Personality regarding mathematical matchmaking one of node education, amplitude out of local vibration and you may directionality of relations

After that, this new directionality anywhere between all the local node character is counted utilizing the led phase lag index (dPLI), hence exercises the latest stage direct and you may lag matchmaking anywhere between a couple oscillators (pick Product and techniques to own detail by detail definition)

The new central function of this study was to select an over-all dating regarding circle topology, regional node dynamics and directionality during the inhomogeneous companies. I proceeded of the building an easy combined oscillatory circle design, playing with a good Stuart-Landau model oscillator so you can depict the latest neural mass populace pastime from the per node of your own community (come across Product and methods, and you may S1 Text message having info). The fresh Stuart-Landau design is the typical style of the brand new Hopf bifurcation, meaning that simple fact is that best design trapping the most top features of the device close to the bifurcation point [22–25]. New Hopf bifurcation seems commonly for the biological and agents systems [24–33] that will be will always investigation sito solo incontri gamer oscillatory decisions and you can brain fictional character [twenty five, 27, 31, 33–36].

I very first went 78 coupled Stuart-Landau models on the a size-100 % free model system [37, 38]-that is, a system that have a diploma shipments following the a power rules-where coupling energy S between nodes is varied given that control parameter. The natural regularity of every node was randomly pulled out-of a beneficial Gaussian distribution for the indicate in the ten Hz and you will practical departure of 1 Hz, simulating the latest alpha data transfer (8-13Hz) away from person EEG, and we methodically varied the new coupling strength S out of 0 in order to 50. We together with ranged the amount of time slow down factor across the an over-all range (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 identify new relationship ranging from community topology (node degree), node personality (amplitude) and you will directionality anywhere between node personality (dPLI) (get a hold of S1 Text message to have done derivation)

dPLI between two nodes a and b, dPLI_ab, 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 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 .