Dynamical Systems in Neuroscience:

488 Synchronization (see www.izhikevich.com)rre iFigure 10.33: The Kuramoto synchronization index(10.21) describes the degree of coherence inthe network (10.20).10.3.4 Mean-field approximationsSynchronization of the phase model (10.15) with randomly distributed frequency deviationsω i can be analyzed in the limit n → ∞, often called thermodynamic limit byphysicists. We illustrate the theory using the special case, H(χ) = sin χ (Kuramoto1975)ϕ ′ i = ω i + K n∑sin(ϕ j − ϕ i ) , ϕ i ∈ [0, 2π] , (10.20)nj=1where K > 0 is the coupling strength and the factor 1/n ensures that the modelbehaves well as n → ∞. The complex-valued sum of all phases,re iψ = 1 nn∑e iϕ j(Kuramoto synchronization index), (10.21)j=1describes the degree of synchronization in the network. The parameter r is oftencalled order parameter by physicists. Apparently, the in-phase synchronized stateϕ 1 = · · · = ϕ n corresponds to r = 1, with ψ being the population phase. In contrast,the incoherent state with all ϕ i having different values randomly distributed on theunit circle, corresponds to r ≈ 0. (The case r ≈ 0 can also correspond to two or moreclusters of synchronized neuron, oscillating anti-phase or out-of-phase and cancelingeach other). Intermediate values of r correspond to a partially synchronized or coherentstate, depicted in Fig. 10.33. Some phases are synchronized forming a cluster, whileothers roam around the circle.Multiplying both sides of (10.21) by e −iϕ iand considering only the imaginary parts,we can rewrite (10.20) in the equivalent formϕ ′ i = ω i + Kr sin(ψ − ϕ i ) ,which emphasizes the mean-filed character of interactions between the oscillators: Theyare all pulled into the synchronized cluster (ϕ i → ψ) with the effective strength proportionalto the cluster size r. This pull is offset by the random frequency deviationsω i , which pull away from the cluster.