G14TNS: Theoretical Neuroscience

Starts Tue 22nd, 2pm in Coates A1. All welcome. Timetable.

S Coombes


The aim of this module is to study the dynamics of neural networks using the techniques of dynamical systems theory. The emphasis will be on the modern approach to modelling the single neuron and to introduce the mathematical techniques appropriate for analysing the rich behaviour of systems of interacting neurons.


On completion of this module, students should be able to analyse mathematical models of single neurons and neural systems using phase plane analysis, bifurcation theory, the method of averaging and geometric singular perturbation theory.

Contents and notes | Reading list | Problem sheets | ODE files | Neural web sites



XPP Tutorial and Worsksheet

  1. The Single Neuron

    Hodgkin-Huxley model

    Nonlinear integrate-and-fire models

    Phase response curves

    1. Models of the single neuron.
    2. Mathematical reductions and canonical models.
    3. Analysis of excitable and oscillatory behaviour using dynamical systems techniques.
    4. Phase response curves and isochronal coordinates.
    5. Mode-locking to periodic stimuli.

  2. Neural Systems

    Resonant dendrites

    Weak gap junction coupling

    1. Models of the synapse and dendrite.
    2. Return maps for pulse-coupled oscillators.
    3. Networks of interacting phase oscillators.
    4. Rhythm generation in biological central pattern generators.
    5. Travelling waves in neural systems.
    6. Firing rate models, population models and mean field descriptions for large networks.
    7. Pattern formation in neural systems and drug-induced visual hallucinations.

Reading list


Ermentrout, G. B. and Terman, D. H. (2010). Mathematical Foundations of Neuroscience, Springer.
Wilson, H. (1999). Spikes, Decisions, and Actions: The Dynamical Foundations of Neuroscience, Oxford University Press.
Hoppensteadt F. C. and Izhikevich, E. (1997) Weakly connected neural networks, Springer.
Izhikevich, E (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting,The MIT Press.
Koch, C. (1999). The Biophysics of Computation, Oxford University Press.


Keener, J. and Sneyd, S. (1999) Mathematical Physiology, Springer.
Murray, J. D. (1993) Mathematical Biology (2nd edition), Springer-Verlag.
Nicholls, J. G., Martin, A. R., Wallace, B. G. and Fuchs, P. A. (2001) From Neuron to Brain (fourth edition) Sinauer
Strogatz, S. (1994). Nonlinear Dynamics and Chaos. Addison-Wesley.

Problem Sheets

  1. Problem sheet 1
  2. Problem sheet 2
  3. Problem sheet 3
  4. Problem sheet 4
  5. Problem sheet 5
  6. Problem sheet 6
  7. Problem sheet 7
  8. Problem sheet 8

ODE files

The Hodgkin-Huxley model The integrate-and-fire model Two pulse-coupled integrate-and-fire neurons
The reduced Hodgkin-Huxley model The integrate-and-fire-or-burst model Two synaptically-coupled integrate-and-fire neurons
The FitzHugh-Nagumo model The Izhikevich model Two synaptically-coupled Hodgkin-Huxley neurons
A cortical model Square Wave Bursting Two gap-junction Hodgkin-Huxley neurons
The Pinsky-Rinzel model The McCormick model A Half Center oscillator model
The Wang-Buzsaki model The Morris-Lecar model

Neural web sites

A Virtual Brain Synapse web
Society for Neuroscience The Human Brain Project
The Journal of Neurophysiology online Neuroscience for Kids
Neural Computation Neuroscience on the internet
The Journal of Neuroscience online Scholarpedia


Hodgkin, A. L., Huxley, A. F., and Katz, B. (1952).
Measurement of current-voltage relations in the membrane of the giant axon of Loligo
J. Physiol. 116: 424-448
Full text in PDF format.

Hodgkin, A. L., and Huxley, A. F. (1952a).
Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo
J. Physiol. 116: 449-472
Full text in PDF format.

Hodgkin, A. L., and Huxley, A. F. (1952b).
The components of membrane conductance in the giant axon of Loligo.
J. Physiol. 116: 473-496
Full text in PDF format.

Hodgkin, A. L., and Huxley, A. F. (1952c).
The dual effect of membrane potential on sodium conductance in the giant axon of Loligo
J. Physiol. 116: 497-606
Full text in PDF format.

Hodgkin A.L., and Huxley A.F. (1952d).
A quantative description of membrane current and its application to conduction and excitation in nerve
J. Physiol. 117: 500-544
Full text in PDF format.

S Coombes Jan 2008