Spatial multi-scaled chimera states of cerebral cortex network and its inherent structure-dynamics relationship in human brain

Abstract Human cerebral cortex displays various dynamics patterns under different states, however the mechanism how such diverse patterns can be supported by the underlying brain network is still not well understood. Human brain has a unique network structure with different regions of interesting to perform cognitive tasks. Using coupled neural mass oscillators on human cortical network and paying attention to both global and local regions, we observe a new feature of chimera states with multiple spatial scales and a positive correlation between the synchronization preference of local region and the degree of symmetry of the connectivity of the region in the network. Further, we use the concept of effective symmetry in the network to build structural and dynamical hierarchical trees and find close matching between them. These results help to explain the multiple brain rhythms observed in experiments and suggest a generic principle for complex brain network as a structure substrate to support diverse functional patterns.

In this work, we study how chimera states with multiple scales can emerge from the underlying structural substrates of human cortical network, which can provide new insights into the generic dynamical network principle underlying brain structure-function relationship. For this purpose, we use the data of Ref. [1,2] to construct a network of cerebral cortex. In this data, the cerebral cortex was divided into relatively uniform regions of interests (ROIs) with each representing a network node, and the connections in all possible pairs of ROIs were measured noninvasively by using diffusion spectrum imaging (DSI). In this way, a connection between two ROIs was derived from the number of fibers found by the tractography algorithm, which results in connections and isolated nodes without detected fibers due to resolution limitation of DSI. Furthermore, the cerebral cortex can be parcellated into functional regions [1,2]. In our work, we here remove the isolated nodes, leaving nodes, with nodes in the right hemisphere and nodes in the left hemisphere. The number of cortical regions covered by these nodes is also reduced to . The obtained connection matrix is actually weighted, with the connection weights representing the fiber density between the connected nodes. Fig. S1 shows the weighted connection matrix, where the points represent the presence of links and the color represents the value of weight with .
FIG. S1. The weighted connection matrix for the network of cerebral cortex with nodes and links. Data from Ref. [11,12] where represents the ROI with , the points represent the presence of links and the color represents the weight.
To see the cortical regions of Fig. S1 clear, we put all the nodes on a circle and number them from one cortical area to another one. The regions are equally distributed on the left and right hemispheres, i.e.
on the left hemisphere and on the right hemisphere. We label the nodes in each region consecutively. Then, we put the links correspondingly into the circle. In this way, the links in the same cortical region, between different regions, and between the left and right hemispheres will not be overlapped, so that it is clear to see how the cortical regions are connected. Fig. S2 shows the topology where the names of anatomical cortical regions are labeled on the circle and the green, blue and red lines represent the links among the nodes within cortical regions, between different regions, and between the left and right hemispheres, respectively. The brain network can be also represented by the coarse-grained network of cortical regions. For this purpose, we consider two cortical regions and be connected if there is at least one link between their nodes. The weight of this inter-regional connection will be the average weight for all those links between ROIs in the two regions and . Fig.  S3 shows the coarse-grained network of cortical regions from Fig. S1.
cortex, ST= superior temporal cortex, SMAR= supramarginal gyrus, TP= temporal pole, and TT = transverse temporal cortex. represents the cortical region with , the points represent the presence of links and the color represents the connection weight. Table S1. Structural parameters of all the cortical regions and their local order parameters for six typical cases. The first and second columns represent the index and their functional names of all the cortical regions, respectively, the third to fifth columns represent the number of oscillators in each region-, the average degree of region-, and the average intra-degree for those links within the same region-, respectively, and the columns represent the local of all the cortical regions for six typical cases with the parameter sets of for cases being , , , , , and , respectively.   Fig. 3(a) in the main text shows the local representation of for the case of and and Fig. 3(b) in the main text shows its CS within lFUS. The situation is similar for some other regions in Fig. 3(a) of main text with clearly smaller than and is quite generic for different parameters . We here show one more example corresponding to Fig. 1(b) in the main text, see Fig. S4 where (a) and (b) represent and for all the cortical regions in the left and right hemispheres, respectively. We see that the values of and are different for different cortical regions, suggesting that different regions may show different CS. Furthermore, we take the cortical region (lFUS) with the nodes from Fig. S4(a) as an example. Fig. S4(c) shows its spatiotemporal pattern. It can be seen that some of the oscillators are synchronized and the others are incoherent, indicating a CS. Thus, the chimera state in the case of Fig. 1(b) in the main text also shows the feature of spatial multi-scaled CS.  , and , respectively. It is easy to see that their values of are also distributed, although they are different from case to case. For reference, we here show all the phase diagram of of the cortical regions on the parameter space plane, see Fig. S5 for the results. We see that the distribution of can be quite different for different cortical regions, indicating that they take different roles in brain functions and thus guarantee the diversity of CS patterns.

C. Case of distributed time-delays
The time delay is approximately considered as a constant in the main text. However, in realistic situation, the time delay is not a constant but related to the distance between two nodes. It was pointed out that the axonal conduction delays depend on the distance between neurons in the cerebral cortex and can amount to several tens of milliseconds [3]. Then, an important question is whether the observed results in the main text are robust to the distributed time delay . To answer this question, we first need to know how the time delay is formed in cerebral cortex. Two key elements of time delay are the speed of signal transmission over links and the distance between two connected nodes, i.e. the length of link. For the first aspect, many studies have been done on both human beings and animals and it is found that the speed of signal transmission is finite. This limited speed of signal transmission is the direct reason to give rise to a finite time delay [3]. For animals, it was pointed out that the axonal conduction velocities are of for ipsilateral cortico-cortical connections and for callosal cortico-cortical fibers in the rabbit brain [4]. While for the human brain, it was pointed out that signals travel with approximately for myelinated and for unmyelinated fibers. In sum, the conclusion is that the speed of signal transmission is complicated, i.e. depending on agents. For simplicity, we here approximately consider the same speed for all the links in the network of cerebral cortex. For the second aspect, it is difficult to measure the real distance between two connected nodes because of the complicated structure of cerebral cortex such as the gyrus. Especially, the links between the left and right hemispheres are connected by the corpus callosum, which makes the measure more difficult.
Combining these two aspects, we here use an approximate approach, i.e. Euclidean distance, to measure all the link lengths of the network of cerebral cortex and assume that the time delay for each link is proportional to its link length. In details, we first calculate a Euclidean distance between connected two nodes and . Then, we obtain the time delay for the link between the two nodes and as follows ⑴ where and represent the maximum of and the maximum of , respectively. Fig. S6 shows the local order parameter where the x-axis represents the coupling strength , the y-axis represents the index of local regions, and (a) and (b) represent the cases of and , respectively. It is easy to notice that the two panels of Fig. S6 are similar to each other, indicating that the local order parameter is robust to the delay parameter .
We also notice from the two panels of Fig. S6 that the preference for to take larger values is different for different region . Especially, the regions and have the largest fraction for to take larger values. Recall that we have revealed in the main text that the regions and have the largest values of synchronized . Thus, the regions with the largest fraction of synchronized in Fig. S6 is highly consistent with the regions with the largest values of synchronized in Fig. 4(a) in the main text, confirming that these specific regions have the most probability to take part in the brain functions.
Further, we would like to check how the distributed influences the spatial multiscaled CS. For this purpose, we arbitrary choose a specific case from Fig. S6(a) with and show all the values of for the local regions in Fig. S7(a). We see that the regions and have the largest values of . Then, we choose a region (lMOF) with a middle value of to check its dynamical behaviors, shown in Fig.  S7(b). We see that it is a coexistence of synchronized cluster and disordered oscillators, indicating that it is a CS. Comparing Fig. S7(b) with Fig. 3(b) in the main text, we see that they are similar each other, confirming the robustness of the spatial multi-scaled CS to the distribution of .

D. Hierarchy trees of anatomical and functional networks
By the dissimilarity , Fig. 6(a) and (b) in the main text show the hierarchical trees of both anatomical and functional networks for the case of and , where the hierarchy trees of both anatomical and functional networks are divided into four branches. An interesting finding is that the hierarchy tree of anatomical network is closely related to the hierarchy tree of functional network. That is, some dynamical branches match well with the corresponding structural branches, while other dynamical branches are the combination of a few structural branches. Here we examine the relationship at larger scale by dividing the trees into two branches, see Fig. S8. The results show that ST1 contains the whole right hemisphere and part of left hemisphere, while ST2 completely comes from part of the left hemisphere [ Fig. S8 (c)]. DT1 is mainly from ST2, but also from part of ST1 [ Fig. S8 (e)], while DT2 is mainly from ST1, but also from part of ST2 [ Fig. S8 (f)], and DT1 is solely from the left hemisphere [ Fig. S8 (d)]. The coherence in the whole right hemisphere and incoherence between different dynamical clusters in the left hemisphere is similar to the situation of unihemispheric sleep in human brain. The results suggest that it is possible under suitable dynamical conditions to have only part of the hemisphere to be dynamically segregated (here DT1), while leaving the others in a coherent state, including the combination between left and right hemispheres (DT2). The results further confirm that there is a close relationship between the hierarchy trees of anatomical and functional networks that the structural trees provide both constrains on the dynamical clusters (e.g., a structural branch can dominate a dynamical cluster) and the flexibility of combinations (e.g., the structural branches can split and reunite to from different clusters). Our analysis shows that this close relationship between the hierarchy trees of anatomical and functional networks also holds for other cases of chimera states.