Then, the fresh directionality anywhere between the regional node fictional character is measured by using the brought phase lag directory (dPLI), and that exercises the brand new phase lead and you may slowdown dating anywhere between one or two oscillators (discover Product and methods having detailed definition)
The fresh new central intent behind this research were to choose a general dating off system topology, regional node fictional character and you can directionality for the inhomogeneous companies. I continued by creating an easy combined oscillatory circle model, having fun with good Stuart-Landau model oscillator to help you depict the new sensory mass inhabitants hobby within for each node of community (find Materials and methods, and S1 Text to possess facts). The brand new Stuart-Landau model ‘s the regular form of this new Hopf bifurcation, which means that it is the simplest design capturing many options that come with the system around the bifurcation area [22–25]. The brand new Hopf bifurcation appears widely in the physiological and chemical substances systems [24–33] that’s have a tendency to regularly study oscillatory choices and you will head figure [25, 27, 29, 33–36].
We basic went 78 coupled Stuart-Landau models into the a scale-free model circle [37, 38]-that is, a system which have a degree delivery pursuing the an electrical energy laws-where coupling strength S anywhere between nodes would be varied just like the control factor. This new sheer volume of each node is actually at random removed regarding an excellent Gaussian shipping with the mean at 10 Hz and you can standard deviation of just one Hz, simulating the brand new alpha bandwidth (8-13Hz) away from person EEG, and we also systematically varied the new coupling stamina S out of 0 in order to fifty. We including varied committed decelerate parameter across a broad 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.
We upcoming proceeded to identify the new dating ranging from system topology (node degree), node dynamics (amplitude) and you will directionality ranging from node personality (dPLI) (come across S1 Text message to own 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 https://datingranking.net/de/biracial-dating-de/ (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 .