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Edge-centric functional network representations of human cerebral cortex reveal overlapping system-level architecture

Richard F. Betzel
Network neuroscience has relied on a node-centric network model in which cells, populations and regions are linked to one another via anatomical or functional connections. This model cannot account for interactions of edges with one another. In this study, we developed an edge-centric network model that generates constructs 'edge time series' and 'edge functional connectivity' (eFC). Using network analysis, we show that, at rest, eFC is consistent across datasets and reproducible within the same individual over multiple scan sessions. We demonstrate that clustering eFC yields communities of edges that naturally divide the brain into overlapping clusters, with regions in sensorimotor and attentional networks exhibiting the greatest levels of overlap. We show that eFC is systematically modulated by variation in sensory input. In future work, the edge-centric approach could be useful for identifying novel biomarkers of disease, characterizing individual variation and mapping the architecture of highly resolved neural circuits.
网络神经科学一直依赖于一个以节点为中心的网络模型,其中细胞、群体和区域通过解剖或功能连接相互联系。这种模型无法解释边缘之间的相互作用。在这项研究中,我们开发了一个以边缘为中心的网络模型,生成了“边缘时间序列”和“边缘功能连接”(eFC)的构建。利用网络分析,我们展示了在静息状态下,eFC 在数据集之间保持一致,并在同一人在多次扫描会话中具有可重复性。我们证明,对 eFC 进行聚类会产生将大脑自然划分为重叠簇的边缘社区,感觉运动和注意力网络中的区域展现出最高水平的重叠。我们展示了 eFC 受感觉输入变化的系统调节。在未来的工作中,以边缘为中心的方法可能有助于识别疾病的新生物标志物,描述个体差异并绘制高分辨率神经回路的结构。
N etwork science offers a promising framework for representing and modeling neural systems . From interconnected cells , to neuronal populations , to large-scale brain areas , network analysis has contributed insights into the topological principles that govern nervous system organization and shape brain function. These include small-world architecture , the emergence of integrative hubs and rich clubs , modular structure to promote specialized information processing and tradeoffs between topological features and the material and metabolic costs of wiring .
Central to these and other discoveries in network neuroscience is a simple representation of the brain in which neural elements and their pairwise interactions are treated as the nodes and edges of a network, respectively 9 . This standard model is fundamentally node-centric in that it treats neural elements (nodes) as the irreducible units of brain structure and function. This emphasis on network nodes is further reinforced by the analyses carried out on brain networks, which tend to focus on properties of nodes or groups of nodes-for example, their centralities or community affiliations .
A limitation of the node-centric approach is that it cannot capture potentially meaningful features or patterns of inter-relationships among edges. In other scientific domains, prioritizing network edges-for example, by modeling and analyzing edge-edge interactions as a graph-has provided important insights into the organization and function of complex systems . Nonetheless, network neuroscience has remained largely focused on nodal features and partitions, paralleling a rich history of parceling, mapping and comparing cortical and subcortical gray matter regions . On the other hand, several recent studies have begun modeling brain networks from the perspective of interacting edges, including one foundational study that applied graph theoretic measures to a 'line graph'12 of interrelated white matter tracts . Although highly novel, line graphs were never adopted widely, as their construction requires users to first specify and then apply a sparsity threshold to a connectivity matrix.
Here we present a novel modeling framework for investigating functional brain network data from an edge-centric perspective. Our approach can be viewed as a temporal 'unwrapping' of the Pearson correlation measure-the metric commonly used for estimating the strength of functional connectivity between pairs of brain regions -thereby generating interpretable time series for each edge that express fluctuations in its weight across time. Importantly, edge time series allow the estimation of edge correlation structure, a construct we refer to as eFC. High-amplitude eFC indexes strong similarity in the co-fluctuation of two edges across time, whereas low-amplitude eFC indicates co-fluctuation patterns that are largely independent.
在这里,我们提出了一个新颖的建模框架,用于从边缘中心的角度研究功能性脑网络数据。我们的方法可以被视为对Pearson相关性测量的时间“展开” - 这是一种常用于估计脑区域间功能连接强度的度量标准 - 从而为每个边缘生成可解释的时间序列,表达其权重在时间上的波动。重要的是,边缘时间序列允许估计边缘相关结构,我们称之为eFC。高振幅的eFC指数表示两个边缘在时间上共同波动的相似性很强,而低振幅的eFC表示共同波动模式在很大程度上是独立的。
From a neuroscientific perspective, eFC can be viewed both as an extension of and a complement to traditional node-centric representations of brain networks. In node-centric network models, functional connections represent the temporal correlation of activity recorded from spatially distinct regions and often interpreted as a measure of inter-regional communication . That is, strong functional connections are thought to reflect the time-averaged strength of 'communication' between brain regions. eFC, on the other hand, tracks how communication patterns evolve over time and ultimately assesses whether similar patterns are occurring in the brain simultaneously (Supplementary Fig. 1).
In this study, we demonstrate that eFC is highly replicable given sufficient amounts of data, stable within individuals across multiple scan sessions and consistent across datasets. Next, we apply data-driven clustering algorithms to , which result in partitions of the eFC network into communities of co-fluctuating edges. Each community can be mapped back to individual nodes, yielding overlapping regional community assignments. We find that brain regions associated with sensorimotor and attention networks participate in disproportionately many communities compared to other brain systems, but that, relative to one another, those same regions participate in similar sets of communities. Finally, we compare the organization of eFC at rest and during passive viewing of movies and find that eFC is consistently and reliably modulated by changes in sensory input.
在这项研究中,我们展示了在有足够数据的情况下,eFC 是高度可复制的,在多个扫描会话中在个体内是稳定的,并且在数据集之间是一致的。接下来,我们应用数据驱动的聚类算法对 进行处理,将 eFC 网络划分为共同波动边缘的社区。每个社区可以映射回个体节点,产生重叠的区域社区分配。我们发现与感觉运动和注意力网络相关的大脑区域参与了比其他大脑系统更多的社区,但相对于彼此,这些区域参与了类似的社区集合。最后,我们比较了静息状态下和 passively 观看电影期间的 eFC 组织,并发现 eFC 受感觉输入变化的影响是一致且可靠的。

Results 结果

In this section, we analyze eFC estimated using functional MRI (fMRI) data from three independently acquired datasets: 100 unrelated participants from the Human Connectome Project (HCP) , ten participants scanned ten times as part of the Midnight Scan Club (MSC) and ten participants scanned multiple times as part of the Healthy Brain Network (HBN) Serial Scanning Initiative .
在本节中,我们分析使用功能性磁共振成像(fMRI)数据估计的 eFC,这些数据来自三个独立获取的数据集:来自人类连接组计划(HCP)的 100 名无关参与者,作为午夜扫描俱乐部(MSC)的一部分被扫描了十次的十名参与者,以及作为健康大脑网络(HBN)串行扫描倡议的一部分被多次扫描的十名参与者。
eFC. Many studies have investigated networks whose nodes and edges represent brain regions and pairwise functional interactions, respectively . Here we extend this framework to consider interactions not between pairs of brain regions but between pairs of edges.
Extant approaches for estimating edge-edge connectivity matrices include construction of line graphs or calculating edge overlap indices . Although suitable for sparse networks with positively weighted edges, these approaches are less appropriate for functional neuroimaging data, where networks are typically fully weighted and signed. Here we introduce a method that is well suited for these types of data, operates directly on time series and is closely related to the Pearson correlation coefficient typically used to assess strength of inter-regional functional connections. We refer to the matrices obtained using this procedure as .
Beginning with regional time series, calculating eFC can be accomplished in three steps, starting by -scoring the time series (Fig. la,d). Next, for all pairs of brain regions, we calculate the element-wise product of their -scored time series (Fig. 1b,e). This results in a new set of time series, referred to as 'edge time series', whose elements represent the instantaneous co-fluctuation magnitude between pairs of brain regions and whose average across time is exactly equal to the Pearson correlation coefficient (Fig. 1c) . Co-fluctuation values are positive when activity of two regions deflects in the same direction at precisely the same moment in time, are negative when activity deflects in the opposite direction and are zero when activity is close to baseline. Importantly, the magnitude of these edge time series is not systematically related to in-scanner motion (Supplementary Fig. 2). The third and final step involves calculating the element-wise product between pairs of edge time series. When repeated over all pairs of edges, the result is an edge-by-edge matrix whose elements are normalized to the interval (Figs. If and 2a; see Methods for additional details on eFC construction).
从区域时间序列开始,可以通过三个步骤完成计算 eFC,首先是对时间序列进行评分(图1a,d)。接下来,对所有脑区域的配对,我们计算它们评分时间序列的逐元素乘积(图1b,e)。这将产生一组新的时间序列,称为“边缘时间序列”,其元素表示脑区域之间瞬时共同波动的大小,其时间平均值恰好等于皮尔逊相关系数(图1c)。当两个区域的活动在同一时刻完全朝着相同方向偏离时,共同波动值为正,当活动朝相反方向偏离时为负,当活动接近基线时为零。重要的是,这些边缘时间序列的大小与扫描中的运动没有系统关联(附图2)。第三和最后一步涉及计算边缘时间序列之间的逐元素乘积。当在所有边缘对上重复时,结果是一个边对边矩阵,其元素被归一化到区间内。 如果和 2a; 请参阅有关 eFC 构建的详细信息的方法)。
Although eFC is, to our knowledge, a novel construct, we note that the first two steps in calculating eFC are the same as those used to calculate nodal functional connectivity ( ); the mean value of any co-fluctuation time series is simply the Pearson correlation coefficient. Given that eFC is mathematically related to , we first asked whether it was possible to approximate eFC using estimates of . This is an important question. Whereas the calculation of eFC can be implemented efficiently, performing certain operations on the eFC matrix can prove computationally expensive (it is a fully weighted matrix, where , and is the number of nodes; Fig. 2b). However, a direct comparison of and is not possible owing to differences in dimensionality. Still, we can approximate eFC using nFC edge weights. Perhaps the simplest approach is to model the edge connection between region pairs and as a linear combination of the six edges that can be formed between those regions (Methods). Although this model performs poorly (correlation of observed and approximated eFC; edge-edge pairs), we can improve on its performance by including interaction terms based on node connectivity-that is, edge-edge pairs; Fig. 2c). Collectively, these observations suggest that eFC is not well approximated using linear combinations of , but, with nonlinear transformations and inclusion of interaction terms, can approximate eFC. However, these transformations are unintuitive, and the approximation still fails to fully explain variance in .
尽管据我们所知,eFC 是一种新颖的构造,但我们注意到计算 eFC 的前两个步骤与计算节点功能连接的步骤相同( );任何共同波动时间序列的均值仅是皮尔逊相关系数。鉴于 eFC 在数学上与 有关,我们首先询问是否可能使用 的估计来近似 eFC。这是一个重要的问题。虽然可以有效地计算 eFC,但对 eFC 矩阵执行某些操作可能会在计算上昂贵(它是一个完全加权的 矩阵,其中 ,而 是节点数;图 2b)。然而,由于维度不同,无法直接比较 。尽管如此,我们可以使用 nFC 边缘权重来近似 eFC。也许最简单的方法是将区域对 之间的边缘连接建模为这些区域之间可以形成的六个边缘的线性组合(方法)。 尽管这个模型表现不佳(观察到的和近似的eFC之间的相关性; 边-边对),但我们可以通过包含基于节点连接性的交互项来改善其性能-也就是说, 边-边对;图2c)。总的来说,这些观察结果表明,使用线性组合无法很好地近似eFC,但是,通过非线性转换和包含交互项, 可以近似eFC。然而,这些转换是不直观的,而且近似仍然无法完全解释 中的方差。
Next, we explored variation of eFC across acquisitions and processing decisions. We found that weights are similar across three independently acquired datasets (Supplementary Fig. 3) and that the omission of global signal regression from our pre-processing pipeline induced a consistent upward shift of eFC weights, analogous to its effect on nFC (Supplementary Fig. 4). Additionally, we found that the overall pattern of eFC calculated using edge time series estimated from observed data was uncorrelated with the pattern of eFC calculated using edge time series estimated from phase-randomized surrogate time series (Supplementary Fig. 5).
接下来,我们探讨了在获取和处理决策过程中 eFC 的变化。我们发现在三个独立获取的数据集中, 权重相似(附图3),并且在我们的预处理流程中省略全局信号回归导致 eFC 权重一致上升,类似于其对 nFC 的影响(附图4)。此外,我们发现使用从观察数据估计的边缘时间序列计算的 eFC 的整体模式与使用从相位随机化替代时间序列估计的边缘时间序列计算的 eFC 模式不相关(附图5)。
Next, we asked whether eFC exhibits any clear spatial dependence, as is known to decay as a function of Euclidean distance . We assessed the spatial dispersion of eFC with the surface area of the quadrilateral formed by the centroids of the brain region pairs (we explore an alternative edge-level distance metric in Supplementary Fig. 6). We found evidence of a weak negative relationship between surface area and eFC ; edge-edge pairs; Fig. 2d), suggesting that, unlike traditional , whose connection weights are more strongly influenced by spatial relationships of brain areas to one another, eFC is less constrained by the brain's geometry.
Finally, we asked whether eFC bears the imprint of nFC communities-brain regions whose activity is highly correlated with members of its own community but weakly correlated or anti-correlated with members of other communities? To address this question, we classified every edge in the nFC network according to whether it fell within or between brain systems , resulting in three possible combinations of connections in the eFC graph: eFC could link edges that both fell within a community, edges that both fell between communities or an edge that fell within and an edge that fell between communities. In general, we found that eFC was significantly stronger for within-community edges compared to the other two categories (Fig. 2e). Interestingly, we found that eFC could be distinguished further by dividing within-community edges by cognitive system (one-way analysis of variance (ANOVA); ; Fig. 2f).
eFC is stable within individuals. In this section, we describe the robustness of eFC to scan duration - that is, how much data are required before eFC stabilizes and whether eFC is consistent across repeated scans of the same individual. To address these questions, we leveraged the within-individual design of the MSC dataset. For each participant, we aggregated their fMRI data across all scan sessions and estimated a single eFC matrix. Then, we sampled smaller amounts of temporally contiguous data, thus approximately preserving the auto-correlation structure, and estimated , which we compared against the aggregated eFC matrix (this procedure was repeated 25 times). Similarly to other studies , we found that, with small amounts of data, eFC was highly variable (Fig. 3a). However, we observed a monotonic increase in similarity as a function of time, so that, with of data, the similarity was much greater edge-edge pairs). This is of practical significance; like traditional nFC, it implies that eFC estimated using data from short scan sessions might not deliver accurate representations of an individual's edge network organization. We note that this relationship is strengthened when data are sub-sampled randomly and uniformly edgeedge pairs; Supplementary Fig. 7).
eFC在个体内是稳定的。在本节中,我们描述了eFC对扫描持续时间的稳健性 - 即在eFC稳定之前需要多少数据,以及eFC在同一参与者的重复扫描中是否一致。为了回答这些问题,我们利用了MSC数据集的个体内设计。对于每个参与者,我们汇总了他们在所有扫描会话中的fMRI数据,并估计了一个单一的eFC矩阵。然后,我们对时间上连续的较小数据进行抽样,从而大致保留了自相关结构,并估计了

Fig. 1 | Derivation of the eFC matrix. Each element of the eFC matrix is calculated based on the fMRI BOLD activity time series from four nodes (brain regions). a,d, We show four representative times series from regions , u and v. c, nFC is typically calculated by standardizing ( -scoring) each time series, performing an element-wise product (dot product) of time series pairs and calculating the average value of a product time series (actually the sum of each element divided by , where is the number of observations). To calculate eFC, we retain the vectors of element-wise products for every pair of regions. b,e, We show product time series for the pairs and , respectively. The elements of these product time series represent the magnitude of time-resolved co-fluctuation between region pairs (or edges in the nFC graph). We can calculate the magnitude of eFC by performing an element-wise multiplication of the product time series and normalizing the sum by the squared root standard deviations of both time series, ensuring that the magnitude of eFC is bounded to the interval . , The resulting value is stored in the eFC matrix.
图1 | eFC矩阵的推导。 eFC矩阵的每个元素是基于四个节点(脑区)的fMRI BOLD活动时间序列计算得出的。a、d,我们展示了来自区域 ,u和v的四个代表性时间序列。c,nFC通常通过对每个时间序列进行标准化(标准分数化),执行时间序列对的逐元素乘积(点乘)并计算产品时间序列的平均值(实际上是每个元素除以 ,其中 是观察次数)。为了计算eFC,我们保留了每对区域的逐元素乘积的向量。b、e,我们分别展示了对应于配对 的产品时间序列。这些产品时间序列的元素表示了区域对之间的时间相关共同波动的幅度(或nFC图中的边)。我们可以通过对产品时间序列进行逐元素乘法并将总和标准化为两个时间序列的平方根标准差来计算eFC的幅度,确保eFC的幅度被限制在间隔 内。 ,得到的值存储在 eFC 矩阵中。
Next, we examined the reliability of eFC over multiple scan sessions. That is, if we imaged an individual on separate days, would their eFC on those days be more similar to each other than to that of a different individual? We estimated eFC and calculated the pairwise similarity (Pearson correlation) between all pairs of MSC participants and scans. We found eFC to be highly reliable in the MSC dataset, where the mean within-participant similarity was compared to between participants (two-sample -test; Fig. ). Indeed, we found that, for each eFC matrix, the matrix to which it was most similar belonged to the same participant ( accuracy). Additionally, eFC exhibited slightly greater differential identifiability compared to to 0.210 -calculated as the difference between mean within- and between-participant similarity . In Fig. , we show the results of applying multi-dimensional scaling to the similarity matrix from Fig. 3b. We found similar results in the HBN and HCP datasets (Supplementary Fig. 8).
接下来,我们检查了 eFC 在多次扫描会话中的可靠性。也就是说,如果我们在不同的日子对一个人进行成像,他们在那些日子的 eFC 是否更相似,而不是与另一个人的 eFC 更相似?我们估计了 eFC 并计算了所有 MSC 参与者和扫描之间的成对相似性(皮尔逊相关性)。我们发现在 MSC 数据集中,eFC 非常可靠,其中参与者内部相似性的平均值为 ,而参与者之间的相似性为 (双样本 -检验; )。事实上,我们发现,对于每个 eFC 矩阵,它最相似的矩阵属于同一参与者( 准确度)。此外,与 相比,eFC 表现出稍微更大的差异可识别性,为 0.210 -计算为参与者内部和参与者之间相似性之间的平均差异 。在图 中,我们展示了将图 3b 中的相似性矩阵应用于多维缩放的结果。我们在 HBN 和 HCP 数据集中发现了类似的结果(附图 8)。
Collectively, these findings suggest that eFC exhibits a high level of participant specificity and captures idiosyncratic features of an individual, provided that eFC was estimated over a sufficiently long time period. This observation serves as an important validation of and suggests that eFC might be useful in future applications as substrate for biomarker generation and 'fingerprinting .
总的来说,这些发现表明,eFC 表现出高度的参与者特异性,并捕捉到个体的特异特征,前提是 eFC 是在足够长的时间段内估计的。这一观察结果是对 的重要验证,并暗示 eFC 可能在未来的应用中作为生物标志物生成和“指纹化 ”的基质有用。
The overlapping community structure of human cerebral cortex. Although many studies have investigated the brain's community structure , most have relied on methodology that forces each brain region into one and only one community. However, partitioning brain regions into non-overlapping communities clashes with evidence suggesting that cognition and behavior require contributions from regions that span multiple node-defined communities and systems . Accordingly, a definition of communities is needed that more closely matches the brain's multifunctional nature and the pervasive overlap of its community structure .
人类大脑皮层的重叠社区结构。尽管许多研究已经调查了大脑的社区结构 ,但大多数依赖于将每个脑区强制分配到一个且仅一个社区的方法。然而,将脑区划分为不重叠的社区与证据相冲突,该证据表明认知和行为需要来自跨越多个节点定义的社区和系统的区域的贡献 。因此,需要一个更符合大脑多功能性质和其社区结构普遍重叠的社区定义
Although deriving overlapping communities of brain regions can be challenging when using , overlap is inherent (indeed, pervasive ) within the eFC construct. Clustering the eFC matrix assigns each edge to a community. Each edge is associated with two brain regions (the nodes it connects). Thus, edge community assignments can be mapped back onto individual brain regions and, because every region is associated with edges, allow regions to simultaneously maintain a plurality of community assignments.
尽管在使用 时,推导出大脑区域的重叠社区可能具有挑战性,但在 eFC 构造中重叠是固有的(确实,是普遍的 )。对 eFC 矩阵进行聚类将每个边分配给一个社区。每个边与两个大脑区域(它连接的节点)相关联。因此,边的社区分配可以映射回单个大脑区域,并且因为每个区域与 条边相关联,允许区域同时保持多个社区分配。
In this section, we cluster eFC matrices to discover overlapping communities in human cerebral cortex. More specifically, we use a modified -means algorithm to partition the eFC matrix into non-overlapping communities and map the edge assignments back to individual nodes.
在本节中,我们对人类大脑皮层中的 eFC 矩阵进行聚类,以发现重叠的社区。更具体地,我们使用修改后的 -means 算法将 eFC 矩阵划分为非重叠的社区,并将边的分配映射回单个节点。
In Fig. 4, we show representative communities obtained with (Supplementary Figs. 9 and 10 show examples with different numbers of communities). To demonstrate that the communities capture meaningful variance in our data, we show the edge co-fluctuation time series, the eFC matrix and the community co-assignment matrix reordered according to the derived communities (Fig. ). Here the elements of the co-assignment matrix represent the probability that two edges were assigned to the same community across partitions as we varied the number of communities from to .
在图4中,我们展示了使用 获得的代表性社区(附图9和10展示了具有不同社区数量的示例)。为了证明这些社区捕捉到了我们数据中的有意义的变化,我们展示了边缘共同波动时间序列、eFC矩阵和根据派生社区重新排序的社区共分配矩阵(图 )。这里,共分配矩阵的元素表示两条边被分配到相同社区的概率,随着我们将社区数量从 变化到 的分区。

Fig. 2 I Organization of the eFC matrix. a, Force-directed layout of the eFC matrix (largest connected component after thresholding away weak connections). Nodes in this graph represent edges in the traditional nFC matrix. Here nodes are colored according to whether the corresponding edge fell within or between cognitive systems. Within-system edges are encircled in black. b, eFC matrix in which rows and columns correspond to pairs of brain regions. c, Two-dimensional histogram of the relationship between and the product of edges' respective weights. , Two-dimensional histogram of the relationship between eFC and the surface area of the quadrilateral defined by the four nodes. e, Mean eFC among edges where both edges fall between systems (between; ), where one edge falls within and the other between systems (mixture; ) and where both edges fall within systems (within; ). f, Mean eFC among edges within 16 cognitive systems ). All results presented in this figure are derived from HCP data. Box plots, shown in green and overlaid on data points in and , depict the interquartile range (IQR) and the median value of the distribution. Whiskers extend to the nearest points IQR above and below the 25th and 75th percentiles. Note that, in e, not all points can be displayed owing to the large number of edge-edge connections.
图2:eFC矩阵的组织。a,eFC矩阵的力导向布局(在去除弱连接后的最大连通组件)。该图中的节点代表传统nFC矩阵中的边。这里的节点根据相应边是否在认知系统内或系统间而着色。系统内边用黑色圈起。b,eFC矩阵,其中行和列对应于大脑区域对。c, 与边的 权重乘积之间关系的二维直方图。 ,eFC与由四个节点定义的四边形的表面积之间关系的二维直方图。e,两个边都在系统间(系统间; )之间、一个边在系统内另一个在系统间(混合; )以及两个边都在系统内(系统内; )的边之间的平均eFC。f,16个认知系统内边的平均eFC )。本图中所有结果均来自HCP数据。 箱线图,显示为绿色并叠加在 的数据点上,描述了分布的四分位距(IQR)和中位数值。须至最近的点延伸到第25和第75百分位数上下 的IQR。请注意,在e中,由于边缘连接的数量较多,无法显示所有点。

Fig. 3 | Intra- and inter-participant similarity of eFC across scan sessions. a, Correlation of session-averaged eFC matrices with eFC estimated using different amounts of data; the mean value is shown as a black line. b. Similarity of eFC within and between participants. Each block corresponds to data from a single participant; participants are also identifiable by the color of the rectangle alongside each block. c, Violin plots of within- and between-participant similarity values from the matrix shown in and within- and between-participant comparisons (two-sample -test; ). Box plots, shown in green and overlaid on data points in , depict the IQR (box) and the median value of the distribution. Whiskers extend to the nearest points IQR above and below the 25 th and 75 th percentiles. d, Scan sessions plotted according to coordinates generated by performing a two-dimensional multidimensional scaling (MDS) analysis of the matrix in b. Note that scans from the same participant (shown here with the same color) are located near each other. All panels from this figure were generated using data from the MSC.
图3 | 扫描会话中 eFC 的参与者内部和参与者间相似性。a,会话平均 eFC 矩阵与使用不同数据量估算的 eFC 的相关性;平均值显示为黑线。b,参与者内部和参与者间的 eFC 相似性。每个块对应于单个参与者的数据;参与者也可通过每个块旁边矩形的颜色进行识别。c,来自 矩阵的参与者内部和参与者间相似性值的小提琴图(两样本 -test; )。绿色的箱线图叠加在 中的数据点上,描述了分布的 IQR(箱体)和中位数值。须端延伸至最近的点 IQR 在第25和第75百分位数之上和之下。d,根据在b中执行的二维多维尺度(MDS)分析生成的坐标绘制的扫描会话。请注意,来自同一参与者的扫描(此处显示为相同颜色)位于彼此附近。本图中的所有面板均使用 MSC 的数据生成。
Although the communities detected here are defined at the level of edges rather than nodes, we can project edge communities back onto brain regions. This was accomplished by extracting the edges associated with each community, determining which nodes were at the endpoints of each edge (the 'stubs') and counting the number of times that each node was represented in this stub list. We show these results in matrix form in Fig. 4d. In this panel, rows and columns represent nodes ordered according to the canonical system labels described in ref. .
尽管此处检测到的社区是在边的级别上定义的,而不是节点,但我们可以将边的社区投影回大脑区域。这是通过提取与每个社区相关联的边,确定哪些节点位于每个边的端点('存根')并计算每个节点在此存根列表中出现的次数来实现的。我们在图 4d 中以矩阵形式展示这些结果。在此面板中,行和列表示根据参考文献中描述的规范系统标签排序的节点。
The overlapping nature of communities is made clearer in Fig. 4e, in which communities are represented topographically. The edges associated with the same visual nodes are all involved in communities 7, 8, 9 and 10 to some extent, thereby linking the visual system to multiple other brain systems. In community 8 , for example, edges incident upon nodes in the visual and somatomotor systems are clustered together, whereas, in community 9, edges incident upon visual and control network nodes are assigned to the same community.
在图 4e 中,社区的重叠性质更加清晰,其中社区以地形方式表示。与相同视觉节点相关联的边在某种程度上都涉及到社区 7、8、9 和 10,从而将视觉系统与多个其他大脑系统连接起来。例如,在社区 8 中,涉及到视觉和体感运动系统节点的边被聚集在一起,而在社区 9 中,涉及到视觉和控制网络节点的边被分配到同一个社区。
Community overlap and functional diversity of cognitive systems. In the previous section, we showed that the human cerebral cortex could be partitioned into overlapping communities based on its edge correlation structure. This observation leads to a series of additional questions. For instance, which brain areas participate in many communities? Which participate in few? If we changed the scale of inquiry-the number of detected communities-do the answers to these questions change? Do the answers depend on which dataset we analyze? In this section, we explore these questions in detail.
One strategy for assessing community overlap is to simply count the number of different communities to which each nodes' edges are assigned . A more nuanced measure that accounts for the distribution of edge community assignments is the normalized entropy, which indexes the uniformity of a distribution. We therefore calculated normalized entropy for every brain region while varying the number of communities from to . In this section, we focus on results with .
评估社区重叠的一种策略是简单地计算每个节点的边被分配到不同社区的数量。考虑到边社区分配的分布的更微妙的度量是标准化熵,它指数了分布的均匀性。因此,我们在改变社区数量从 的同时,为每个脑区计算了标准化熵。在本节中,我们将重点关注 的结果。
We found that normalized community entropy was greatest within sensorimotor and attentional systems and lowest within regions traditionally associated with control and default mode networks (Fig. 5a-c). Notably, we obtained similar results from the MSC and HBN datasets (Supplementary Fig. 11), at the level of individual participants (Supplementary Fig. 12), as we varied the number of clusters (Supplementary Fig. 13) and when using different parcellation schemes (Supplementary Fig. 14). These observations seemingly contradict previous reports in which functional overlap was greatest in control networks and lowest in primary sensory systems (Fig. 5d) .
Is it possible to reconcile these seemingly opposed viewpoints? To address this question, we calculated a second measure of functional diversity. Whereas normalized entropy was defined at the level of individual brain regions based on the edge communities in which they participated, this second measure was defined at the level of brain systems as a whole and assessed the average similarity of edge community assignments among the system regions (Fig. 6a,b and Methods). Intuitively, functionally diverse systems are comprised of brain regions whose edge community assignments are unique and dissimilar from one another. We found that regions within sensorimotor networks, which exhibited among the highest levels of entropy, exhibited the greatest levels of within-system similarity (Fig. 6c). In contrast, sub-networks that make up the control network exhibited the lowest levels of within-system similarity, whereas their constituent nodes had among the lowest entropy (Fig. 6c).
In the Supplementary Material, we explore the relationship of normalized entropy with more familiar measures of overlap derived from , including participation coefficient, dynamic flexibility and versatility (Supplementary Figs. 15 and 16). We also compare patterns of normalized entropy derived from eFC community structure with entropy patterns obtained using alternative methods, including line graphs, the affiliation graph model, Bayesian non-negative matrix factorization and mixed-membership stochastic block models (Supplementary Fig. 17).
在补充材料中,我们探讨了标准化熵与更熟悉的重叠度量之间的关系,这些重叠度量来自于包括参与系数、动态灵活性和多功能性在内的方法(补充图 15 和 16)。我们还比较了从 eFC 社区结构中得出的标准化熵模式与使用替代方法获得的熵模式,包括线图、从属图模型、贝叶斯非负矩阵分解和混合成员随机块模型(补充图 17)。
eFC is modulated by changes in sensory input. In the previous sections, we demonstrated that is a reliable marker of an individual and that by clustering eFC we naturally obtain overlapping communities. We leveraged this final observation to demonstrate that sensorimotor and attentional systems participate in disproportionately more communities than association cortices. Analogous to previous studies documenting the effect of task on nodal FC, we expect that eFC is also modulated by task.
eFC 受感觉输入变化的调节。在前面的部分中,我们证明了 是个体的可靠标记,并且通过对 eFC 进行聚类,我们自然地获得了重叠的社区。我们利用这一最终观察结果来证明感觉运动和注意系统参与的社区比关联皮层多得多。类似于先前研究记录任务对结节 FC 的影响的研究,我们预期 eFC 也受任务调节。
To address this question, we analyzed fMRI data from the HBN Serial Scanning Initiative recorded during rest and while participants passively viewed the movie 'Raiders of the Lost Ark'. We estimated group-averaged eFC separately for each of the movie and rest scans.
为了解决这个问题,我们分析了 HBN Serial Scanning Initiative 记录的 fMRI 数据,包括休息时和参与者被动观看电影《夺宝奇兵》时的数据。我们分别估计了每个电影和休息扫描的群体平均 eFC。
In general, we found that eFC during movie watching was highly correlated with eFC estimated during rest (Fig. 7a). Across six movie
总的来说,我们发现在观看电影时的 eFC 与休息时估计的 eFC 高度相关(图 7a)。在六部电影中,

Fig. 4 | Edge communities. We applied similarity-based clustering to eFC from the HCP dataset. Here we show results with the number of clusters fixed at . a, Here we reordered edge time series according to the detected community assignments. Horizontal lines divide communities from each other. The colors to the left of the time series plots identify each of the ten communities. , We also reordered the rows and columns of the eFC matrix to highlight the same ten communities. Note that, on average, within-community eFC is greater than between-community eFC. c, We calculated the probability that pairs of edges (node pairs) were co-assigned to the same community. Here we show the co-assignment matrix with rows and columns reordered according to community assignments. Note that, in general, the range of co-assignment probabilities goes to 1. Here we truncate the color limits to emphasize the ten-community partition (Supplementary Fig. 9 shows the same co-assignment matrix at different values of and with non-truncated color limits). We present two visualizations of the edge communities projected back to brain regions (nodes). d, We depict overlapping communities in matrix form. Each column in this matrix represents one of ten communities. For each community and for each node, we calculated the proportion of all edges assigned to the community that included that node as one of its endpoints ('stubs'), indicated by the color and brightness of each cell. Dark colors indicate few edges; bright colors indicate many. e, Topographic distribution of communities. Note that many of the communities resemble known intrinsic connectivity networks. However, because the communities here can overlap, it is possible for nodes associated with a particular intrinsic connectivity network to participate in multiple edge communities.
图4 | 边缘社区。我们对来自HCP数据集的eFC应用基于相似性的聚类。这里我们展示了在固定聚类数为 时的结果。a,我们根据检测到的社区分配重新排序了边缘时间序列。水平线将各个社区分开。时间序列图左侧的颜色标识了十个社区中的每一个。b,我们还重新排序了eFC矩阵的行和列,以突出相同的十个社区。请注意,平均而言,社区内的eFC大于社区间的eFC。c,我们计算了边缘对(节点对)被分配到同一社区的概率。这里我们展示了根据社区分配重新排序的共分配矩阵的行和列。请注意,一般而言,共分配概率的范围为1。这里我们截断了颜色限制以强调十个社区的划分(附图9显示了在不同值的 和未截断颜色限制下的相同共分配矩阵)。我们提供了两种将边缘社区投影回脑区域(节点)的可视化。 d, 我们以矩阵形式描绘重叠的社区。该矩阵中的每一列代表十个社区中的一个。对于每个社区和每个节点,我们计算了分配给该社区的所有边中包含该节点作为其端点之一('stubs')的比例,由每个单元格的颜色和亮度表示。深色表示边较少;明亮的颜色表示边较多。e, 社区的地形分布。请注意,许多社区类似于已知的内在连接网络。然而,由于这里的社区可以重叠,因此与特定内在连接网络相关联的节点可能参与多个边社区。
scans, the mean correlation with resting eFC was (all edge-edge pairs). When we compared the pairwise similarity of all movie-watching scans with rest, we found that similarity of eFC was greater within a given task than between tasks , uniform and random permutation of movie and rest conditions; Fig. 7b). To better understand what was driving this effect, we generated representative matrices for rest and movie conditions and computed the element-wise difference between these matrices. We contrasted these differences with those estimated after randomly permuting scan (condition) labels and found that of all edge connections exhibited significant changes between conditions (permutation test; ; uncorrected). Although eFC differences were widespread, the most pronounced effects were associated with two specific communities (Fig. 7c), one of which exhibited a decrease in its within-module eFC, whereas both decreased eFC with respect to each other. These communities consisted of edges associated with somatomotor and visual systems (Fig. 7d). To confirm that these system-level effects were statistically significant, we compared the mean within- and between-system eFC differences against a constrained null model in which edges communities were randomly permuted repetitions; Supplementary Fig. 18 shows a detailed schematic illustrating
扫描结果显示,与静息状态的eFC的平均相关性为 (所有 边缘对)。当我们比较所有观影扫描与静息状态之间的成对相似性时,我们发现在给定任务内的eFC相似性大于任务间的相似性 ,均匀和随机排列观影和静息条件;图7b)。为了更好地理解这一效应的驱动因素,我们生成了静息和观影条件的代表性矩阵,并计算了这些矩阵之间的逐元素差异。我们将这些差异与随机排列扫描(条件)标签后估计的差异进行对比,发现所有边缘连接中有 在条件之间出现显著变化(排列检验; ;未校正)。尽管eFC的差异普遍存在,但最显著的效应与两个特定社区相关联(图7c),其中一个社区的模内eFC减少,而两者之间的eFC均减少。这些社区包括与体感运动和视觉系统相关的边缘(图7d)。 为了确认这些系统级效应在统计上显著,我们将系统内和系统间的平均 eFC 差异与一个受限的空模型进行比较,在该模型中,边缘社区被随机置换了 次;附图。18 显示了详细的示意图

Fig. 5 | Edge community entropy and overlap. a, Topographic distribution of normalized entropies. Normalized entropy, in this case, measures the uniformity of a node's community assignments across all communities and serves as a measure of overlap. In general, higher entropy corresponds to greater levels of overlap. b, Brain systems associated with the highest levels of normalized entropy. These include visual, attentional, somatmotor and temporoparietal systems. c, Entropy values for all brain systems; brain regions. Box plots, shown in green and overlaid on data points, depict the IQR (box) and the median value of the distribution. Whiskers extend to the nearest points IQR above and below the 25th and 75th percentiles. , Here we highlight communities in which somatomotor (red) and visual (blue) systems are represented.
图 5 | 边缘社区熵和重叠。a,标准化熵的地形分布。在这种情况下,标准化熵衡量了节点在所有社区中的分配的均匀性,并作为重叠的度量。一般来说,更高的熵对应于更高水平的重叠。b,与最高标准化熵水平相关的大脑系统。这些包括视觉、注意、体感和颞顶系统。c,所有大脑系统的熵值; 大脑区域。箱线图,显示为绿色并叠加在数据点上,描述了分布的 IQR(箱线图)和中位数值。须延伸到最近的点 IQR 在第 25 和 75 百分位数之上和之下。 ,在这里我们突出显示了体感运动(红色)和视觉(蓝色)系统被代表的社区。
Fig. 6 | System-level similarity of edge communities. a, Edge communities can be mapped into a matrix. The element at row and column of the edge community matrix denotes the community label of edge . , We can then calculate the similarity of edge communities involving nodes and by comparing the values of columns and . This matrix depicts the similarity for all pairs of nodes. c, Within-system similarity values for each of the 16 pre-defined brain systems; within-system similarity values. Box plots, shown in green and overlaid on data points, depict the IQR (box) and the median value of the distribution. Whiskers extend to the nearest points IQR above and below the 25th and 75th percentiles.
图 6 | 边缘社区的系统级相似性。a,边缘社区可以映射到一个 矩阵中。边缘社区矩阵中第 行第 列的元素表示边缘 的社区标签。 ,然后我们可以通过比较列 的值来计算涉及节点 的边缘社区的相似性。该矩阵描述了所有节点对的相似性。c,16 个预定义脑系统中每个系统的内部相似性值; 内部相似性值。绿色显示的箱线图叠加在数据点上,显示了分布的 IQR(箱体)和中位数值。须延伸到最接近的点 IQR,超过第 25 和第 75 百分位数以下和以上的点。
this procedure). As expected, the eFC involving systems 5 and 6 was significantly decreased from rest to movie (permutation test; false discovery rate fixed at ). Supplementary Fig. 19a shows the complete list of condition differences.
此过程)。如预期的那样,涉及系统 5 和 6 的 eFC 从休息到电影时显著减少(置换检验;假发现率固定在 )。附图 19a 显示了条件差异的完整列表。

The differences in the connection weights of eFC between movie watching and rest strongly suggested that the locations of high and low cluster overlap might also differ between conditions. To test this, we used the same clustering algorithm described earlier to partition
电影观看和休息状态之间的 eFC 连接权重差异强烈暗示高低聚类重叠位置可能在条件之间也不同。为了测试这一点,我们使用了早期描述的相同聚类算法来划分


movie minus rest
Fig. 7 | Effect of passive movie watching on eFC. a, Two-dimensional histogram of eFC estimated at rest with eFC estimated during movie watching. b, Similarity of whole-brain eFC estimated at rest with movie watching. Note that within-condition similarity is greater for both conditions.
图 7 | 被动观看电影对 eFC 的影响。a、在休息状态下估计的 eFC 与在观看电影期间估计的 eFC 的二维直方图。b、在休息状态下估计的整个大脑 eFC 与观看电影的相似性。请注意,对于两种条件,条件内相似性更大。
c, Community-averaged differences in eFC. Communities 5 and 6 are associated with the largest magnitude differences, on average. Note that these are communities estimated from HBN data and are not identical to those shown in Fig. 4, which were estimated from HCP data. d, Topographic distribution of communities 5 and 6 . Note that these communities involve edges associated with visual and somatomotor systems. e, Averaged differences in community overlap (normalized entropy); brain regions whose entropy scores were compared across rest and movie-watching conditions (permutation test; mean difference in paired samples; ). f, Similarity of whole-brain normalized entropy estimated at rest with movie watching. , Violin plot showing system-specific differences in normalized entropy. Note that some of the greatest increases in entropy are concentrated with control and default mode networks; brain regions. , Topographic distribution of differences in entropy. Box plots, shown in green and overlaid on data points in and , depict the IQR (box) and the median value of the distribution. Whiskers extend to the nearest points IQR above and below the 25th and 75th percentiles.
c,社区平均的eFC差异。社区5和6平均关联着最大幅度的差异。请注意,这些社区是根据HBN数据估计的,并不完全等同于图4中显示的那些,后者是根据HCP数据估计的。d,社区5和6的地形分布。请注意,这些社区涉及与视觉和体感系统相关的边缘。e,社区重叠的平均差异(归一化熵);对比了休息和观影条件下大脑区域的熵分数(置换检验;成对样本的平均差异;)。f,休息状态下大脑整体归一化熵与观影的相似性。小提琴图显示系统特定的归一化熵差异。请注意,一些熵增加最大的区域集中在控制和默认模式网络;大脑区域。熵差异的地形分布。箱线图,显示为绿色并叠加在和的数据点上,描述了分布的IQR(箱体)和中位数值。 胡须延伸到第25和第75百分位数上下四分位范围的最近点。
node pairs into non-overlapping clusters and, based on these clusters, calculated each node's cluster overlap as a normalized entropy. We found that, compared to rest, entropy increased during movie watching (permutation test; mean difference in paired samples; ), indicating increased overlap between communities (Fig. 7e), and that the brain-wide pattern of entropy also differed (permutation test; Fig. ). We performed analogous tests at the level of individual brain regions and found that of brain regions passed statistical testing (permutation test; false discovery rate fixed at 5%; ; Supplementary Fig. 19b). We further tested whether these differences exhibited system-specific effects by calculating the mean change in entropy for each system and comparing it against mean changes after randomly and uniformly permuting system labels. We found that seven systems exhibited such effects, with increases concentrated within control and salience/ventral attention networks and decreases in dorsal attention temporal-parietal and visual systems (permutation of system labels; false discovery rate fixed at 5%; Fig. and Supplementary Fig. 19c).
将节点对转化为不重叠的簇,并基于这些簇计算每个节点的簇重叠作为归一化熵。我们发现,与休息状态相比,在观看电影期间熵增加(排列测试;成对样本的平均差异;),表明社区之间的重叠增加(图7e),并且整个大脑的熵模式也不同(排列测试;图)。我们在个体大脑区域的水平上进行类似的测试,发现个体大脑区域中的个体通过了统计测试(排列测试;虚发现率固定在5%;;附图19b)。我们进一步测试这些差异是否表现出系统特异性效应,通过计算每个系统的熵变化均值,并将其与随机和均匀排列系统标签后的均值变化进行比较。我们发现七个系统表现出这种效应,增加集中在控制和显著性/腹侧注意网络中,而在背侧注意颞顶叶和视觉系统中减少(系统标签的排列;虚发现率固定在5%;图)。 和补充图 19c)。
Collectively, these results suggest that, like is reconfigurable and can be modulated by sensory inputs. The observed changes in eFC, which implicated two clusters associated with both somatomotor and visual systems, are in close agreement with past studies of passive movie watching that documented changes in activity and in similar . We also found increased overlap in areas associated with control and default mode networks, which agrees with evidence that activity throughout these areas is sensitive to movie narrative structure . An important area of future research involves systematically assessing the effect of different cognitively demanding tasks on eFC.
总的来说,这些结果表明,像 一样,是可重构的,并且可以通过感觉输入进行调节。观察到的 eFC 变化涉及与运动感觉和视觉系统相关的两个簇,与过去关于被动观影的研究结果高度一致,这些研究结果记录了类似 的活动和 的变化。我们还发现了与控制和默认模式网络相关的区域之间的重叠增加,这与证据一致,即这些区域的活动对电影叙事结构 敏感。未来研究的一个重要领域涉及系统评估不同认知需求任务对 eFC 的影响。

Discussion 讨论

Here we presented a network model of human cerebral cortex that focused on edge-edge interactions. The network formed by these interactions - a construct we referred to as eFC-was similar across datasets and more similar within individuals than between them. When clustered, eFC provided a natural estimate of pervasively overlapping community structure. We found that the amount of overlap varied across the cortex but peaked in sensorimotor and attention networks. We found that brain regions associated with sensorimotor and attention networks participated in disproportionately many communities compared to other brain systems, but that, relative to one another, those same regions participated in similar sets of communities. Lastly, we showed that eFC and community overlap varied systematically during passive viewing of movies.
在这里,我们提出了一个关注边缘-边缘相互作用的人类大脑皮层网络模型。这些相互作用形成的网络 - 我们称之为 eFC - 在数据集中是相似的,并且在个体内比个体间更相似。当进行聚类时,eFC 提供了广泛重叠社区结构的自然估计。我们发现重叠量在大脑皮层中有所变化,但在感觉运动和注意力网络中达到峰值。我们发现与感觉运动和注意力网络相关的大脑区域参与了比其他大脑系统更多的社区,但相对于彼此,这些区域参与了相似的社区。最后,我们展示了在观看电影时,eFC 和社区重叠在系统上有所变化。
Edge-centric perspective on functional network organization. Node-centric representations have dominated the field of network neuroscience and have served as the basis for nearly every discovery within that field . The edge-centric representation shifts focus away from dyadic relationships between nodal activations and, instead, onto the interactions between edges (similarity in patterns of co-fluctuation, a potential hallmark of communication). Although

related models have been explored in other scientific domains , including neuroscience, where they were first used in a study to represent interacting white matter , they require, as input, sparse node-node connectivity matrices and are poorly suited for continuous-valued time series data.
In this study, we developed a novel edge-centric representation of functional neuroimaging data that operates directly on observed time series. Our method for estimating connection weights between edges can be viewed as a temporal 'unwrapping' of the familiar Pearson correlation-the measure frequently used to estimate the magnitude of between pairs of brain regions. Whereas the Pearson correlation coefficient calculates the time-averaged co-fluctuation magnitude for node pairs, we simply omit the averaging step, yielding 'edge time series', which represent the co-fluctuation magnitude between two nodes at every instant in time. This simple step enables us to track fluctuations in edge weight across time and, critically, allow for dyadic relationships between edges, creating an edge-centric representation of nervous system architecture (Fig. 1). If we interpret edge time series as a temporal unwrapping of , which is thought to reflect the aggregate effect of communication processes between neural element , then edge times series track, with high temporal resolution, the communication patterns between distributed neural elements.
在这项研究中,我们开发了一种新颖的功能性神经影像数据的边缘中心表示,该表示直接操作观察到的时间序列。我们估计边缘之间连接权重的方法可以被看作是对熟悉的Pearson相关性的时间“展开” - 这个测量经常用来估计大脑区域对之间关系的幅度。而Pearson相关系数计算节点对的时间平均共同波动幅度,我们简单地省略了平均步骤,得到“边缘时间序列”,它代表了在每个时间点上两个节点之间的共同波动幅度。这一简单的步骤使我们能够跟踪边缘权重在时间上的波动,并且至关重要的是,允许边缘之间的双向关系,从而创建了神经系统结构的边缘中心表示(图1)。如果我们将边缘时间序列解释为对神经元素之间通信过程的总体效果的时间展开,那么边缘时间序列可以以高时间分辨率跟踪分布式神经元素之间的通信模式。
We note that our edge-centric approach is conceptually similar to several existing methods. For instance, 'multiplication of temporal derivatives calculates the element-wise products using differenced activity time series for all pairs of nodes. These time series are then convolved with a kernel to generate smooth estimates of time-varying . Although similar, our approach relies on untransformed activity to estimate edge time series, thereby preserving the relationship between static and the mean value of each edge time series. Another related method is 'co-activation patterns' (CAPs) , which extracts and clusters voxel- or vertex-level activity during high-activity frames. Because a voxel can be co-active under different contexts, the cluster centroids spatially overlap with one another. Although both CAPs and eFC result in overlapping structures, they operate on distinct substrates, with CAPs focusing on activity and eFC focusing on similarity of co-activity. Although CAPs requires the specification of additional parameters compared to eFC-for example, the threshold for a high-activity frameCAPs might scale better owing to the focus on activity rather than connectivity.
我们注意到我们的边缘中心方法在概念上与几种现有方法类似。例如,“时间导数的乘法”通过使用节点对的差异活动时间序列计算元素级乘积。然后,这些时间序列与核卷积以生成时间变化的平滑估计。尽管类似,我们的方法依赖于未转换的活动来估计边缘时间序列,从而保留静态关系和每个边缘时间序列的均值之间的关系。另一个相关方法是“共同激活模式”(CAPs),它在高活动帧期间提取和聚类体素或顶点级活动。由于一个体素可以在不同情境下共同激活,聚类中心在空间上会相互重叠。尽管CAPs和eFC都会产生重叠结构,但它们作用于不同的基质,CAPs关注活动,而eFC关注共同激活的相似性。 尽管 CAPs 需要相对于 eFC 指定额外的参数,例如,高活动帧的阈值,但由于其专注于活动而不是连接性,CAPs 可能更好地扩展。
Finally, we note that and are both frameworks for investigating pairwise relationships from neural time series. Critically, however, and eFC differ in terms of what elements are being related to one another and how we interpret those relationships. In the case of , correlations refer to similarities in the activity of individual neural elements, often interpreted as two parts of the brain 'talking' to one another. In the case of eFC, on the other hand, correlations express similarities in co-fluctuations along edges, which might loosely be interpreted as 'conversations' between node pairs (Supplementary Fig. 1). In other words, nFC focuses on co-activation between nodes whereas focuses on co-fluctuation along edges. In this way, and should be viewed as complementary approaches that can reveal unique organizational features of nervous systems.
最后,我们注意到 都是用于研究神经时间序列中成对关系的框架。然而,关键在于, 和 eFC 在被关联的元素以及我们如何解释这些关系方面存在差异。在 的情况下,相关性指的是个别神经元活动的相似性,通常被解释为大脑的两个部分在“交流”。另一方面,在 eFC 的情况下,相关性表示边缘上的共同波动的相似性,这可能被宽泛解释为节点对之间的“对话”(附图 1)。换句话说,nFC 关注节点之间的共同激活,而 关注边缘上的共同波动。通过这种方式, 应被视为可以揭示神经系统独特组织特征的互补方法。
Overlapping communities extend understanding of system-level cortical organization. Here we demonstrated that clustering eFC using community detection methods naturally leads to communities that overlap when mapped back to the level of brain regions and nodes. Past investigations of cortical organization have focused almost exclusively on non-overlapping communities. The decision to define communities in this way is partially motivated by interpretability but also by limitations of the methods used to detect communities, which assign nodes to one community only . This current view of communities has been profoundly successful'. It provides a low-dimensional description of the brain, it can be used to define node roles and detect hubs , and it can be applied to both anatomical and functional networks with equal success.
The dominant non-overlapping perspective of communities has strongly influenced how we think about brain function. Because functional communities exhibit reliable correspondence with patterns of task-evoked activity , we have come to associate individual communities with specific cognitive domains. For instance, it is not uncommon to refer to communities as primarily processing visual information, enacting cognitive control or performing attentional functions. This localization of brain function to communities, although likely a reasonable first-order approximation, perpetuates a view of brain function in which brain areas, systems and communities are fundamentally unifunctional. Such a view, however, disagrees with observations that many aspects of cognition and behavior transcend these traditional community labels.
Another perspective is that overlap arises from time-varying fluctuations in community structure . That is, at any given instant, communities are non-overlapping but appear 'fuzzy' due to nodes changing their community allegiances over time. The approach developed here is closely aligned with the perspective that brain areas and communities are dynamic and exhibit highly degenerate functionality. Other studies have investigated overlapping and dynamic communities by studying overlap in co-activation or through the use of sliding window analysis and multi-layer models to detect flexible regions that change their community assignment over time. Our approach, however, is distinct, emphasizing a state of pervasive overlap in which nodes belong to several communities instantaneously.
Limitations. One of the most important limitations concerns the estimation of edge time series from functional imaging data. To calculate edge time series, we first -scored regional time series. Here, the -score is appropriate only if the time series has a temporally invariant mean and s.d. If there is a sustained increase or decrease in activity-for example, the effect of a blocked taskthen the -scoring procedure can result in a biased mean and s.d., resulting in poor estimates of fluctuations in activity. In future work, investigation of task-evoked changes in eFC could be investigated with already common pre-processing steps-for example, constructing task regressors to remove the first-order effect of tasks on activity .
限制。其中一个最重要的限制涉及从功能成像数据中估计边缘时间序列。为了计算边缘时间序列,我们首先对区域时间序列进行了 -score。在这里, -score 只有在时间序列具有时间不变的均值和标准差时才是合适的。如果活动有持续增加或减少的情况-例如,受阻任务的影响-那么 -score 过程可能导致偏倚的均值和标准差,从而导致对活动波动的估计不准确。在未来的工作中,可以通过已经常见的预处理步骤来研究任务诱发的eFC变化-例如,构建任务回归器以消除任务对活动的一阶影响
Another limitation concerns the scalability of eFC. Calculating eFC given for a brain divided into parcels results in an matrix of dimensions . This means that an increase in the number of parcels results in a squared increase in the dimensionality of eFC. If the number of parcels is large, this can result in massive, fully weighted matrices that require large amounts of memory to store and manipulate. In the future, however, it might be necessary to explore dimension reduction methods to retain the most relevant sub-graphs for a given task or set of behaviors.
另一个限制涉及 eFC 的可扩展性。计算将大脑划分为 个区块的 eFC 会导致一个 维度为 的矩阵。这意味着区块数量的增加会导致 eFC 维度的平方增加。如果区块数量很大,这可能导致需要大量内存来存储和操作的大型、完全加权的矩阵。然而,未来可能需要探索降维方法,以保留给定任务或行为集的最相关子图。
Future directions. Although eFC characterizes interactions between edges rather than nodes, it can still be analyzed using the same methods previously applied to . We can use graph theory to detect its hubs and communities (Supplementary Fig. 20 shows examples), estimate edge gradients and compare eFC connection weights across individuals and conditions . On the other hand, eFC affords many new opportunities, beginning with the edge time series used to estimate eFC. Essentially, edge time series offer a moment-to-moment assessment of how strongly two nodes (brain regions) co-fluctuate with one another, providing an estimate of time-varying without the requirement that we specify a window . This overcomes one of the main limitations of sliding window estimates of time-varying , namely that the use of a
未来的方向。虽然 eFC 描述的是边之间的相互作用,而不是节点,但仍然可以使用之前应用于 的相同方法进行分析。我们可以使用图论来检测其中心和社区 (附图 20 显示示例),估计边缘梯度 并比较个体 和条件 下的 eFC 连接权重。另一方面,eFC 提供了许多新机会,从用于估计 eFC 的边缘时间序列开始。基本上,边缘时间序列提供了两个节点(脑区域)如何随时间共同波动的瞬时评估,提供了时间变化的估计 ,而无需指定窗口 。这克服了滑动窗口估计时间变化的 的主要限制之一,即使用窗口的需求。

window leads to a 'blurring' of events across time 42 . Other directions for future work include developing whole-brain functional atlases with overlapping system labels and applications to specific brain areas and sub-systems for constructing fine-grained overlapping atlases . We note, also, that, because the derivation of eFC is based on Pearson correlations, it would be straightforward to estimate analogs of eFC based on lagged and partial relationships.
eFC might be useful in applications of machine learning and classification of neuroimaging data . The dimensionality of the eFC matrix is much greater than that of a typical nFC matrix. We speculate that some of the added dimensions might be useful for studying brain-behavior relationships-for example, by identifying manifolds along which individuals, clinical cohorts or behaviors naturally separate, enhancing classification accuracy (the results of exploratory analyses of brain-behavior relationships based on eFC are shown in Supplementary Figs. 21-23). On the other hand, the increased dimensionality of eFC requires special considerations, as it presents statistical and interpretational challenges. Multivariate methods , such as canonical correlation analysis or partial least squares, both of which can help circumvent multiple comparison issues, might prove useful and should be investigated in future brain-behavior analysis involving eFC.
Additionally, future studies should investigate appropriate null models for eFC. Like nFC, eFC is correlation based, and the weights of edge-edge connections are not independent of one another . This means that rewiring-based null models (which treat connections as independent) are not appropriate. Consideration should be given to other classes of null models, including time-series-based surrogates. Appropriate null models might help clarify brainbehavior relationships in future studies.
The framework proposed here for investigating interactions between pairs of nodes can be generalized to study mutual interactions between many more nodes by simply calculating the element-wise product of node triplets, quartets and quintets . This extension is, in some respects, analogous to recent applications of algebraic topology