Dynamical Systems in Neuroscience:

72 One-Dimensional Systemsλ(V-Veq)VeqF(V)VFigure 3.21: Hartman-Grobman theorem:The non-linear system ˙V = F (V )is topologically equivalent to the linearone ˙V = λ(V −V eq ) in the local (shaded)neighborhood of the hyperbolic equilibriumV eq .our analysis to a small neighborhood of the line R, e.g., a neighborhood of the restingstate or of the threshold, and study behavior locally in this neighborhood.An important tool in the local analysis of dynamical systems is the Hartman-Grobman theorem, which says that a non-linear one-dimensional system˙V = F (V )sufficiently near an equilibrium V = V eq is locally topologically equivalent to the linearone˙V = λ(V − V eq ) (3.8)provided that the eigenvalueλ = F ′ (V eq )at the equilibrium is non-zero, i.e., the slope of F (V ) is non-zero. Such an equilibriumis called hyperbolic. Thus, nonlinear systems near hyperbolic equilibria behave as ifthere were linear, as in Fig. 3.21.It is easy to find the exact solution of the linearized system (3.8) with an initialcondition V (0) = V 0 . It is V (t) = V eq + e λt (V 0 − V eq ) (check by differentiating).If the eigenvalue λ < 0, then e λt → 0 and V (t) → V eq as t → ∞, so that theequilibrium is stable. Conversely, if λ > 0, then e λt → ∞ meaning that the initialdisplacement, V 0 − V eq , grows with the time, and the equilibrium is unstable. Thus,the linearization predicts qualitative dynamics at the equilibrium and quantitative rateof convergence/divergence to/from the equilibrium.If the eigenvalue λ = 0, then the equilibrium is non-hyperbolic, and analysis ofthe linearized system ˙V = 0 cannot describe the behavior of the nonlinear system.Typically, non-hyperbolic equilibria arise when the system undergoes a bifurcation,i.e., a qualitative change of behavior, which we consider next. To study stability, weneed to consider higher-order terms of the Taylor series of F (V ) at V eq .3.3.3 BifurcationsThe final and the most advanced step in the qualitative analysis of any dynamicalsystem is the bifurcation analysis. In general, a system is said to undergo a bifurcationwhen its phase portrait changes qualitatively. For example, the energy landscape inFig. 3.22 changes so that the system is no longer bistable. The precise mathematicaldefinition of a bifurcation will be given later.