## Subsequently, new directionality between every regional node dynamics is measured by using the directed stage slowdown list (dPLI), and that calculates the newest phase direct and slowdown dating anywhere between several oscillators (come across Product and methods having in depth meaning)

The new main purpose of this study would be to select a standard relationships regarding network topology, regional node figure and directionality during the inhomogeneous communities. I proceeded because of the developing a straightforward combined oscillatory system design, playing with good Stuart-Landau design oscillator so you’re able to show the newest neural size population passion in the for every node of one’s community (find Product and methods, and you may S1 Text for info). This new Stuart-Landau design ‘s the typical variety of the newest Hopf bifurcation, for example it’s the easiest design capturing the quintessential attributes of the system around the bifurcation area [22–25]. The Hopf bifurcation looks extensively during the physiological and you can agents possibilities [24–33] that is often familiar with studies oscillatory conclusion and mind dynamics [twenty-five, twenty seven, 29, 33–36].

We first went 78 combined Stuart-Landau patterns into the a size-100 % free model system [37, 38]-that is, a system with a degree shipping following the an electrical power law-in which coupling strength S between nodes would be varied due to the fact manage factor. The brand new absolute regularity of each node was randomly drawn regarding a beneficial Gaussian delivery for the indicate from the ten Hz and you may fundamental deviation of 1 Hz, simulating this new leader data transfer (8-13Hz) of human EEG, and in addition we methodically varied the newest coupling strength S regarding 0 so you can fifty. I along with ranged enough time delay parameter all over an over-all range (dos

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.

## We following went on to determine new relationships between network topology (node knowledge), node fictional character (amplitude) and you may directionality ranging from node personality (dPLI) (select S1 Text to own over 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 (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 .