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Supplementary Materials 1 Supplementary Information 2 Theory of weakly coupled oscillators 3 Detailed reviews and mathematical descriptions of the theory, also its extensions and limitations, 4 can be found in a number of publications1–19. In essence, many oscillatory phenomena in the 5 natural world represent dynamic systems with a limit-cycle attractor2. Even though the 6 underlying system might be complex (e.g. a neuron or neural population), the dynamics of the 7 system can be reduced to a phase-variable if the interaction strength among oscillators is weak. 8 In general, coupled oscillators can interact through adjustments of their amplitudes and phases, 9 yet if interaction strength is weak, amplitude changes are small and play a minor role in the 10 oscillatory dynamics (no strong deviation from the limit-cycle) and only the adjustments of the 11 phase are essential for understanding the behaviour. The transformation of a complex system to a 12 phase variable, if valid, reduces the dimensions of the problem, thereby allowing exact 13 mathematical investigation. The manner by which mutually coupled oscillators adjust their 14 phases, either by phase-delay or phase-advancement, is described by the phase response curve 15 (also called phase resetting curve), the PRC7,9,14–16,19–21. The PRC is important, because if the 16 PRC of a system can be described, the synchronization behaviour of the system can be 17 understood and predicted. Many biological oscillators are inherently noisy or chaotic. Therefore, 18 it is important to take this variability of the dynamics into account for a better understanding of 19 biological data. Cortical neurons in vivo exhibit irregular spiking patterns24 and neural networks 20 oscillations also show significant variability over time25. This type of variation is referred to as 21 phase noise and is distinct from measurement noise with the latter being unrelated to the 22 dynamics of the system. 23 According to the theory of weakly coupled oscillators, the synchronization of two coupled 24 oscillators can be predicted from the forces they exert on each other as a function of their 25 instantaneous phase difference. This function is referred to as the phase-response curve (PRC). 26 Accordingly, the phase evolution of two given cortical V1 locations is reduced to: 27 28 1 112121121 ε H 29 2 2 2ε 12H 12 2 1 2 30 31 where φ1,2 is the phase, 1,2 its temporal derivative, ω1,2 is the preferred frequency, ε12 and ε21 are 32 the interaction strengths, H12 and H21 are the single PRCs and ƞ1,2 is a phase-noise term with ƞ1,2 33 ~ N(0, σ2) N being the normal distribution. The two equations, as given in the main text equation 34 1, can be further simplified to: 35 36 3 εG 37 1 38 where θ = φ1 –φ2 is the phase difference, ∆ω = ω1 – ω2 the detuning, εG(θ)= ε21H21(θ)-ε12H12(-θ) 39 the combined interaction term with ε being the interaction strength and G(θ) the mutual PRC 2 40 and ƞ= ƞ1 –ƞ2 the phase noise with η ~ N(0, 2σ ). 41 42 Analytical derivation of phase-locking and mean phase difference 43 Equation 3 is a stochastic differential equation (a Langevin equation) and was solved as 44 described in 1. Equation 3 can be rewritten in the form of a Fokker-Planck equation that has been 45 developed to give an analytical solution for the evolution of a probability distribution P of a 46 particle influenced by a drag force (first term on the right side of the equation) and a random 47 Gaussian noise process (second term). The drag force is here the combined systematic force of 48 detuning ∆ω and the interaction function εG(θ): 49 ∆ 50 4 51 52 The stationary (time-independent) solution of the Fokker-Planck equation which is: ′ 1 ′ 53 5 exp 54 ∆ ′ 1 ′ 55 6 exp 56 where C is a normalization constant defined by 1. V(θ) represents the influence 57 of systematic force as a function of phase difference. is the phase difference probability 58 distribution and describes how likely a particular phase difference is to occur. A uniform 59 distribution means that every phase difference is equally likely and the oscillator are hence 60 asynchronous. If the distribution approximates a delta distribution (meaning only one phase 61 difference has non-zero probability), then the oscillators are completely synchronized. All other 62 distributions in between signify intermittent (partial) synchronization (also called cycle slipping 63 or phase walk-through, 1,2). To quantifying the narrowness of the distribution, we use the phase- 64 locking value (the mean resultant vector length,3) defined here as: 65 7 | | 66 Further, we were also interested in the mean phase difference, also described as the preferred 67 phase difference, defined here as: 68 8 arg 2 69 A phase difference between oscillators in neural networks implies spike timing differences. It has 70 been shown that spike-timing is an important characteristic in addition to spike synchrony 4–9. 71 72 General behaviour of the model in term of PLV and mean phase difference 73 In Fig.S1 the model’s behaviour is illustrated as a function of detuning ∆ω and interaction 74 strength ε. To understand how the PLV and the mean phase difference change, it is illustrative to 75 consider the noise-free case first. In the noise-free case one needs to solve the equation 3 for 76 zero-points (root or equilibrium points), meaning that the time derivative of the phase difference 77 is zero, ( 0, i.e. zero frequency difference). Let us assume here for illustration that G(θ) is a 78 sine function, as is for example used for the Kuramoto model 10,11. It then becomes (also called 79 the Adler equation, 1): 80 9 0 ∆ 81 The zero-point or the equilibrium is stable if the derivative of the PRC is negative (stable if 82 < 0). The sin function has two zero crossing (at 0 and π with Δω =0), but only one is stable. 83 In the case ε = 0 (no interaction), only the condition of ∆ω = 0 (no detuning) can lead to 84 synchrony. If ε = 0 and ∆ω ≠ 0 (no interaction with detuning), the oscillators will be 85 asynchronous and there will be linear phase precession with the speed determined by the ∆ω 86 value. If ε > 0 (there is interaction), then an equilibrium can only exist if the detuning is not 87 stronger than the interaction strength, denoted by |∆ω|<= ε. This is because for equilibrium the 88 detuning needs to be counter-balanced by the interaction. In the case of |∆ω|> ε (detuning 89 exceeding interaction strength), the interaction cannot counterbalance the detuning and the 90 oscillators start to phase precess. 91 If the oscillators do phase precess but are coupled (ε>0), the phase-precession is non-linear. The 92 precession rate is determined by the detuning (∆ω), the modulation shape (G(θ)), and the 93 modulation amplitude (ε). In this case, the oscillators are in the intermittent synchronization 94 regime. The transition point ∆ω = ε defines the border between high PLV values and low PLV 95 values and defines the borders of the so-called Arnold tongue, the synchronization region in the 96 ∆ω- ε parameter space 1,2. The larger the ε, the larger the detuning can be for oscillators to still 97 synchronize. This leads to a triangle shape of the synchronization region in the ∆ω- ε space as 98 illustrated in Fig.S1A. The mean (preferred) phase difference can also be derived from equation 99 6. For reaching equilibrium the detuning ∆ω and interaction term (εsin(θ)) need to be 100 counterbalanced. For different detuning ∆ω values the counterbalance will depend on different 101 phase difference values of the interaction term εsin(θ). It can be represented graphically 102 (Fig.S1B,2) as the intersection of a horizontal line (∆ω) with the graph of εsin(θ). Notice that the 103 stronger the interaction strength ε is, the smaller the slope of the detuning-to-phase difference 104 translation becomes. Outside of the Arnold tongue (∆ω> ε), there will be phase precession yet 105 with a preference (minimal precession rate) for a phase difference where the PRC is min or max. 106 Phase noise has important effects on the synchronization behavior1. Strictly speaking, the 107 condition of complete synchrony for noisy oscillators does not exist as θ’ cannot remain stable 108 all the time (there will always be small fluctuations). If ∆ω+ƞ < ε, meaning that in spite of 109 frequency variability detuning can be counterbalanced by interaction strength, then oscillators 110 remain close to the equilibrium point and they have high phase-locking. Yet, when noise is 3 111 larger, it is likely that the interaction term cannot counterbalance all the time and the oscillators 112 will phase precess from time to time. However, also in this case, the transition point ∆ω = ε still 113 determines the Arnold tongue border and outlines the transition between the regions of high 114 phase-locking and regions of low phase-locking, but the transition is smoother1,2. In Fig.S1 we 115 show the Arnold tongue in terms of PLV and mean phase-difference for coupled oscillators 116 having moderate levels of white noise (σ=15Hz). 117 118 Biophysical modeling of coupled gamma-generating neural networks 119 To demonstrate that the results from the phase-oscillator equations are generalizable to more 120 biophysically realistic neuronal network oscillations12, we simulated two coupled excitatory- 121 inhibitory spiking neural networks generating pyramidal-interneuron gamma (PING,13) 122 oscillations.
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