Figure 1

Synchronization discrimination for the coupled phase oscillators (1) with: ${\omega }_1=1.2$, ${\omega }_2=0.8$, ${\epsilon }_1=0.1$, ${\epsilon }_2=0.35$, $f_i({\varphi }_i)=0$, $g_1=\sin ({\varphi }_2-{\varphi }_1)$, $g_2=\sin ({\varphi }_1-{\varphi }_2)$, and noise strengths $E_{11}=E_{22}=2$. (a) Schematic Arnold tongue to illustrate synchronization [36]. (b) Phase difference, exhibiting two phase slips. (c) Map of $M({\psi }_e)$ for (b) demonstrating that a root of $M(\psi )=\psi$ exists, i.e., that the state is, in fact, synchronized.Reuse & Permissions

Figure 2

Extraction of time-varying parameters, synchronization, and coupling functions from numerical data created by (5). The plots show the results inferred for the numerical values of constants listed in the text. The frequency $f_1(t)$ and coupling ${\epsilon }_2(t)$ are independently varied: (a) ${\omega }_1(t)={\omega }_1+{\tilde{A}}_1\sin ({\tilde{\omega }}_{1}t)$; (b) ${\epsilon }_2(t)={\epsilon }_2+{\tilde{A}}_2\sin ({\tilde{\omega }}_{2}t)$. The dotted and full lines plot the parameters when the two oscillators are synchronized for part of the time (${\epsilon }_1=0.3$), and not synchronized at all (${\epsilon }_1=0.1$), respectively. The regions of synchronization, found by calculation of the synchronization index, are indicated by the gray shaded regions. (c)–(f) show the coupling functions $q_1({\varphi }_1,{\varphi }_2)$ and $q_2({\varphi }_1,{\varphi }_2)$ for time windows centered at different times: (c) and (d) at $t=350s$; (e) and (f) at $t=1000s$. The window length $t_w=50s$, and ${\epsilon }_1=0.1$ in both cases. Note the similarity in forms of (c) and (e), and of (d) and (f). The other parameters were: ${\epsilon }_2=0.1$, ${\omega }_1=2\pi 1$, ${\omega }_2=2\pi 1.14$, ${\tilde{A}}_1=0.2$, ${\tilde{A}}_2=0.13$, ${\tilde{\omega }}_1=2\pi 0.002$, ${\tilde{\omega }}_2=2\pi 0.0014$ and noise $E_{11}=E_{22}=0.1$. The phases were estimated as ${\varphi }_i=\arctan (y_i/x_i)$.Reuse & Permissions

Figure 3

Synchronization, directionality, and coupling functions in the cardiorespiratory interaction. (a) Standard $1:N$ synchrogram. (b) Synchronization index for ratios $1:4$, $1:5$, and $1:6$, as indicated. The light-gray dotted line represents the mean, and the dark-gray dashed line the mean $+2$ SD, of synchronization indices calculated from 100 surrogate [37] realizations. (c) The time-varying respiration frequency (note the downward ramp due to pacing). The gray areas in (c) represent $\pm 2$ SD from the mean value. (d) Directionality index (full curve); the light-gray dotted line represents the mean directionality index calculated from 100 surrogate realizations, and the dark-gray dashed line represents the mean $+2$ SD. (e)–(g) coupling functions $q_1({\varphi }_1,{\varphi }_2)$ calculated at different times, as indicated by the gray arrows.Reuse & Permissions