Figure 1

Waveform and response function of the oscillations in metal dissolution experiments. (a) Electrode potential $e(t)=V-i(t)R$ as a function of phase $[\varphi (t)]$ for a smooth oscillator; $\varphi (t)=2\pi t/T$, where $T=2.26\mathrm{s}$ is the period, $t$ is the time from the maximum of the oscillation, and $V=1.105\mathrm{V}$ is the circuit potential. (b) Response function (or infinitesimal phase response curve) ($Z$) of a smooth oscillator at $V=1.105\mathrm{V}$. The response function is obtained from the phase response curve ($\Delta \Phi$ vs $\varphi$). Phase advance ($\Delta \Phi$) of the oscillations due to a short (8 ms) perturbation of the circuit potential (with an amplitude of $-300\mathrm{mV}$, resulting in $\Delta e=40\mathrm{mV}$ change in electrode potential) as a function of phase was obtained. The phase advance was calculated from the inherent (free-running) period and the period observed during the perturbation cycle ($T_{\mathrm{pert}}$): $\Delta \Phi =2\pi (1-T_{\mathrm{pert}}/T)$. The response function is $Z(\varphi )=\Delta \Phi (\varphi )/\Delta e$. (c) Electrode potential as a function of phase for a relaxation oscillator ($V=1.275\mathrm{V}$, $T=2.77\mathrm{s}$). (d) Response function ($Z$) of a relaxation oscillator ($V=1.275\mathrm{V}$).Reuse & Permissions

Figure 2

Experimental interaction function and its prediction of mutual entrainment. (a)–(c) Interaction function $H(\Delta \varphi )$ (left panel) and its odd part (right panel) at $V=1.105\mathrm{V}$ (smooth oscillator), 1.245, and 1.275 V (relaxation oscillator), respectively. ($K=1$.) (d) Stability of the balanced clusters (${\lambda }_{\max }$) as a function of the circuit potential. Solid circle: one-cluster. Open square: two-cluster. ${\lambda }_{\max }$ is the largest (nonzero) value of eigenvalues ${\lambda }_p$ of the cluster solutions of Eq. (1). The eigenvalues of a balanced $M$ cluster state of a system of $N$ globally coupled oscillators with interaction function $H(\Delta \varphi )$ are obtained [2, 9] as: for $p=0,1,\dots ,M-1\mathrm{\text{:}}{\lambda }_p=1/M{\Sigma }_{k=0}^{M-1}{F}^{\prime }(2\pi k/M)[1-\exp (-2\pi ikp/M)]$; for $p=M,M+1,\dots ,N-1\mathrm{\text{:}}{\lambda }_M=1/M{\Sigma }_{k=0}^{M-1}{F}^{\prime }(2\pi k/M)$. $F(\Delta \varphi )=H(-\Delta \varphi )$.Reuse & Permissions

Figure 3

Mutual entrainment of smooth and relaxation oscillators with positive (excitatory) and negative (inhibitory) coupling in experiments. (a) In-phase entrainment of two smooth oscillators with positive coupling. $K=0.25$ ($K$ is the strength of electrical coupling), $V=1.105\mathrm{V}$. Without coupling the frequencies of the oscillators are ${\omega }_1=0.471\mathrm{Hz}$ and ${\omega }_2=0.466\mathrm{Hz}$. (b) Antiphase entrainment of smooth oscillators with negative coupling. $K=-0.26$, $V=1.105\mathrm{V}$. [Negative coupling was achieved with a negative differential resistance ($R_{\mathrm{s}}<0$) feedback circuit built in the potentiostat.] (c) Antiphase entrainment of two relaxation oscillators with positive coupling. $K=0.30$, $V=1.300\mathrm{V}$. ${\omega }_1=0.157\mathrm{Hz}$, ${\omega }_2=0.198\mathrm{Hz}$. (A $30\Omega$ additional resistance was added to the electrode with longer period; this additional resistance decreases the period disparity between the oscillators.) (d) In-phase entrainment of two relaxation oscillators with negative coupling. $K=-0.32$, $V=1.300\mathrm{V}$. (e) Stable one-cluster (left panel) and two-cluster (right panel) state of a population of globally coupled oscillators under the same experimental condition. $K=1.5$, $V=1.265\mathrm{V}$. The two-cluster state was obtained from the one-cluster state by a perturbation of the circuit potential at $\varphi =0.5$. (Amplitude: $-600\mathrm{mV}$, length: 250 ms.)Reuse & Permissions

Figure 4

Numerical simulations: interaction functions obtained with an ODE model. (a)–(c) Interaction function (left panel) and its odd part (right panel) at $V=15$ (smooth oscillator), 24.77, and 26.3 (relaxation oscillator), respectively. The interaction function for the model was calculated with the software xppaut [21].Reuse & Permissions