Dynamical systems in the context of neuron models

In this module we analyze neuronal models using dynamical systems. We examine the case of continuous variables, with continuous and discrete time, and hybrid systems. The focus is on qualitative description and numerical exploration. In continuous variable systems, we study equilibria/fixed points, stability, bifurcations, periodic, homoclinic and heteroclinic orbits and dynamic bifurcations. We explain neuronal activity, in particular spiking and bursting, using these concepts. We introduce sophisticated numerical tools used for the computation and visualization of dynamic objects. Hybrid systems, combining continuous and discrete variables, are becoming an important modeling paradigm in neuroscience. One relevant context is when the continuous state approximation fails, when there are too few ion channels for continuous variable modeling, for example. The evolution of the discrete variable may be stochastic. We discuss hybrid systems in the context of neuroscience. Students learn and are able to apply the methods of dynamical systems in neuroscience.

C. Borgers "An introduction to modeling neuronal dynamics", Springer, New York, 2017, B. Ermentrout and D. Terman "Mathematical Foundations of Neuroscience", Springer, New York, 2010, E. Izhikievich "Dynamical Systems in Neuroscience", MIT Press, Boston 2007.