We derive a two-layer multiplex Kuramoto model from Wilson-Cowan type physiological equations that describe neural activity on a network of interconnected cortical regions. This is mathematically possible due to the existence of a unique, stable limit cycle, weak coupling, and inhibitory synaptic time delays. We study the phase diagram of this model numerically as a function of the inter-regional connection strength that is related to cerebral blood flow, and a phase shift parameter that is associated with synaptic GABA concentrations. We find three macroscopic phases of cortical activity: background activity (unsynchronized oscillations), epileptiform activity (highly synchronized oscillations) and resting-state activity (synchronized clusters/chaotic behaviour). Previous network models could hitherto not explain the existence of all three phases. We further observe a shift of the average oscillation frequency towards lower values together with the appearance of coherent slow oscillations at the transition from resting-state to epileptiform activity. This observation is fully in line with experimental data and could explain the influence of GABAergic drugs both on gamma oscillations and epileptic states. Compared to previous models for gamma oscillations and resting-state activity, the multiplex Kuramoto model not only provides a unifying framework, but also has a direct connection to measurable physiological parameters.

Social interactions take place simultaneously through different interaction types, such as communication, friendship, trade, exchange, enmity, revenge, etc. These interactions can be conveniently described with time-dependent multilayer networks. Little is known about the dynamics of social network formation on single layers. How the dynamics on one layer is coupled to and influences the dynamics on another layer is a completely unexplored territory. This is mainly due to the lack of comprehensive microscopic interaction data on time-dependent multilayer networks. In this work, we study a unique dataset of 350,000 odd players in a massive multi-player online game, for which we know practically every social interaction event. We focus on the dynamics of friendship interactions and how they are coupled to the dynamics of enmity interactions. We are able to identify the driving processes behind the joint network formation of friendship and enmity links. The essential mechanisms turn out to be specific triadic closure rules. We propose a simple dynamical model that allows us to predict not only the correct levels of social balance but also the detailed simultaneous structural properties of the friendship and enmity networks, including their degree distributions, clustering coefficients and nearest neighbor degrees. While the formation of new friendship links can be largely understood on the basis of structural features of the friendship network alone, this is not true for enmity networks. The formation of enmity links is driven by the need to socially balance triadic relations that contain negative and positive interactions. Networks of enmity relations can only be understood structurally in the context of the positive social ties they are embedded in.

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