Dynamical Systems in Neuroscience:

Two-Dimensional Systems 101stableunstable4.1.3 Limit cyclesFigure 4.9: Limit cycles (periodic orbits).A trajectory that forms a closed loop is called a periodic trajectory or a periodic orbit(the latter is usually reserved for mappings, which we do not consider here). Sometimesperiodic trajectories are isolated, as in Fig. 4.9, sometimes they are part of a continuum,as in Fig. 4.13, left. An isolated periodic trajectory is called a limit cycle. The existenceof limit cycles is a major feature of two-dimensional systems that cannot exist in R 1 .If the initial point is on a limit cycle, then the solution (x(t), y(t)) stays on the cycleforever, and the system exhibits periodic behavior; i.e.,x(t) = x(t + T ) and y(t) = y(t + T ) (for all t)for some T > 0. The minimal T for which this equality holds is called the periodof the limit cycle. A limit cycle is said to be asymptotically stable if any trajectorywith the initial point sufficiently near the cycle approaches the cycle as t → ∞. Suchasymptotically stable limit cycles are often called limit cycle attractors, since they“attract” all nearby trajectories. The stable limit cycle in Fig. 4.9 is an attractor.The limit cycle in Fig. 4.10 is also an attractor; It corresponds to the periodic (tonic)spiking of the I Na,p +I K -model (4.1, 4.2). The unstable limit cycle in Fig. 4.9 is oftencalled a repeller, since it repels all nearby trajectories. Notice that there is always atleast one equilibrium inside any limit cycle on a plane.In Fig. 4.11 we depict limit cycles of three types of neurons recorded in vitro.Since we do not know the state of the internal variables, such as the magnitude of theactivation and inactivation of Na + and K + currents, we plot the cycles on the (V, V ′ )-plane, where V ′ is the time derivative of V . The cycles look jerky because of the poordata sampling rate during each spike.4.1.4 Relaxation OscillatorsMany models in science and engineering can be reduced to two-dimensional fast/slowsystems of the formẋ = f(x, y) (fast variable)