Fig 1.
Methodological flow of the study.
(A) The methodology of the study is shown sequentially. We simulated oscillators zj(t) on model complex networks, then derived the analytical result. We applied the same simulation scheme for the human anatomic network and empirically validated the result from human EEG analysis. We made predictions by applying the simulation scheme to the human brain networks. (B) The simulation scheme for networks is shown. Stuart-Landau oscillators zj(t) were applied to the node of each network. We measured whether the signals from each oscillator would phase lead or lag compared to other oscillators using dPLI. (C) We analytically demonstrate that for oscillators zj(t) on networks with sufficient coupling strength S and small time delay τjk, if degree of node m is larger than degree of node n, the amplitude will be larger and phase lag n. (D) From 64 channel human EEG data, we constructed a connectivity network between each channel and measured phase lead/lag relationships by dPLI.
Fig 2.
Distinct local dynamics of hub and peripheral nodes.
A coupled Stuart-Landau model oscillator was simulated on a scale-free network with 78 nodes; distinct local dynamics at hub nodes (green triangle: defined as nodes with degree above the group standard deviation), and peripheral nodes (yellow circle: defined as nodes with degree 1) are found as coupling strength S is varied. (A) Mean phase coherence (PC), (B) amplitude, and (C) averaged dPLI for the two groups of nodes are presented. (D) The average coupling strength of j, Kj, is shown as function of amplitude . Here,
is the order parameter. We analytically identified that
is a monotonic increasing function of
, such that
. For the simulation, the time delay between each node was given as 10ms.
Fig 3.
Relationships of node degree, amplitude and dPLI in inhomogeneous model networks.
The Stuart-Landau model was simulated on two different inhomogeneous networks, a random network (A, C) and a scale-free network (B, D). The dPLI and amplitude have strong correlations with node degree, which demonstrate the relationship between network topology (node degree) and local node dynamics (i.e., phase and amplitude modulation). Larger node degrees have phase lag (dPLI <0) and larger amplitude, while smaller node degrees have phase lead (dPLI >0) and smaller amplitude, irrespective of the type of inhomogeneous network. Average dPLI for each node was calculated by averaging the dPLI values of each node with respect to all other nodes. For the simulation, the time delay between each node was given as 10ms. The coupling strength S was set to 1.5, where the separation of activities between hub nodes and peripheral nodes begins to emerge.
Fig 4.
Relationships of node degree, amplitude and dPLI in human neuroanatomical networks.
The Stuart-Landau model was simulated on the human anatomical brain network before ((A), (C) and (E)) and after ((B), (D) and (F)) perturbation with preferential disruption of hub nodes. The general relationship of node degree, amplitude and dPLI is also demonstrated in this modeled human brain network. The strong negative correlations between node degree and dPLI in (A) and the strong positive correlation between node degree and amplitude in (C) disappear in the perturbed homogeneous network ((B) and (D)). Average dPLI for each node was calculated by averaging the dPLI values of each node with respect to all other nodes. The anatomical connectivity of different brain regions are presented in (E) and (F) ring plots together with average dPLI value for each region. The nodes are aligned in groups: frontal lobe, central regions (including motor and somatosensory cortex), parietal lobe, occipital lobe, temporal lobe, limbic region, and Insula (Ins). Red arrow in (E) points to left and right precuneus. Color of each node shows the average dPLI values with respect to other nodes, from red (dPLI = 1) to blue (dPLI = -1). Average dPLI for each group is also shown in color. The inset within the ringplot shows connections between nodes, highlighted by darker color if the node has a higher degree of connections. Only the links from hub nodes (node with degree value within top 30%) are colored. In the simulation, the time delay between each node was given proportional to the delay, with propagation speed of 6m/s. The coupling strength S was given as 3. The full names for the cortical regions of the human brain network are available in Gong et al. [47].
Fig 5.
Comparison between model network and EEG network.
Results were compared from the Stuart-Landau model on the human anatomical brain network of Gong et al. (A) and functional networks reconstructed from EEG (B). For each case, degree, average dPLI, and amplitude for each node is plotted with standard error, before and after perturbing the anatomic network for (A), and in wake (eyes closed) and anesthetized states for (B). In the graphs of the first row, the degree of nodes in red/blue circle is from PLI of the functional network constructed for each case. For the case of the model, (A), the degree of anatomical network is also shown in gray triangle for comparison. The nodes are aligned in a way that they are grouped by regions and span from frontal lobe to parietal lobe. For (A), nodes 1~22 are from frontal lobe, nodes 23~30 are from central regions (motor and somatosensory cortex), and nodes 31~40 are from parietal lobe. For (B), nodes 1~18 are from frontal lobe, nodes 19~29 are from central regions, and nodes 31~39 are from parietal lobe. For the simulation, time delay between each node was given proportional to the delay, with propagation speed of 6m/s. The coupling strength S was given as 1.5.