Introduction to Mathematical Neuroscience: neuronal models and their bifurcations

DATES: 18 - 22 June 2018 (5 sessions)
TIME: 10:00 - 12:00 (a total of 10 hours)

This mini-course proposes an introduction to dynamical systems modelling in neuroscience. Special care will be given to numerical aspects so that every theoretical aspect introduced along the way will be also studied through computer simulations showcased during each sessions with the aid of the software package XPPAUT [1]. Theoretical aspects will be introduced around neuronal models and will aim to initiate the analysis of their excitability properties.

PROGRAMME (numerical aspects in blue)
1. Neuronal modelling:
Basic circuit theory, Nernst potential, ion channels, the Hodgkin-Huxley model, its 2D reductions (Rinzel, Morris-Lecar) and main 2D caricature, the FitzHugh-Nagumo system (FHN);
Introduction to XPPAUT and its functionalities.
2. Aspects of the qualitative theory of dynamical systems:
Phase portraits, equilibria, stability, periodic orbits, separatrices;
Simulations of ODE models, tracing nullclines, finding equilibria and their stability boundaries.
3. Codimension-one bifurcations:
Fold, pitchfork, transcritical, Hopf and homoclinic;
Phase portraits, numerical continuation & its use to compute bifurcation diagrams.
4. Planar neuronal slow-fast systems (1 fast and 1 slow variables):
Excitability, type I/II neurons, slow and fast subsystems and their application to spiking solutions.
Computing excitability thresholds as "canards", neuronal response to an applied current I, f-I curves.
5. Bursting neurons as 3D slow-fast systems (2 fast and 1 slow variables):
Bistability, dynamic bifurcations, slow-fast dissection, Rinzel & Izhikevich´s classifications, study of the Hindmarsh-Rose (HR) system;
Numerical bifurcation study of several bursters (HR, Morris-Lecar-Terman model, Plant model).

PREREQUISITES
Linear Algebra, Differential Equations.

REFERENCES
[1] G. B. Ermentrout, Simulating, analysing and animating dynamical systems: a guide to XPPAUT for researchers and students, SIAM, 2008.
[2] G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, Springer, 2010.
[3] E. Izhikevich, Dynamical Systems in Neuroscience: the Geometry of Excitability and Bursting, The MIT Press, 2007.
[4] L. Perko, Differential Equations and Dynamical Systems, 3rd Edition, Springer-Verlag, 2000.

*Registration is free, but inscription is required before 13th June: So as to inscribe send an e-mail to reception@bcamath.org. Student grants are available. Please, let us know if you need support for travel and accommodation expenses.

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