Order in Complex Systems of Nonlinear Oscillators: Phase Locked Subspaces

Any order parameter quantifying the degree of organisation in a physical system can be studied in connection to source extraction algorithms. Independent component analysis (ICA) by minimising the mutual information of the sources falls into that line of thought, since it can be interpreted as searching components with low complexity. Complexity pursuit, a modification minimising Kolmogorov complexity, is a further example. In this article a specific case of order in complex networks of selfsustained oscillators is discussed, with the objective of recovering original synchronisation pattern between them. The approach is put in relation with ICA. 1 Interactions in complex systems Synchronisation is a commonly encountered phenomenon in complex systems consisting of many interacting nonlinear oscillatory elements, each with a stable limit cycle [1]. Such dynamic systems are present in nature, e.g., nervous systems [2] or chemical oscillators [3], and in technical devices like Josephson’s junction [4]. Inference about the interactions in complex systems is also related to such omnipresent phenomena as self-organisation and the 1 f noise [5]. The quantification of synchronisation in complex systems from empirical measurements is accompanied with a difficulty: access to the individual oscillators is crucial otherwise spurious synchronisation will be measured [6]. In many fields these are not easily available, but one rather deals with superpositions of several elementary oscillators [7]. Below it is addressed how the original oscillators could be regained by postulating a general synchronisation structure for the system. 2 Superpositions and phase synchrony Phase synchrony between two oscillators uj(t) and uk(t) with arbitrary phase lag can be quantified with the phase locking factor (PLF), the circular variance of the phase lag between them. Formally it is defined as the amplitude ̺jk ∈ R of the complex variable ̺jk e iΨjk = 〈