Anti-phase collective synchronization with intrinsic in-phase coupling of two groups of electrochemical oscillators

Abstract

The synchronization of two groups of electrochemical oscillators is investigated during the electrodissolution of nickel in sulfuric acid. The oscillations are coupled through combined capacitance and resistance, so that in a single pair of oscillators (nearly) in-phase synchronization is obtained. The internal coupling within each group is relatively strong, but there is a phase difference between the fast and slow oscillators. The external coupling between the two groups is weak. The experiments show that the two groups can exhibit (nearly) anti-phase collective synchronization. Such synchronization occurs only when the external coupling is weak, and the interactions are delayed by the capacitance. When the external coupling is restricted to those between the fast and the slow elements, the anti-phase synchronization is more prominent. The results are interpreted with phase models. The theory predicts that, for anti-phase collective synchronization, there must be a minimum internal phase difference for a given shift in the phase coupling function. This condition is less stringent with external fast-to-slow coupling. The results provide a framework for applications of collective phase synchronization in modular networks where weak coupling between the groups can induce synchronization without rearrangements of the phase dynamics within the groups.

This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.

1. Introduction

There are various examples of biological and mechanical systems which are composed of nonlinear units, and, through their interactions, a variety of coherent collective dynamical states emerge [1–6]. The collective behaviour of such systems is derived from the individual dynamics of each element in the system, the unavoidable heterogeneities of the elements across the population and the type of interactions. Therefore, a fundamental problem is revealing how the dynamics of the components give rise to the collective dynamics. Analysis of emerging synchronization can be greatly facilitated by a phase model approach [1,2,7,8]. The oscillators are described by a single variable, the phase, and a model can be constructed by determining the phase responses to perturbations related to the coupling [2,7,9]. Then, usually by summations of pairwise interactions, the emergence of synchronization can be characterized for globally coupled elements and complex networks [4,10–13].

In addition to being a powerful theoretical tool, phase models can also be derived from direct experiments [9]. Several techniques exist for such an approach, e.g. determining the phase response curve and oscillation waveform [9,14], direct measurement of the instantaneous frequencies and phase differences of two weakly coupled oscillators [15–26], one oscillator with a delayed self-feedback [27–29] or direct reconstruction from oscillator populations [30–32]. The extracted phase models can be used to predict synchronization patterns: the formation of cluster [9,14,33,34] and chimera states [35,36], anomalous phase synchronization [37], echo behaviour [38] and the effect of time delay of self-feedback on multi-rhythmicity [39] were analysed using experiment-based phase models. Phase models also facilitated the design of synchronization patterns with closed- and open-loop techniques [26,40,41]; for example, stable [25,27] and slow switching [26] cluster states, desynchronization [26,42] and chimera [28] states were designed with a polynomial delayed feedback, and waveforms for external entrainments were obtained for wide Arnold tongue regions [43,44], fast entrainment [45], desynchronization [46] and phase assignment [47].

Many networks are modular in structure [12]: there are domains that can be considered as single units, and the domains can exhibit synchronization patterns. For example, the brain of a cat can be classified into 53 functional units [48]. The dynamical behaviours exhibited by interacting groups of globally coupled phase oscillators have been intensively investigated [49–58]. The appearance of the Ott–Antonsen ansatz [59–61] has considerably facilitated theoretical investigations on interacting groups of noiseless non-identical phase oscillators with global sinusoidal coupling. To study the phase synchronization between groups, a theory was formulated for the collective phase description emerging from coupled phase oscillators for coherent states in globally coupled noisy identical oscillators [62–64], partially phase-locked states in globally coupled noiseless non-identical oscillators [65] and fully phase-locked states in networks of coupled noiseless non-identical oscillators [66–68]. The theory can describe the dynamics with the use of a collective phase coupling function, which determines the dynamics of the collective phase difference between the groups. The theory plays an important role in exploring how intrinsic dynamical properties can result, in some cases, in quite counterintuitive collective-level behaviour. For example, with interaction that results in in-phase synchronization between two oscillators and a one-cluster synchronized state for a globally coupled population, anti-phase collective oscillations were predicted for two groups of globally coupled populations [63–65,67].

In this paper, inspired by the phase model methodology for collective synchronization, we explore whether two weakly coupled groups of electrochemical oscillators can produce anti-phase collective synchronization with intrinsic in-phase interactions. First, the theory [66] is adapted to two groups of oscillators with two elements in each group. The groups consist of a fast and a slow oscillator. The conditions for anti-phase collective synchronization are determined in terms of the phase difference between the fast and slow oscillators in the groups and the shift in the phase coupling. As a new theoretical contribution, the importance of the links between the fast and slow oscillators between the two groups is analysed. Second, experiments are performed with four electrochemical oscillators arranged in two groups of two elements. The experiments analyse the group phase difference for isochronous (phase coupling without shift) and non-isochronous (phase coupling with shift) interactions, and for comparison of all-to-all external coupling vs. coupling between the fast and slow oscillators only.

2. Experimental methods

The experiments were performed with four nickel wires, which exhibit oscillatory metal dissolution. The schematic of the experimental set-up is shown in figure 1a. A standard electrochemical cell was used with 3 M H₂SO₄ as the electrolyte. The counter electrode was a platinum-coated titanium rod, the reference electrode was Hg/Hg₂SO₄/sat. K₂SO₄ and the working electrode was an array of four 1.00 mm diameter nickel wires embedded in epoxy. The temperature was held at 10°C. Each nickel electrode in the array was connected to the potentiostat (ACM Instruments GillAC) through an individual resistance (R_ind); at the beginning of the experiment, R_ind = 1 k ohm. A constant potential (V ) was applied by the potentiostat and the oscillatory current of each electrode in the array was digitized with a data acquisition rate of 200 Hz. The oscillation arises through a Hopf bifurcation; the circuit potential (V ) was set to about 20 mV above the bifurcation point. Groups 1 and 2 consisted of wires (1, 2) and (3, 4), respectively.

Figure 1.

Schematics of the experimental set-up and the network. (a) Experimental set-up with four Ni working electrodes (1 − 4), a Pt counter (CE) and a Hg/Hg₂SO₄/sat. K₂SO₄ reference (RE) electrode. The Ni wires are connected to the working point of a potentiostat (WE) through individual resistors (R_ind). The wires are coupled; internal (strong) and external (weak) coupling are induced by coupling resistors R_i and R_e, respectively. (b) General schematic of the network topology of two weakly interacting groups of two phase oscillators. The internal and external couplings are represented by the solid and dotted arrows, respectively. Superscript is group number and subscript is element number of the particular group. In the experiments, ϕ⁽¹⁾₁, ϕ⁽¹⁾₂, ϕ⁽²⁾₁ and ϕ⁽²⁾₂ correspond to the phase of the current oscillations measured for wires 1, 2, 3, 4, respectively. The coupling strength K corresponds to the conductance through the internal coupling resistance (R_i). The external coupling corresponds to the conductance of the external coupling resistance (R_e). (Online version in colour.)

The natural frequency of each electrode was adjusted by its individual resistance, setting two electrodes (1 and 3) to a high natural frequency (by removing 50 ohm from R_ind) and the remaining two electrodes (2 and 4) to a low natural frequency (by adding 50 ohm to R_ind). The electrodes were coupled into two groups of two, a high and low frequency in each, using the cross-resistance method; a coupling resistance was inserted between two electrode pairs [17]. The strength of interaction, the coupling strength (K), was the inverse of the coupling resistance. The two groups were obtained by two different strengths of couplings. Internal coupling (K_i, coupling between wires 1 − 2 and 3 − 4) was achieved by coupling resistors R_i, with coupling strength K_i = 1/R_i. External coupling (K_e) was obtained with coupling resistors R_e, and thus K_e = 1/R_e. (These experimentally calculated coupling strengths have units of mS and can be considered proportional to the coupling strengths in the phase model.) The external coupling can be added in the all-to-all configuration as shown in figure 1a, or with only selected electrodes, e.g. external coupling between only the high- and low-frequency elements can be achieved by adding R_e between wires 1 − 4 and 2 − 3.

Weak coupling through the resistances can be described with a phase model with a sinusoidal interaction function (isochronous coupling) [17]. Non-isochronicity (i.e. phase shift in the interaction function) can be achieved through the delay of the coupling current by the addition of capacitance parallel to coupling resistances [17]. The level of non-isochronicity induced by the capacitance in a pair of oscillators was studied in great detail in a previous publication [17]. With a coupling time constant RC ≈ 2 s, the phase shift was α =  − 0.97. Therefore, for non-isochronous coupling, parallel capacitors C_i and C_e were applied, such that C_iR_i = C_eR_e ≈ 2.0 s. Further experimental details on coupling the electrochemical oscillators with cross resistances and capacitances can be found in the earlier publication [17].

3. Results and discussions

(a). Theoretical results

We consider two weakly interacting groups of two phase oscillators described by the following equation (see figure 1b):

$${\dot{\varphi }}_j^{(\sigma )}(t)={\omega }_j+K\mathit{\Gamma }({\varphi }_j^{(\sigma )}-{\varphi }_k^{(\sigma )})+ϵK_j\mathit{\Gamma }({\varphi }_j^{(\sigma )}-{\varphi }_j^{(\tau )})+ϵK_3\mathit{\Gamma }({\varphi }_j^{(\sigma )}-{\varphi }_k^{(\tau )}),$$ 3.1

for (j, k) = (1, 2), (2, 1) and (σ, τ) = (1, 2), (2, 1), where ϕ^(σ)_j(t) is the phase of the j-th oscillator at time t in the σth group and ω_j is the natural frequency of the j-th phase oscillator. The two groups are identical, i.e. the oscillators with the same value of j have the same parameters, independent of the group index, σ. Γ is the intrinsic phase coupling function. The second term on the right-hand side of equation (3.1) represents the internal coupling within the same group, while the third and fourth terms represent the external coupling between the different groups. The characteristic intensity of the external coupling is denoted by ϵ≥0. When the external coupling is absent, i.e. ϵ = 0, equation (3.1) is assumed to have the following stable phase-locked solution:

$${\varphi }_j^{(\sigma )}(t)={\mathit{\Theta }}^{(\sigma )}(t)+{\psi }_j,$$ 3.2

where Θ^(σ)(t) is the collective phase at time t for the σ-th group and the constants ψ_j represent the relative phases of the individual oscillators for the fully phase-locked state. In this state, the phases of the oscillators (and, consequently, the collective phases) increase with time at a rate of Ω, the collective frequency. When the external coupling is sufficiently weak, i.e. ϵ≪1, the phase arrangement of the elements within each group remains the same, but the collective phases can adjust; we can approximately derive a collective phase equation in the following form [66]:

$${\dot{\mathit{\Theta }}}^{(\sigma )}(t)=\mathit{\Omega }+ϵ\gamma ({\mathit{\Theta }}^{(\sigma )}-{\mathit{\Theta }}^{(\tau )}).$$ 3.3

Here, the collective phase coupling function γ(Θ) is given by the following equation:

$$\gamma \left(\mathit{\Theta }\right)=U_1^{\ast }[K_1\mathit{\Gamma }\left(\mathit{\Theta }\right)+K_3\mathit{\Gamma }(\mathit{\Theta }+{\psi }_1-{\psi }_2)]+U_2^{\ast }[K_2\mathit{\Gamma }\left(\mathit{\Theta }\right)+K_3\mathit{\Gamma }(\mathit{\Theta }+{\psi }_2-{\psi }_1)],$$ 3.4

where the left zero eigenvector U*_j is normalized as

$$U_1^{\ast }+U_2^{\ast }=1.$$ 3.5

More details and other applications of this collective phase description method are given in [66–68].

Now, we consider the special case in which K₁ = K₂ and K₃ = 0. In this case, as found from equations (3.4) and (3.5), the collective phase coupling function is essentially the same as the intrinsic phase coupling function as follows:

$$K_1=K_2,K_3=0,\gamma \left(\mathit{\Theta }\right)=K_1\mathit{\Gamma }\left(\mathit{\Theta }\right).$$ 3.6

That is, anti-phase collective synchronization can never occur in this case.

Hereafter, we consider the case of the following intrinsic sinusoidal phase coupling function corresponding to the experiment [17]:

$$\mathit{\Gamma }\left(\varphi \right)=-\sin (\varphi +\alpha )+\sin \alpha ,\left|\alpha \right|<\frac{\pi }{2},$$ 3.7

which results in in-phase synchronization with a pair of oscillators and satisfies Γ(0) = 0; the latter property is not essential for results. First, the phase difference of the stable phase-locked solution is given by

$$\sin (\mathrm{\Delta }\psi )=\eta ,\mathrm{\Delta }\psi \equiv {\psi }_1-{\psi }_2,\left|\mathrm{\Delta }\psi \right|<\frac{\pi }{2},\eta \equiv \frac{{\omega }_1-{\omega }_2}{2K\cos \alpha },\left|\eta \right|<1.$$ 3.8

Second, the left zero eigenvector is given by

$$U_1^{\ast }=\frac{1+\tan (\alpha )\tan (\mathrm{\Delta }\psi )}{2}\mathrm{and}{U}_2^{\ast }=\frac{1-\tan (\alpha )\tan (\mathrm{\Delta }\psi )}{2}.$$ 3.9

Finally, the collective phase coupling function is given by

$$\begin{array}{rl}\gamma \left(\mathit{\Theta }\right)& =K_1{U}_1^{\ast }\mathit{\Gamma }\left(\mathit{\Theta }\right)+K_2{U}_2^{\ast }\mathit{\Gamma }\left(\mathit{\Theta }\right)+K_3[U_1^{\ast }\mathit{\Gamma }(\mathit{\Theta }+\mathrm{\Delta }\psi )+U_2^{\ast }\mathit{\Gamma }(\mathit{\Theta }-\mathrm{\Delta }\psi )]\\ & \equiv -\rho \sin (\mathit{\Theta }+\delta )+k\sin \alpha .\end{array}$$ 3.10

Therefore, the collective phase coupling function can be found from the following equation:

$$\rho \cos \delta =K_1\frac{\cos (\alpha )+\sin (\alpha )\tan (\mathrm{\Delta }\psi )}{2}+K_2\frac{\cos (\alpha )-\sin (\alpha )\tan (\mathrm{\Delta }\psi )}{2}+K_3\frac{{\cos }^2(\alpha )-{\sin }^2(\mathrm{\Delta }\psi )}{\cos (\alpha )\cos (\mathrm{\Delta }\psi )}.$$ 3.11

We note that ρcosδ > 0 and ρcosδ < 0 indicate stable in-phase and anti-phase collective synchronization, respectively. On the basis of equation (3.11), we study the following four representative cases.

(i). Isochronous coupling

We consider the isochronous case

$$\alpha =0,\rho \cos \delta =\frac{K_1+K_2}{2}+K_3\cos (\mathrm{\Delta }\psi ).$$ 3.12

In this case, anti-phase collective synchronization can never occur.

(ii). All-to-all external coupling

With all-to-all external coupling K₁ = K₂ = K₃,

$$K_1=K_2=K_3,\rho \cos \delta =K_1\frac{(1+\cos (\mathrm{\Delta }\psi )){\cos }^2(\alpha )-{\sin }^2(\mathrm{\Delta }\psi )}{\cos (\alpha )\cos (\mathrm{\Delta }\psi )},$$ 3.13

which corresponds to the case studied in Ref. [67]. In this case, anti-phase collective synchronization can occur. As found from equation (3.13), the stable anti-phase synchronization condition is given by the following inequality:

$${\cos }^2(\alpha )<\frac{{\sin }^2(\mathrm{\Delta }\psi )}{1+\cos (\mathrm{\Delta }\psi )}.$$ 3.14

The phase diagram associated with equation (3.14) is shown in figure 2a. For example, in the experiment, the phase shift is α =  − 0.97; therefore, the internal phase difference Δψ should be larger than 0.82 rad for anti-phase collective synchronization between the groups.

Figure 2.

Coupling topology (top) and phase diagram (bottom) for in- and anti-phase collective synchronization. The phase diagram shows the collective synchronization in the |α| versus Δψ parameter space. (a) All-to-all external coupling. Phase diagram according to equation (3.14). (b) External coupling between high- and low-frequency elements. The transition line is given by the following equation: |α| + |Δψ| = π/2. In the network topologies, triangles and circles denote high- and low-frequency elements, respectively. (Online version in colour.)

(iii). External coupling between high-to-high- and low-to-low-frequency elements only

When coupling can only occur between elements of the same natural frequencies K₁ = K₂ and K₃ = 0,

$$K_1=K_2,K_3=0,\rho \cos \delta =K_1\cos (\alpha ),$$ 3.15

which corresponds to equation (3.6). In this case, anti-phase collective synchronization can never occur.

(iv). External coupling between the high- and low-frequency elements only

When the external coupling occurs between the high- and low-frequency elements only, K₁ = K₂ = 0. We note that this coupling topology is also equivalent to a ring network with alternating natural frequencies and coupling strengths along the ring,

$$K_1=K_2=0,\rho \cos \delta =K_3\frac{{\cos }^2(\alpha )-{\sin }^2(\mathrm{\Delta }\psi )}{\cos (\alpha )\cos (\mathrm{\Delta }\psi )},$$ 3.16

which represents the external coupling between the faster oscillator in the σ-th group and the slower oscillator in the τ-th group. In this case, stable anti-phase collective synchronization can occur. As found from equation (3.16), the effective anti-phase coupling condition is given by the following inequality:

$${\cos }^2(\alpha )<{\sin }^2(\mathrm{\Delta }\psi ).$$ 3.17

That is, the transition line is given by the following equation:

$$|\alpha |+|\mathrm{\Delta }\psi |=\frac{\pi }{2}.$$ 3.18

The phase diagram associated with equation (3.17) is shown in figure 2b. When the phase shift is α =  − 0.97, the internal phase difference Δψ should be larger than 0.60 rad for anti-phase collective synchronization between the groups. Note that the required internal phase difference for this coupling topology is less than that required for all-to-all coupling (0.82 rad).

Therefore, the theory reveals the importance of connections between the high- and low-frequency elements in the external coupling for anti-phase collective synchronization. With coupling only between the high-to-high- and low-to-low-frequency elements, anti-phase collective synchronization cannot occur. With coupling through high- and low-frequency elements, anti-phase synchronization occurs in a relatively large parameter region. In all-to-all external coupling, i.e. with coupling through high-to-high-, low-to-low- and high-to-low-frequency elements, anti-phase synchronization can occur but in a smaller parameter region than with coupling through high- and low-frequency elements (e.g. compare figure 2a,b).

(b). Experimental results

The experiments aim to find the counterintuitive anti-phase collective synchronization behaviour following the guidelines of the theoretical results.

According to the theory, the anti-phase behaviour of the collective dynamics is observed when both α and Δψ are large. The phase shift for the experiments is set to α =  − 0.97, which, according to equation (3.14), means that the phase difference between the elements in each group needs to be maintained between the interval 0.82 < |Δψ| < π/2 for all-to-all external coupling. The phase differences are established by keeping the difference of the natural frequencies similar for each group and setting the internal coupling K_i such that (without external coupling) the internal phase difference (Δψ) falls within this interval. Lastly, the weak external coupling between the groups was added, K_e = ϵK_i, to yield the required topology. The group phases are denoted ϑ_G1 and ϑ_G2, respectively, and calculated from the angle of the group order parameter

$${\vartheta }_{\mathrm{G}\sigma }=\arg \left(\frac{1}{2}\sum _{j=1}^2\exp \left(i{\varphi }_j^{(\sigma )}\right)\right),$$ 3.19

where σ = (1, 2). The group phase difference, Δϑ = ϑ_G1 − ϑ_G2, was measured in the experiments to show the presence of the anti-phase synchrony between the groups. When collective synchronization was established, Δϑ was equivalent to Θ⁽¹⁾ − Θ⁽²⁾ in the theory.

The experimental approach is as follows. First, we consider a globally coupled population (ϵ = 1) with or without a phase shift in the intrinsic coupling function. Then, the effect of weakening the external coupling is demonstrated (ϵ < 1). Finally, the features of collective synchronization are compared for the links in all-to-all and the high- to low-frequency configurations.

(i). Uniform coupling

To emphasize the importance of network behaviour and to eliminate trivial anti-phase synchronization in the experiments, first uniform, all-to-all coupling (K_i = K_e, ϵ = 1) was considered. The coupling schematic is depicted in figure 3a. (Group 1 and 2 elements are represented with filled and hollow nodes, respectively, for the high-frequency (triangles) and low-frequency (circles) elements.) This topology corresponds to K = K₁ = K₂ = K₃ and ϵ = 1 in the theory.

Figure 3.

In-phase collective synchronization with uniform global coupling (K_i = K_e). (a) Depiction of the coupling topology, where triangles represent high-frequency elements and circles represent low-frequency elements. (b,c) Global resistive coupling; V = 1095 mV, K_i = 0.04 mS, α = 0. (b) The group phase difference for 45 cycles. Dashed lines are the anti-phase threshold. (c) Current time series, where colour distinguishes the groups while the solid and dash-dot lines indicate the high- and low-frequency elements, respectively. (d,e) Global resistive and capacitive coupling; V = 1105 mV, K_i = 0.0068 mS, C_i = 13.3 μF, α = − 0.97. (d) The group phase difference for 45 cycles. (e) Current time series. (Online version in colour.)

Without any coupling, the natural frequency difference between the high- and low-frequency elements was 29 mHz. With this heterogeneity, the internal coupling K_i = 0.04 mS induced synchronization within the groups with Δψ₁ = Δψ₂ = 0.7 rad without any external coupling.

Upon adding all-to-all external coupling K_e = K_i = 0.04 mS (with α = 0, i.e. purely resistive coupling), in-phase collective synchronization was observed with a constant group phase difference of Δϑ = 0.09 rad (figure 3b). For easier visual determination of in- or anti-phase behaviour, a threshold was set at π/2 (dashed lines); the group phase difference thus clearly indicates the nearly in-phase collective synchronization. The inset in the figure is a snapshot of the phase for each individual oscillator; the high- and low-frequency elements of both groups are practically the same, hence the near-zero group phase difference. A current time series is shown in figure 3c; the similar frequency elements have nearly identical variations in each group. (The groups are delineated by colour, solid lines are the high-frequency and the dash-dot lines are the low-frequency oscillators.) Note that the high-frequency oscillators have larger amplitudes; the change of individual resistance, which resulted in the large natural frequency difference, also impacted the amplitude of the oscillations. Since the coupling was all-to-all uniform, the synchronization pattern simply reflects the natural frequencies of the elements. Elements with similar natural frequency establish similar phases. The system thus synchronized to clusters based on natural frequencies with a constant phase difference between the fast and slow elements.

When uniform coupling was added with non-isochronicity (α =  − 0.97, i.e. through a combination of a resistance and a capacitance), in-phase synchronization of the collective dynamics was observed with a constant group phase difference of Δϑ = 0.24 rad (figure 3d). In this experiment, the natural frequency difference between the high and low frequencies prior to coupling was 18 mHz, which provided initial phase differences Δψ₁ = 0.7 rad and Δψ₂ = 0.6 rad. The inset snapshot of the phases shows that, regardless of the non-isochronicity, the system exhibited a similar behaviour to the isochronous case with the group phase of the two groups in phase synchronized while the fast and slow frequency elements exhibited similar phases. A plot of the current time series also shows this similarity (figure 3e).

Uniform coupling, as expected, generates synchronization patterns with nearly zero group phase difference. In this example, the collective synchronization is similar to the intrinsic synchronization of the oscillations.

(ii). All-to-all external coupling

Weakened all-to-all external coupling can be applied by changing ϵ such that K_i > K_e, as depicted in figure 4a,b. The internal coupling strength is stronger than the external. This situation corresponds to K = K₁ = K₂ = K₃ and ϵ < 1 in the theory.

Figure 4.

In- and anti-phase collective synchronization with weak coupling between the groups (K_i≫K_e). (a) Depiction of the coupling topology where triangles represent high-frequency elements and circles represent low-frequency elements. Line thickness represents coupling strength. (b,c) Weak resistive coupling between groups; V = 1095 mV, K_i = 0.10 mS, K_e = 0.005 mS, α = 0. (b) The group phase difference for 45 cycles. Dashed lines are the anti-phase threshold. (c) Current time series, where colour distinguishes the groups while the solid and dash-dot lines indicate the high- and low-frequency elements, respectively. (d,e) Weak resistive and capacitive coupling between groups; V = 1105 mV, K_i = 0.0182 mS, K_e = 6.25 × 10⁻⁴ mS, C_i = 36.4 μF, C_e = 1 μF, α =  − 0.97. (d) The group phase difference for 45 cycles. (e) Current time series. (Online version in colour.)

In this coupling configuration with isochronicity (α = 0), the natural frequency difference between the high and low frequencies was 30 mHz, and Δψ₁ = 0.7 rad, Δψ₂ = 0.8 rad. With external coupling, ϵ = 0.05, the group phase difference was Δϑ = 0.89 rad, as shown in figure 4b. While the group phase difference is larger than the uniform coupling case, it is still below the threshold value for anti-phase behaviour at ± π/2 rad (dashed lines). The inset snapshot of phases shows that the initial group phase differences are well maintained, but the group 1 oscillators are somewhat ahead of the corresponding group 2 oscillators. The current time series shows a similar constant phase difference between the two fast frequency and the two slow frequency elements of the respective groups (figure 4c). At weaker K_e, the system is desynchronized and the two groups exhibit phase slips; at stronger K_e, the system further synchronized, reducing the group phase difference. Again, as expected, from the theory, anti-phase collective synchronization behaviour was not observed.

With non-isochronous coupling (α =  − 0.97), the natural frequency difference between the high and low frequencies was found to be 20 mHz prior to the coupling, while the group phase differences were Δψ₁ = Δψ₂ = 0.8 rad. The system was observed to synchronize at ϵ = 0.034, with a group phase difference of Δϑ = 2.3 rad (figure 4d); this phase difference is above the threshold for anti-phase synchronization (|Δϑ| > π/2). The inset snapshot of phases shows that the two groups are positioned on opposing sides of the phase space, indicative of anti-phase behaviour. The current time series in figure 4e displays the well-separated synchronized peaks. For stronger coupling than those shown in figure 4d,e, both the group phase differences and the internal phase differences in the groups decreased. A decrease in external coupling resulted in phase slips between the two groups.

Therefore, signatures of the phenomenon of anti-phase collective synchronization, predicted by the theory, were observed in the experiments; as expected, large phase difference (>π/2) could only be observed in the experiments with weak cross connections in the presence of the non-isochronous coupling.

(iii). External coupling through high- and low-frequency elements only

The theory predicted that all-to-all cross coupling of the elements in the two populations is not required to observe the anti-phase collective behaviour. The phenomenon could exist as long as coupling exists between the high- and low-frequency elements. For two groups with two elements in each group, this translates to a network for a ring of elements, where the natural frequencies and the coupling strengths alternate between high and low values (figure 5a). In the theory, this corresponds to K₃ = ϵK, K₁ = K₂ = 0. Prior to coupling (α =  − 0.97), the natural frequency difference was 21 mHz, and the phase differences within the groups were Δψ₁ = 0.7 rad and Δψ₂ = 0.6 rad. With a cross coupling of ϵ = 0.037, the group phase difference was Δϑ = 2.7 rad (figure 5b), which is very close to the anti-phase configuration. The inset snapshot of the phases and the current time series (figure 5c) clearly depicts the nearly anti-phase positioning. The very robust anti-phase synchronization could be attributed to the removal of coupling between the similar frequency connections, which would shepherd the system towards in-phase collective behaviour.

Figure 5.

Anti-phase collective synchronization on a ring. V = 1095 mV, K_i = 0.0172 mS, K_e = 6.25 × 10⁻⁴ mS, C_i = 34.5 μF, C_e = 1 μF, α = − 0.97. (a) Depiction of the coupling topology where triangles represent high-frequency elements and circles represent low-frequency elements. Line thickness represents coupling strength. (b) Current time series, where colour distinguishes the groups while the solid and dash-dot lines indicate the high- and low-frequency elements, respectively. (c) The inter-group phase difference for 45 cycles. Dashed lines are the anti-phase threshold. (Online version in colour.)

4. Conclusion

The experiments thus confirm that two pairs of electrochemical oscillators with strong internal and weak external coupling can exhibit anti-phase collective synchronization when non-isochronicity is present. One important feature of the experiment is that, in each group, the natural frequencies of the oscillations are different. With internal coupling, there is a phase difference between the oscillations within each group; this phase difference allows the collective synchronization to differ from the intrinsic one. Such a phase difference typically occurs close to the onset of synchronization in the given example. Therefore, the anti-phase collective synchronization phenomenon, in the studied configuration, is a property of the system close to the synchronization transition due to the internal coupling.

Other requirements include the presence of a phase shift in the interaction function. In the experiments, the intrinsic phase coupling functions have been quantified in a large parameter space [17]. These studies allowed us to find experimental conditions (e.g. total resistance, circuit potential) and coupling type (combination of resistance and capacitance) where non-trivial collective synchronization could occur. We have chosen a circuit potential where, with coupling through the resistance, the intrinsic interaction function is predominantly a sinusoidal wave. Then, the coupling signal was delayed with a capacitance to obtain a phase-shifted coupling function. In other systems (or in the same system with different conditions [17]), the phase shift can be an intrinsic property of the interactions.

Finally, we showed that external coupling between fast and slow oscillators facilitates the anti-phase entrainment. Intuitively, one would think that coupling among fast and slow oscillators brings the phases together. However, because the fast oscillators are typically ahead of the slow oscillators, the coupling attempts to align the phases of fast and slow oscillators in the different groups, which supports different phases between the two fast (and slow) oscillators, and thus the collective phase difference. While the importance of fast-to-slow links was identified with two groups of two oscillators, similar phenomena could be expected for large populations; here additional effects could impact the behaviour, e.g. the shape of the probability density function of the natural frequency distribution and the correlations between link strengths with natural frequencies.

The anti-phase collective synchronization observed in the experiment is a network-induced behaviour. Two coupled oscillators [17] or a globally coupled population (figure 3) would exhibit in-phase synchronization. However, network interactions created anti-phase synchronization patterns. While the experiments were performed with electrochemical oscillators, interacting groups of global oscillators have been experimentally realized using other system types, e.g. with Belousov–Zhabotinsky beads with an optical feedback [69] and mechanical oscillators [70]. Therefore, collective synchronization effects in these other systems could be important. Applications could be especially enticing in brain dynamics, where neural cross-frequency coupling functions have been derived [32]. In this example, one challenge is that the coupling function can exhibit time dependence [71]. Theoretical development is needed to better characterize experiment-based phase models; promising approaches include the introduction of amplitude dependence [72] or factoring in slow relaxation processes resulting in changes in natural frequencies [73].

Data accessibility

Current time series data presented in the figures are available on Figshare (doi:10.6084/m9.figshare.8080964).

Authors' contributions

I.Z.K. and Y.K. conceived the study. Y.K. carried out the theoretical component of the research, with feedbacks from I.Z.K. and M.S. I.Z.K., Y.K. and M.S. designed the experiments. M.S. and A.M.N. performed all the experiments. M.S. analysed the data and created the figures. I.Z.K. and Y.K. prepared the first draft of the manuscript. All authors provided critical feedback and helped shape the research, analysis and manuscript.

Competing interests

We declare we have no competing interests.

Funding

This material is based upon the work supported by the National Science Foundation under grant no. CHE-1465013 and 1900011. Y.K. acknowledges financial support from JSPS (Japan) KAKENHI under grant nos. JP18H03205, JP17H03279 and JP16K17769.