Synchronization in Simple Network Motifs with Negligible Correlation and Mutual Information Measures

Miguel C. Soriano, Guy Van der Sande, Ingo Fischer, Claudio R. Mirasso
2012 Physical Review Letters  
Can different or even identical coupled oscillators be completely uncorrelated and still be synchronized? What can be concluded from the absence of correlations or even mutual information in networks of dynamical elements about their connectivity? These are fundamental and far-reaching questions arising in many complex systems. In this manuscript we address these two questions and demonstrate in simple and generic network motifs that synchronized behavior in the generalized sense can be
more » ... and constructed such that no correlations and even negligible mutual information remain. Our findings raise new questions, in particular whether and to what extent indirect connections are being underestimated, since the related collective behavior and even synchronization is less likely to be detected. PACS numbers: 05.45.Xt,05.45.Pq, 05.45.Jn, 05.45.Tp The study of synchronization in coupled dynamical systems dates back to the works of Huygens in the 17th century [1] . When addressing synchronization in coupled complex dynamical systems, often the focus is on the particular case of an identically synchronized state [2] . However, synchronization in a generalized sense can still exist, even if there is no one-to-one relationship between the responses of the coupled systems [3] . Generalized synchronization requires a relationship between the dynamics of the constituting systems, no matter how complex this relationship is. If such a relationship exists, the response of one system is completely determined by the other one. The interaction between the systems can be either bidirectional or unidirectional. In the latter case, also known as drive-response configuration, the dynamics of the response system is, after transients, fully reproducible for the repetition of the drive signals, i.e. generalized synchronization boils down to the notion of consistency [4] . From an information theory perspective, synchronization requires a minimum amount of information to be transferred between the coupled elements. This minimum information has been determined precisely for the example of a system of coupled chaotic oscillators [5] . The dynamics of coupled systems has been extensively studied in biological networks [6, 7] , lasers [8], neural networks [9] , and many other self-organizing systems. One of the key ingredients in many of these studies is the network topology in which the dynamical elements are embedded. The behavior that emerges from the interaction strongly depends on the underlying network. From a theoretical point of view different topologies have been extensively considered and analyzed [10] . In real-world systems, however, the underlying network topology is often unknown and only measured time series of a subset of elements or of a mean field are available. Prominent examples are climate modeling [11], ecological modeling [12] and neuroscience [13], among others. In all these areas correlation measures are being extensively used to deduce functional or, in some cases, effective connectivity. Functional connectivity assumes statistical dependencies between distinct units of the system while effective connectivity refers to causal interactions among the constituents. Functional connectivity, in turn, is used to develop models and to conclude on the abilities of a network [14] . The problem of network reconstruction is particularly important in neuroscience, where it has been identified that several diseases and impairments are related to changes in the network topology [15] . In human neurophysiology, mostly functional magnetic resonance (fMRI), electroencephalography (EEG) or magneto encephalography (MEG) data are available. These data boil down to a measurement of average activity of an already large ensemble of neurons. To unveil the connectivity information, several techniques have been used, particularly cross-correlation and mutual entropy are the most widely considered (see e.g. [13] and references therein). However, as we will show below, these two indicators can, under certain circumstances, underestimate and completely miss indirect connections. It is our aim to show in this Letter, via modeling and experiments, that for some simple configurations of coupled dynamical elements negligible correlation or mutual information are observed, although the elements are synchronized and determine each other's behaviors completely. We have chosen different topologies, which can be classified into two general categories: bidirectional and unidirectional coupling schemes. Figure 1(a) illustrates a mutual interaction scheme while Fig. 1(b) depicts a driveresponse configuration. The square boxes in Fig. 1 account for different coupling interactions between dynamical elements A and B. For illustration purposes, we
doi:10.1103/physrevlett.108.134101 pmid:22540702 fatcat:kfxt7diqqffrjpwayiml557jm4