Abstract
This work introduces the phenomenon of Collective Almost Synchronisation (CAS), which describes a universal way of how patterns can appear in complex networks for small coupling strengths. The CAS phenomenon appears due to the existence of an approximately constant local mean field and is characterised by having nodes with trajectories evolving around periodic stable orbits. Common notion based on statistical knowledge would lead one to interpret the appearance of a local constant mean field as a consequence of the fact that the behaviour of each node is not correlated to the behaviours of the others. Contrary to this common notion, we show that various well known weaker forms of synchronisation (almost, time-lag, phase synchronisation, and generalised synchronisation) appear as a result of the onset of an almost constant local mean field. If the memory is formed in a brain by minimising the coupling strength among neurons and maximising the number of possible patterns, then the CAS phenomenon is a plausible explanation for it.
Highlights
Spontaneous emergence of collective behaviour is common in nature [1,2,3]
It is a natural phenomenon characterised by a group of individuals that are connected in a network by following a dynamical trajectory that is different from the dynamics of their own
Since the work of Kuramoto [4], the spontaneous emergence of collective behaviour in networks of phase oscillators with full bidirectionally connected nodes or with nodes connected by some special topologies [5] is analytically well understood
Summary
Spontaneous emergence of collective behaviour is common in nature [1,2,3]. It is a natural phenomenon characterised by a group of individuals that are connected in a network by following a dynamical trajectory that is different from the dynamics of their own. If hi is the variable describing the phase of an oscillator i in the network, and h represents the mean field defined as h~ N i~1 hi, collective behaviour appears in the network because every node becomes coupled to the mean field. Peculiar characteristics of this collective behaviour is that hi=h and nodes evolve in a way that cannot be described by the evolution of only one individual node, when isolated from the network
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