Dynamical Systems in Neuroscience:

500 Synchronization (see www.izhikevich.com)the jump. Let τ 1 (d) be the time needed to slide from point b ′ to point d ′ , i.e., thephase difference after the jump. The phase difference between the oscillators duringthe jump changes by the factor C(d) = τ 1 (d)/τ 0 (d) called the compression function.The difference decreases when the compression function C(d) < 1 uniformly for all dnear the left knee a. This condition has a simple geometrical meaning: The rate ofchange of y(t) is slower before the jump than after it, so that y(t) has a “scalloped”shape, as in Fig. 10.40c. As an exercise, prove that C(d) → |g(a)|/|g(b ′ )| as d → a.If the compression function at the right (upper) knee is also less than 1, then thein-phase synchronization is stable. Indeed, the phase difference does not change whilethe oscillators slide along the nullclines, and it decreases geometrically with each jump.In fact, it suffices to require that the product of compression factors at the two kneesbe less than 1, so that any expansion at one knee is compensated by even strongercompression at the other knee.10.4.5 Bursting oscillatorsLet us consider bursting neurons coupled weakly through their fast variables:ẋ i = f(x i , y i ) + εp(x i , x j ) , (10.27)ẏ i = µg(x i , y i ) , (10.28)i = 1, 2 and j = 2, 1. Since bursting involves two time scales, fast spiking and slowtransition between spiking and resting, there are two synchronization regimes: spikesynchronization and burst synchronization, illustrated in Fig. 9.51 and discussed inSect. 9.4. Below we outline some useful ideas and methods of studying both regimes.Our exposition is not complete, but it rather lays the foundation for a more detailedresearch program.Spike synchronizationTo study synchronization of individual spikes within the burst, we let µ = 0 to freezethe slow subsystem (10.28) and consider the fast subsystem (10.27) describing weaklycoupled oscillators. When y i ≈ y j , the fast variables oscillate with approximately equalperiods, so (10.27) can be reduced to the phase model˙ϕ i = εH(ϕ j − ϕ i , y i ) ,where y i = const parameterize the form of the connection function. For example,during the ”circle/Hopf” burst, the function is transformed from H(χ) = sin 2 χ or1 − cos χ at the beginning of the burst (saddle-node on invariant circle bifurcation)to H(χ) = sin χ at the end of the burst (supercritical Andronov-Hopf bifurcation).Changing y i slowly, one can study when spike synchronization appears and when itdisappears during the burst. When the slow variables y i have different values, fastvariables typically oscillate with different frequencies, so one needs to look at low-orderresonances (see next section) to study the possibility of spike synchronization.