UNIVERSITY OF CALIFORNIA Santa Barbara. Graph Theoretical Analysis of Dynamic Brain Functional Networks

Transcription

1 UNIVERSITY OF CALIFORNIA Santa Barbara Graph Theoretical Analysis of Dynamic Brain Functional Networks A dissertation submitted in partial satisfaction of the requirements for Honors in the degree Bachelor of Science in Physics by Elizabeth N. Davison Thesis Advisor: Professor Jean M. Carlson June 2014

2 The dissertation of Elizabeth N. Davison is approved by Dr. Jean M. Carlson Dr. Scott T. Grafton Dr. Joan-Emma Shea Dr. Harry N. Nelson

3 1 Acknowledgments First of all, I would like to thank Professor Jean Carlson and Professor Danielle Bassett for giving me an incredible project and assisting me throughout the research process. I could not ask for more inspiring mentors and am grateful for the time and energy they invested in helping me grow as a scientist. I am grateful for the illuminating discussions on neurological implications of our results we were able to have with Professors Scott Grafton and Michael Miller. Thanks also to Dr. Ben Turner for all of the help with data visualization, and to Kimberly Schlesinger and Mary-Ellen Lynall for valuable contributions to the project. I would also like to thank other members of the Complex Systems Group at UCSB, Dr. Sean Stromberg, Charles Lieou, and Edwin Yuan, for their advice, encouragement, and excellent conversations. I am very thankful for the support of my family and friends. I cannot thank you enough for everything you have done for me. I would also like to thank all of the professors at UCSB who gave me opportunities and mentored me throughout my time here. A special thank you to Professor Harry Nelson, Professor Philip Lubin, and Professor Leon Balents. Lastly, the Worster family funded my research in Summer 2013 and their generous contribution made much of this work possible. This work was also supported by the David and Lucile Packard Foundation and the Institute for Collaborative Biotechnologies through contract no. W911NF-09-D-0001 from the U.S. Army Research Office. iii

4 2 Abstract The human brain moves between diverse states to meet the demands of our dynamic environment. Fundamental principles guiding these transitions remain poorly understood. Here we capitalize on recent advances in network science to treat the pattern of functional interactions between brain regions as evolving networks. We use this representation to probe the landscape of brain reconfigurations that accompany task performance both within and between four task states: a task-free resting state, an attention-demanding state, and two memory-demanding states. Using the formalism of hypergraphs, we identify the presence of groups of functional interactions that fluctuate with one another in strength over time both within (task-specific) and across (task-general) brain states. In contrast to prior emphases on the complexity of many dyadic (region-to-region) relationships, these results demonstrate that brain adaptability can be described by common processes that drive the dynamic integration of cognitive systems. Moreover, our results establish the hypergraph as an effective measure for understanding functional brain dynamics, which may also prove useful in examining cross-task, cross-age, and cross-cohort functional change. iv

5 Contents 1 Acknowledgments iii 2 Abstract iv 3 Introduction Graph Theory Dynamic Graph Theory Functional Magnetic Resonance Imaging Functional Connectivity Task States Hypergraph Analysis Overview Methods Imaging Functional Connectivity Time Windows for Temporal Network Construction Hyperedge Construction Co-Evolution Network Construction Hypergraph Diagnostics Null Models Behavioral Measures Methodological Considerations Results Hypergraph Analysis and Statistics Community Detection Task-Specific Hyperedges Rest Attention Memory for Words Memory for Faces Individual Hypergraph Properties Discussion 30 7 Conclusion 33 8 References 34 List of Figures 1 Word and Face Task Stimuli v

6 2 Hyperedge Construction Schematic Construction of the Hyperedge Co-Evolution Network and 60 TR Size Distributions Individual Hypergraph Individual Hypergraph Hyperedge Size Distribution Hyperedge Node Degree and Co-Evolution Network Community Detection Results Edge Time Series and Average Correlations Task-Specific Size Distributions Task-specific Co-Evolution Networks and Hyperedge Node Degrees Task-specific network statistics List of Tables 1 Brain Regions Statistics for Hypergraph Node Degree Results vi

7 3 Introduction 3.1 Graph Theory Graph theory is a field in mathematics wherein systems of interest are represented by sets of nodes (components of the system) and edges (connections or relevant interactions between the vertices) [1]. By adopting this framework, simple and complex systems can be characterized mathematically and the extraction of quantitative properties is possible. The applicability of graph theory is one of its most appealing features. Relevant systems span the disciplines of physics, biology, social science, and engineering. Some examples include transportation networks, social groups, proteins, quantum phase transitions, the Internet, and granular materials. Application of graph theoretical analyses to these systems can increase understanding of their fundamental properties and provides a mathematically rigorous method for investigating their structure Dynamic Graph Theory Complex systems can often be represented as temporal networks, where nodes are connected by time-dependent edges [2]. Dynamic graph theory allows us to use mathematical methods to investigate intrinsic properties of relevant complex systems as they change over time [3]. My research incorporates methods from dynamic graph theory to assist in understanding and characterizing functional properties of the brain. 3.2 Functional Magnetic Resonance Imaging In the past two decades, developments in brain imaging technology have produced significant insight into mechanisms of brain function [4]. An outcome of scientific progress over the past century, functional magnetic resonance imaging (fmri) has provided a wealth of information for brain scientists and catalyzed significant progress in functional brain mapping. The motion of atoms in water subject to a strong magnetic field was studied by 1

8 Bloch and Purcell in 1946 [5, 6], and magnetic resonance imaging (MRI) was developed from these principles several decades later as a method to study tissues in the body [7]. While the knowledge that blood flow corresponds to function in the brain has been accepted since the 19th century, the implications for imaging were not recognized until the 1990s [8]. The concept that increased blood flow is correlated with higher blood oxygen content and a groundbreaking experiment by Ogawa et. al. that established MRI s potential to detect a blood oxygen level dependent contrast (BOLD) signal in rodents [9] led to the development of fmri, which relies on the BOLD contrast to identify active brain regions. Three articles published in 1992 investigated the applicability of BOLD fmri to humans and established the technique as a useful method for identifying brain regions implicated in task performance [10 12]. 3.3 Functional Connectivity Identifying structural and functional networks are two applications of graph theoretic methods that have uncovered new information about the brain. Investigations into these networks holds a wealth of future directions for research. Diffusion tensor imaging (DTI) data, which maps the diffusion of water through white matter tracts, is used to construct structural networks. Nodes in these networks correspond to brain regions, while edges are drawn when two regions are physically connected. Functional networks are constructed from fmri data, which contains information about the activity patterns of regions. Graph theory has already proved useful in the characterization of functional activity across the whole brain and a temporal description will reveal further novel attributes of the system [13, 14]. We construct a graph where nodes are brain regions and edges correspond to correlations in blood oxygen levels, which provide a measure of functional connectivity [15, 16]. There has been progress on determining how structural and functional networks are related and whether it is possible to infer structure from function or function from structure [17, 18]. Furthermore, there is an ongoing effort in our group 2

9 to model appropriate functional behavior by using the structural network as a base. 3.4 Task States Our brains are constantly balancing complex cognitive tasks, transitioning fluidly in response to changing environmental stimuli. Despite the prevalence of this activity, we lack an understanding of the fundamental mechanisms for switching between task states. This work uses the dynamic graph theoretical measure of hypergraphs to investigate how the brain is altered as it switches between tasks. Differences in functional brain activity between cognitive states have previously been quantified in terms of task-positive and task-negative networks [19, 20]. At rest, the brain continues to display correlated activity, a default mode that is a consistent baseline observed in the adult brain [21, 22]. Deviations in the strongly correlated functional network from this state are grouped into task positive regions that are activated during a cognitive task and task-negative regions that are deactivated during task performance. These functional networks have been analyzed in attention [19] and memory [23] tasks; results from these investigations have increased understanding of intrinsic functional organization and clarified aspects of memory encoding and retrieval. Other analyses of task-rest networks found individual changes in resting state network could be used to predict task-related brain activity [24]. Unlike these methods, our hypergraph analysis provides a broader view of functional activity by considering all functional connections regardless of strength. We further examine task-motivated dynamics by considering how edges change together in functional brain networks as rest, attention and memory tasks are executed. 3.5 Hypergraph Analysis Recent developments in dynamic graph theory have resulted in further understanding of integral brain network properties [14]. A wide range of approaches, from principal 3

10 component analysis to dynamic community detection, have been employed to investigate temporal structure [25, 26]. Dynamic community detection divides the nodal time series and groups the nodes together in communities during each time window. These communities are then allowed to evolve over time, elucidating the dynamic modular properties of the network and allowing scientific to quantify measures such as flexibility (how often nodes switch between communities) and network centrality over time. A recently developed dynamical graph theory measure, the hypergraph, has proved useful in describing real and simulated complex dynamical systems by investigating the cross-linked structure of their network representations. Novel network co-evolution characteristics have been discovered in several systems through the implementation of this technique [27]. Instead of analyzing the time-dependent behavior of groups of nodes, the hypergraph investigation considers the edge weight time series, where edges with statistically significant similarities in their temporal profiles are grouped into hyperedges. Analysis of hyperedges in functional brain networks has provided insight into modulation of functional connectivity [28]. The edge-based hypergraph analysis promises to provide complimentary information to community detection, a node-based method. Additionally, hypergraphs are constructed without thresholding the edge weights, providing a comprehensive view of the changing functional structure of the brain. 3.6 Overview The rest of this thesis is organized as follows. We first define methods to explore the hyperedge network topology and spatial properties over all individuals. We compare the resulting distributions of hypergraph properties with those of two null models. From this and investigations into hyperedge temporal profiles, we find there are hyperedges that are more correlated in one task and hyperedges that have a distinct profile across the tasks. Task-specific hypergraphs are constructed from hyperedge groups specific to 4

11 rest, attention, memory for words, and memory for faces. The graphs for memory tasks show a higher degree of occipital lobe co-evolution probability than the attention and rest tasks, and the differences in structure indicate that the hypergraph representation of memory tasks is very different from rest or attention. We then probe statistical differences between the co-evolution networks and find that the hypergraph analysis shows significant distinctions between task states. Lastly, we perform a simple analysis on the task performance measures and show that there are significant correlations between performance and certain hyperedge properties. 4 Methods 4.1 Imaging Functional MRI data was acquired at the UCSB Brain Imaging Center from 116 healthy adult participants using a phased array 3T Siemens TIM Trim with a 12 channel head coil. We excluded a total of 30 participants for one or more of the following reasons: excessive head motion, missing over 10% of memory trials, or failing to follow task instructions. For all analyses reported in this thesis, we therefore studied a total of 86 participants. Each participant engaged in the following four tasks: rest (task-free), two separate iterations of the same attention-demanding task, a memory task with lexical stimuli, and a memory task with face stimuli. In both memory tasks, 180 previously studied stimuli and 180 novel stimuli were presented to the subjects, who were asked to determine whether the stimulus was old or new. All tasks were divided into a section where the stimuli were previously seen (or there is a stimulus present, in the attention task) 70% of the time and a section where the stimuli were only 30% old. Figure 1 depicts examples of the memory task stimuli, where color corresponds to probability that the stimulus has been seen. For additional experimental details see [29]. This analysis combines the two separate trials of the same attention task, which required the subjects to detect the 5

12 Figure 1: Word and Face Task Stimuli: Examples of the task stimuli used in the word memory and face memory tasks. The color of the stimulus represents the probability that it has been seen before. presence or absence of a target stimulus [17]. During the resting state, participants were asked to lie still and look at a blank screen. 4.2 Functional Connectivity Specific frequencies of oscillations in the BOLD signal have been associated with different cognitive functions. We focus our investigation on low frequency ( Hz) oscillations in the BOLD signal that have proven useful for examining resting [30, 31] and task-based functional connectivity [26]. The task-related oscillations are posited to be specific to this frequency range, possibly due to a bandpass-filter-like effect from the hemodynamic response function [32]. We examine scale two wavelet coefficients, obtained from a wavelet decomposition of the nodal time series, to isolate this frequency band [33]. Our sampling period (TR) was 2s, providing scale two wavelet coefficients in approximately Hz [34]. To construct a functional brain network, we partitioned the brain into 194 regions of roughly equal voxel number, which are represented by network nodes. The x, y, and z positions of each node are given by the center of voxels which comprise the node. This 194 region parcellation was chosen for comparability across subjects and sensitivity to individual surface structure [35]. Edges in the functional brain network are computed by taking Pearson s correlations between the scale two coefficients within a defined time 6

13 period for each pair of nodes [36]. 4.3 Time Windows for Temporal Network Construction Dynamic networks were constructed by taking the scale two wavelet coefficients in temporal windows of 30 TRs and computing a N N adjacency matrix of nodal correlations for each time window, where N = 194 is the number of nodes. Each of these N N adjacency matrices represents the functional network over the 30 TRs in question. From this set of networks, we extract the edge weight time series by considering the correlation strength in each sequential network. We let E = N(N 1)/2 = be the total number of edges between all 194 nodes and construct an E E adjacency matrix X, where X ab gives the Pearson correlation coefficient between the time series of edge weight for edges a and b. The entries of the E E adjacency matrix represent pairs of edges with correlated weight time series [27]. 4.4 Hyperedge Construction The cross-linked network structure, which contains information about groups of edges with similar time series (hyperedges), can be extracted from the edge-edge correlation matrix X [27]. Figure 2 provides a schematic illustration of the process of determining the cross-linked structure of a network. To exclude entries of X that are not statistically significant, we threshold X by evaluating the p-values for the Pearson coefficient R for each edge-edge correlation using a false-positive correction for multiple comparisons [37]. If the p-value for an entry X ij satisfies the false discovery rate correction threshold, we set ξ ij = R(i, j) for our thresholded matrix ξ. We set the thresholded entry of all other 7

14 Table 1: Brain Regions The 194 regions used in the hyperedge analysis, listed in order and including the number of regions in left and right hemispheres. Region Name L R lateralorbitofrontal 2 2 parsorbitalis 1 1 medialorbitofrontal 1 1 parstriangularis 1 1 parsopercularis 2 2 rostralmiddlefrontal 5 6 superiorfrontal 9 8 caudalmiddlefrontal 3 2 precentral 7 6 paracentral 1 1 rostralanteriorcingulate 1 1 caudalanteriorcingulate 0 1 posteriorcingulate 2 2 isthmuscingulate 1 1 postcentral 7 5 supramarginal 5 4 superiorparietal 7 7 inferiorparietal 5 6 precuneus 5 5 cuneus 1 1 pericalcarine 1 1 lateraloccipital 5 5 lingual 2 3 fusiform 3 3 parahippocampal 1 1 inferiortemporal 1 0 middletemporal 3 4 bankssts 1 1 superiortemporal 5 5 transversetemporal 1 1 insula 2 2 thalamusproper 1 1 caudate 1 1 putamen 1 1 pallidum 1 1 accumbensarea 1 1 hippocampus 1 1 amygdala 1 1 8

15 Figure 2: Hyperedge Construction: A schematic illustration of the method used to identify hyperedges. A set of node-node edges (A) and their time series (B), of which three [green, pink and orange traces, (B)] exhibit strong pairwise temporal correlations. These edges are cross-linked (C) by temporal covariance in edge weight time series, and thereby form a hyperedge (D) of size three on six nodes. The final [blue] edge is a singleton, forming a hyperedge of size one. elements X ij to zero. We binarize this thresholded matrix and obtain ξ ij, where 1, if ξ ij > 0; ξ ij = 0, if ξ ij = 0. (1) Each connected component in ξ represents a hyperedge, a set of edges that have significantly correlated temporal profiles. The groups of nodes in Figure 2 (D) are examples of such connected components. The set of all hyperedges defined in ξ produces an individual hypergraph, illustrated for one subject in Figure 5. Hyperedges contain information about edge dynamics without restricting the analysis to edges with strong weights. 4.5 Co-Evolution Network Construction We develop a co-evolution network to consolidate hypergraph results into a single graph that illustrates be precise where hyperedges are most likely to be physically located over an ensemble of individuals. Here, we use the co-evolution network to collapse over all tasks and individuals, and then look at separate co-evolution networks for each task that encompass all individual data. Figure 3 illustrates a schematic of our construction. We 9

16 begin by constructing a matrix, C, of probabilities that edges are included in a hyperedge over a set of hypergraphs. Again, nodes correspond to brain regions and connections correspond to inter-region associations, but here the weight of a connection joining nodes i and j is the matrix entry C i,j. This results in a static network encompassing the dynamics of hyperedge activity, where connection weight corresponds to the probability that the two nodes are co-evolving over all of the hypergraphs considered. In later sections, we Figure 3: Schematic Construction of the Hyperedge Co-Evolution Network: We analyze edge time series and group edges exhibiting similar temporal profiles into a hyperedge (as in Figure 1). We construct hypergraphs for each subject and find the matrix of probabilities that two nodes are in the same hyperedge over all subjects and hyperedges (C). This matrix is then used to create a co-evolution network, where the weight for an edge connecting nodes i and j corresponds to the entry C i,j. Here, node colors are used to indicate individual nodes and the edge color indicates distinct edges. refer to co-evolution connection strength, which we define as the magnitude of the probability matrix entry corresponding to that connection. 4.6 Hypergraph Diagnostics We use several methods to extract statistical features from individual hypergraphs and the group co-evolution network. We define the size s(h) of a hyperedge h, as the number 10

17 of edges contained in it, s(h) = i,j h ξ i,j, (2) where the sum is performed over the upper triangular elements of ξ, and ξ is the binarized edge-edge adjacency matrix defined above. We define the hyperedge degree of a node to be the number of hyperedges that contain that node. We examine the size distribution and hyperedge node degree as a spatial distribution over the subjects as a group to understand characteristic hypergraph properties. Previous work identified regions with task-specific activity in rest, attention, and memory tasks [17]. Further understanding of the regions that have a correlation structure unique to one class can give us insight into network structure differences between tasks. To investigate the task-specific hyperedge structure, we first group hyperedges that exhibit a significantly higher correlation within one task into task-specific sets. Permutation tests are used to test the significance of several measures. The task-specificity of hyperedges was calculated by comparing the correlation within single tasks to the correlation within a time series of the same length as the task, but with each time point randomly chosen from one of the four tasks. Additionally, diagnostics for co-evolution networks were evaluated by comparing the correlation between the diagnostic and connection strength to the correlation performed on the same number of randomly chosen hyperedges from a pair of tasks. All pairwise combinations were tested. The final permutation test utilized in this analysis was carried out on the task performance measures. The correlation between task performance and the individual hypergraph measure of interest was compared to the same correlation with individual performance measures shuffled randomly. All of these tests used a Bonferroni correction for false positives due to multiple comparisons [38]. To quantitatively probe the differences in co-evolution network properties for the 11

18 four tasks, we investigated two summary correlations that show significant variation across tasks. Choice of these measures was motivated by simple visual differences in co-evolution networks and there was no consideration of anatomical significance involved in this selection. The first correlation is comparing the strength of a connection and Cartesian distance between the two nodes linked by the connection (physical length) over the co-evolution network. The Cartesian distance is computed by taking the x, y, and z coordinates of each node and calculating the square root of the differences squared. This correlation gives us a sense of the geometric properties of the network, as well as a coarse estimate of the length of the strongest connections. We further evaluated the co-evolution networks by considering the correlation between connection strength and the average anterior-posterior position of the two nodes. A measure of anterior-posterior position for each connection was found by taking the average y-position of the two nodes in the connection. The task-specific co-evolution networks with a larger negative correlation between connection strength and y-position should correspond to the ones with more occipital lobe hyperedge activity. 4.7 Null Models In this analysis, we compare our results with two statistical null models based on measures for dynamic networks [39]. Hyperedges are formed from correlated edge time series; consequentially the overall null model randomly shuffles each edge time series over all experiments. This null model was designed to ensure that the hyperedges identified in our analysis can be attributed to the dynamics of the system, rather than some overall statistical property of the data set. The other null test we performed, which we will refer to as the within-task null model, reorders each edge time series within each task, keeping tasks distinct. This was constructed in order to determine whether there are specific differences in the data between tasks. 12

19 4.8 Behavioral Measures To evaluate task performance, we used two scores given in the task data from the UCSB Brain Imaging Center. The dprime score describes how well the task was performed. Another matric, the criterion switching score, measured how well the subject adapted their strategy between portions of the task that are explicitly identified to the subject to be low or high probability. In this thesis, we restrict our analysis to outliers of the behavioral measures, which we define as scores at least one standard deviation from the mean. We can then compare high and low performing subjects. 4.9 Methodological Considerations While constructing dynamic networks, we considered temporal window sizes of 20, 30, and 60 TRs. We found that our results for hyperedge size and spatial distributions are robust to the window size in this range. Figure 4 depicts the cumulative size distribution results for these window sizes. There was very little variation between the 20 and 30 TR window sizes, while the 60 TR window begin to lose information about the distribution. Figure 4: Size Distribution Results for 20 and 60 TRs: Cumulative size distributions for time windows of 20 and 60 TRs. The 60 TR window size begins to lose definition because the window size is considering too large a segment of the shorter tasks. The distribution from the 20 TR window is very similar to that of the 30 TR window size. 13

20 5 Results We compile the results from the hypergraph analysis for each of the subjects and combine these results to obtain a size distribution, anatomical node degree distribution, and coevolution network for the group. We then divide the data into task-specific hypergraphs, on which we perform the same sets of analyses. Measures from individual hypergraphs are then compared to individual task performance scores. 5.1 Hypergraph Analysis and Statistics We construct a hypergraph for each individual and examine the cumulative distribution of hyperedge sizes (s(h) from Equation 2), shown in Figure 7. There is a distinct break between two branches of the distribution occurring at a size of approximately 30, which we use to distinguish between large and small hyperedges. The total number of small hyperedges appears to roughly follow a power law with an exponent of approximately 2.5. The total number of large hyperedges also appears to roughly follow a power law but with a smaller exponent of approximately 0.1. In Figure 7, the sharp drop off in the distribution at large hyperedge sizes reflects the system size limitation on hyperedge size; the hyperedges of this size are the outliers near the upper bound mentioned above. There is a distinct partition in most individual frequency versus sizes distributions; one large hyperedge (s(h) > 30), and many small hyperedges that peak at the smallest size (s(h) < 30). There are significant outliers, including subjects who have hypergraphs near the upper bound on hyperedge size for the system (s(h) = E). The other extreme consists of individuals (6 total) who have networks that are constructed solely of small hyperedges. The individual hypergraph of a subject with this distribution is shown in Figure 5. The hypergraph of a subject with typical size distribution has thousands of edges between nodes as part of the largest hyperedge, which obscures the locations of smaller hyperedges. A more typical hypergraph is shown in Figure 6. Due to the high 14

21 Figure 5: Individual Hypergraph: Hypergraph for Subject 24, with maximum hyperedge size of 9. In this representation, different colors represent separate hyperedges, and only edges of size three or greater are shown. A more typical hypergraph would consist of many edges in the same hyperedge that are the main feature of the graph, but is hard to visualize because it obscures any structure attributed to the smaller hyperedges. amount of individual variability, we assess hypergraph properties over the subjects as a group. The null overall model shuffles the data over all tasks. As can be seen in Figure 7, there are essentially no hyperedges greater than size one. We performed no further analysis on this null model because it only found singletons or no statistically significant connections. The fact that no significant hyperedges were found in the null overall model validates the statistical significance of our results. The null within-task model shuffles the data but ensures that task data stays specific to each task. The size distribution of hyperedges from the null within-task model is shown in Figure 7. There are fewer hyperedges than in the original data, but the shape of the distribution is similar. This indicates there is co-evolution structure across tasks. This structure corresponds to changes in edge states between two or more tasks. For example, if groups of edges have an overall high correlation in one task and a significantly lower correlation in another, it would induce a hyperedge across the tasks irregardless of how the within-task time series are shuffled. In the true task-specific hypergraphs, there are 15

22 Figure 6: Individual Hypergraph: Hypergraph for Subject 2, with maximum hyperedge size of 626 (in teal). In this representation, different colors represent separate hyperedges, and only edges of size three or greater are shown. This is a more typical hyperedge size distribution (the maximum hyperedge size is still less than most of the individuals), but it is difficult to visualize the structure. fewer hyperedges than in the unshuffled data, indicating that a majority of the hyperedges are a result of within-task correlation. Examining the cumulative hyperedge size distribution provides information about the network topology but does not supply descriptive spatial information. Because the brain is a physically embedded object, we want to know which anatomical locations in the brain participate in hyperedges, which indicates a possible differential role in taskinduced co-evolution. To investigate this, we group all individual hypergraphs together and count the total number of times a node appeared in any hyperedge. We denote the resulting value as hyperedge node degree. There is a large variation in the hyperedge node degree over the brain, as visualized in Figure 8A on a natural log scale. The densest regions are located in posterior portions of the cortex, primarily in visual areas. A second set of dense regions is located in the frontal cortex. We construct a co-evolution network, as illustrated schematically in Figure 3, where connection weight corresponds to the probability that two nodes participate in the same hyperedge. In Figure 8B we present this co-evolution network over all individuals and all tasks. Edge color corresponds to threshold percentage value, where only the top 1% of co-evolution probabilities are shown. Within this 1%, brown connections comprise the 16

23 Figure 7: Hyperedge Size Distribution: Cumulative frequency distribution of hyperedge sizes. The small hyperedges appear to roughly follow a power law with an exponent of approximately 2.5, while the large group appears to follow a power law with an exponent of approximately 0.1. Results for the null overall model, where the data is shuffled over all tasks, are shown in red. In the null overall model, there is only one hyperedge of size 2 and significantly fewer hyperedges of size 1 than in the unshuffled data. Results for the null within-task model where the data is shuffled within each task, are in green. highest 0.2% of probabilities, red connections cover 0.2% to 0.4%, orange connections are 0.4% to 0.6%, gold connections represent probabilities from 0.6% to 0.8%, and yellow connections delineate 0.8% to 1%. The graph is highly clustered with sparse long-range connections between regions that are densely connected. A high degree of bilateral symmetry indicates that corresponding nodes in the left and right hemispheres have a high likelihood of being placed together in a hyperedge. Dense areas of the graph include primary visual areas, portions of prefrontal cortex, and primary motor cortex Community Detection Community detection is a method in network science to determine how to partition a set of network nodes into communities, where there is a higher degree of connectedness 17

24 Figure 8: Hyperedge Node Degree and Co-Evolution Network: In (A), we show hyperedge node degree on a natural log scale. The cumulative number of hyperedges at each vertex over all individuals is plotted on the brain, where higher values at a node correspond to more hyperedges that include the node. (B) depicts a sagittal view of the co-evolution network. The edge strength represents the probability that the edge will be in a hyperedge over all individuals. Only the top 1% of connections are shown, with darker colors depicting higher probability for co-evolution pairs. 18

25 within communities than between communities [40]. Identification of these groups is accomplished by optimization of a quality function, a measure that quantifies the relative density of intra-community and inter-community edges [41]. To identify structure within the co-evolution network, we perform community detection using a locally greedy Louvain algorithm [42]. The resolution parameter, γ, determines the amount of communities the algorithm will return. We performed community detection on the co-evolution network with values of γ ranging from to 5, and chose γ = 1 because it consistently returned seven communities that visually corresponded with areas of higher density in the coevolution network. The results of our community detection investigation on the 1% of most probable connections in the co-evolution network can be seen in figure 9; we use a resolution parameter of γ = 1 and identify the consistent communities in an ensemble of 100 optimizations. Figure 9: Community Detection Results: Here we see the seven communities that partition the co-evolution network. there is more community structure towards the anterior and posterior, reflecting the increased co-evolution network complexity in those regions. 19

26 5.2 Task-Specific Hyperedges In our hyperedge construction, we observed that some hyperedges appeared to have similar edge dynamics in one task more so than the others. We computed the average within-task correlation and found that in some cases, edge correlation spanned the tasks, while in other hyperedges, a strong correlation between edges within one task drives the hyperedge. An example of this task-specific correlation structure can be seen in Figure 10. In the average within-task correlation on the left, we see that there is a stronger average correlation in the word memory task than in any other task. Furthermore, the edge time series in the first hyperedge indicates it is driven mainly by a correlation within the word memory task. Figure 10: Left: Average hyperedge correlation in each task for three hyperedges (where hyperedges with small sizes are chosen for illustrative purposes). Right: Correlation (absolute value) time series for the same three hyperedges. The colored lines represent each edge, while the black line is the average edge time series. Each time point represents the static network over 30 TRs. These results display the task-specificity of hyperedges, where significant correlations in the hyperedge are restricted to one task. The attention task is broken into two sections because two separate iterations of the same task were combined in this analysis. To investigate this further, we construct task-specific co-evolution networks, where the hyperedges have a significantly stronger average correlation in one task than the others (see Methods). For each task, we perform a permutation test on the edge weights, 20

27 as described in the methods section, and compare the total correlation within the task to the expected values. If the hyperedge is significantly correlated (determined by the Bonferroni correction on the p-values from the permutation test) in only one task, we label it as a task-specific hyperedge. Figure 11: Task-Specific Size Distributions: Cumulative frequency distribution as a function of hyperedge size for all task specific groups. The results are compared to the overall distribution of hyperedges (dark blue), previously illustrated in Figure reference earlier ones2. The hyperedges specific to attention and rest tasks are primarily in the small regime, while the memory tasks have a greater number of large hyperedges. Figure 11 illustrates the size distributions of all the task-specific results alongside the overall hyperedge size distribution. The sizes and spatial distributions of single task-driven hyperedges vary across tasks and incorporate significant information about functional network organization with respect to changing cognitive states. There is an obvious lack of large attention and rest hyperedges, while the memory tasks more closely mimic the overall distribution. The distinction in the distributions indicates that the tasks can be characterized by differing complexities of edge co-variations, and potentially also by differences in computational complexity. The spatial distributions of hyperedge node degree in each task, along with taskspecific co-evolution networks are shown in Figure 12. There are clear distinctions between the co-evolution networks for each task, which we characterize with two statistics 21

28 based on observable differences in co-evolution network structure. The first is correlation between connection length and strength in the co-evolution network. The second measure we consider is correlation between connection position (anterior-posterior) and strength. The results of this analysis over the full unthresholded co-evolution network are in Figure 13. All correlation values are negative, indicating that, in all tasks, stronger connections in the co-evolution network are located in posterior portions of cortex and are physically shorter. Figure 12: Task-specific Co-Evolution Networks and Hyperedge Node Degrees: A: Distribution of task-specific hyperedge node degree on the brain. Here, the log of the total number of hyperedges containing each node is represented on the brain. B: Coevolution networks for each task. Edge strength corresponds to the probability that a hyperedge will contain the edge over all individual hypergraphs. Color represents a threshold in percentage value, as in Figure 8, and the top 1% of co-evolution probabilities are shown. We compared these values across tasks by performing pairwise permutation tests to determine which networks have statistically different properties. Figure 13 depicts the p-values from these tests, where the horizontal axis represents the statistic being tested and the vertical axis corresponds to the task being tested against. The black squares in this figure represent significant values, which are described in the following: 1. The rest task has a significantly less strong correlation between y-postition and strength than the word memory and face tasks (rest is less likely to have the strong 22

29 connections at the back of the brain). 2. The rest task has a stronger correlation between physical length and strength than the word memory task. 3. The attention task is less strongly correlated than the word memory task in terms of y-position and the rest task in terms of physical length. 4. The word memory task has a weaker correlation between physical length and strength than either the rest or face memory tasks. 5. The face memory task has a stronger negative correlation between physical length and strength than the word memory task. These results delineate significant differences in co-evolution network structure between the tasks, confirming that the hypergraph analysis is a useful method for distinguishing between task states Rest Resting-state brain activity contains correlated patterns that are used to construct a default mode network, a system that is engaged during internal cognition [43, 44]. Certain brain regions active at rest are consistently deactivated during goal-oriented tasks, indicating that they comprise a functional mode that is rest-specific [21]. Rest-specific hyperedges are primarily represented in the small range of the size distribution in Figure 11. The result that no large groups of nodes share similar coevolution properties could be a result of the specificity of correlated resting state regions, or a simplicity intrinsic to resting state function that does not necessitate more concerted efforts involving numerous brain regions. There are fewer rest-specific hyperedges than in any other task, so the hyperedge node degree plot in Figure 12A has the lowest overall magnitude across task states. The 23

30 Figure 13: Task-specific network statistics: Values for the correlation between the y position (the front-back location of an edge) and co-evolution network strength for the four tasks (blue), and physical length of an edge and co-evolution network strength of that edge are depicted in A. B shows p-values for the pairwise statistical permutation test between tasks, where black denotes a significant value, after a Bonferroni correction for multiple comparisons. Values are obtained for correlations between y-position and strength and physical length and strength. areas that have the highest degree of hyperedge activity are in the posterior portions of the brain, a configuration that is consistent across tasks. This suggests there is an underlying pattern of hyperedge generation centered in the occipital lobe. The rest co-evolution network is highly clustered in the most probable 0.2% of coevolution pairs, but lower thresholds show very little structure, as visualized in Figure 12B. These high probability clusters are centered in previously mentioned regions, but the top 1% of connections is far more randomized in rest than any other task-specific co-evolution network we have observed. There is relatively little lateral symmetry and few visible core areas with high hyperedge node degree. After performing pairwise permutation tests for each pair of tasks and including a Bonferroni correction for false positives, we found the negative correlation between physical connection length and strength was significantly stronger for the rest task than the 24

31 word memory task. This indicates that the strongest connections in the co-evolution network for the rest task are short, reflecting our initial observations in Figure 12B. The rest co-evolution network has the smallest negative correlation between connection position and strength, which the permutation test (Figure 13B) confirmed to be significantly smaller than the word or face memory tasks. Our analysis returns a spatial co-evolution distribution with structures of higher probability in brain regions traditionally associated with the resting state. Dense areas of the network with high probabilities of being in the same hyperedge include the inferior parietal lobule, superior frontal gyrus, precuneus, and posterior cingulate cortex. These regions have been identified as integral components of the default mode network; the posteromedial cortex includes the precuneus and posterior cingulate cortex and performs a particularly pivotal role in awareness and memory retrieval [45 47] Attention Two attention systems exist in the human brain: a top-down network controls goaldirected attention, while a bottom-up group of brain regions detect and orient attention to relevant sensory stimuli that are generally novel or unexpected [48, 49]. Our task probes the former, as subjects are asked to focus on repetitive stimuli in a controlled environment. This requires an executive control network, a bilateral dorsal system that governs guided attention and working memory [50]. The attention specific network solely consists of small hyperedges, visualized in the distribution in Figure 11. This composition may account for the disorganization in the overall co-evolution structure of networks made of only small hyperedges. In Figure 12A, the hyperedge node degree plot for the attention task looks qualitatively similar to the rest task. However, the values are significantly larger here, corresponding to the greater number of hyperedges that are present in the attention task. In Figure 12B, the co-evolution structure specific to the attention task is depicted. 25

32 The overall network of most probable hyperedge pairs is similar in the attention and rest specific networks, but the structure of the attention network is more organized in terms of bilateral symmetry and node degree distribution. There are multiple prefrontal cortical regions that are likely to cohesively evolve with several other nodes. Numerous links between rostral and caudal brain regions are a unique feature of the attention network. The negative correlation between physical length and strength in the attention coevolution network is significantly less strong than the rest task, as we can verify in Figure 12B by observing that the attention co-evolution network has strong connections that are significantly longer than the rest task. Additionally, the attention task has a weaker correlation between anterior-posterior position and connection strength than the word memory task. Regions of high clustering in the most probable threshold include the lateral parietal occipital lobe, the superior frontal cortex, and dorsal parietal cortex. The parietal and frontal areas are involved in attention control and localization, specifically in visual attention tasks [48, 51]. Activation of the superior frontal cortex occurs in attention tasks, but a significant increase in activity occurs during tasks where peripheral attention is required [52, 53]. The dorsal parietal cortex also performs a central role in the executive control network; patients with lesions in the dorsal parietal cortex showed significant impairment in goal-directed attention tasks [54] Memory for Words Existence of a dedicated visual word processing network has been a topic of frequent discussion in neuroscience. The visual word form area (vwfa), located in the occipitotemporal cortex, is consistently activated by orthographic stimuli [55] and is invariant to changes in case, size, font, or type of visual stimulation [56, 57]. Moreover, there is evidence from a multitude of studies that the vwfa is not preferential towards words; other visual stimuli produce similar levels of activation [58]. The vwfa has also been 26

33 shown as functionally linked to the dorsal attention network in resting state fmri data, indicating that it fulfills a complex cognitive role [59]. Hyperedges specifically correlated in the word memory task mirror the overall hyperedge distribution more than any other task-specific group. There are many large hyperedges present, indicating that a significant portion of hyperedges in this regime are driven by correlations in the word memory task. The word memory task-specific hypergraph has the most hyperedges and contains many large hyperedges. The resulting hyperedge node degree distribution has higher node degrees than any of the other task-specific hypergraphs. Although similar areas have the highest concentration of hyperedges, there is a marked increase in node degree of posterior cortices (seen in the face memory and word memory node degree plots in Figure 12A). Visual orthographic and face processing have a common reliance on central vision [60] and share neural circuitry [61]. The resemblance of the co-evolution networks for the two tasks, especially when compared with the very different graph structure of the attention and rest networks, indicates a similarity in the hypergraph representation of the memory tasks. This in turn signifies a correspondence in brain dynamics specific to memory. The correlation between connection length and strength is the least strong for the word memory network. We found that these two variables were significantly less correlated than in the rest or face memory tasks. This indicated that the many connections from the occipital to frontal lobes are a distinguishing characteristic of the word co-evolution network. In the word memory co-evolution network in Figure 12B, low strength co-evolution pairs are more broadly distributed throughout the brain, while the strength and number of bilateral links is diminished. This is consistent with the understood activation structure of working memory tasks [62]. Another compelling feature of the word memory coevolution network is the absence of dorsal attention areas, which we would expect to see 27