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Dynamical Systems Modelling:  2019-2020




This course is aimed at PRS and DPhil students only and is open to students from across the University who may find it intertesting for their Postgradaute research degree. The course is offered by Prof. Alan Garfinkel, Professor of Medicine (Cardiology) and Integrative Biology and Physiology UCLA. 2019-2020 Newton Abraham Visiting Professor, University of Oxford. Prof. Garfinkel can be contacted at

This course introduces the physiologist to the essential concepts of modeling and dynamics, with applications at various levels of physiology. The level of maths preparation is assumed to be modest.

Please email to register your interest in attending this course.


Week 1. The basic machinery. State spaces, vector fields, trajectories, attractors. Models as differential equations that generate vector fields. Trajectories, theoretical and computed. Equilibrium Points.  Stability and instability of equilibrium points. Principle of linear stability analysis in 1D.

Reading: Modeling Life (Chapters 1,3)


Week 2. Multiple equilibria. Phase plane methods. Systems with multiple equilibrium points (1D and 2D). Bistability and biological ‘switches’. How positive feedback creates bistable systems.

Reading: Modeling Life (Chapter 3)

Monod, J, and F Jacob. “General conclusions: teleonomic mechanisms in cellular metabolism, growth, and differentiation.” Cold Spring Harbor symposia on quantitative biology 1961

Ferrell Jr, JE. "Self-perpetuating states in signal transduction: positive feedback, double-negative feedback and bistability." Current Opinion Cell Biol 2002

Lisman, J. “A mechanism for memory storage insensitive to molecular turnover: A bistable autophosphorylating kinase’. PNAS 1985


Week 3. The saddle-node bifurcation. Mathematics of the saddle-node bifurcation. Applications: the commitment of pluripotent cells to specific lineages. Maturation of Oocytes as a saddle-node bifurcation

Reading:  Modeling Life (Chapter 3)

Ferrell, JE, and EM Machleder. "The biochemical basis of an all-or-none cell fate switch in Xenopus oocytes." Science 1998

Lai, K, MJ Robertson, and DV Schaffer. "The sonic hedgehog signaling system as a bistable genetic switch." Biophysical Journal 2004

Ahrends, R, Asuka O, KM Kovary, T Kudo, BO Park, and Mary N Teruel. "Controlling low rates of cell differentiation through noise and ultrahigh feedback." Science 2014.                  

Ferrell, JE, J R Pomerening, SY Kim, NB Trunnell, W Xiong, CY Frederick Huang, and EM Machleder. "Simple, realistic models of complex biological processes: positive feedback and bistability in a cell fate switch and a cell cycle oscillator." FEBS letters 2009

Gardner, T.S., Cantor, C.R. and Collins, J.J. “Construction of a genetic toggle switch in Escherichia coli’. Nature, 2000


Week 4. Negative feedback and oscillation. Oscillation in physiology. Beyond homeostasis to limit cycle attractors. Mechanisms of oscillation: role of steep negative feedback and time delays. Hopf bifurcation. Mathematical Theory. Negative feedback control of hormones. Respiratory control of CO2. Negative feedback control of gene expression.

Reading: Modeling Life (Chapter 4.1-4.3)

Mackey, Michael C, and Leon Glass. "Oscillation and chaos in physiological control systems." Science 1977

Lewis, Julian "Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator." Current Biology 2003

Hirata, H, S Yoshiura, T Ohtsuka, Y Bessho, T Harada, K Yoshikawa, and R Kageyama. "Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop." Science 2002

Boiteux, A., Goldbeter, A., & Hess, B. “Control of oscillating glycolysis ofyeast by stochastic, periodic, and steady source of substrate: a model and experimental study.” PNAS 1975


Week 5. The neuron. Theory behind modeling cell electrophysiology. Electrical circuits and cell membranes: the differential equations. The Fitzhugh model of the neuron: phase plane analysis. The Hodgkin Huxley model. Model reduction: from the Hodgkin-Huxley to the Fitzhugh equations. Bifurcations in neurons: periodic firing and bursting.

Reading: Modeling Life (Chapter 4.4)

Keener, J and J Sneyd Mathematical Physiology (Chapter 5)


Week 6. Chaos. Attractors of the third kind: chaotic attractors as models of disordered or irregular phenomena. The discrete time logistic model and routes to chaos. Dripping taps and cardiac arrhythmias. Diagnosing chaos in real world phenomena.

Reading: Modeling Life (Chapter 5)

May, R. M. “Simple mathematical models with very complicated dynamics” Nature 1976.

H Hayashi, S Ishizuka, M Ohta, and K Hirakawa, “Chaotic behavior in the Onchidium giant neuron under sinusoidal stimulation,” Physics Letters 1982


Week 7. Spatial Phenomena. Introduction to Partial Differential Equations. The gradient and diffusion operators. From an agent-based model of random walkers to a diffusion equation. Euler and Sophie Germain: modeling using PDEs. The PDE for electrical conduction in neural and cardiac tissue.

            Reading: DE2: But what is a partial differential equation?  3 blue1brown

Maini, Philip K., and Thomas E. Woolley. "The Turing Model for Biological Pattern Formation." The Dynamics of Biological Systems. Springer, 2019. 189-204.

vanCapelle FJL, Durrer D. “Computer simulation of arrhythmias in a network of coupled excitable elements.” Circ. Res. 1980



Week 8. The Turing bifurcation.  Emergence of Spatial Structure. Traffic jams, spotty arterial calcification, the formation of hair follicles, development of fingers. Prospects of the Turing paradigm in physiology.

            Reading: Garfinkel, A., Y. Tintut, D. Petrasek, K. Bostrom and L. L. Demer (2004). "Pattern formation by vascular mesenchymal cells." Proc Natl Acad Sci U S A

Sick S, Reinker S, Timmer J, Schlake T.WNT and DKK determine hairfollicle spacing through a reaction-diffusion mechanism.” Science. 2006

Raspopovic J, Marcon L, Russo L, Sharpe J. “Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients.”  Science. 2014

Guo, T. H. Chen, X. Zeng, D. Warburton, K. I. Bostrom, C. M. Ho, Zhao, A. Garfinkel, “Branching Patterns Emerge in a Mathematical Model of the Dynamics of Lung Development.” J Physiol 2014.

Maini, P. K., Woolley, T. E., Baker, R. E., Gaffney, E. A., & Lee, S. S. (2012). Turing's model for biological pattern formation and the robustness problem. Interface focus, 2(4), 487-496.


Note: The first 6 weeks of the course rely extensively on the text Modeling Life (Springer 2017). Free print copies will be handed out at the first lecture, and a free PDF can be downloaded at Springerlink from any Oxford IP address or VPN.