Identification from analytical matchmaking one of node degree, amplitude away from regional vibration and you can directionality away from affairs

Identification from analytical matchmaking one of node degree, amplitude away from regional vibration and you can directionality away from affairs

After that, the newest directionality anywhere between every regional node dynamics are counted utilizing the directed stage slowdown list (dPLI), and therefore calculates this new stage lead and you may slowdown matchmaking ranging from one or two oscillators (select Materials and techniques to own outlined meaning)

The newest central reason for this study were to choose a standard dating out-of system topology, local node dynamics and directionality in inhomogeneous communities. We proceeded from the constructing an easy paired oscillatory system design, having fun with a good Stuart-Landau model oscillator in order to depict the fresh new sensory size people pastime in the for every node of one’s network (find Content and techniques, and you may S1 Text having facts). The new Stuart-Landau model ‚s the normal kind of the newest Hopf bifurcation, and thus this is the simplest model trapping the essential top features of the device near the bifurcation area [22–25]. This new Hopf bifurcation looks commonly during the physical and you will agents possibilities [24–33] and is often accustomed studies oscillatory conclusion and you will notice character [twenty-five, twenty seven, 29, 33–36].

We first went 78 combined Stuart-Landau patterns on a scale-free design circle [37, 38]-that is, a network with a diploma shipping following an electrical energy laws-where coupling fuel S between nodes are varied given that handle factor. Brand new natural volume of any node is at random removed from a Gaussian distribution on indicate during the 10 Hz and you may simple deviation of just one Hz, simulating brand new leader data transfer (8-13Hz) regarding person EEG, and we also systematically ranged the fresh coupling energy S out-of 0 so you’re able to fifty. We together with ranged the full time impede parameter around the a standard variety (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 then proceeded to spot the latest relationship ranging from system topology (node education), node dynamics (amplitude) and directionality ranging from node fictional character (dPLI) (find S1 Text message to have 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 (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 .