Dynamical Systems in Neuroscience:

Synchronization (see www.izhikevich.com) 479membranepotential (mV)phasephasedeviationphasedifference200-20-40-60-80T0T0TV 1 V 22122 1100 5 10 15 20 25 30 35 40 45 50time (ms)Figure 10.25: The relationship between membrane potential oscillation of two neurons,V 1 (solid) and V 2 (dashed), their phases, phase deviations, and phase difference. Shownare simulation of two I Na + I K -models with parameters as in Fig. 10.3 and coupledsymmetrically via gap junctions 0.1(V j − V i ) (see Sect. 2.3.4).10.2.5 Phase model for coupled oscillatorsNow consider n weakly coupled oscillators of the formp i (t){ }} {n∑ẋ i = f i (x i ) + ε g ij (x i , x j ) , x i ∈ R m , (10.11)j=1and assume that the oscillators, when uncoupled (ε = 0), have equal free-runningperiods T 1 = · · · = T n = T . Applying any of the three methods above to such a weaklyperturbed system, we obtain the corresponding phase model˙ϑ i = 1 + ε Q i (ϑ i ) ·p i (t){ }} {n∑g ij (x i (ϑ i ), x j (ϑ j )) , (10.12)j=1where each x i (ϑ i ) is the point on the limit cycle having phase ϑ i . Note that (10.11) isdefined in R nm , whereas the phase model (10.12) is defined on the n-torus, denoted asT n .To study collective properties of the network, such as synchronization, it is convenientto represent each ϑ i (t) asϑ i (t) = t + ϕ i , (10.13)