Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 4590.60.5V-nullclineK + activation gate, n K + activation gate, n0.40.30.20.100.60.50.40.30.20.1n-nullcliney 0y(t 1 )x(t 1 )x 0isochrony(t 3 )y(t 2 )x(t 3 )x(t 2 )0-80 -70 -60 -50 -40 -30 -20 -10 0 10 20membrane potential, V (mV)Figure 10.2: Top: An isochron, or a stable manifold, of a point x 0 on the limit cycleattractor is the set of all initial conditions y 0 such that y(t) → x(t) as t → +∞.Bottom: Isochrons of the limit cycle attractor in Fig. 10.1 corresponding to 40 evenlydistributed phases nT/40, n = 1, . . . , 40.phase space R 2 in Fig. 10.1b given by ϑ ↦→ x(ϑ).We could put the initial point x 0 corresponding to the zero phase anywhere else onthe limit cycle, and not necessarily at the peak of the spike. The choice of the initialpoint introduces an ambiguity in parameterizing the phase of oscillation. Differentparametrizations, however, are equivalent up to a constant phase shift, i.e., translationin time. In the rest of the chapter, ϑ always denotes the phase of oscillation, theparameter T denotes the period of oscillation, and ϑ = 0 corresponds to the peak ofthe spike unless stated otherwise. If the system has two or more co-existing limit cycleattractors, then a separate phase variable needs to be defined for each attractor.10.1.2 IsochronsThe phase of oscillation can also be introduced outside the limit cycle. Consider, forexample, point y 0 in Fig. 10.2, top. Since the trajectory y(t) is not on a limit cycle,