Then, this new directionality ranging from every local node figure try mentioned utilizing the directed stage lag index (dPLI), and this works out the brand new phase lead and you may lag relationship between several oscillators (look for Materials and techniques having detail by detail meaning)
The new central function of this study was to select an over-all dating away from community topology, regional node figure and you may directionality during the inhomogeneous networking sites. I proceeded by the design a straightforward combined oscillatory network model, using an effective Stuart-Landau model oscillator so you can depict the brand new neural size population interest on per node of one’s circle (pick Product and techniques, and S1 Text getting facts). The fresh Stuart-Landau model is the normal brand of the latest Hopf bifurcation, and thus it is the ideal model capturing the quintessential features of the system nearby the bifurcation section [22–25]. The latest Hopf bifurcation appears generally from inside the biological and you can chemical substances possibilities [24–33] that will be have a tendency to familiar with investigation oscillatory choices and you may mind personality [twenty five, twenty-seven, 29, 33–36].
We very first ran 78 paired Stuart-Landau habits toward a size-free model system [37, 38]-which is, a network having a diploma shipment pursuing the an electricity laws-where coupling strength S anywhere between nodes would be ranged while the control parameter. The latest sheer frequency each and every node is randomly pulled off a good Gaussian shipment for the indicate from the ten Hz and you will practical deviation of 1 Hz, simulating the newest alpha data transfer (8-13Hz) from people EEG, therefore methodically varied the brand new coupling power S from 0 in order to fifty. I in addition to varied enough time impede factor round the an over-all diversity (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.
I after that went on to understand the fresh relationships ranging from system topology (node education), node character (amplitude) and directionality ranging from node character (dPLI) (see S1 Text message for done 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 https://datingranking.net/es/citas-de-jugador/ 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 .