Dynamical Systems in Neuroscience:

482 Synchronization (see www.izhikevich.com)frequency lockingin-phaseentrainment(1:1 frequency locking)synchronizationphase lockinganti-phaseFigure 10.28: Various degrees of locking of oscillators.The coupled oscillators above are said to be frequency locked when there is a periodictrajectory on the 2-torus, which is called a torus knot. It is said to be of type (p, q)if ϑ 1 makes p rotations while ϑ 2 makes q rotations, and p and q are relatively primeintegers, i.e., do not have a common divisor greater than 1. Torus knots of type (p, q)produce p:q frequency locking, e.g., 2:3 frequency locking in Fig. 10.27. A 1:1 frequencylocking is called entrainment. There could be many periodic orbits on the torus, withstable orbits between unstable ones. Since the orbits on the 2-torus cannot intersect,they all are knots of the same type, resulting in the same p:q frequency locking.Let us follow a trajectory on the torus and count the number of rotations of thephase variables. The limit of the ratio of rotations as t → ∞ is independent on thetrajectory we follow, and it is called the rotation number of the torus flow. It is rationalif and only if there is a (p, q) periodic orbit, in which case the rotation number is p/q.An irrational rotation number implies there are no periodic orbits, and it corresponds toa quasi-periodic or multifrequency torus flow. Oscillators exhibit phase drifting in thiscase. Denjoy (1932) proved that such coupled oscillators are topologically equivalentto the uncoupled system ˙ϑ 1 = r, ˙ϑ2 = 1 with irrational r.Suppose the oscillators are frequency locked; that is, there is a p : q limit cycleattractor on the torus. We say that the oscillators are p : q phase locked ifqϑ 1 (t) − pϑ 2 (t) = conston the cycle. The value of the constant determines whether the locking is in-phase(const= 0), anti-phase (const= T/2, half-period), or out-of-phase. Frequency lockingdoes not necessarily imply phase locking: The (2, 3) torus knot in Fig. 10.27b correspondsto phase locking, whereas that in Fig. 10.27c does not. Frequency lockingwithout phase locking is called phase trapping. Finally, synchronization is a 1:1 phaselocking. The phase difference ϑ 2 − ϑ 1 is also called phase lag or phase lead. Therelationships between all these definitions are shown in Fig. 10.28.Frequency locking, phase locking, entrainment, and synchronization of a networkof n > 2 oscillators is the same as pair-wise locking, entrainment, and synchronization