Chapter 23 - Low-Dimensional Nonlinear Dynamics: Fixed Points, Limit Cycles, and Bifurcations

Abstract

In this chapter we introduce a number of characteristics associated with low-dimensional nonlinear dynamics, and we illustrate this type of nonlinear behavior using models inspired by neuronal excitability. First we introduce the analysis of nonlinear systems in continuous time as well as discrete time, and we review the typical behaviors that may be exhibited by these systems: trajectories governed by fixed points, limit cycles, or a chaotic attractor. Next we introduce bifurcations, a sudden change of a system's state caused by a change of its parameters. This chapter is not a comprehensive text on nonlinear dynamics and bifurcations, but rather an introduction to the concepts of the effect of nonlinearity and parameter changes in nonlinear systems and how this may affect the behavior of neuronal systems. The MATLAB® examples in this chapter are based on the application of one- and two-dimensional dynamics to modeling of neurons: linear and quadratic versions of the integrate-and-fire model (IF and QIF), and examples of reduced versions of the Hodgkin and Huxley formalism.