the stochastic asymptotic phase for noisy oscillators.'' Biological Cybernetics, 116.2 (2022): 219-234.
  • Konstantin Holzhausen, Lukas Ramlow, Shusen Pu, Peter J. Thomas, Benjamin Lindner. ``Mean-return-time phase of a stochastic oscillator provides an approximate renewal description for the associated point process.'' Biological Cybernetics, 116.2, (2022): 235251.
  • Konstantin Holzhausen, Peter J. Thomas, Benjamin Lindner. ``An analytical approach to the Mean-Return-Time Phase of isotropic stochastic oscillators.'' Physical Review E, 105.2 (2022): 024202.
  • Alberto Perez-Cervera, Benjamin Lindner, and Peter J. Thomas. ``Isostables for Stochastic Oscillators.'' Physical Review Letters, Vol. 127, No. 25, 17 December 2021 (published online 14 December 2021).
  • Shusen Pu and Peter J. Thomas. ``Resolving Molecular Contributions of Ion Channel Noise to Interspike Interval Variability through Stochastic Shielding.'' Biological Cybernetics, Vol. 115, pp. 267302, (22 May, 2021). ArXiv link.
  • Shusen Pu, Peter J. Thomas. ``Fast and Accurate Langevin Simulations of Stochastic Hodgkin-Huxley Dynamics". Neural Computation, 32(10), Oct. 2020. Authors' PDF. ArXiv link.
  • Alexander Cao, Benjamin Lindner, Peter J. Thomas. ``A Partial Differential Equation for the Mean--Return-Time Phase of Planar Stochastic Oscillators". SIAM Journal on Applied Mathematics 80(1) 422-447. Authors' PDF. Journal link. ArXiv link.
  • Peter J. Thomas and Benjamin Lindner. ``Phase descriptions of a multidimensional Ornstein-Uhlenbeck process". Physical Review E (2019). PDF. ArXiv link.
  • Jordi Giner-Baldo, Peter J. Thomas, and Benjamin Lindner. ``Power Spectrum of a Noisy System Close to a Heteroclinic Orbit". Journal of Statistical Physics (2017). PDF.
  • Bressloff, P. C., Ermentrout, B., Faugeras, O., and Thomas, P. J. (2016). ``Stochastic Network Models in Neuroscience: A Festschrift for Jack Cowan. Introduction to the Special Issue.'' Journal of Mathematical Neuroscience 6:4. Open Access Link.
  • David F. Anderson, 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): 67-82 (Jan. 2015). PDF.
  • Peter J. Thomas and Benjamin Lindner, ``Asymptotic Phase for Stochastic Oscillators", Physical Review Letters, 113(25), 2014: 254101. PDF.
  • J. Vincent Toups, Jean-Marc Fellous, Peter J. Thomas, Terrence J. Sejnowski, Paul H. Tiesinga, ``Multiple Spike Time Patterns Occur at Bifurcation Points of Membrane Potential Dynamics.'' PLoS Computational Biology 8(10):e1002615 (2012). PDF.
  • 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. PDF.
  • 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). PDF.
  • 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). 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. PDF.
  • 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. PDF.
  • 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. PDF.

Integrating Neural Dynamics and Biomechanics -- Brain-Body Interaction and Control of Rhythmic Motor Patterns

Nonsmooth isochrons for a limit cycle with sliding components, see Wang et al. 2021.
Arterial oxygen homeostasis under closed-loop respiratory control, see Diekman et al 2017.
Robustness of a motor pattern to increase in applied load, see Lyttle et al 2016.

The central nervous system is strongly coupled to the body. Through peripheral receptors and effectors, it is also coupled to the constantly changing outside world. A chief function of the brain is to close the loop between sensory inputs and motor output. It is through the brain's effectiveness as a control mechanism of the body and the external world that it facilitates long-term survival. As part of an ongoing collaboration with the Chiel laboratory, which studies biomechanics and neuromotor control in the marine mollusk Aplysia californica, and the Wilson laboratory, which studies generation and modulation of respiratory rhythms in the mammalian central nervous system, we pursue analysis of feedback control and other mechanisms that endow organisms with robust and flexible rhythm generation.

  • Chris Fietkiewicz, Robert A. McDougal, David Corrales Marco, Hillel J. Chiel, and Peter J. Thomas, ``Tutorial: Using NEURON for Neuromechanical Simulations.'' Frontiers in Computational Neuroscience. Volume 17. Doi: 10.3389/fncom.2023.1143323. In press, 2023. Link to journal.
  • Zhuojun Yu, Jonathan E. Rubin, and Peter J. Thomas. ``Sensitivity to Control Signals in Triphasic Rhythmic Neural Systems: a Comparative Mechanistic Analysis via Infinitesimal Local Timing Response Curves.'' Neural Computation, 35 (6): 1028-1085, 12 May, 2023. On arXiv. At journal.
  • Yangyang Wang, Jeffrey P. Gill, Hillel J. Chiel, and Peter J. Thomas. ``Variational and phase response analysis for limit cycles with hard boundaries, with applications to neuromechanical control problems.'' Biological Cybernetics (116) 687710 (2022).
  • Zhuojun Yu and Peter J. Thomas. "A Homeostasis Criterion for Limit Cycle Systems Based on Infinitesimal Shape Response Curves.'' Journal of Mathematical Biology, 84:24, (online) Feb. 25, 2022.
  • Shane Riddle, William Nourse, Zhuojun Yu, Peter J. Thomas, and Roger D. Quinn. ``A Synthetic Nervous System with Coupled Oscillators Controls Peristaltic Locomotion." In Biomimetic and Biohybrid Systems. Living Machines 2022. Lecture Notes in Computer Science, vol 13548. Springer, Cham.; pp. 249261 (2022).
  • Wenhuan Sun, Mengdi Xu, Jeffrey P. Gill, Peter J. Thomas, Hillel J. Chiel, and Victoria A. Webster-Wood. ``GymSlug: Deep Reinforcement Learning toward Biologically Interpretable Control based on Aplysia californica Feeding.'' In Biomimetic and Biohybrid Systems. Living Machines 2022. Lecture Notes in Computer Science, vol 13548. Springer, Cham.; pp. 326-248 (2022).
  • Yangyang Wang, Jeffrey P. Gill, Hillel J. Chiel, and Peter J. Thomas. ``Shape versus timing: linear responses of a limit cycle with hard boundaries under instantaneous and static perturbation'', SIAM Journal on Applied Dynamical Systems (2021). ArXiv link.
  • Zhuojun Yu and Peter J. Thomas. ``Dynamical Consequences of Sensory Feedback in a Half-Center Oscillator Coupled to a Simple Motor System". Biological Cybernetics (in press 2021).
  • Victoria A. Webster-Wood, Jeffrey P. Gill, Peter J. Thomas and Hillel J. Chiel. ``Control for Multifunctionality: Bioinspired Control Based on Feeding in Aplysia californica". Biological Cybernetics (2020). ArXiv link.
  • Youngmin Park, Kendrick M. Shaw, Hillel J. Chiel, and Peter J. Thomas. ``The Infinitesimal Phase Response Curves of Oscillators in Piecewise Smooth Dynamical Systems". European Journal of Applied Mathematics, special issue on analysis and applications of nonsmooth dynamics (2018). Arxiv link. Simulations available on Github.
  • Casey O. Diekman, Peter J. Thomas, and Christopher G. Wilson. ``Eupnea, tachypnea, and autoresuscitation in a closed-loop respiratory control model." Journal of Neurophysiology (2017). PDF.
  • David N. Lyttle, Jeff P. Gill, Kendrick M. Shaw, Peter J. Thomas, and Hillel J. Chiel. ``Robustness, flexibility, and sensitivity in a multifunctional motor control model.'' Biological Cybernetics (2016) Dec. 21: 1-23. PDF.
  • Kendrick M. Shaw, David N. Lyttle, Jeffrey P. Gill, Miranda J. Cullins, Jeffrey M. Mc- Manus, Hui Lu, Peter J. Thomas, and Hillel J. Chiel. ``The Significance of Dynamical Architecture for Adaptive Responses to Mechanical Loads During Rhythmic Behavior'', Journal of Computational Neuroscience, appeared online 04 Sep. 2014. PDF.
  • Casey O. Diekman, Christopher G. Wilson, Peter J. Thomas, ``Spontaneous Autoresuscitation in a Model of Respiratory Control'' (2012 Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC)). PDF
  • 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. PDF. See the movies profiled on DS Web! Simulation codes are available on the Chiel Lab's github site. See also coauthor Jeff Gill's NEURON Reconstruction of Susswein et al. 2002 (on github).
  • Hillel J. Chiel and Peter J. Thomas, ``Applied Neurodynamics: From Neural Dynamics to Neural Engineering''. Journal of Neural Engineering 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 2011.

Gradient Sensing, Signal Transduction, and Information Theory

Biochemical signal transduction (two-state BIND channel, Thomas & Eckford (2016) Trans. Info. Thy.).

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? In collaboration with Prof. Andrew Eckford of York University, 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. 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.

  • Alexander S. Moffett, Peter J. Thomas, Michael Hinczewski, Andrew W. Eckford. ``Cheater suppression and stochastic clearance through quorum sensing." PLoS Computational Biology, 18(7): e1010292, July 28, 2022.
  • Tyler S. Barker, Massimiliano Pierobon, Peter J. Thomas. ``Subjective Information and Survival in a Simulated Biological System," Entropy, 24 (63), May 2, 2022.
  • Tyler Barker, Peter J. Thomas, and Massimiliano Pierobon. ``Subjective Information in Life Processes: A Computational Case Study.'' (2021). Proceedings of ACM Conference (ACM NanoCom21). ACM, New York, NY, USA.
  • Gregory R. Hessler*, Andrew W. Eckford and Peter J. Thomas. ``Linear Noise Approximation of Intensity-Driven Signal Transduction Channels.'' Proceedings of IEEE GLOBECOM 2019, December 9-13, in Waikoloa, Hawaii, USA (accepted). Authors' PDF.
  • Andrew W. Eckford and Peter J. Thomas. ``The Channel Capacity of Channelrhodopsin and Other Intensity-Driven Signal Transduction Receptors.'' IEEE Transactions on Molecular, Biological and Multi-Scale Communications 4.1 (2018): 27-38. ArXiv link.
  • Andrew W. Eckford, Benjamin Kuznets-Speck*, Michael Hinczewski, and Peter J. Thomas, ``Thermodynamic Properties of Molecular Communication.'' International Society for Information Theory 2018. PDF. ArXiv link.
  • Peter J. Thomas and Andrew W. Eckford, ``Capacity of a simple intercellular signal transduction channel'', IEEE Transactions on Information Theory (2016). PDF. Simulation codes are available on Andrew Eckford's Github page.
  • Peter J. Thomas and Andrew W. Eckford, ``Shannon Capacity of Signal Transduction for Multiple Independent Receptors'', International Society for Information Theory 2016. PDF. ArXiv link.
  • Andrew W. Eckford, Kenneth A. Loparo, and Peter J. Thomas, ``Finite-State Channel Models for Signal Transduction in Neural Systems'', 2016 IEEE International Conference on Acoustics, Speech and Signal Processing. PDF.
  • Andrew W. Eckford and Peter J. Thomas, ``Information Theory of Intercellular Signal Transduction'', 2015 49th Asilomar Conference on Signals, Systems and Computers. PDF.
  • Peter J. Thomas and Andrew W. Eckford, ``Capacity of a Simple Intercellular Signal Transduction Channel'', submitted 2014. ArXiv link.
  • Andrew W. Eckford and Peter J. Thomas, ``Capacity of a Simple Intercellular Signal Transduction Channel'', International Society for Information Theory 2013. PDF.
  • 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.

Applications of Graph Theory and Statistical Mechanics to Biological Networks

An Erdos-Renyi network with 50 nodes, connection probability 0.5, and nodes labelled '1' (black) or '0' (gray) with equal probability.

Distribution of edge importance measure (Schmidt and Thomas 2014).

Mathematical models of cellular physiological mechanisms often involve random walks on graphs representing transitions within networks of functional states. Schmandt and Galán recently introduced a novel stochastic shielding approximation as a fast, accurate method for generating approximate sample paths from a finite state Markov process in which only a subset of states are observable. For example, in ion-channel models, such as the Hodgkin-Huxley or other conductance-based neural models, a nerve cell has a population of ion channels whose states comprise the nodes of a graph, only some of which allow a transmembrane current to pass. The stochastic shielding approximation consists of neglecting fluctuations in the dynamics associated with edges in the graph not directly affecting the observable states. In Schmidt and Thomas 2014 we consider the problem of finding the optimal complexity reducing mapping from a stochastic process on a graph to an approximate process on a smaller sample space, as determined by the choice of a particular linear measurement functional on the graph. The partitioning of ion-channel states into conducting versus nonconducting states provides a case in point. In addition to establishing that Schmandt and Galán’s approximation is in fact optimal in a specific sense, we use recent results from random matrix theory to provide heuristic error estimates for the accuracy of the stochastic shielding approximation for an ensemble of random graphs. Moreover, we provide a novel quantitative measure of the contribution of individual transitions within the reaction graph to the accuracy of the approximate process.

In collaboration with Alan Lerner, director of the Brain Health and Memory Center at CWRU, and Prof. Wojbor Woyczynski we have also applied elementary graph theoretic measures to the analysis of word association networks used as diagnostic tools related to Alzheimer's Disease and other cognitive impairments.

  • Danny Chen*, Alexander G. Strang, Andrew W. Eckford, and Peter J. Thomas, ``Explicitly Solvable Continuous-time Inference for Partially Observed Markov Processes." IEEE Transactions on Signal Processing (2022). Preprint arXiv:2301.00843.
  • Alexander S. Moffett, Guiying Cui, Peter J. Thomas, William D. Hunt, Nael A. McCarty, Ryan S. Westafer, and Andrew W. Eckford. ``Permissive and Nonpermissive Channel Closings in CFTR Revealed by a Factor Graph Inference Algorithm.'' Biophysical Reports (2) 4: 100083, December 14, 2022.
  • Alexander G. Strang, Karen C. Abbott, and Peter J. Thomas. ``The Network HHD: Quantifying Cyclic Competition in Trait-Performance Models of Tournaments.'' SIAM Review 64 (2) pp. 360-391, 05 May, 2022.
  • Alexander G. Strang, Karen C. Abbott, and Peter J. Thomas, ``How to avoid an extinction time paradox." Theoretical Ecology 12, 467–487 (2019). Journal link. Authors' PDF.
  • Deena R. Schmidt, Roberto F. Galán, and Peter J. Thomas, ``Stochastic shielding and edge importance for Markov chains with timescale separation'' PLoS computational biology 14 (6), e1006206. June 18, 2018. Open Access at PLoS.
  • Deena R. Schmidt and Peter J. Thomas, ``Measuring edge importance: a quantitative analysis of the stochastic shielding approximation for random processes on graphs'', Journal of Mathematical Neuroscience, 4:6, April 17, 2014. PDF
  • Meyer DJ*, Messer J*, Singh T*, Thomas PJ, Woyczynski WA, Kaye J, Lerner AJ, ``Random local temporal structure of category fluency responses.'' J Comput Neurosci. 2011 Jul 8. Pubmed. PDF.
  • Lerner AJ, Ogrocki PK, Thomas PJ. ``Network graph analysis of category fluency testing.'' Cogn Behav Neurol. 2009 Mar;22(1):45-52. Pubmed. 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.

  • Zily Burstein, David Reid, Peter J. Thomas, and Jack Cowan. ``Pattern Forming Mechanisms of Color Vision." Network Neuroscience (in press 2023).
  • Paul C. Bressloff, Bard Ermentrout, Olivier Faugeras, and Peter J. Thomas. ``Stochastic Network Models in Neuroscience: A Festschrift for Jack Cowan. Introduction to the Special Issue.'' The Journal of Mathematical Neuroscience 6, no. 1 (2016): 1-9. PDF.
  • 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, 65.6-7 (2012): 1149-1186. 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 Disease Modeling

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. Recent efforts include development of a method for disambiguating birth and death rates from population time series data, with applications to distinguishing mechanisms of drug resistance.

  • Linh Huynh, Jacob G. Scott, and Peter J. Thomas, ``Inferring Density-Dependent Population Dynamics Mechanisms through Rate Disambiguation for Logistic Birth-Death Processes'', Journal of Mathematical Biology, 86 50, 3 March 2023. On bioRxiv. On arXiv. At journal.
  • Celine Barnadas, David Kent*, Lincoln Timinao, Jonah Iga, Laurie Gray, Peter Siba, Ivo Mueller, Peter J. Thomas, and Peter A. Zimmerman, ``A new high through-put method for simultaneous detection of mutations associated with Plasmodium vivax drug resistance in pvdhfr, dhps and mdr1 genes.'' Malaria Journal, 2011, 10(282). PDF.
  • J.T. Da Re, D.P. Kouri*, P.A. Zimmerman, and P.J. Thomas, ``Differentiating Plasmodium falciparum alleles by transforming Cartesian X,Y data to polar coordinates'', BMC Genetics, 2010, 11:57, doi:10.1186/1471-2156-11-57. PDF.


Allosteric model for single-stranded polymer formation. From Miraldi et al 2008. Every theorist should have some lab experience.
  • E.R. Miraldi*, P.J. Thomas, L. Romberg. ``Allosteric models for cooperative polymerization of linear polymers.'' Biophysical Journal, Volume 95, Issue 5, 1 September 2008, Pages 2470-2486.
  • P.J. Thomas*, B.E. Wendelburg, S.E. Venuti, G.M. Helmkamp Jr. ``Mature rat testis contains a high molecular weight species of phosphatidylinositol transfer protein.'' Biochimica et Biophysica Acta, Volume 982, Issue 1, 26 June 1989, Pages 24-30.

Geophysical Dynamics

Benthic foraminifera $\delta^{18}$O, a proxy for global ice volume and ocean temperature, over the past 3~Myr. From Lisiecki & Raymo, 2005. Redrawn from Omta & Kooi, 2016. Note the change in frequency marked by the arrow. Nonlinear oscillations occur in geological as well as biological systems! In joint work with Earth, Environmental and Planetary Sciences professor Anne Willem Omta we used dynamical systems analysis to evaluate a hypothesis about a possible mechanism for the change in ice-age frequency that occurred roughly one million years ago.
  • John Shackleton*, Mick Follows, Peter J. Thomas, and Anne Willem Omta. ``The Mid-Pleistocene Transition: A delayed response to an increasing positive feedback?" Climate Dynamics, November 4 (2022).



My students and I are grateful to the following funding sources:

Grants to PJT

Grants to laboratory members


Computational Biomathematics Laboratory members, summer 2021.
Top row: D. Chen, P. Thomas, Z. Yu.
Middle row: M. Kreider, L. Huynh, L. Zheng.
Bottom row: L. Arudchandran, R. Hao, A. Perez (visiting postdoc).

Computational Biomathematics Laboratory members, summer 2020.
Top row: L. Zheng, P. Thomas, T. Burleyson.
Middle row: M. Kreider, L. Huynh, A. Strang.
Bottom row: H. Louh, Z. Yu, S. Pu.

Postdoctoral trainees

Doctoral students

Magisterial students

Undergraduate students (CWRU)

Undergraduate students (Oberlin College)

Undergraduate students (external)

For more information, please contact Dr. Thomas.

Updated: July 26, 2023

Lab Retreats


Computational Biomathematics Laboratory personnel as of June 2019.
Left to right: H-D. Louh, L. Huynh, S. Pu, G. Hessler, A. Strang, and P. Thomas.
Photo credit: D. Preston, Presto Photo.


Computational Biomathematics Laboratory personnel as of June 2018.
Left to right: L. Huynh, S. Pu, N. Barendregt, P. Thomas, J. Austrow, A. Strang, and G. Hessler.
Photo credit: D. Preston, Presto Photo.

Miscellaneous Gallery

Computational Biomathematics Laboratory reunion at the 2019 Society for Mathematical Biology meeting in Montreal.
Left to right: D. Schmidt, S. Pu, Y. Park, N. Barendregt, and Y. Wang.