Coupling strength versus coupling impact in nonidentical bidirectionally coupled dynamics

Abstract

The understanding of interacting dynamics is important for the characterization of real-world networks. In general real-world networks are heterogeneous in the sense that each node of the network is a dynamics with different properties. For coupled non-identical dynamics symmetric interactions are not straightforwardly defined from the coupling strength values. Thus, a challenging issue is whether we can define a symmetric interaction in this asymmetric setting. To address this problem we introduce the notion of the coupling impact. The coupling impact considers not only the coupling strength but also the energy of the individual dynamics which is conveyed via the coupling. To illustrate this concept, we follow a data-driven approach by analyzing signals from pairs of coupled model dynamics using two different connectivity measures. We find that the coupling impact, but not the coupling strength, correctly detects a symmetric interaction between pairs of coupled dynamics regardless of their degree of asymmetry. Therefore, this approach allows to reveal the real impact that one dynamics has on the other and hence to define symmetric interactions in pairs of non-identical dynamics.

Detection of time delays and directional interactions based on time series from complex dynamical systems

“…where K(x 1 ) and L(y 2 ) designate bidirectional couplings between a Lorenz oscillator in the variable x 1 and a Rössler system in y 2 , and α is a scaling factor [40,41]. For discrete delays, we set K(x 1 ) = Cx 1 (t − τ 1 ) and L(y 2 ) = Dy 2 (t − τ 2 ) with parameters C and D, the coupling strengths, and τ 1 = 3 and τ 2 = 4, the delays.…”

Seizure forecasting by tracking cortical response to electrical stimulation

“…In addition, it has been shown that in the phenomenon of “critical slowing down” in which dynamical systems take longer time to return to equilibrium after perturbations there is an increase in the signal variance and autocorrelation [Scheffer_2009]. Furthermore, it has been shown in theoretical models that the variance is a metric that captures the energy between signals [Laiou_2017]. Additional features that were employed in the logistic regression classifier were the cumulative average of variance and autocorrelation to allow the model to consider the history and the evolution of those features.…”

Robustness and versatility of a nonlinear interdependence method for directional coupling detection from spike trains

“…In general, the values of L are not monotonically increasing with an increase of the coupling value. As emphasized in [54], the effect of the coupling depends not only on the coupling strength but also on the energy of the individual dynamics. As a consequence, the estimate of the coupling is not independent from the structure of the data, for example in bursting or spiking regimes.…”

“…For example, if the spike trains are from two coupled stochastic point processes, the measure L could have trouble with the detection of directionality, even if the presence of a coupling is still detectable. Another caveat is the possibility to have a small but significant difference in the values of L in the two directions also for bidirectionally coupled dynamics [54]. As a consequence, in the interpretation of experimental results, more caution is necessary to assume directionality if L(X|Y ) ≈ L(Y |X).…”