Then, the brand new directionality ranging from most of the regional node character was measured making use of the directed stage slowdown index (dPLI), hence works out new stage head and slowdown dating between a couple oscillators (look for Content and methods for in depth meaning)

The fresh central aim of this study was to pick a broad matchmaking out of network topology, regional node personality and you may directionality when you look at the inhomogeneous communities. We continued by developing a simple paired oscillatory community model, using a good Stuart-Landau design oscillator to portray the new sensory mass society interest in the for each and every node of the network (get a hold of Content and techniques, and you will S1 Text having information). This new Stuart-Landau design ‘s the regular style of brand new Hopf bifurcation, and therefore it is the best design trapping one particular options that come with the computer close to the bifurcation area [22–25]. Brand new Hopf bifurcation appears commonly inside biological and you will agents systems [24–33] and is tend to regularly studies oscillatory conclusion and you will mind character [twenty-five, 27, 30, 33–36].

I very first ran 78 combined Stuart-Landau patterns for the a scale-totally free design circle [37, 38]-which is, a network that have a degree distribution after the an energy law-where coupling strength S anywhere between nodes will likely be ranged because the control parameter. Brand new natural frequency of each node is actually randomly drawn off an excellent Gaussian delivery on mean at ten Hz and you may practical deviation of just one Hz, simulating the latest leader data transfer (8-13Hz) of human EEG, therefore systematically varied the fresh new coupling energy S out of 0 so you can fifty. I also varied enough time reduce parameter round the an over-all diversity (2

50ms), but this did not yield a qualitative difference in the simulation results as long as the delay was less than a quarter cycle (< 25 ms) of the given natural frequency (in this case, one cycle is about 100 ms since the frequency is around 10Hz). The simulation was run 1000 times for each parameter set.

I after that continued to spot this new relationships anywhere between circle topology (node knowledge), node fictional character (amplitude) and you will directionality between node figure (dPLI) (come across S1 Text message to possess done derivation)

dPLI between two nodes a and b, dPLI_ab, can be interpreted as the time average of the sign of phase difference . It will yield a positive/negative value if a is phase leading/lagging b, respectively. dPLI was used as a surrogate measure for directionality between coupled oscillators . Without any initial bias, if one node leads/lags in phase and therefore has a higher/lower dPLI value than another node, the biased phases reflect the directionality of interaction of coupled local dynamics. dPLI was chosen as the measure of analysis because its simplicity facilitated the analytic derivation of the relationship between topology and directionality. However, we note that we also reach qualitatively similar conclusions with our analysis of other frequently-used measures such as Granger causality (GC) and symbolic transfer entropy (STE) (see S1 Text and S1 Fig for the comparison) [39–41].

Fig 2A–2C demonstrates how the network topology is related to the amplitude and phase of local oscillators. Fig 2A shows the mean phase coherence (measure of how synchronized the oscillators are; see Materials and Methods for details) for two groups of nodes in the network: 1) hub nodes, here defined as nodes with a degree above the group standard deviation (green triangles, 8 out of 78 nodes); and 2) peripheral nodes, here defined as nodes with a degree of 1 (yellow circles, 33 out of 78 nodes). When the coupling strength S is large enough, we observed distinct patterns for each group. For example, at the coupling strength of S = 1.5, which represents a state in between the extremes of a fully desynchronized and a fully synchronized network (with the coherence value in the vicinity of 0.5), the amplitudes of node activity are plitudes, and peripheral nodes, with smaller amplitudes (Fig 2B). More strikingly, the phase lead/lag relationship is clearly differentiated between the hub and peripheral nodes: hub nodes phase lag with dPLI <0, while the peripheral nodes phase lead with dPLI >0 (Fig 2C). Fig 3 shows the simulation results in random and scale-free networks, which represent two extreme cases of inhomogeneous degree networks. This figure clearly demonstrates that larger degree nodes lag in phase with dPLI <0 and larger amplitude (see S2 Fig for various types of networks: scale free, random, hierarchical modular and two different human brain networks) even at the coupling strength S = 1.5, where the separation of activities between hub nodes and peripheral nodes just begins to emerge. To explain these simulation results, we utilized Ko et al.'s mean-field technique approach to derive the relationships for the coupled Stuart-Landau oscillators with inhomogeneous coupling strength, which in turn can be applied to inhomogeneous degree networks by interpreting inhomogeneous coupling strength as inhomogeneous degree for each oscillator .