Subsequently, the newest directionality anywhere between all of the regional node dynamics is actually measured utilising the directed stage slowdown index (dPLI), and that computes brand new phase lead and you will slowdown matchmaking between a few oscillators (discover Information and methods for outlined meaning)
The latest central aim of this study were to pick a broad dating of network topology, local node personality and you will directionality from inside the inhomogeneous channels. I went on from the developing an easy combined oscillatory system model, using a great Stuart-Landau model oscillator so you’re abdominalle to show the sensory mass inhabitants passion from the for each and every node of your own circle (find Content and methods, and you can S1 Text getting info). The fresh new Stuart-Landau design ‘s the regular types of the brand new Hopf bifurcation, meaning that it’s the best model capturing by far the most features of the system around the bifurcation section [22–25]. This new Hopf bifurcation seems extensively for the biological and you will agents solutions [24–33] and is tend to accustomed research oscillatory behavior and you can head figure [twenty five, twenty seven, 29, 33–36].
We basic ran 78 coupled Stuart-Landau habits on a level-free model community [37, 38]-that’s, a system having a degree shipment adopting the an electricity legislation-where coupling energy S anywhere between nodes would be ranged because the control factor. Brand new natural regularity of every node is at random removed of an excellent Gaussian shipping for the imply at 10 Hz and you can basic departure of 1 Hz, simulating the fresh leader bandwidth (8-13Hz) out-of human EEG, and now we systematically ranged new coupling power S away from 0 in order to fifty. I including varied the full time decrease factor round the a broad 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 determine the latest matchmaking ranging from network topology (node education), node personality (amplitude) and you can directionality ranging from node dynamics (dPLI) (see S1 Text message to possess 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 citas perro. 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 .