I am an Associate Professor of Applied Mathematics in the School of Mathematical Sciences at the University of Nottingham.

Prior to that I was an Assistant Professor and I held a Leverhulme Trust Early Career fellowship at the University of Nottingham.


My research focuses on mathematical cell physiology and computational cell biology with a strong emphasis on intracellular calcium signalling. I am particularly interested in how the spatial organisation of intracellular signalling cascades and intrinsic molecular fluctuations shape the formation of signalling micro-domains and whole cell responses. I have developed models for intracellular calcium waves and oscillations using bottom-up as well as top-down approaches. While the former is based on partial differential equations with appropriate descriptions of the calcium signalling toolkit, the latter employs Bayesian ideas and stochastic point processes.

From a more mathematical perspective I have been investigating non-smooth dynamical systems both deterministically and stochastically. Non-smooth dynamical systems provide a natural language for describing a wide variety of real world phenomena ranging from engineering to neuroscience. At the same time, they allow for a deep mathematical analysis, which often requires the generalisation of techniques for smooth dynamical systems such as the master stability function.

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Phd students

  • Sunil Modhara - Pattern formation in biological neural networks with rebound currents
  • Tomas Dean - Multiscale modelling of signalling microdomains
  • Jake Powell - Heterogeneity driven cell signalling
  • Mustafa Sayli - Nonsmooth network dynamics
  • Hayley Mills - Shaping sustainable food production through understanding plant calcium dynamics
  • Joshua Veasy - Subcellular calcium patterns in ventricular myocytes (2018)
  • Emma McIvor - Modelling store operated calcium entry  (2018)
  • Agnė Tilūnaitė - Modelling of intracellular calcium dynamics (2017)
  • Wilhelm Braun - First passage dynamics in neuron models with stochastic threshold (2015)
  • Kyle Wedgwood - Dynamical systems techniques in the analysis of neural systems (2013)


  • Y. M. Lai, S. Coombes, R. Thul , Calcium buffers and L-type calcium channels as modulators of cardiac subcellular alternans , Communications in Nonlinear Science and Numerical Simulation (2020), accepted
  • J. Powell, M. Falcke, A. Skupin, T. C. Bellamy, T. Kypraios, R. Thul , A statistical view on calcium oscillations, Advances in Experimental Medicine and Biology, 1131, 799 (2020)
    DoiDOI - eprints
  • M. Sayli, Y. M. Lai, R. Thul, S. Coombes, , Synchrony in networks of Franklin bells, IMA Journal of Applied Mathematics, 85, 1001 (2019)
    DoiDOI - eprints
  • Y. M. Lay, J. Veasy, S. Coombes, R. Thul , A master stability function approach to cardiac alternans, Applied Network Science, 4, 1812 (2019)
    DoiDOI - eprints
  • J. Veasy, Y.M. Lai, S. Coombes, R. Thul, Complex patterns of subcellular cardiac alternans, Journal of Theoretical Biology, 478, 102 (2019)
    DoiDOI - eprints
  • Y. M. Lai, R. Thul, S. Coombes, Analysis of networks where discontinuities and nonsmooth dynamics collide: understanding synchrony, European Physical Journal Special Topics, 227, 1251 (2018)
  • E. McIvor, S. Coombes, R. Thul, Three-dimensional spatio-temporal modelling of store operated Ca2+ entry: insights into ER refilling and the spatial signature of Ca2+ signals, Cell Calcium, 73, 11 (2018)
  • M. Falkce, M. Moein, A. Tilunaite, R. Thul, A. Skupin, On the phase space structure of IP3 induced Ca2+ signalling and concepts for predictive modeling, Chaos, 28, 045115 (2018)
  • S. Coombes, Y. M. Lay, M. Sayli, R. Thul, Networks of piecewise linear neural mass models, European Journal of Applied Mathematics (2018)
  • A. Tilunaite, W. Croft, N. Russell, T. C. Bellamy, R. Thul, A Bayesian approach to modelling heterogeneous calcium responses in cell populations, PLoS Computational Biology, 13, e1005794 (2017)
  • W. Braun, R. Thul, A. Longtin, Evolution of moments and correlations in non-renewal escape-time processes, Physical Review E, 95, 052127 (2017)
  • W. Braun, R. Thul, Sign changes as a universal concept in first-passage-time calculations, Physical Review E, 95, 012114 (2017)
  • S. Coombes, R. Thul, Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function, European Journal of Applied Mathematics, 27, 904 (2016)
  • R. Thul, S. Coombes, C. R. Laing, Neural Field Models with Threshold Noise, The Journal of Mathematical Neuroscience 6, 1 (2016)
  • W. Croft, K. Reusch, A. Tilunaite, N. Russel, R. Thul, T. C. Bellamy, Probabilistic encoding of stimulus strength in astrocyte global calcium signals, Glia. 64, 537 (2016)
  • W. Braun, P. C. Matthews, R. Thul, First-passage times in integrate-and-fire neurons with stochastic thresholds, Physical Review E, 91, 052701 (2015)
  • D. S. Barrack, R. Thul, M. Owen, Modelling cell cycle synchronisation in networks of coupled radial glial cells, Journal of Theoretical Biology, 377, 85 (2015)
  • R. Thul, K. Rietdorf, M. D. Bootman, S. Coombes, Unifying principles of calcium wave propagation — insights from a three-dimensional model for atrial myocytes, Biochimica et Biophysica Acta - Molecular Cell Research, 1853, 2131 (2015)
  • M Zachariou, R. Thul, Cannabinoid-mediated short-term plasticity in hippocampus, Journal of Computational Neuroscience, 37, 533 (2014)
  • D. S. Barrack, R. Thul, M. Owen, Modelling the coupling between intracellular calcium release and the cell cycle during cortical brain development, Journal of Theoretical Biology, 347, 17 (2014)
  • R. Thul, Translating intracellular calcium signaling into models, Cold Spring Harbor Protocols, pdb.top066266 (2014)
  • K.C. Wedgwood, K. K. Lin, R. Thul, S. Coombes, Phase-amplitude descriptions of neural oscillator models, The Journal of Mathematical Neuroscience, 3 (2013)
  • M. D. Bootman, R. Thul, Calcium: A Local and Global Messenger in Cells, in V. N. Uversky, R. H. Kretsinger, E. A. Permyakov (eds), Encyclopedia of Metalloproteins, 493, (2013)
  • R. Thul, S. Coombes, M. D. Bootman, Persistence of pro-arrhythmic spatio-temporal calcium patterns in atrial myocytes: a computational study of ping waves, Frontiers in Computational Physiology and Medicine, 3, 279 (2012)
  • R. Thul, S. Coombes, H. L. Roderick, M. D. Bootman, Subcellular calcium dynamics in a whole-cell model of an atrial mycoyte, Proceedings of the National Academy of Sciences, 209, 2150 (2012)
  • K. Thurley, A. Skupin, R. Thul, M. Falcke, Fundamental properties of Ca2+ signals, Biochimica et Biophysica Acta, 1820, 1185 (2012)
  • T. Schendel, R.Thul, J. Sneyd, M. Falcke, How does the ryanodine receptor in the ventricular myocyte wake up: by a single or by multiple open L-type Ca2+ channels? European Biophysics Journal, 41, 27 (2012)
  • S. Coombes, R. Thul, K. C. A. Wedgwood, Nonsmooth dynamics in spiking neuron models, Physica D, 241, 2042 (2012)
  • S. Coombes, R. Thul, J. Laudanski, A. R. Palmer, C. J. Sumner, Neuronal spike-train responses in the presence of threshold noise, Frontiers in Life Sciences, 5, 91 (2011)
  • M. D. Bootman, I. Smyrnias, R. Thul, S. Coombes, H. L. Roderick, Atrial cardiomyocyte calcium signalling, Biochimica et Biophysica Acta, 1813, 922 (2011)
  • M. Fink, S. A. Niederer, E. M. Cherry, F. H. Fenton, J. T. Koivumäki, G. Seemann, R. Thul, H. Zhang, F. B. Sachse, D. Beard, E. J. Crampin, N. P. Smith, Cardiac cell modelling: Observations from the heart of the cardiac physiome project, Progress in Biophysics and Molecular Biology, 104, 2 (2011)
  • R. Thul, S. Coombes, Understanding cardiac alternans: a piecewise linear modelling framework, Chaos, 20, 045102 (2010)
  • R. Thul, S. Coombes, G.D. Smith, Sensitisation waves in a bidomain fire-diffuse-fire model of intracellular Ca2+ dynamics, Physica D, 238, 2142 (2009)
  • R. Thul, K. Thurley, M. Falcke, Torward a predictive model of Ca2+ puffs, Chaos, 19, 037108 (2009)
  • R. Thul, T. C. Bellamy, H. L. Roderick, M. D. Bootman, S. Coombes, Calcium Oscillations, in Miguel Maroto and Nick Monk (eds), Cellular Oscillatory Mechanisms, Springer, Advances in Experimental Medicine and Biology (2009)
  • R. Thul, G. D. Smith, S. Coombes, A bidomain threshold model of propagating calcium waves, Journal of Mathematical Biology, 56, 435 (2008)
  • R. Thul, M. Falcke, Waiting time distributions for clusters of complex molecules, Europhysics Letters, 79, 38003 (2007)
  • R. Thul, M. Falcke, Building oscillations bottom up: elemental time scales of intracellular calcium dynamics ,in L. Schimansky-Geier, B. Fiedler, J. Kurths and E. Schöll (eds), Analysis and Control of Complex Nonlinear Processes in Physics, Chemistry and Biology, World Scientific (2006)
  • R. Thul, M. Falcke, Frequency of elemental events of intracellular Ca2+ dynamics, Physical Review E, 73, 061923 (2006)
  • R. Thul, M. Falcke, Reactive clusters on a membrane, Physical Biology, 2, 51 (2005)
  • R. Thul, M. Falcke, Stability of membrane bound reactions, Physical Review Letters, 93, 188103 (2004)
  • R. Thul, M. Falcke, Release currents of IP3 receptor channel clusters and concentration profiles, Biophysical Journal, 86, 2660 (2004)
  • J. Bünemann, F. Gebhard, R. Thul, Landau-Gutzwiller quasiparticles, Physical Review B, 67, 075103 (2003)

Work with me

Surfing the calcium wave

Pineapple All of us know calcium. We get a healthy dose of it with every sip of milk – and doctors tell us that it is good for us. But there is another facet to it. Without calcium, our heart would stop beating, information would not be relayed in our brain and our pancreas would stop producing insulin. The reason for this is that calcium controls the behaviour of many cells in our body. For example, it determines the behaviour of millions of muscle cells in the heart, telling them to contract more than one billion times during an average human lifespan – which is what we perceive as a heartbeat.

The question that fascinates me is: how does calcium do all of this? It turns out that at the single cell level, the concentration of calcium changes over space and time. It can be in high concentration in only one part of a cell for a small period of time, or it can travel through a cell in the form of a wave –  similar to the rippling patterns that we observe when dropping a pebble in a pond. Depending on these different dynamics, cells exhibit different behaviour.

I construct and analyse mathematical models that describe the rich dynamical repertoire of intracellular calcium. I am interested in how cellular geometry shapes calcium signals, and how the random activation and inactivation of molecular switches further controls the intracellular calcium concentration. This requires the application of mathematical techniques from a variety of fields, including network theory. And sometimes we need to develop novel mathematical approaches to unravel the dynamics of this truly universal controller of cellular life.

Potential projects include

The eyes provide a window into the mind

eyetracking Eye-tracking technology can in part make this statement a reality. It allows us to monitor people’s eye-movements to determine what they are attending to and for how long while reading, which allows us to make inferences about how much cognitive effort they expend doing so. Importantly, eye-tracking provides a rich moment-to-moment data source that tells us that it takes us longer to read words that we are not familiar with, or when there are ambiguities.

One of the things that we can investigate with eye-tracking is the processing of formulaic language, which is the umbrella term for constructs such as idioms (“break the ice”), binomials (“salt and pepper”), collocations (“strong coffee”), etc. More than half of our daily discourse is made up formulaic language. Eye-tracking demonstrates that reading is faster for formulaic language (“break the ice” vs. “crack the ice”) and particularly for their final highly predictable word(s) – “the straw that broke…. the camel’s back”.

I have become interested in modelling reading behaviour. Given a text, I would like to know how we can predict where our eyes land, and how long we fixate a word. This research is conducted in close collaboration with Dr Kathy Conklin from the School of English, who is an expert in eye-tracking for reading research.

Potential projects include