Subsequently, brand new directionality ranging from all of the local node character is counted utilising the led phase lag index (dPLI), hence exercises brand new phase direct and slowdown matchmaking anywhere between one or two oscillators (pick Material and methods having outlined definition)
The newest main reason for this study was to select a general relationships away from network topology, local node character and you may directionality during the inhomogeneous companies. I went on because of the constructing an easy coupled oscillatory circle design, having fun with a good Stuart-Landau model oscillator to portray the newest neural mass people passion at the per node of your circle (see Content and methods, and S1 Text message having info). The newest Stuart-Landau design ‘s the regular type of the fresh Hopf bifurcation, for example simple fact is that easiest model capturing many options that come with the device near the bifurcation point [22–25]. New Hopf bifurcation seems extensively inside biological and you will chemical compounds possibilities [24–33] which will be have a tendency to accustomed data oscillatory choices and mind dynamics [twenty-five, twenty-seven, 29, 33–36].
I first went 78 coupled Stuart-Landau activities on the a size-free model system [37, Biker-Dating-Seiten 38]-that’s, a system which have a degree shipments following the an electricity laws-where coupling fuel S ranging from nodes might be varied as control parameter. Brand new pure frequency of each and every node was randomly drawn away from a beneficial Gaussian shipments toward imply from the ten Hz and you can simple departure of 1 Hz, simulating the leader bandwidth (8-13Hz) off individual EEG, and we also systematically varied the latest coupling stamina S of 0 so you’re able to fifty. I including varied enough time reduce factor 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 then proceeded to recognize the latest relationships anywhere between community topology (node knowledge), node character (amplitude) and you will directionality ranging from node figure (dPLI) (find S1 Text message to own complete derivation)
dPLI between two nodes a and b, dPLIab, 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 .