Multiple Timescale Dynamics in Neuroscience


In this module we will analyse neuronal models using dynamical systems approach with a specific emphasis on exploiting the multiple timescales that are naturally present (explicitly or implicitly) in neural activity models. After recalling elements of dynamical systems theory (equilibria, limit cycles, phase portrait), we will introduce more advanced tools (connecting orbits, bifurcations) that are pertinent to analysing neuronal dynamics and excitability. We will then introduce key concepts related to multiple-timescale ("slow-fast") systems, which are ubiquitous in neuronal models and useful for the classification of experimentally-observed neuronal oscillations. We will explain neuronal activity, in particular "spiking" and "bursting", using these concepts. In parallel, we will introduce sophisticated numerical tools allowing for the computation and visualisation of the dynamical objects studied in both biophysical and phenomenological neural models.
All computations will be done using the software package XPPAUT, which students should install on their machine. Hybrid systems, combining continuous and discrete variables, are becoming an important modeling paradigm in neuroscience, in particular for large-scale simulation within the framework of "integrate-and-fire (IF) neuron models". We will review simple IF models and study their excitability properties.
The module will end by a few case studies of neural activity models (in the context of epileptic seizure, firing rate models & population bursting, and sleep regulatory networks), where multipletimescale analysis can unveil key properties of the system, in link with experimental data.


  • Christoph Börgers, An introduction to modeling neuronal dynamics, Springer, New York, 2017, published version
  • G. Bard Ermentrout, Simulating, Analysing and Animating Dynamical Systems, SIAM, 2002, published version, online tutorial
  • G. Bard Ermentrout and David H. Terman, Mathematical Foundations of Neuroscience, Springer, New York, 2010, published version [Book available in the M4NC library]
  • Eugene M. Izhikevich, Dynamical Systems in Neuroscience: The geometry of excitability and bursting, MIT Press, Boston, 2007, draft version, published version [Book available in the M4NC library]
teacher face to face hours working hours ects
Mathieu Desroches (MathNeuro team, Inria Centre at UCA) 30 60 6