Dynamical Systems in Neuroscience:

70 One-Dimensional Systems+ - - + + - - + + - - + + -VFunction F(V)Attraction domainsVPhase PortraitFigure 3.18: Phase portrait of a one-dimensional system ˙V = F (V ).F (V)1VF (V)2VFigure 3.19: Two “seemingly different”dynamical systems ˙V = F1 (V ) and˙V = F 2 (V ) are topologically equivalent,hence they have qualitativelysimilar dynamics.3.3 Phase PortraitsAn important component in the qualitative analysis of any dynamical system is reconstructionof its phase portrait. For this one depicts all stable and unstable equilibria(as black and white circles respectively), representative trajectories, and correspondingattraction domains in the system’s state/phase space, as we illustrate in Fig. 3.18. Thephase portrait is a geometrical representation of system dynamics. It depicts all possibleevolutions of the state variable and how they depend on the initial state. Looking atthe phase portrait, one immediately gets all important information about the system’squalitative behavior without even knowing the equation for F .