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Professor Stephen Coombes

Professor of Applied Mathematics, Head of Section Applied Mathematics, Faculty of Science

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Biography

I am Professor of Applied Mathematics and Head of the Applied Mathematics Section in the School of Mathematical Sciences at the 糖心原创, where I have worked since 2003. I hold a BSc in Theoretical Physics and a PhD in Neurocomputing.

My research lies in mathematical neuroscience, drawing on nonlinear dynamics and statistical physics to understand neural systems across scales, from individual cells to networks and brain tissue. Particular interests include neural oscillations, synchronisation, travelling waves, spatial patterns and the influence of transmission delays on brain dynamics. My work combines the development of mathematical methods with collaborations in experimental and clinical neuroscience.

I am co-author, with Kyle C. A. Wedgwood, of (Springer, 2023), and have co-edited volumes on neural bursting and neural field theory. I have also contributed to the development of the mathematical neuroscience community through research networks, editorial work and the organisation of international conferences.

Expertise Summary

My expertise is in mathematical neuroscience and nonlinear dynamical systems, particularly the development and analysis of models linking cellular mechanisms to collective neural activity.

Research areas include: 鈥� Spiking neural networks and event-based dynamics. 鈥� Neural mass and neural field models, including next-generation population models. 鈥� Synchronisation, oscillations, travelling waves and localised activity. 鈥� Delay differential equations and networks with multiple, distributed and adaptive delays. 鈥� Phase and phase-isostable reduction of coupled oscillators. 鈥� Nonsmooth dynamical systems, bifurcation theory and stability analysis. 鈥� Structural and functional brain connectivity. 鈥� Thalamo-cortical dynamics and applications in systems and clinical neuroscience.

A developing strand of my research concerns white-matter plasticity: how activity-dependent changes in axonal conduction can alter communication delays, reorganise network dynamics and contribute to learning. The spiking Haken Lighthouse model provides a tractable framework for investigating these questions.

Research Summary

My research lies in mathematical neuroscience and neurodynamics, with an emphasis on using ideas from nonlinear dynamics, pattern formation and statistical physics to understand the collective… read more

Selected Publications

  • BYRNE, AINE, AVITABILE, DANIELE and COOMBES, STEPHEN, 2019. PHYSICAL REVIEW E. 99(1),
  • COOMBES, STEPHEN and BYRNE, AINE, 2019. NONLINEAR DYNAMICS IN COMPUTATIONAL NEUROSCIENCE. 1-16
  • S COOMBES, Y-M LAI, M SAYLI and R THUL, 2018. Euro. Jnl of Applied Mathematics. 22
  • VISSER, S., NICKS, R., FAUGERAS, O. and COOMBES, S., 2017. PHYSICA D-NONLINEAR PHENOMENA. 349, 27-45

Neurodynamics: An Applied Mathematics Perspective

Stephen Coombes and Kyle C. A. Wedgwood Springer, 2023. Texts in Applied Mathematics, Volume 75.

Our book introduces the mathematical study of neural dynamics, progressing from individual neurons, axons, dendrites and synapses to networks and spatially extended neural tissue. It develops models and analytical methods for understanding excitability, oscillations, synchronisation, waves and pattern formation, including the effects of delays and noise. A recurring theme is the use of tractable models to obtain explicit insight into neural behaviour. Written for readers with a background in mathematics, physics or engineering, it assumes no prior knowledge of neuroscience and is suitable for Master's-level teaching and postgraduate researchers entering mathematical neuroscience. Exercises throughout support independent study.

Current Research

My research lies in mathematical neuroscience and neurodynamics, with an emphasis on using ideas from nonlinear dynamics, pattern formation and statistical physics to understand the collective behaviour of neural systems.

A major theme of my work is the development and analysis of mathematical models that bridge different spatial and temporal scales of brain activity. This includes neural mass and neural field models for the dynamics of neuronal populations, oscillator and phase-reduced descriptions of neural networks, and models that retain the discrete timing of neuronal spikes. I am particularly interested in oscillations, waves, synchrony, localised states and other forms of emergent spatio-temporal activity.

Recent work has focused on next-generation neural mass and field models, which provide a principled connection between microscopic spiking dynamics and population-level descriptions of brain activity. These models are being used to investigate the relationship between structural and functional brain connectivity, the role of electrical and chemical coupling, and the generation of experimentally observable brain rhythms.

Transmission delays are another important strand of my research. Axonal conduction times can strongly influence synchronisation, wave propagation and pattern formation in large-scale neural networks. I have developed mathematical approaches for studying networks with multiple and distributed delays, including phase and phase-isostable reduction techniques.

Alongside this theoretical work, I collaborate on applications in systems and clinical neuroscience. Current projects include mathematical models of thalamo-cortical dynamics, epilepsy and seizure transitions, cortical beta activity, and the interpretation of neuroimaging data. A recurring aim is to develop models that are sufficiently biophysically grounded to connect with experiment while remaining sufficiently tractable for mathematical analysis.

Future Research

An important direction for my future research is to develop mathematical descriptions of neural computation in which the timing of communication between brain regions is itself dynamic and adaptive.

I am particularly interested in the Haken Lighthouse model as a tractable model of a spatially extended spiking neural network. Unlike conventional firing-rate descriptions, the model retains individual spike times while still allowing analytical treatment of collective phenomena such as synchronisation, travelling waves, localised activity and pattern-forming instabilities. Current work is developing reductions that connect its event-based spiking dynamics to lower-dimensional descriptions of population activity.

This provides a natural framework for investigating the computational role of white matter. Axonal conduction velocities, and hence communication delays, are not necessarily fixed: they can change through activity-dependent modification of myelin and other axonal properties. I aim to understand how such plasticity in conduction delays can reorganise spike timing, phase relationships and network attractors. This raises the possibility that white-matter plasticity provides a form of temporal learning, in which neural systems adapt not only the strength of their connections but also the relative arrival times of signals.

More broadly, I am interested in developing mathematical theories that link adaptive delays, spiking dynamics and large-scale brain organisation. The longer-term goal is to understand how changes in the physical substrate of neural communication can shape computation, learning and coherent brain dynamics across multiple spatial and temporal scales.

  • COOMBES, STEPHEN and BYRNE, AINE, 2019. NONLINEAR DYNAMICS IN COMPUTATIONAL NEUROSCIENCE. 1-16
  • VEASY, JOSHUA, LAI, YI MING, COOMBES, STEPHEN and THUL, RUDIGER, 2019. JOURNAL OF THEORETICAL BIOLOGY. 478, 102-114
  • BYRNE, AINE, AVITABILE, DANIELE and COOMBES, STEPHEN, 2019. PHYSICAL REVIEW E. 99(1),
  • S COOMBES, Y-M LAI, M SAYLI and R THUL, 2018. Euro. Jnl of Applied Mathematics. 22
  • 脕 BYRNE, S COOMBES and P LIDDLE, 2018. A neural mass model for abnormal beta-rebound in schizophrenia. In: Handbook of Multi-scale Models of Brain Disorders Springer.. (In Press.)
  • A G脰K脟E, D AVITABILE and S COOMBES, 2018. Neural fields: Localised states with piece-wise constant interactions In: NeuroMath, INdAM Series, Springer. (In Press.)
  • R NICKS, L CHAMBON and S COOMBES, 2018. Physical Review E. 97, 032213
  • TEWARIE, PREJAAS, HUNT, BENJAMIN A E, O'NEILL, GEORGE C, BYRNE, AINE, AQUINO, KEVIN, BAUER, MARKUS, MULLINGER, KAREN J, COOMBES, STEPHEN and BROOKES, MATTHEW J, 2018. Relationships Between Neuronal Oscillatory Amplitude and Dynamic Functional Connectivity. Cerebral cortex (New York, N.Y. : 1991).
  • MCIVOR, EMMA, COOMBES, STEPHEN and THUL, RUDIGER, 2018. CELL CALCIUM. 73, 11-24
  • TEWARIE P, ABEYSURIYA R, BYRNE 脕, O'NEILL GC, SOTIROPOULOS SN, BROOKES MJ and COOMBES S, 2018. NeuroImage. 186, 211-220 (In Press.)
  • LAI, YI MING, THUL, RUDIGER and COOMBES, STEPHEN, 2018. EUROPEAN PHYSICAL JOURNAL-SPECIAL TOPICS. 227(10-11), 1251-1265
  • BYRNE, AINE, BROOKES, MATTHEW J and COOMBES, STEPHEN, 2017. Journal of computational neuroscience.
  • VISSER, S., NICKS, R., FAUGERAS, O. and COOMBES, S., 2017. PHYSICA D-NONLINEAR PHENOMENA. 349, 27-45
  • A GOKCE, D AVITABILE and S COOMBES, 2017. Journal of Mathematical Neuroscience. (In Press.)
  • M BONILLA-QUINTANA, K C A WEDGWOOD, R O'DEA and S COOMBES, 2017. Journal of Mathematical Neuroscience. 7(9),
  • ASHWIN, PETER, COOMBES, STEPHEN and NICKS, RACHEL, 2016. Journal of mathematical neuroscience. 6(1), 2
  • S COOMBES and 脕 BYRNE, 2016. . In: Lecture Notes in Nonlinear Dynamics in Computational Neuroscience: from Physics and Biology to ICT Springer. (In Press.)
  • S COOMBES and R THUL, 2016. European Journal of Applied Mathematics. (In Press.)
  • R THUL, S COOMBES and C R LAING, 2016. Journal of Mathematical Neuroscience. 6(3),
  • WOJTAK, WERONIKA, COOMBES, STEPHEN, BICHO, ESTELA and ERLHAGEN, WOLFRAM, 2016. ARTIFICIAL NEURAL NETWORKS AND MACHINE LEARNING - ICANN 2016, PT I. 9886, 411-418
  • THUL R, RIETDORF K, BOOTMAN MD and COOMBES S, 2015. Biochimica et biophysica acta. (In Press.)
  • S COOMBES, 2015. Mathematical Neuroscience: from neurons to networks In: EDP-Normandie Proceedings of the International Conference Symposia in Applied Mathematics.
  • COOMBES, STEPHEN, 2015. Validating Neuro-Computational Models of Neurological and Psychiatric Disorders Foreword VALIDATING NEURO-COMPUTATIONAL MODELS OF NEUROLOGICAL AND PSYCHIATRIC DISORDERS. 14, V-V
  • MEIJER HGE and COOMBES S, 2014. Journal of mathematical biology. 68(5), 1249-1268,
  • Y TIMOFEEVA and S COOMBES, 2014. . In: H CUNTZ, M REMME and B TORBEN-NIELSEN, eds., The Computing Dendrite: From Structure to Function Springer. 449-464
  • H G E MEIJER and S COOMBES, 2014. EPJ Nonlinear Biomedical Physics. 2(3),
  • S COOMBES, P BEIM GRABEN and R POTTHAST, 2014. . In: S COOMBES, P BEIM GRABEN and R POTTHAST, eds., Neural Fields: Theory and Applications Springer. 1-43
  • S COOMBES, H SCHMIDT and D AVITABILE, 2014. . In: S COOMBES, P BEIM GRABEN, R POTTHAST and J J WRIGHT, eds., Neural Fields: Theory and Applications Springer. 187-211
  • LIN KK, WEDGWOOD KCA, COOMBES S and YOUNG L, 2013. Journal of mathematical biology. 66, 139-161
  • ZACHARIOU, M., ALEXANDER, S.P.H., COOMBES, S. and CHRISTODOULOU, C., 2013. PLoS One. 8(3), e58926
  • WEDGWOOD KC, LIN KK, THUL R and COOMBES S, 2013. Journal of Mathematical Neuroscience. 3(2),
  • TIMOFEEVA Y, COOMBES S and MICHIELETTO D, 2013. Journal of mathematical neuroscience. 3(1), 15 (In Press.)
  • COOMBES S and TIMOFEEVA Y, 2013. Journal of mathematical neuroscience. 3(1), 10 (In Press.)
  • P C BRESSLOFF and S COOMBES, 2013. Cognitive Computation. 5(3), 281-294
  • ZACHARIOU M, ALEXANDER SPH, COOMBES S and CHRISTODOULOU C, 2013. PloS one. 8(3), e58926
  • TAXIDIS, J., COOMBES, S., MASON, R. and OWEN, M.R., 2012. Hippocampus. 22(5), 995-1017
  • HORACIO G. ROTSTEIN, STEPHEN COOMBES and ANA MARIA GHEORGHE, 2012. SIAM J. on Applied Dynamical Systems. 11(1), 135-180
  • THUL, R., COOMBES, S., RODERICK, H.L. and BOOTMAN, M.D., 2012. Proceedings of the National Academy of Sciences of the United States of America. 109(6), 2150-2155
  • SVENSSON, C., COOMBES, S. and PEIRCE, J.W., 2012. Neuroinformatics. 10(2), 199-218
  • COOMBES, S., THUL, R. and WEDGWOOD, K.C.A., 2012. Physica D: Nonlinear Phenomena. 241(122), 2042-2057
  • S COOMBES, C R LAING, H SCHMIDT, N SVANSTEDT and J A WYLLER, 2012. Discrete and Continuous Dynamical Systems - Series A. 32, 2951 - 2970
  • COOMBES, S., SCHMIDT, H. and BOJAK, I., 2012. Journal of Mathematical Neuroscience. 2(1), 9
  • HLINKA J and COOMBES S, 2012. The european journal of neuroscience. 36(2), 2137-45
  • COOMBES S and TERRY JR, 2012. The european journal of neuroscience. 36(2), 2118-20
  • R THUL, S COOMBES and M D BOOTMAN, 2012. Frontiers in Computational Physiology and Medicine. 3, 279
  • LANGDON, ANGELA J., BREAKSPEAR, MICHAEL and COOMBES, STEPHEN, 2012. PHYSICAL REVIEW E. 86(5), 061903(1-12)
  • BOOTMAN, MARTIN D, SMYRNIAS, IOANNIS, THUL, R脺DIGER, COOMBES, STEPHEN and RODERICK, H LLEWELYN, 2011. Biochimica et Biophysica Acta. 1813, 922-934
  • COOMBES, S. and LAING, C.R., 2011. Physical review E: Statistical, Nonlinear, and Soft Matter Physics. 83(1), 011912
  • S COOMBES, R THUL, J LAUDANSKI, A R PALMER and C J SUMNER, 2011. Frontiers in Life Science. 5, 91-105
  • COOMBES, S., 2010. NeuroImage. 52, 731-739
  • LAUDANSKI, J., COOMBES, S., PALMER, A. R. and SUMNER, C. J., 2010. Journal of Neurophysiology. 103, 1226-1237
  • COOMBES, S. and TIMOFEEVA, Y., 2010. Physica D - Nonlinear Phenomena. 239(9), 475-578
  • HLINKA, J. and COOMBES, S., 2010. Physical Review Letters. 104(6), 068101
  • COOMBES, S. and SCHMIDT, H., 2010. Discrete and Continuous Dynamical Systems Series A. 28(4), 1369-1379
  • LAUDANSKI, J., SUMNER, C. J. and COOMBES, S., 2010. Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. 82, 011924
  • THUL, R. and COOMBES, S., 2010. Chaos. 20(4), 045102
  • COOMBES, S. and ZACHARIOU, M., 2009. . In: JOSIC, K., RUBIN, J, MATIAS, M. and ROMO, R., eds., Coherent Behavior in Neuronal Networks Springer.
  • COOMBES, S. and LAING, C.R., 2009. Physica D: Nonlinear Phenomena. 238(3), 264-272
  • COOMBES, S. and LAING, C.R., 2009. Philosophical Transactions of the Royal Society A: Mathematical, Physical, and Engineering Sciences. 367(1891), 1117-1129
  • SVENSSON, C-M, and COOMBES, S., 2009. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. 19(8), 2597-2607
  • ZACHARIOU, M., DISSANAYAKE, D. W., COOMBES, S., OWEN, M. R. and MASON, R., 2008. COGNITIVE NEURODYNAMICS. VOL 2(NUMBER 2), 159-170
  • THUL, R, BELLAMY, T.C., RODERICK, H.L., BOOTMAN, M.D. and COOMBES, S, 2008. . In: MIGUEL MAROTO AND NICK MONK, ed., Cellular Oscillatory Mechanisms Springer.
  • THUL, R., SMITH, G.D. and COOMBES, S., 2008. Journal of Mathematical Biology. 56(4), 435-463
  • COOMBES, S., 2008. SIAM Journal on Applied Dynamical Systems. 7(3), 1101-1129
  • TIMOFEEVA, Y., COOMBES, S., COX, S.J. and JOSIC, K., 2008. Journal of Computational Neuroscience. 25, 228-244
  • ZACHARIOU, M., DISSANAYAKE, D.W.N., OWEN, M.R., MASON, R. and COOMBES, S, 2007. Neurocomputing. 70(10-12), 1902-1906
  • BUTT, I.A. and WATTIS, J.A.D., 2007. Physica D - Nonlinear Phenomena. 231(2), 165-179
  • COOMBES, S, and OWEN, M.R., 2007. . In: BOTELHO, F., HAGEN, T. and JAMISON, J., eds., Fluids and Waves: Recent Trends in Applied Analysis 440. American Mathematical Society. 123-144
  • OWEN, M.R., LAING, C.R. and COOMBES, S., 2007. New Journal of Physics. 9, 378
  • S COOMBES, 2007. Journal of Mathematical Biology. 54(2), 305-307
  • COOMBES, S, VENKOV, N A, SHIAU, L, BOJAK, I, LILEY, D T J and LAING, C R, 2007. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 76(5 Pt 1), 051901
  • COOMBES, S., TIMOFEEVA, Y., SVENSSON, C-M., LORD, G.J., JOSIC, K., COX, S.J. and COLBERT, C.M., 2007. Biological Cybernetics. 97(2), 137-149
  • COOMBES, S., DOIRON, B., JOSI膯, K. and SHEA-BROWN, E., 2006. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical, and Engineering Sciences. 364(1849), 3301-18
  • TIMOFEEVA, Y., LORD, G. J. and COOMBES, S., 2006. Neurocomputing. VOL 69(NUMBER 10-12), 1058-1061
  • LAING, CARLO and COOMBES, STEPHEN, 2006. Network (Bristol, England). 17(2), 151-72
  • S COOMBES, 2006. Available at: <http://www.scholarpedia.org/article/Neural_Fields>
  • TIMOFEEVA, YULIA, LORD, GABRIEL J and COOMBES, STEPHEN, 2006. Journal of computational neuroscience.. 21(3), 293-306
  • CHOW, C. C. and COOMBES, S., 2006. SIAM Journal on Applied Dynamical Systems. 5(4), 552-574
  • P C BRESSLOFF AND S COOMBES, 2006. Mathematical Medicine & Biology. 23(2), 77-78
  • BOOTMAN, MARTIN D, HIGAZI, DANIEL R, COOMBES, STEPHEN and RODERICK, H LLEWELYN, 2006. Journal of Cell Science. 119(Pt 19), 3915-25
  • COOMBES, S. and OWEN, M.R., 2005. Physical Review Letters. VOL 94(NUMB 14), 148102
  • COOMBES,S. and BRESSLOFF,P.C., eds., 2005. Singapore; World Scientific Publishing.
  • COOMBES, S., 2005. Biological Cybernetics. VOL 93(NUMBER 2), 91-108
  • LAING, C. R. and COOMBES, S., 2005. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. VOL 15(NUMB 4), 1433-1444
  • COOMBES, S. and OWEN, M.R., 2004. SIAM Journal on Applied Dynamical Systems. 3(4), 574-600
  • COOMBES, S., HINCH, R. and TIMOFEEVA, Y., 2004. Progress in Biophysics and Molecular Biology. 85(2-3), 197-216
  • TIMOFEEVA, Y. and COOMBES, S., 2004. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. VOL 70(NUMB 6), 062901
  • CHIK, D. T. W., COOMBES, S. and WANG, Z. D., 2004. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. VOL 70(PART 1), 011908
  • COOMBES, S. and BRESSLOFF, P.C., 2003. Physical Review Letters. 91(2), 028102
  • COOMBES, S., LORD, G. J. and OWEN, M. R., 2003. Physica D - Nonlinear Phenomena. VOL 178(NUMBER 3-4), 219-241
  • COOMBES, S., 2003. Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. VOL 67(PART 4), 041910
  • TIMOFEEVA, Y. and COOMBES, S., 2003. A Stochastic Fire-Diffuse-Fire Model of Ca^2^+ Release FORMA -TOKYO-. VOL 18(PART 1), 53-58
  • COOMBES, S. and TIMOFEEVA, Y., 2003. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. VOL 68(PART 2), 021915
  • DENMAN-JOHNSON, M. and COOMBES, S., 2003. Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. VOL 67(PART 5), 051903
  • JAMES, M. P., COOMBES, S. and BRESSLOFF, P. C., 2003. Physical Review E: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics. VOL 67(PART 5), 051905
  • BRESSLOFF, P.C. and COOMBES, S., 2003. Synchronization of synaptically-coupled neural oscillators. In: Epilepsy as a dynamic disease Springer-Verlag, Heidelberg.
  • TIMOFEEVA, Y. and COOMBES, S., 2003. Journal of Mathematical Biology. VOL 47(PART 3), 249-269
  • LORD, G. J. and COOMBES, S., 2002. Physica D - Nonlinear Phenomena. VOL 161(NUMBER 1-2), 1-20
  • BRESSLOFF,P.C. and COOMBES,S., 2001. Mathematical reduction techniques for modeling biophysical neural networks. In: POZNANZKI, R.R., ed., Biophysical Neural Networks: Foundations of integrative neuroscience Mary Ann Liebert, Inc., New York, USA. 215-260
  • COOMBES, S, OWEN, M R and SMITH, G D, 2001. Mode locking in a periodically forced integrate-and-fire-or-burst neuron model. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. 64(4 Pt 1), 041914
  • COOMBES, S., 2001. Physica D - Nonlinear Phenomena. VOL 160(NUMBER 3-4), 173-188
  • COOMBES, S., 2001. Mathematical Biosciences. VOL 170(NUMBER 2), 155-172
  • COOMBES, S., 2001. Bulletin of Mathematical Biology. VOL 63(PART 1), 1-20
  • COOMBES, S. and BRESSLOFF, P. C., 2000. SIAM Journal on Applied Mathematics. VOL 61(PART 2), 432-453
  • BRESSLOFF, P. C. and COOMBES, S., 2000. Neural Computation. VOL 12(PART 1), 91-130
  • BRESSLOFF, P. C. and COOMBES, S., 2000. SIAM Journal on Applied Mathematics. VOL 60(PART 3), 820-841
  • COOMBES, S. and OSBALDESTIN, A. H., 2000. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics. VOL 62(PART 3), 4057-4066

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