Dynamical Systems in Neuroscience:

One-Dimensional Systems 71F (V)1? F (V)2VVFigure 3.20: Two “seemingly alike”dynamical systems ˙V = F1 (V ) and˙V = F 2 (V ) are not topologicallyequivalent, hence they do not havequalitatively similar dynamics. (Thefirst system has three equilibria, whilethe second system has only one.)3.3.1 Topological equivalencePhase portraits can be used to determine qualitative similarity of dynamical systems.In particular, two one-dimensional systems are said to be topologically equivalent whenthe phase portrait of one of them treated as a piece of rubber can be stretched orshrunk to fit the other one, as in Fig. 3.19. Topological equivalence is a mathematicalconcept that clarifies the imprecise notion of “qualitative similarity”, and its rigorousdefinition is provided, e.g., by Guckenheimer and Holmes (1983).The stretching and shrinking of the “rubber” phase space are topological transformationsthat do not change the number of equilibria or their stability. Thus, twosystems having different number of equilibria cannot be topologically equivalent, hencethey have qualitatively different dynamics, as we illustrate in Fig. 3.20. Indeed, the topsystem is bistable because it has two stable equilibria separated by an unstable one.The evolution of the state variable depends on which attraction domain the initial conditionis in initially. Such a system has “memory” of the initial condition. Moreover,sufficiently strong perturbations can switch it from one equilibrium state to another.In contrast, the bottom system in Fig. 3.20 has only one equilibrium, which is a globalattractor, and the state variable converges to it regardless of the initial condition. Sucha system has quite primitive dynamics, and it is topologically equivalent to the linearsystem (3.1).3.3.2 Local equivalence and the Hartman-Grobman theoremIn computational neuroscience, we usually face quite complicated systems describingneuronal dynamics. A useful strategy is to substitute such systems by simpler oneshaving topologically equivalent phase portraits. For example, both systems in Fig. 3.19are topologically equivalent to ˙V = V − V 3 (please, check this), which is easier to dealwith analytically.Quite often we cannot find a simpler system that is topologically equivalent to ourneuronal model on the entire state line R. In this case, we make a sacrifice: we restrict