General framework for phase synchronization through localized sets

We present an approach which enables one to identify phase synchronization in coupled chaotic oscillators without having to explicitly measure the phase. We show that if one defines a typical event in one oscillator and then observes another one whenever this event occurs, these observations give rise to a localized set. Our result provides a general and easy way to identify PS, which can also be used to oscillators that possess multiple time scales. We illustrate our approach in networks of chemically coupled neurons. We show that clusters of phase synchronous neurons may emerge before the onset of phase synchronization in the whole network, producing a suitable environment for information exchanging. Furthermore, we show the relation between the localized sets and the amount of information that coupled chaotic oscillator can exchange.

Figure 1

(Color online) An illustration of the Definition 1. The set ${\mathcal{D}}_j$ does not intersect the neighborhood ${\Lambda }_j$, therefore, ${\mathcal{D}}_j$ is a localized set of ${\Phi }_j$.

Figure 2

Illustration of the dynamics near a UPO.

Figure 3

(Color online) PS onset in two coupled Rössler oscillators. In (a,c) we depict the attractor of the oscillator ${\Sigma }_1$ and in (b,d) the attractor of ${\Sigma }_2$ in light gray, the sets $\mathcal{D}$ are depicted in black. The bars on (a) and (c) represent the segment ${\mathcal{S}}_1$, while in (b) and (d) the segment ${\mathcal{S}}_2$. In (a) and (b) the sets ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ spread over the attractor of the oscillator ${\Sigma }_1$ and ${\Sigma }_2$, respectively; and there is no PS, the phase difference diverges (e). The parameters are $ϵ=0.001$ and $\mathrm{\Delta }\alpha =0.001$. In (c) and (d) the sets ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$, respectively, are localized and there is PS; the phase difference is bounded (f). The parameters are $ϵ=0.011$ and $\mathrm{\Delta }\alpha =0.001$.

Figure 4

(Color online) Illustration of a localized set in a Rössler-like attractor. The attractor can be approximated by a disk with major radius $r_M$ and minor radius $r_m$. The sets $\mathcal{D}$ are confined within an angle $\xi$.

Figure 5

$H_{jk}$ is depicted for the two coupled Rösslers, Eq. (6), with $ϵ=0.001$. The estimator $H_{jk}$ is computed by means of Eq. (13), whenever $H_{ij}=1$ there is no PS. At $\mid \delta \alpha \mid \approx 0.0009$ the coupled Rösslers undergo a transition to PS, and therefore, $H_{jk}<1$, which shows the presence of PS.

Figure 6

(Color online) Illustration of two possible sections on the torus $T^2$. In (a) the section $\mu$ takes into account only the dynamics of $\omega$, the synchronized scale. In (b) on the other hand, the dynamics of the synchronized time scale is ruled out on the section $\rho$. Therefore, using this particular section one cannot observe localized sets $\mathcal{D}$, since the synchronized time scale is not taken into account.

Figure 7

(Color online) In (a) we present the time series of the membrane potential of two neurons ${\mathcal{N}}_k$ and ${\mathcal{N}}_j$ in light gray and black, respectively. We show the threshold, in dashed line, for the burst occurrence, and in light gray dots, for the spike occurrence. While the bursting scale is synchronized the spiking scale is not. However, both scales can be used to construct the sets $\mathcal{D}$ and they will be localized due to the synchronization in the bursting scale. In (b), we show the set ${\mathcal{D}}_k$ constructed using the spiking scale (black circles) and the set ${\mathcal{D}}_k$ constructed using the bursting scale (black squares). As one can see both are localized.

Figure 8

(Color online) PS between two HR neurons coupled via inhibitory synapses. We analyze the effect of different threshold levels on the detection of PS. We depicted the attractor projection $(x,y)$ in gray, and the set ${\mathcal{D}}_k$ in black, for (b) and (d). In (a) the time series of the membrane potentials (full lines) and the threshold $x_b=-1.3$ (dashed line) are depicted. In (b) the set ${\mathcal{D}}_k$, construct by means of the threshold $x_b$, is localized; showing the presence of PS. The time series of the membrane potentials (full lines) and the threshold $x_s=1.1$ (dashed line) are depicted in (c). The spikes are not in PS. With our method, even for this threshold one can obtain a localized set. In (d), the set ${\mathcal{D}}_k$ construct by using the threshold $x_s=1.1$ is localized.

Figure 9

Networks generated randomly. In (a) $n=16$ and $k=4$, while in (b) $n=9$ and $k=3$.

Figure 10

The appearance of PS clusters within the network. In (a) we show the number of clusters as a function of the synaptic strength. In (b) we plot the normalized ${\sigma }^{*}$, where ${\sigma }^{*}=\sigma ∕112.8$. Note that when the clusters are formed $\sigma$ becomes small, but bounded away from zero, which means that the neurons within the network undergo a transition where they have almost the same frequency.

Figure 11

The average bursting time on the ensemble of neurons. We plot the normalized ${\sigma }^{*}$, where ${\sigma }^{*}=\sigma ∕12$. The inset numbers show the amount of clusters for a given parameter $g_{\mathrm{syn}}$.

Figure 12

Illustration of Eq. (A9).