How synchronized human networks escape local minima

Spreading the phase

The first dynamic we observe is when the players are spreading their phases to escape from the in-phase local minimum into a vortex-phase state global minimum. This dynamic is enabled by the players’ ability to reduce the coupling between them until they find a stable state.

Figure 2a(top) presents the measured phase of N = 8 players situated on a ring with unidirectional coupling as a function of time t and delay Δt = t/30. Fig. 2(a)(bottom) presents the measured phase difference between each player and its delayed neighbor. As seen, during the first 3 seconds all phase difference converges to near zero, indicating that all players are synchronized in phase, and remain so until t = 8s. This first stage is marked as stage I. The average phase of each player during this stage, presented as the blue circles in the inset of Fig. 2a, confirms that all the players have nearly the same phase. When the delay is further increased, one of the players starts to ignore its neighbor, indicated by the linearly increasing blue curve in the phase difference in Fig. 2a(bottom). Thus, the other players can freely change their phase to follow the increasing delay between them. This is marked as stage II. The phase of the players spread until reaching a total phase difference of 2π between the first and the last player, thus forming a stable vortex solution in the first vortex order³⁹, shown as the black circles in Fig. 2a. This vortex solution remains stable as the delay is further increased, indicated by a constant phase difference between all coupled neighbors (marked as stage III). The average phase of all players in stage III is shown as red circles in the inset of Fig. 2a, and verifies the expected linear phase of the vortex state.

a (top) Measured phase of each player as a function of time as we increase the coupling delay between the players. Each color denotes the phase of a player according to the inset scheme. Black circles denote the vortex order of the system as a function of time. (bottom) The difference between the phase of each player and the phase of its delayed neighbor. (inset) The phase of all players in a state of in-phase synchronization (stage I, blue circles) and the phase of all players in a vortex state (stage III, red circles). b The effective potential of the system as a function of time and coupling delay. In stage II the effective coupling strength is reduced (see text) thus eliminating the potential barrier between the in-phase and the vortex states to enable the system to reach the global minimum vortex state. The trajectory of the system is denoted by the white dashed curve.

Following^19,20,23,24, we use the Kuramoto model for N oscillators on a ring with unidirectional time-delayed coupling^45,46,47 and uniform distributed random frequency ω_n to introduce an effective potential whose local and global minima dictate the players’ dynamics. The phase of each oscillator φ_n(t) follows:

with κ, and Δt denoting the strength and delay of the coupling. The periodic boundary conditions, φ_i+N = φ_i dictate \(\mathop\sum _n=1^N\Delta \varphi _n=0\), where Δφ_n(t) = φ_n+1(t) − φ_n(t). Assuming uniformity Δφ_n = Δφ, we obtain:

where the effective potential governing the dynamics of the system is:

where Ω = ∑(ω_n+1 − ω_n)/N, and ω = ∑ω_n/N. The same derivation holds for non-uniform phase differences, as long as two adjacent phase differences are similar. The detailed analytical derivation is presented in Supplemental Materials Section S1.1.

The calculated effective potential for constant tempo is shown in Fig. 1d as a function of Δt and Δφ. For Δt = 0 it reveals a global minimum at Δφ = 0, where all oscillators have the same frequency and phase, namely, an in-phase state of synchronization. However, when the delay increases beyond ωΔt > π/(2N), the Δφ = 0 becomes a local minimum and the vortex state emerges as a new global minimum at Δφ = − 2π/N. Between the in-phase state and the vortex state, there is a potential barrier. Therefore, increasing the delay adiabatically (slowly), so the network remains in the in-phase state of synchronization, transfers the system to a local minimum.

To account for the players’ dynamics observed in Fig. 2, we assume that the coupling strength in stage II is reduced by the player’s tendency to ignore frustrating inputs. Specifically, we assume \(\kappa (t)=\kappa (t=0)\,\cos ^2(\Delta tN\pi /T)\). The resulting modified effective potential is shown in Fig. 2b. The dashed curve follows the system trajectory in the phase space. The system starts in an in-phase state of synchronization and as the delay increases, the coupling drops leading to a lower potential barrier. Thus, the system can evolve into the vortex state. Finally, the coupling increases back to its original value κ(t = 0).

When the delay between the players is further increased, the first-order vortex becomes unstable and the next-order vortex becomes stable. We observed such multiple transitions to higher-order vortex states with N = 16 coupled violin players on a unidirectional ring. The measured phases of all the players together with the vortex order of the network are shown in Fig. 3a. In the inset, we show the phase of the players at four representative times, 1.5, 13.5, 24, and 34 seconds, with vortex orders n = 0, 1, 2, and 3, respectively. As the players’ phases spread over a wider range, the network reaches a higher vortex state, denoted by a higher vortex order. We also calculate the effective potential as a function of time, again assuming \(\kappa (t)=\cos ^2(\Delta tN\pi /T)\), showing how the system can evolve from the in-phase to each order of vortex.

a The measured phase of N = 16 coupled violin players together with the vortex order of the network as we increase the delay of the coupling between them, showing higher-order vortex states. (Inset) the phase of the 16 players at t = 1.5, 13.5, 24, and 34 s, with vortex order n = 0, 1, 2, and 3, respectively. b The effective potential of the system as a function of time and coupling delay shows how the system evolves.

Slowing the tempo

The second dynamic we observe is the slowing down of the players’ tempo as an alternative strategy to maintain a stable in-phase synchronization state in the presence of delayed coupling⁴⁹. The measured phase results as a function of time for six coupled players are shown in Fig. 4a. From these results, we evaluate the tempo of the players by calculating the average derivative of the phase. As seen, the initial (natural) tempo of 60 bpm slows down significantly for long delay times reaching about 6 bpm for Δt > 1s. This slowing down lets players maintain the in-phase state of synchronization by keeping the coupling delay small relative to the tempo.

a Experimental measurements of the phase and average tempo as we increase the delayed coupling between them. Each color denotes the phase of a player according to the inset scheme. b Numerical calculations of the system. c The effective potential of the system as a function of time when the tempo of the players is following Eq. (5) indicating that the in-phase state stays the global minima for arbitrary large delays.

To quantitatively analyze the tempo slowing, we resort again to the Kuramoto model^45,46,47. Assuming Δt < T/N, we can expand φ_n(t − Δt) ≈ φ_n(t) − Δt∂φ_n/∂t and \(\sin (\alpha )\approx \alpha\) to obtain from Eq. (1) the phase difference as a function of time for Δφ_N−1:

where Ω is the average tempo difference between the players. Then we obtain,

where φ = ∑φ_n/N is the average phase of all oscillators. Thus, the tempo of the coupled oscillators slows down as long as the players stay phase-locked, ensuring that the condition Δt < T/N is satisfied and indicating that our assumptions are valid. Using Eq. (5), to fit the measured tempo (blue curve in Fig. 4a) yields excellent agreement, where the fit parameters ω = 0.29Hz and κ = 0.63 are consistent with our system. The detailed analytical derivation is presented in the Supplemental Materials Section S1.2.

Next, We perform numerical simulations of the Kuramoto model (Eq. (1)) for six coupled players with a coupling strength of κ = 0.6. The calculated phase of all the players as a function of time/delay together with the average tempo are shown in Fig. 4b. As evident, the exact numerical simulations agree with the measured experimental results as well as with the analytical approximation of Eq. (5) (blue curve in Fig. 4b).

Finally, We calculate the effective potential as a function of time with the tempo obtained from Eq. (5). As evident, the in-phase state of synchronization remains the global minimum even though the coupling delay increases. Therefore, the system follows the dashed line and remains in the in-phase synchronization state for all coupling delays.

Oscillation death and amplitude death

In this section, we present two additional mechanisms that have been observed in other networks on coupled nonlinear oscillators: oscillation death⁴⁸ and amplitude death⁵⁰, and show how both enable the human network to escape from a local minimum (in-phase synchronization) into a global one (vortex state).

When the tempo slows too much, the players can get stuck in a state of oscillation death⁴⁸ where all the players are playing the same note indefinitely, thereby maintaining a degenerate form of synchronization. Representative results of such oscillation death for four coupled violin players are shown in Fig. 5a. The four players are slowing down their tempo, and after 40 s they all play the same note for 20 seconds. Then, the players spontaneously revive the oscillation when one of the players stopped following its neighbor. Reviving oscillations after oscillation death typically requires external perturbation to the system⁵⁰, but here we demonstrate that human networks can revive the oscillations spontaneously. In addition, they revive the oscillations into the stable vortex state as indicated by the vortex order which jumps to n = 1 when the oscillations revive.

a Coupled violin players reach a state of oscillation death. The players slow down their tempo until all the players play the same note. After 20 seconds, the players revive the oscillation spontaneously into a vortex state. Each color denotes the phase of a player according to the inset scheme. b Coupled violin players show the amplitude death of one player. The upper graph shows the phase of all the players together with the vortex order as a function of time. The lower graph shows the playing amplitude of Player 1 as a function of time. We see that Player 1 stops playing around t = 5s until the system finds the stable first-order vortex state.

Violin players can adjust not just the phase and tempo but also their playing amplitude. In highly frustrating situations one or more of the players can reduce their amplitude significantly, known as amplitude death⁵⁰. Representative measurements of such amplitude death are shown in Fig. 5b. Here, the coupling delay of four violin players is increased until one of the players, denoted as Player 1, cannot synchronize with Player 4 leading to frustration. Therefore, this player stops playing, as evidenced by its nearly vanishing amplitude, shown in the lower graph in Fig. 5b. During this time, the player did produce some noise, but no notes were detected. When one of the players stops playing, the closed ring switches into an open ring topology where all the other players are free to shift their phases according to the coupling delay. In an open ring, the players are not limited by the periodic boundary conditions, so the network is stable for any value of delay. After a few seconds, they find the first-order vortex state, and Player 1 resumes playing.