A Q-analysis package for higher-order interactions analysis in Python and its application in network physiology

Previous neuroimaging studies have revealed abnormal brain networks in patients with major depressive disorder (MDD) in emotional processing. While any cognitive task consists of a series of stages, little is yet known about the topology of functional brain networks in MDD for these stages during emotional face recognition. To address this problem, electroencephalography (EEG)-based functional brain networks of MDD patients at different stages of facial information processing were investigated in this study. First, EEG signals were collected from 16 patients with MDD and 18 age-, gender-, and education-matched normal subjects when performing an emotional face recognition task. Second, the global field power (GFP) method was employed to divide group-averaged event-related potentials into different stages. Third, using the phase transfer entropy (PTE) approach, the brain networks of MDD patients and normal individuals were constructed for each stage in negative and positive face processing, respectively. Finally, we compared the topological properties of brain networks of each stage between the two groups using graph theory approaches. The results showed that the analyzed three stages of emotional face processing corresponded to specific neurophysiological phases, namely, visual perception, face recognition, and emotional decision-making. It was also demonstrated that depressed patients showed abnormally decreased characteristic path length at the visual perception stage of negative face recognition and normalized characteristic path length in the stage of emotional decision-making during positive face processing compared to healthy subjects. Furthermore, while both the MDD and normal groups' brain networks were found to exhibit small-world network characteristics, the brain network of patients with depression tended to be randomized. Moreover, for patients with MDD, the centro-parietal region may lose its status as a hub in the process of facial expression identification. Together, our findings suggested that altered emotional function in MDD patients might be associated with disruptions in the topological organization of functional brain networks during emotional face recognition, which further deepened our understanding of the emotion processing dysfunction underlying MDD.

Complex networks have become the main paradigm for modelling the dynamics of interacting systems. However, networks are intrinsically limited to describing pairwise interactions, whereas real-world systems are often characterized by higher-order interactions involving groups of three or more units. Higher-order structures, such as hypergraphs and simplicial complexes, are therefore a better tool to map the real organization of many social, biological and man-made systems. Here, we highlight recent evidence of collective behaviours induced by higher-order interactions, and we outline three key challenges for the physics of higher-order systems.

Despite the recent successes of vanilla Graph Neural Networks (GNNs) on various tasks, their foundation on pairwise networks inherently limits their capacity to discern latent higher-order interactions in complex systems. To bridge this capability gap, we propose a novel approach exploiting the rich mathematical theory of simplicial complexes (SCs) - a robust tool for modeling higher-order interactions. Current SC-based GNNs are burdened by high complexity and rigidity, and quantifying higher-order interaction strengths remains challenging. Innovatively, we present a higher-order Flower-Petals (FP) model, incorporating FP Laplacians into SCs. Further, we introduce a Higher-order Graph Convolutional Network (HiGCN) grounded in FP Laplacians, capable of discerning intrinsic features across varying topological scales. By employing learnable graph filters, a parameter group within each FP Laplacian domain, we can identify diverse patterns where the filters' weights serve as a quantifiable measure of higher-order interaction strengths. The theoretical underpinnings of HiGCN's advanced expressiveness are rigorously demonstrated. Additionally, our empirical investigations reveal that the proposed model accomplishes state-of-the-art performance on a range of graph tasks and provides a scalable and flexible solution to explore higher-order interactions in graphs. Codes and datasets are available at https://github.com/Yiminghh/HiGCN.

The study of irreducible higher-order interactions has become a core topic of study in complex systems, as they provide a formal scaffold around which to build a quantitative understanding of emergence and emergent properties. Two of the most well-developed frameworks, topological data analysis and multivariate information theory, aim to provide formal tools for identifying higher-order interactions in empirical data. Despite similar aims, however, these two approaches are built on markedly different mathematical foundations and have been developed largely in parallel - with limited interdisciplinary cross-talk between them. In this study, we present a head-to-head comparison of topological data analysis and information-theoretic approaches to describing higher-order interactions in multivariate data; with the goal of assessing the similarities, and differences, between how the frameworks define “higher-order structures.” We begin with toy examples with known topologies (spheres, toroids, planes, and knots), before turning to more complex, naturalistic data: fMRI signals collected from the human brain. We find that intrinsic, higher-order synergistic information is associated with three-dimensional cavities in an embedded point cloud: shapes such as spheres and hollow toroids are synergy-dominated, regardless of how the data is rotated. In fMRI data, we find strong correlations between synergistic information and both the number and size of three-dimensional cavities. Furthermore, we find that dimensionality reduction techniques such as PCA preferentially represent higher-order redundancies, and largely fail to preserve both higher-order information and topological structure, suggesting that common manifold-based approaches to studying high-dimensional data are systematically failing to identify important features of the data. These results point towards the possibility of developing a rich theory of higher-order interactions that spans topological and information-theoretic approaches while simultaneously highlighting the profound limitations of more conventional methods.

Altered cortical functional network in major depressive disorder: A resting-state electroencephalogram study.

The fronto-limbic network is implicated in the neurobiology of major depressive disorder. However, no studies are designed to assess directly the abnormalities of regional activity and network homogeneity of this network in major depressive disorder. A total of 44 drug-naive major depressive disorder patients and 44 healthy controls participated in the study, and resting-state functional magnetic resonance imaging data were obtained. The fractional amplitude of low-frequency fluctuations and network homogeneity methods were employed to analyze the data. Compared with the controls, the patients exhibited reduced fractional amplitude of low-frequency fluctuations in the right middle frontal gyrus (orbital part) and decreased network homogeneity in the left middle frontal gyrus. There was no correlation between abnormal fractional amplitude of low-frequency fluctuations/network homogeneity and clinical variables. Our findings suggest that decreased regional activity and network homogeneity in the frontal cortex may be the key impairment of the fronto-limbic network in major depressive disorder, and thus highlight the importance of the fronto-limbic network in the neurobiology of major depressive disorder.

Recent developments in graph theory have heightened the need for investigating the disruptions in the topological structure of functional brain network in major depressive disorder (MDD). In this study, we employed resting-state functional magnetic resonance imaging (fMRI) and graph theory to examine the whole-brain functional networks among 42 MDD patients and 42 healthy controls. Our results showed that compared with healthy controls, MDD patients showed higher local efficiency and modularity. Furthermore, MDD patients showed altered nodal centralities of many brain regions, including hippocampus, temporal cortex, anterior cingulate gyrus and dorsolateral prefrontal gyrus, mainly located in default mode network and cognitive control network. Together, our results suggested that MDD was associated with disruptions in the topological structure of functional brain networks, and provided new insights concerning the pathophysiological mechanisms of MDD.

Neuroimaging studies have shown topological disruptions of both functional and structural whole-brain networks in major depressive disorder (MDD). This study examined common and specific alterations between these two types of networks and whether the alterations were differentially involved in the two hemispheres. Multimodal MRI data were collected from 35 MDD patients and 35 healthy controls, whose functional and structural hemispheric networks were constructed, characterized, and compared. We found that functional brain networks were profoundly altered at multiple levels, while structural brain networks were largely intact in patients with MDD. Specifically, the functional alterations included decreases in intra-hemispheric (left and right) and inter-hemispheric (heterotopic) functional connectivity; decreases in local, global and normalized global efficiency for both hemispheric networks; increases in normalized local efficiency for the left hemispheric networks; and decreases in intra-hemispheric integration and inter-hemispheric communication in the dorsolateral superior frontal gyrus, anterior cingulate gyrus and hippocampus. Regarding hemispheric asymmetry, largely similar patterns were observed between the functional and structural networks: the right hemisphere was over-connected and more efficient than the left hemisphere globally; the occipital and partial regions exhibited leftward asymmetry, and the frontal and temporal sites showed rightward lateralization with regard to regional connectivity profiles locally. Finally, the functional–structural coupling of intra-hemispheric connections was significantly decreased and correlated with the disease severity in the patients. Overall, this study demonstrates modality- and hemisphere-dependent and invariant network alterations in MDD, which are helpful for understanding elaborate and characteristic patterns of integrative dysfunction in this disease.

Background: Understanding the neural basis for major depressive disorder (MDD) is essential for its diagnosis and treatment. Aberrant activation and functional connectivity of the default mode network (DMN), salience network (SN) and dorsal attention network (DAN) have been found consistently in patients with MDD. However, whether effective connectivity within and between these networks is altered in MDD remains unknown. The primary objective of this study was to investigate the effective connectivity of the 3 networks in patients with MDD at rest. Methods: We included 63 patients with MDD (35 first-episode and 28 recurrent) and 74 healthy controls, and collected resting-state functional MRI data for all participants. We defined 15 regions of interest from the 3 functional brain networks of interest using group independent component analysis. We estimated the coupling parameters that reflected the causal interactions among these regions using spectral dynamic causal modelling. We used parametric empirical Bayes to determine commonalities across groups, differences between patients with MDD and healthy controls, and differences between patients with recurrent and first-episode MDD. Results: We found positive (excitatory) connections within each network, negative (inhibitory) connections from the SN and DAN to the DMN, and positive connections from the DAN to the SN across groups. Compared to healthy controls, patients with MDD showed increased positive connections within the DMN, a decreased absolute value of negative connectivity from the SN to the DMN, and increased positive connections from the SN to the DAN. We also found that patients with recurrent MDD showed remarkably different effective connections compared to patients with first-episode MDD, especially related to the DAN. Limitations: Because of the relatively small sample size and the unclear medication history of the MDD sample, the present findings are in need of replication. Conclusion: These findings suggest that effective connectivity among high-order brain functional networks during rest was disrupted in patients with MDD. Moreover, patients with recurrent MDD exhibited different effective connections compared to patients with first-episode MDD. These differences in effective connectivity might provide new insights into the neural substrates of MDD.

Higher-order interactions in functional brain networks in major depressive disorder

Abnormal structural covariance network in major depressive disorder: Evidence from the REST-meta-MDD project

Disease-specific alterations of effective connectivity across anti-correlated networks in major depressive disorder and bipolar disorder.

Major depressive disorder (MDD) has been characterized by abnormal brain activity and interactions across the whole-brain functional networks. However, the underlying alteration of brain dynamics remains unclear. Here, we aim to investigate in detail the temporal dynamics of brain activity for MDD, and to characterize the spatiotemporal specificity of whole-brain networks and transitions across them. We developed a hidden Markov model (HMM) analysis for resting-state functional magnetic resonance imaging (fMRI) from two independent cohorts with MDD. In particular, one cohort included 127 MDD patients and 117 gender- and age-matched healthy controls, and the other included 44 MDD patients and 33 controls. We identified brain states characterized by the engagement of distinct functional networks that recurred over time and assessed the dynamical configuration of whole-brain networks and the patterns of activation of states that characterized the MDD groups. Furthermore, we analyzed the community structure of transitions across states to investigate the specificity and abnormality of transitions for MDD. Based on our identification of 12 HMM states, we found that the temporal reconfiguration of states in MDD was associated with the high-order cognition network (DMN), subcortical network (SUB), and sensory and motor networks (SMN). Further, we found that the specific module of transitions was closely related to MDD, which were characterized by two HMM states with opposite activations in DMN, SMN, and subcortical areas. Notably, our results provide novel insights into the dynamical circuit configuration of whole-brain networks for MDD and suggest that brain dynamics should remain a prime target for further MDD research.

Complex networks have been successfully used to describe the spread of diseases in populations of interacting individuals. Conversely, pairwise interactions are often not enough to characterize social contagion processes such as opinion formation or the adoption of novelties, where complex mechanisms of influence and reinforcement are at work. Here we introduce a higher-order model of social contagion in which a social system is represented by a simplicial complex and contagion can occur through interactions in groups of different sizes. Numerical simulations of the model on both empirical and synthetic simplicial complexes highlight the emergence of novel phenomena such as a discontinuous transition induced by higher-order interactions. We show analytically that the transition is discontinuous and that a bistable region appears where healthy and endemic states co-exist. Our results help explain why critical masses are required to initiate social changes and contribute to the understanding of higher-order interactions in complex systems.

Some real-world phenomena and human-made problems have been modeled as networks where the objects form pairwise interactions. However, this is a limited approach when the existence of high-order interactions is inherent in a system, such as the brain, social networks and ecosystems. The way in which these high-order interactions affect the collective behavior of a complex system is still an open question. For this reason, it is necessary to analyze theoretically, numerically and experimentally the consequences of higher-order interactions in complex systems. Here, we provide experimental datasets of the dynamics of three nonlinear electronic oscillators, namely, Rössler oscillators, interacting into a simplicial complex whose connections rely on both linear (diffusive) and nonlinear (high-order) coupling. It is well-known that Rössler systems only achieve the synchronization when they are coupled by means of x or y variable. Considering this fact, we designed our experiment considering four scenarios. The first one, when both linear and nonlinear coupling functions are introduced through the x variable. The second one, occurring when linear coupling is introduced through the x variable and the nonlinear coupling through the y variable. The third case happens when the linear coupling is introduced through the y variable whereas nonlinear coupling goes through the x variable. The last case, when both linear and nonlinear coupling are introduced through the y variable. For each scenario, we acquired 10000 times series when both the linear and nonlinear coupling strengths were modified. Each time series contained 30000 temporal points. These datasets are useful to corroborate the conditions to reach the synchronized state varying the linear/non-linear coupling strengths and to test new metrics for better understanding the effects of higher-order interactions in complex networks.