Directed Flow of Information in Chimera States

TL;DR

This paper analyzes how information flows between coupled subpopulations in chimera states using delayed mutual information, revealing a directed influence from desynchronized to synchronized groups in phase oscillator networks.

Contribution

It introduces a method to quantify directed information flow in chimera states through discretized mutual information estimates, applicable to neural-like data.

Findings

Delayed mutual information peaks at zero delay within subpopulations.

Between subpopulations, mutual information peaks at non-zero delay.

Desynchronized subpopulations drive synchronized ones.

Abstract

We investigated interactions within chimera states in a phase oscillator network with two coupled subpopulations. To quantify interactions within and between these subpopulations, we estimated the corresponding (delayed) mutual information that -- in general -- quantifies the capacity or the maximum rate at which information can be transferred to recover a sender's information at the receiver with a vanishingly low error probability. After verifying their equivalence with estimates based on the continuous phase data, we determined the mutual information using the time points at which the individual phases passed through their respective Poincar\'{e} sections. This stroboscopic view on the dynamics may resemble, e.g., neural spike times, that are common observables in the study of neuronal information transfer. This discretization also increased processing speed significantly, rendering it particularly suitable for a fine-grained analysis of the effects of experimental and model parameters. In our model, the delayed mutual information within each subpopulation peaked at zero delay, whereas between the subpopulations it was always maximal at non-zero delay, irrespective of parameter choices. We observed that the delayed mutual information of the desynchronized subpopulation preceded the synchronized subpopulation. Put differently, the oscillators of the desynchronized subpopulation were 'driving' the ones in the synchronized subpopulation. These findings were also observed when estimating mutual information of the full phase trajectories. We can thus conclude that the delayed mutual information of discrete time points allows for inferring a functional directed flow of information between subpopulations of coupled phase oscillators.

Figures

Equations

$$d{\varphi }_{j,\mu }(t)=\left[{\omega }_{j,\mu }+\sum _{\nu =1}^M\sum _{k=1}^{N_{\mu }}{C}_{kj,\mu \nu }\sin \left({\varphi }_{k,\nu }(t)-{\varphi }_{j,\mu }(t)-\alpha \right)\right]dt+d{W}_{j,\mu }(t),$$

$$C=\frac{1}{N}\left(\begin{array}{cc}{C}_{\zeta }& C_{\eta }\\ C_{\eta }& C_{\zeta }\end{array}\right)\in {\mathbb{R}}^{2N\times 2N}$$

$$I(X;Y)$$

$$=H(X)+H(Y)-H(X,Y).$$

$$I_{XY}(\tau )$$

$$=H_X+H_Y-H_{XY}(\tau )$$

$${\tilde{I}}_{\mu \nu }(\tau )=\tilde{I}\left(u_{\mu }(t);u_{\nu }(t+\tau )\right).$$

$$p_{\mu \nu }(t,t+\tau )$$

$$=P\left(t\in S_{\mu }\right)+P\left(t+\tau \in S_{\nu }\right)$$

$$-P\left(\left\{t\in S_{\mu }\right\}\cup \left\{t+\tau \in S_{\nu }\right\}\right).$$

$$I_{\mu \nu }(\tau )$$

$$=H_{\mu }+H_{\nu }-H_{\mu \nu }(\tau ).$$

Full text

Keywords

Chimeras, phase oscillators, coupled networks, mutual information, information flow

I Introduction

Oscillatory units are found in a spectacular variety of systems in nature and technology. Examples in biology include flashing fireflies Ermentrout and Rinzel (1984), cardiac pacemaker cells Guevara et al. (1981, 1989); Glass and Mackey (1988); Zeng et al. (1990); Glass and Shrier (1991), and neurons Singer (1999); Gray et al. (1989); MacKay (1997); Singer and Gray (1995); Stern et al. (1998); in physics one may think of Josephson junctions Jain et al. (1984); Saitoh and Nishino (1991); Valkering et al. (2000), electric power grids Menck et al. (2014); Nardelli et al. (2014); Motter et al. (2013); Dörfler et al. (2013); Lozano et al. (2012); Susuki and Mezic (2011); Rohden et al. (2012), and, of course, pendulum clocks Huygens (1673). Synchronization plays an important role in the collective behavior of and the communication between individual units Pikovsky et al. (2003); Buzsaki (2006); Strogatz (2003). In the last two decades or so, many studies addressed the problem of synchronization in networks with complex structure, such as networks of networks, hierarchical networks and multilayer networks Boccaletti et al. (2014); Arenas et al. (2008); Kivelä et al. (2014); Rodrigues et al. (2016). Alongside efforts studying synchronization on networks, a new symmetry breaking regime coined chimera state has been observed. In a chimera state an oscillator population ‘splits’ into two parts, one being synchronized and the other being desynchronized Kuramoto and Battogtokh (2002); Abrams and Strogatz (2004); Montbrió et al. (2004), or more generally, different levels of synchronization Martens et al. (2016a). This state is a striking manifestation of symmetry breaking, as it may occur even if oscillators are identical and coupled symmetrically; see Panaggio and Abrams (2015); Schöll (2016) for recent reviews. Chimera states have spurred much interest resulting in many theoretical investigations, but they have also been demonstrated in experimental settings using, e.g., mechanical and (electro-)chemical oscillators or lasers Martens et al. (2013); Tinsley et al. (2012); Hagerstrom et al. (2012); Wickramasinghe and Kiss (2013), and electronic circuits implementing FitzHugh-Nagumo neurons Gambuzza et al. (2014).

Chimera states can be considered patterns of localized synchronization. As such they may contribute to the coding of information in a network. This is particularly interesting since systems with chimera states typically display multi-stability, i.e., different chimera configurations may co-exist for identical parameters Martens (2010a, b); Shanahan (2010); Bick (2018). Such a network may hence be able to encode different stimuli through different chimera states without the need for adjusting parameters or altering network structure. It is even possible to dynamically switch between different synchronization patterns, thus allowing for dynamic coding of information Bick and Martens (2015); Martens et al. (2016b).

The mere existence of different levels of synchronization in the same network can also facilitate the transfer of information across a network, especially between subpopulations. Coherent oscillations between neurons or neural populations have been hypothesized to provide a communication mechanism while full neural synchronization is usually considered pathological Buzsaki (2006); Fries (2005); Fell and Axmacher (2011). A recent study reported chimera-like states in neuronal networks in humans, more specifically in electro-encephalographic patterns during epileptic seizures Andrzejak et al. (2016). Admittedly, our understanding of these mechanism is still in its infancy also because the (interpretations of) experimental studies often lack mathematical rigor, both regarding the description of synchronization processes and the resulting implications for the network’s capacity for information processing. What is the information transfer within and between synchronized and desynchronized populations? We investigated the communication channels between two subpopulations in a system of coupled phase oscillators. Since we designed that system to exhibit chimera states, we expected non-trivial interactions between the subpopulations. To tackle this, we employed the delayed version of mutual information which measures the rate at which information can be sent and recovered with vanishingly low probability of error Cover and Thomas (2012). Furthermore, assuming symmetry of our configuration, any directionality found in the system should be regarded of functional nature. With this we sought not only to answer the aforementioned question but to contribute to a more general understanding of how information can be transferred between subpopulations of oscillators. In view of potential applicability in other (experimental) studies, we also investigated whether we could gain sufficient insight into this communication by looking at the passage times through the oscillators’ Poincaré sections rather than evaluating the corresponding continuous-time data. Such data may resemble, e.g., spike trains in neurophysiological assessments.

Our paper is structured as follows. First, we introduce our model in the study of chimera states Abrams et al. (2008); Montbrió et al. (2004); Panaggio et al. (2016); Bick et al. (2018). It consists of two subpopulations that either can be synchronized or desynchronized. We generalize this model by including distributed coupling strengths among the oscillatory units as well as additive Gaussian white noise. We briefly sketch the conditions under which chimera states can exist. Second, we outline the concept of delayed mutual information and detail how to estimate this for ‘event-based’ data where we define events via Poincaré sections of the phase oscillators’ trajectories. With this tool at hand, we finally investigate the flow of information within and between the two subpopulations, and characterize its dependency on the essential model parameters.

II Model

We build on a variant of the noisy Kuramoto-Sakaguchi model Acebrón et al. (2005), generalized to $M$ subpopulations of phase oscillators Bick et al. (2019). The phase $\phi_{j,\mu}(t)$ of oscillator $j=1,\ldots,N$ in population $\mu=1,\ldots,M$ evolves according to

[TABLE]

where $\omega_{j,\mu}$ is the natural frequency of the oscillator $j$ in subpopulation $\mu$. Throughout our study $\omega_{j,\mu}$ is drawn from a zero-centered Gaussian distribution with variance $\sigma_{\omega}^{2}$. Oscillator $j$ in population $\mu$ and oscillator $k$ in population $\nu$ interact sinusoidally with coupling strength $C_{kj,\mu\nu}$ and a phase lag $\alpha$. The phase lag $\alpha$ varies the interaction function between more cosine ($\alpha\to\pi/2$) or more sine like-behavior ($\alpha\to 0$) and can be interpreted as a (small) transmission delay between units Panaggio and Abrams (2015). The additive noise term $dW_{j,\mu}(t)$ represents mean-centered Gaussian noise with variance (strength) $\sigma^{2}_{W}$, i.e., $\langle dW_{j,\mu}(t)dW_{k,\nu}(t^{\prime})\rangle_{t}=\sigma^{2}_{W}\delta_{jk}\delta_{\mu\nu}\delta(t-t^{\prime})$.

For the sake of legibility, we restrict this model to the case of $M=2$ subpopulations of equal size $N_{1}=N_{2}=N$. In the absence of noise and in the case of identical oscillators, uniform phase lag and homogeneous coupling, i.e., for $\sigma_{W}=0$, $\sigma_{\omega}=0$, $\alpha_{\mu\nu}\coloneqq\alpha$, and $\sigma_{C}=0$, respectively, the aforementioned studies establishes corresponding bifurcation diagrams Abrams et al. (2008); Montbrió et al. (2004); Panaggio et al. (2016).

To generalize the model towards more realistic, experimental setting, we included distributed coupling strengths via a non-uniform, (statistically) symmetric coupling between subpopulations. This can be cast in the following coupling matrix:

[TABLE]

with block matrices $C_{\zeta},C_{\eta}\in R^{N\times N}$. The self-coupling within the subpopulation and the neighbor coupling between subpopulations are represented by $C_{\zeta}$ and $C_{\eta}$, respectively. Each block $C_{\zeta}$ (or $C_{\eta}$) is composed of random numbers drawn from a normal distribution with mean $\zeta$ (or $\eta$) and variance $\sigma_{C}^{2}$. By this, we can tune the heterogeneity in the network. We would like to note that, in general, the chosen coupling is only symmetric for $\sigma_{C}>0$ in the sense of a statistical average, and that the expected coupling values $\zeta$ and $\eta$ in each block only can be retrieved by the mean in the limit of large numbers. Although the symmetry between the two subpopulations $1\leftrightarrow 2$ is broken and only preserved in a statistical sense in the limit of large numbers, we verified that chimera states appeared in both configurations. That is, subpopulation 1 is synchronized and subpopulation 2 is desynchronized (SD) and subpopulation 2 is synchronized and subpopulation 1 is desynchronized (DS), for a particular choice of parameters using numerical simulations. Another consequence of distributed coupling strengths is that only a very few oscillators may have experienced negative (inhibitory) coupling, which can be neglected.

Following Abrams et al. (2008), we parametrize the relation between strengths by $A=\zeta-\eta$ with $\zeta+\eta=1$ and the phase lag $\alpha_{\mu,\nu}$ by $\beta_{\mu,\nu}=\frac{\pi}{2}-\alpha_{\mu,\nu}$ which here can be kept homogeneous for the entire network, i.e., $\beta_{\mu,\nu}\coloneqq\beta$. By this, we may recover the results reported in Abrams et al. (2008); Montbrió et al. (2004) in the limit $\sigma_{C}\to 0$, under the proviso of $\sigma_{W}=\sigma_{\omega}=0$. In brief, increasing $A$ from [math] while fixing $\alpha$ (or $\beta\coloneqq\pi/2-\alpha$) yields the following bifurcation scheme: a “stable chimera” is born through a saddle-node bifurcation and becomes unstable in a supercritical Hopf bifurcation with a stable limit cycle corresponding to a “breathing chimera”, which eventually is destroyed in a homoclinic bifurcation. For all parameter values, the fully synchronized state exists and is a stable attractor; see also Fig. 4 in Appendix A. Subsequent studies demonstrated robustness of chimera states against non-uniformity of delays ($\alpha_{\mu\nu}\neq\alpha$) Martens et al. (2016a), heterogeneity of oscillator frequencies ($\sigma_{\omega}>0$) Laing (2009), and additive noise Laing (2012). Although non-uniform phase lags lead to less degenerate dynamics and give room for more complex dynamics Martens et al. (2016a); Bick et al. (2018), we restrict ourselves to the case of uniform phase lags without compromising the essential phenomenology of chimera states.

The macroscopic behavior, i.e. the synchronization of the two populations, may be characterized by complex-valued order parameters, which are either defined on the population level, i.e., $Z_{\mu}\coloneqq N^{-1}\sum_{j=1}^{N}e^{i\theta_{\mu,j}}$, or globally, i.e., $Z=(2N)^{-1}\sum_{\mu=1}^{M}\sum_{j=1}^{N}e^{i\theta_{\mu,j}}$. As common in the studies of coupled phase oscillators, the level of synchrony can be given by $R_{\mu}\coloneqq|Z_{\mu}|$. Thus, $R_{\mu}=1$ implies that oscillators in population $\mu$ are perfectly phase synchronized (S), while for $R_{\mu}<1$ the oscillators are imperfectly synchronized or de-synchronized (D). For our two-subpopulation case, full synchrony (SS) hence occurs when $R_{1}\lessapprox 1$ and $R_{2}\lessapprox 1$, and chimera states are present if $R_{1}<1$ and $R_{2}\lessapprox 1$ or vice versa. The angular order parameter, $\Phi_{\mu}\coloneqq\arg{{Z}_{\mu}}$ keeps track of the average phase of the (sub)population. Fluctuations inherent to the model may affect the order parameter as illustrated in Fig. 1(b) (please refer to section III.1 for the numerical specifications). We therefore always considered averages over time, ${\langle R_{\mu}\rangle_{t}}$, when discussing the stability of a state. In fact, in our model chimera states remain stable for relatively large coupling heterogeneity, $\sigma_{C}>0$ presuming $\sigma_{\omega}>0$ and $\sigma_{W}>0$, as is evidenced by numerical simulations; see also Appendix A, Fig. 4. The perfect synchronization manifold with $R=1$ cannot be achieved; see Fig. 1(b). Further aspects of these noisy dynamics will be presented elsewhere.

Adding noise and heterogeneity to the system may alter its dynamics. In the present work we concentrated on parameter regions characterized by the occurrence of dynamic states that did resemble stable chimeras, i.e., $\langle R_{1}\rangle_{t}>\langle R_{2}\rangle_{t}$, where $\langle\cdot\rangle_{t}$ denotes the average over a duration of $T=9\cdot 10^{6}$ time steps (after removing a transient of $T=10^{5}$ time steps). Fig. 4 in Appendix A provides an overview and explanation of how parameter points $A$ and $\beta$ were selected.

III Implementation and analysis

III.1 Simulations

For the numerical implementation we employed a Euler-Maruyama scheme with $\Delta{t}=10^{-2}$ for $N=128$ phase oscillators per subpopulation that were evolved for $T=10^{6}$ Acebrón et al. (2005). We varied the coupling parameter $A$ and the phase lag parameter $\beta$, while we fixed the width (standard deviation) of the natural frequency distribution to $\sigma_{\omega}=0.01$ and of the coupling distribution to $\sigma_{C}=0.01$. The additive noise had variance $\sigma_{W}^{2}=0.001$.

III.2 Mutual Information

Mutual information $I(X;Y)$, first introduced by Shannon Shannon and Weaver (1949), is meant to assess the dependence between two random variables $X$ and $Y$. It measures the amount of information that $X$ contains about $Y$. In terms of communication theory, $Y$ can be regarded as output of a communication channel which depends probabilistically on its input, $X$. By construction, $I(X;Y)$ is non-negative; it is equal to zero if and only if $X$ and $Y$ are independent. Moreover, $I(X;Y)$ is symmetric, i.e., $I(X;Y)=I(Y;X)$ implying that it is not directional. The mutual information may also be considered as a measure of the reduction in the uncertainty of $X$(Y) due to the knowledge of $Y$(X) or, in terms of communication, the rate at which information is being shared between the two Cover and Thomas (2012). The maximum rate at which one can send information over the channel and recover it at the output with a vanishingly low probability of error is called channel capacity, $c=\max_{p(x)}I(X;Y)$. In networks with oscillatory nodes, the random variables are the degree of synchronization of different subpopulations, and the rate of information shared will depend on the state of the system for each of the subpopulations.

Mutual information can be defined via the Kullback-Leibler divergence or in terms of entropies,

[TABLE]

where $H(X)$ is the entropy of the random variable $X$, i.e., $H(X)=-\int dx\ p_{X}(x)\log p_{X}(x)$ with $p_{X}(x)$ being the corresponding probability density.

III.3 Delayed mutual information

For time-dependent random variables, one may generalize this definition to that of a delayed mutual information. This variant dates back to Fraser and Swinney Fraser and Swinney (1986), who used the delayed mutual information for estimating the embedding delay in chaotic systems — in that context the delayed auto-mutual information is often considered a ‘generalization’ of the auto-correlation function Abarbanel (2012). Applications range from the study of coupled map lattices Kaneko (1986) via spatiotemporal Vastano and Swinney (1988) and general dependencies between time series Green and Savit (1991) to the detection of synchronization Paluš (1997). With the notation $I_{XY}\coloneqq I(X;Y)$ and $H_{X}\coloneqq H(X)$, the delayed mutual information for a delay $\tau$ may read

[TABLE]

which has the symmetry $I_{XY}(\tau)=I_{YX}(-\tau)$ 111This follows because of: $\tilde{I}_{XY}(\tau)=I\left(X(t);Y(t+\nu)\right)=\tilde{I}\left(X(t-\nu);Y(t)\right)=\tilde{I}\left(Y(t);X(t-\nu)\right)=\tilde{I}_{YX}(-\tau)$.. With this definition, one can measure the rate of information shared between $X$ and $Y$ as a function of time delay $\tau$. In fact, we are not particularly interested in the specific value of the mutual information but rather focus here on the time delay at which the mutual information is maximal. Hence, we define $\tau_{\rm max}\coloneqq\operatorname*{\arg\max}_{\tau}I_{XY}(\tau)$. A positive (negative) value of $\tau_{\rm max}$ implies that $Y$ shares more information with a delayed (advanced) $X$. This means there is an information flow from $X$(Y) to $Y$(X).

III.4 Delayed mutual information between subpopulations

Estimates using continuous-time data.

When the time-dependent random variables are continuous time series $u_{\mu}(t)$ and $v_{\nu}(t)$ associated to populations $\mu$ and $\nu$, the delayed mutual information can be estimated from Eq. (4) for $X(t)=u_{\mu}(t)$ and $Y(t)=u_{\nu}(t)$,

[TABLE]

We determined the probability densities using kernel density estimators with Epanetchnikov kernels with bandwidths given through a uniform maximum likelihood cross-validation search. For our parameter settings (254 oscillators and $10^{6}$ samples), the resulting bandwidths ranged from about $0.10\ \textrm{rad}$ to $0.18\ \textrm{rad}$. This software implementation is part of the KDE-toolbox; cf. Ihler and Mandel (2003) and Thomas et al. (2014) for alternative schemes.

Estimates using event-based time data.

For the aforementioned event signals in the subpopulations $1$ and $2$, i.e., discrete time points are defined as passing moments through the respective Poincaré sections. The probability densities to incorporate when estimating the mutual information are densities of events, or densities of times. We implemented the probability estimates as follows. Let $S_{\mu}$ be a set of event times, i.e., $S_{\mu}=\{t^{(m)}_{\mu,1},\dots,t^{(m^{\prime})}_{\mu,N_{\mu}}\}$ where $t^{(m)}_{\mu,i}$ stands for the time of the $m$-th event of oscillator $i$ in subpopulation $\mu$. Then, the probability density for an event to happen at time $t$ in subpopulation $\mu$ is $p_{\mu}(t)=P(t\in S_{\mu})$ and the probability of an event to happen at time $t$ in subpopulation $\mu$ and time $t+\tau$ in subpopulation $\nu$ is

[TABLE]

The delayed mutual information can be given as

[TABLE]

We again computed the probability densities using kernel density estimators Ihler and Mandel (2003) but now involving Gaussian kernels. We also adjusted the bandwidth selection to a spherical, local maximum likelihood cross-validation due to the sparsity of the data and the resulting bandwidths ranged from about $25$ to $35$ time units. These results appeared robust when using the aforementioned uniform search; again we employed the KDE-toolbox.

III.5 Events defined via Poincaré sections

We analyzed the times $t_{1,\mu},t_{2,\mu},\dots t_{N,\mu}$ at which the individual phases $\phi(t)_{j,\mu}$ passed through their respective Poincaré sections. The latter were defined as $t_{j,\mu}\in{\mathbb{R}}^{+}_{0}:\phi_{j,\mu}(t_{j,\mu})/2\pi\in{\mathbb{Z}}$. As mentioned above, every subpopulation $\mu$ generated an ‘event sequence’ $S_{\mu}=\{t^{(m)}_{\mu,1},\dots,t^{(m^{\prime})}_{\mu,N_{\mu}}\}$; which, as already said, may be considered reminiscent of spike trains; cf. Fig. 1(c).

IV Results

To determine the directionality of the information flow in the network we computed the time lagged mutual information within the subpopulations, $I_{11}(\tau)$ and $I_{22}(\tau)$ and between them, $I_{12}(\tau)=I_{21}(-\tau)$.

The first results for the event data outlined in Section III.5 are shown in Fig. 2(a). Since the recurrence of events mimicked the mean frequency of the phase oscillators, the mutual information turned out periodic. As expected, we observed (average) values of the mutual information that differed between $I_{11}$ (red), $I_{22}$ (blue), $I_{12}$ (black). This relates to the difference in entropy of the subpopulations, with the less synchronized one ($\mu=2$) being more disordered. The latter, hence, contained more entropy. However, since we were not interested in the explicit values of $I_{\mu\nu}(\tau)$, we could rescale the mutual information to maximum value which allowed for a comparative view when zooming in to the neighborhood of $\tau\approx 0$; see panel (c) of Fig. 2. The off-zero peak of $I_{\rm max}\coloneqq I_{12}(\tau_{\rm max})$ in $\tau=\tau_{\rm max}$ clearly indicated a directed information flow, i.e. there is a functional directionality despite the structural symmetry of our model.

When comparing the estimates for the mutual information obtained from the event data with those from the continuous-time we found little to no difference in peak location. That is, the positions $\tau_{\rm max}$ of the maximum peaks, $I_{\rm max}$ and $\tilde{I}_{\rm max}$, were nearly identical using either method as shown in panels (b) and (d) Fig. 2. Thus, to study the effects of varying model parameters on $I_{\mu\nu}(\tau)$, our subsequent analysis may solely rely on the event data approach.

Varying the values of $A$ and $\beta$ revealed a strong relation between the location of the peak of the delayed mutual information $\tau_{\rm max}$ and the relative phase $\arg{(Z_{1}/Z_{2})}$ between the two subpopulations; see Fig. 3(a). This convincingly shows that our approach to analyze the event-based data reveals important information about the (otherwise continuous) dynamics of the subpopulations, here given by the phases (arguments) of the local order parameters. By contrast, the relative strength of local synchronization $|Z_{1}/Z_{2}|$ had little to no effect on $\tau_{\rm max}$; see Fig. 3(b). These dependencies were inverted when looking at the value of mutual information, $I_{\rm max}/I_{\rm max}^{\SS}$. Consequently, the value of mutual information was affected by relative strength of local synchronization $|Z_{1}/Z_{2}|$ — after all the more a subpopulation is synchronized, the lower the corresponding entropy. However, effects were small and probably negligible when transferring to (selected) experimental data. More details about the normalization factor $I_{\rm max}^{\SS}$ are discussed in Appendix B.

V Discussion

The delayed mutual information from event data, here, the passing times through the oscillators’ Poincaré sections, agreed with that of the continuous data. This is an important finding as it shows that studying discrete event time series allows for inferring information-related properties of continuous dynamics. This offers the possibility to bring our approach to experimental studies, be that in neuroscience, where spike trains are a traditional measure of continuous neuronal activity, or economics, where stroboscopic assessments of stocks are common practice. Here, we used this approach to explore the information flow within and between subpopulations of a network of coupled phase oscillators. This information flow turned out to be directed.

Mutual information is the traditional measure of shared information between the two systems Shannon and Weaver (1949). Our findings relied on the introduction of a time delay $\tau$, which readily allows for identifying a direction of flow of information. This concept is much related to the widely used transfer entropy Schreiber (2000); Wibral et al. (2013). In fact, transfer entropy is the delayed conditional mutual information Dobrushin (1959); Wyner (1978). It therefore differs from our approach primarily by a normalization factor. Since here we were not interested in the absolute amount of information (flow), we could simplify assessments and focus on delayed mutual information — adding the conditional part jeopardizes the approximation of probabilities, i.e., estimating transfer entropy in a reliable manner typically requires more data than our delayed mutual information. A detailed discussion of these differences is beyond the scope of the current study.

For our model we found that the delayed mutual information between the synchronized subpopulation and the less synchronized one peaked at a finite delay $\tau_{\rm max}\neq 0$. Hence, there was a directed flow of information between the two subpopulations. Since we found that $\tau_{\rm max}<0$ for $I_{12}(\tau)$, the direction of this flow was from the less synchronized subpopulation to the fully synchronized one, i.e. $2\rightarrow 1$ or D $\rightarrow$ S. In fact, the delay largely resembled the relative phase between the corresponding local order parameters, as shown in panel (a) of Fig. 3. We could not only readily identify the relative phase by encountering event data only, but our approach also allowed for attaching a meaningful interpretation in an information theoretic sense. This is promising since event data are, as stated repeatedly, conventional outcome parameters in many experimental settings. This is particularly true for studies in neuroscience, for which the quest on information processing is often central. There, networks are typically complex and modular. The complex collective dynamics switches at multiple scales, rendering neuronal networks especially exciting when it comes to information routing Kirst et al. (2016). As of yet, our approach does not allow for unraveling dynamics information routing. This will require extending the (time-lagged) mutual information to a time-dependent version, e.g., by windowing the data under study. We plan to incorporate this in future studies.

VI Conclusion

Estimating the delayed mutual information based on time points at which the individual phases passed through their respective Poincaré sections allows for identifying the information flow between subpopulations of networks. If the network displays chimera states, the information flow turns out to be directed. In our model of coupled phase oscillators, the flow of information was directed from the less synchronized subpopulation to the fully synchronized one since the first preceded the latter. Our approach is a first step to study information transfer between spike trains. It can be readily adopted to static well-defined modular networks and needs to be upgraded to a time dependent version to be applied to real, biological data.

Funding

This study received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement #642563 (COSMOS).

Author contributions

ND conducted the simulations and analysis and wrote the manuscript; AD set up the study, incorporated the kernel density estimation and wrote the manuscript, DB contributed to the first concepts of the study and wrote the manuscript; EAM set up the study, conducted simulations and wrote the manuscript.

Acknowledgements

ND and AD want to thank to Rok Cestnik and Bastian Pietras for fruitful discussions.

Appendix A Stability diagram for chimera states

For the study of mutual information, we restricted ourselves to dynamic states most similar to the ’stable chimera states’ reported in Abrams et al. (2008). Due to the various noise sources inherent to the model, the system dynamics undergoes fluctuations, as described in Section II. This poses additional challenges that we addressed by examining specific time averages of the complex-valued order parameter in each subpopulation. Since fluctuations may cause the system to temporarily drift off the synchronized manifold, we may not simply consider $|Z_{1}|=1,|Z_{2}|<1$ to identify chimera states. Instead we require the order to be sufficiently different between the two subpopulations, i.e., we threshold $\langle|Z_{2}|\rangle_{t}/\langle|Z_{1}|\rangle_{t}$, defining the region $R_{1}$ shown in Fig. 4(a).

Furthermore, we presume amplitude variations to be small. To that end, we thresholded $\langle\max(|Z_{2}|)\rangle_{t}-\langle\min(|Z_{2}|)\rangle_{t}$, defining the region $R_{2}$ shown in Fig. 4(b). Here, $\langle\cdot\rangle_{t}$ denotes the average defined over the maxima computed over time windows of length $T=10^{3}$ time steps, over a duration $T=9\cdot 10^{5}$ after removing a transient of $T=10^{5}$ time steps. Finally, chimera states with parameter values $(\beta,A)\in R_{1}\cap R_{2}$ were selected for further analysis of mutual information.

Appendix B Normalization factor $I_{\rm max}^{\SS}$

Our model displays a baseline dependency of the maximum peak of mutual information $I_{\rm max}^{\SS}$ on the parameters $\beta$ and $A$, even when the system is in the fully synchronized state $\SS$, see Fig. 5. We therefore studied the dependency of the maximum peak of the delayed mutual information for chimera states normalized with $I_{\rm max}^{\SS}$, $I_{\rm max}/I_{\rm max}^{\SS}$. For every set of parameters we computed the delayed mutual information on two different solution that have been forced by changing the initial conditions; the maxima of each of these solutions ($\SS$ and ${\rm{SD}}$) are $I_{\rm max}^{\SS}\coloneqq I_{\rm max}^{\SS}(A,\beta)$ and $I_{\rm max}\coloneqq I_{\rm max}^{\rm{SD}}(A,\beta)$. These are the quantities displayed in panels (c) and (d) of Fig. 3.