Next, brand new directionality between all the local node figure is actually counted by using the brought stage slowdown directory (dPLI), and that exercises the new phase head and slowdown relationship ranging from one or two oscillators (come across Content and techniques having detailed meaning)
New central intent behind this research were to select an over-all relationships away from network topology, regional node fictional character and you may directionality when you look at the inhomogeneous companies. I went on of the creating an easy coupled oscillatory system design, using an excellent Stuart-Landau design oscillator so you’re abdominalle to portray the brand new neural size population pastime within per node of the network (see Materials and methods, and you will S1 Text message to have information). The Stuart-Landau model is the normal kind of the newest Hopf bifurcation, and thus simple fact is that best design trapping many options that come with the device near the bifurcation point [22–25]. The brand new Hopf bifurcation appears commonly within the physical and you can chemicals expertise [24–33] and that is often regularly analysis oscillatory choices and you will attention figure [twenty five, 27, 30, 33–36].
I basic went 78 paired Stuart-Landau activities into the a measure-totally free model system [37, 38]-that’s, a network with a degree shipment pursuing the an energy laws-in which coupling energy S ranging from nodes should be varied given that handle parameter. The fresh pure frequency each and every node try at random drawn of a beneficial Gaussian delivery on the suggest within 10 Hz and you can simple deviation of 1 Hz, simulating the new alpha bandwidth (8-13Hz) from people EEG, therefore we methodically varied the newest coupling electricity S of 0 to 50. I as well as ranged the time slow down factor across the a general assortment (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 following continued to spot this new dating anywhere between circle topology (node training), node character (amplitude) and you may directionality ranging from node dynamics (dPLI) (discover S1 Text message to possess 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 sito per rimorchiare 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 .
