A simplex path integral and a simplex renormalization group for high-order interactions.

Modern theories of phase transitions and scale invariance are rooted in path integral formulation and renormalization groups (RGs). Despite the applicability of these approaches in simple systems with only pairwise interactions, they are less effective in complex systems with undecomposable high-order interactions (i.e. interactions among arbitrary sets of units). To precisely characterize the universality of high-order interacting systems, we propose a simplex path integral and a simplex RG (SRG) as the generalizations of classic approaches to arbitrary high-order and heterogeneous interactions. We first formalize the trajectories of units governed by high-order interactions to define path integrals on corresponding simplices based on a high-order propagator. Then, we develop a method to integrate out short-range high-order interactions in the momentum space, accompanied by a coarse graining procedure functioning on the simplex structure generated by high-order interactions. The proposed SRG, equipped with a divide-and-conquer framework, can deal with the absence of ergodicity arising from the sparse distribution of high-order interactions and can renormalize a system with intertwined high-order interactions at thep-order according to its properties at theq-order (p⩽q). The associated scaling relation and its corollaries provide support to differentiate among scale-invariant, weakly scale-invariant, and scale-dependent systems across different orders. We validate our theory in multi-order scale-invariance verification, topological invariance discovery, organizational structure identification, and information bottleneck analysis. These experiments demonstrate the capability of our theory to identify intrinsic statistical and topological properties of high-order interacting systems during system reduction.

Dynamics of simplicial SEIRS epidemic model: global asymptotic stability and neural Lyapunov functions.

The transmission of infectious diseases on a particular network is ubiquitous in the physical world. Here, we investigate the transmission mechanism of infectious diseases with an incubation period using a networked compartment model that contains simplicial interactions, a typical high-order structure. We establish a simplicial SEIRS model and find that the proportion of infected individuals in equilibrium increases due to the many-body connections, regardless of the type of connections used. We analyze the dynamics of the established model, including existence and local asymptotic stability, and highlight differences from existing models. Significantly, we demonstrate global asymptotic stability using the neural Lyapunov function, a machine learning technique, with both numerical simulations and rigorous analytical arguments. We believe that our model owns the potential to provide valuable insights into transmission mechanisms of infectious diseases on high-order network structures, and that our approach and theory of using neural Lyapunov functions to validate model asymptotic stability can significantly advance investigations on complex dynamics of infectious disease.

Genuine high-order interactions in brain networks and neurodegeneration.

Brain functional networks have been traditionally studied considering only interactions between pairs of regions, neglecting the richer information encoded in higher orders of interactions. In consequence, most of the connectivity studies in neurodegeneration and dementia use standard pairwise metrics. Here, we developed a genuine high-order functional connectivity (HOFC) approach that captures interactions between 3 or more regions across spatiotemporal scales, delivering a more biologically plausible characterization of the pathophysiology of neurodegeneration. We applied HOFC to multimodal (electroencephalography [EEG], and functional magnetic resonance imaging [fMRI]) data from patients diagnosed with behavioral variant of frontotemporal dementia (bvFTD), Alzheimer's disease (AD), and healthy controls. HOFC revealed large effect sizes, which, in comparison to standard pairwise metrics, provided a more accurate and parsimonious characterization of neurodegeneration. The multimodal characterization of neurodegeneration revealed hypo and hyperconnectivity on medium to large-scale brain networks, with a larger contribution of the former. Regions as the amygdala, the insula, and frontal gyrus were associated with both effects, suggesting potential compensatory processes in hub regions. fMRI revealed hypoconnectivity in AD between regions of the default mode, salience, visual, and auditory networks, while in bvFTD between regions of the default mode, salience, and somatomotor networks. EEG revealed hypoconnectivity in the γ band between frontal, limbic, and sensory regions in AD, and in the δ band between frontal, temporal, parietal and posterior areas in bvFTD, suggesting additional pathophysiological processes that fMRI alone can not capture. Classification accuracy was comparable with standard biomarkers and robust against confounders such as sample size, age, education, and motor artifacts (from fMRI and EEG). We conclude that high-order interactions provide a detailed, EEG- and fMRI compatible, biologically plausible, and psychopathological-specific characterization of different neurodegenerative conditions.

An Ultrahigh-Dimensional Mapping Model of High-order Epistatic Networks for Complex Traits.

BACKGROUND: Genetic interactions involving more than two loci have been thought to affect quantitatively inherited traits and diseases more pervasively than previously appreciated. However, the detection of such high-order interactions to chart a complete portrait of genetic architecture has not been well explored. METHODS: We present an ultrahigh-dimensional model to systematically characterize genetic main effects and interaction effects of various orders among all possible markers in a genetic mapping or association study. The model was built on the extension of a variable selection procedure, called iFORM, derived from forward selection. The model shows its unique power to estimate the magnitudes and signs of high-order epistatic effects, in addition to those of main effects and pairwise epistatic effects. RESULTS: The statistical properties of the model were tested and validated through simulation studies. By analyzing a real data for shoot growth in a mapping population of woody plant, mei (Prunus mume), we demonstrated the usefulness and utility of the model in practical genetic studies. The model has identified important high-order interactions that contribute to shoot growth for mei. CONCLUSION: The model provides a tool to precisely construct genotype-phenotype maps for quantitative traits by identifying any possible high-order epistasis which is often ignored in the current genetic literature.

An Information-Theoretic Approach to Self-Organisation: Emergence of Complex Interdependencies in Coupled Dynamical Systems.

Self-organisation lies at the core of fundamental but still unresolved scientific questions, and holds the promise of de-centralised paradigms crucial for future technological developments. While self-organising processes have been traditionally explained by the tendency of dynamical systems to evolve towards specific configurations, or attractors, we see self-organisation as a consequence of the interdependencies that those attractors induce. Building on this intuition, in this work we develop a theoretical framework for understanding and quantifying self-organisation based on coupled dynamical systems and multivariate information theory. We propose a metric of global structural strength that identifies when self-organisation appears, and a multi-layered decomposition that explains the emergent structure in terms of redundant and synergistic interdependencies. We illustrate our framework on elementary cellular automata, showing how it can detect and characterise the emergence of complex structures.

A Three-Way Interaction among Maternal and Fetal Variants Contributing to Congenital Heart Defects.

Congenital heart defects (CHDs) develop through a complex interplay between genetic variants, epigenetic modifications, and maternal environmental exposures. Genetic studies of CHDs have commonly tested single genetic variants for association with CHDs. Less attention has been given to complex gene-by-gene and gene-by-environment interactions. In this study, we applied a recently developed likelihood-ratio Mann-Whitney (LRMW) method to detect joint actions among maternal variants, fetal variants, and maternal environmental exposures, allowing for high-order statistical interactions. All subjects are participants from the National Birth Defect Prevention Study, including 623 mother-offspring pairs with CHD-affected pregnancies and 875 mother-offspring pairs with unaffected pregnancies. Each individual has 872 single nucleotide polymorphisms encoding for critical enzymes in the homocysteine, folate, and trans-sulfuration pathways. By using the LRMW method, three variants (fetal rs625879, maternal rs2169650, and maternal rs8177441) were identified with a joint association to CHD risk (nominal P-value = 1.13e-07). These three variants are located within genes BHMT2, GSTP1, and GPX3, respectively. Further examination indicated that maternal SNP rs2169650 may interact with both fetal SNP rs625879 and maternal SNP rs8177441. Our findings suggest that the risk of CHD may be influenced by both the intragenerational interaction within the maternal genome and the intergenerational interaction between maternal and fetal genomes.

A statistical procedure to map high-order epistasis for complex traits.

Genetic interactions or epistasis have been thought to play a pivotal role in shaping the formation, development and evolution of life. Previous work focused on lower-order interactions between a pair of genes, but it is obviously inadequate to explain a complex network of genetic interactions and pathways. We review and assess a statistical model for characterizing high-order epistasis among more than two genes or quantitative trait loci (QTLs) that control a complex trait. The model includes a series of start-of-the-art standard procedures for estimating and testing the nature and magnitude of QTL interactions. Results from simulation studies and real data analysis warrant the statistical properties of the model and its usefulness in practice. High-order epistatic mapping will provide a routine procedure for charting a detailed picture of the genetic regulation mechanisms underlying the phenotypic variation of complex traits.

High-order interactions observed in multi-task intrinsic networks are dominant indicators of aberrant brain function in schizophrenia.

Identifying the complex activity relationships present in rich, modern neuroimaging data sets remains a key challenge for neuroscience. The problem is hard because (a) the underlying spatial and temporal networks may be nonlinear and multivariate and (b) the observed data may be driven by numerous latent factors. Further, modern experiments often produce data sets containing multiple stimulus contexts or tasks processed by the same subjects. Fusing such multi-session data sets may reveal additional structure, but raises further statistical challenges. We present a novel analysis method for extracting complex activity networks from such multifaceted imaging data sets. Compared to previous methods, we choose a new point in the trade-off space, sacrificing detailed generative probability models and explicit latent variable inference in order to achieve robust estimation of multivariate, nonlinear group factors ("network clusters"). We apply our method to identify relationships of task-specific intrinsic networks in schizophrenia patients and control subjects from a large fMRI study. After identifying network-clusters characterized by within- and between-task interactions, we find significant differences between patient and control groups in interaction strength among networks. Our results are consistent with known findings of brain regions exhibiting deviations in schizophrenic patients. However, we also find high-order, nonlinear interactions that discriminate groups but that are not detected by linear, pairwise methods. We additionally identify high-order relationships that provide new insights into schizophrenia but that have not been found by traditional univariate or second-order methods. Overall, our approach can identify key relationships that are missed by existing analysis methods, without losing the ability to find relationships that are known to be important.

A high-order focus interaction model and oral ulcer dataset for oral ulcer segmentation.

Computer-aided diagnosis has been slow to develop in the field of oral ulcers. One of the major reasons for this is the lack of publicly available datasets. However, oral ulcers have cancerous lesions and their mortality rate is high. The ability to recognize oral ulcers at an early stage in a timely and effective manner is a very critical issue. In recent years, although there exists a small group of researchers working on these, the datasets are private. Therefore to address this challenge, in this paper a multi-tasking oral ulcer dataset (Autooral) containing two major tasks of lesion segmentation and classification is proposed and made publicly available. To the best of our knowledge, we are the first team to make publicly available an oral ulcer dataset with multi-tasking. In addition, we propose a novel modeling framework, HF-UNet, for segmenting oral ulcer lesion regions. Specifically, the proposed high-order focus interaction module (HFblock) performs acquisition of global properties and focus for acquisition of local properties through high-order attention. The proposed lesion localization module (LL-M) employs a novel hybrid sobel filter, which improves the recognition of ulcer edges. Experimental results on the proposed Autooral dataset show that our proposed HF-UNet segmentation of oral ulcers achieves a DSC value of about 0.80 and the inference memory occupies only 2029 MB. The proposed method guarantees a low running load while maintaining a high-performance segmentation capability. The proposed Autooral dataset and code are available from  https://github.com/wurenkai/HF-UNet-and-Autooral-dataset .

Assessing the Limitations of Relief-Based Algorithms in Detecting Higher-Order Interactions.

Background: The investigation of epistasis becomes increasingly complex as more loci are considered due to the exponential expansion of possible interactions. Consequently, selecting key features that influence epistatic interactions is crucial for effective downstream analyses. Recognizing this challenge, this study investigates the efficiency of Relief-Based Algorithms (RBAs) in detecting higher-order epistatic interactions, which may be critical for understanding the genetic architecture of complex traits. RBAs are uniquely non-exhaustive, eliminating the need to construct features for every possible interaction and thus improving computational tractability. Motivated by previous research indicating that some RBAs rank predictive features involved in higher-order epistasis as highly negative, we explore the utility of absolute value ranking of RBA feature weights as an alternative method to capture complex interactions. We evaluate ReliefF, MultiSURF, and MultiSURFstar on simulated genetic datasets that model various patterns of genotype-phenotype associations, including 2-way to 5-way genetic interactions, and compare their performance to two control methods: a random shuffle and mutual information. Results: Our findings indicate that while RBAs effectively identify lower-order (2 to 3-way) interactions, their capability to detect higher-order interactions is significantly limited, primarily by large feature count but also by signal noise. Specifically, we observe that RBAs are successful in detecting fully penetrant 4-way XOR interactions using an absolute value ranking approach, but this is restricted to datasets with a minimal number of total features. Conclusions: These results highlight the inherent limitations of current RBAs and underscore the need for enhanced detection capabilities for the investigation of epistasis, particularly in datasets with large feature counts and complex higher-order interactions.

Statistical Approaches to Identify Pairwise and High-Order Brain Functional Connectivity Signatures on a Single-Subject Basis.

Keeping up with the shift towards personalized neuroscience essentially requires the derivation of meaningful insights from individual brain signal recordings by analyzing the descriptive indexes of physio-pathological states through statistical methods that prioritize subject-specific differences under varying experimental conditions. Within this framework, the current study presents a methodology for assessing the value of the single-subject fingerprints of brain functional connectivity, assessed both by standard pairwise and novel high-order measures. Functional connectivity networks, which investigate the inter-relationships between pairs of brain regions, have long been a valuable tool for modeling the brain as a complex system. However, their usefulness is limited by their inability to detect high-order dependencies beyond pairwise correlations. In this study, by leveraging multivariate information theory, we confirm recent evidence suggesting that the brain contains a plethora of high-order, synergistic subsystems that would go unnoticed using a pairwise graph structure. The significance and variations across different conditions of functional pairwise and high-order interactions (HOIs) between groups of brain signals are statistically verified on an individual level through the utilization of surrogate and bootstrap data analyses. The approach is illustrated on the single-subject recordings of resting-state functional magnetic resonance imaging (rest-fMRI) signals acquired using a pediatric patient with hepatic encephalopathy associated with a portosystemic shunt and undergoing liver vascular shunt correction. Our results show that (i) the proposed single-subject analysis may have remarkable clinical relevance for subject-specific investigations and treatment planning, and (ii) the possibility of investigating brain connectivity and its post-treatment functional developments at a high-order level may be essential to fully capture the complexity and modalities of the recovery.

Gradients of O-information highlight synergy and redundancy in physiological applications.

The study of high order dependencies in complex systems has recently led to the introduction of statistical synergy, a novel quantity corresponding to a form of emergence in which patterns at large scales are not traceable from lower scales. As a consequence, several works in the last years dealt with the synergy and its counterpart, the redundancy. In particular, the O-information is a signed metric that measures the balance between redundant and synergistic statistical dependencies. In spite of its growing use, this metric does not provide insight about the role played by low-order scales in the formation of high order effects. To fill this gap, the framework for the computation of the O-information has been recently expanded introducing the so-called gradients of this metric, which measure the irreducible contribution of a variable (or a group of variables) to the high order informational circuits of a system. Here, we review the theory behind the O-information and its gradients and present the potential of these concepts in the field of network physiology, showing two new applications relevant to brain functional connectivity probed via functional resonance imaging and physiological interactions among the variability of heart rate, arterial pressure, respiration and cerebral blood flow.

State-switching and high-order spatiotemporal organization of dynamic functional connectivity are disrupted by Alzheimer's disease.

Spontaneous activity during the resting state, tracked by BOLD fMRI imaging, or shortly rsfMRI, gives rise to brain-wide dynamic patterns of interregional correlations, whose structured flexibility relates to cognitive performance. Here, we analyze resting-state dynamic functional connectivity (dFC) in a cohort of older adults, including amnesic mild cognitive impairment (aMCI, N = 34) and Alzheimer's disease (AD, N = 13) patients, as well as normal control (NC, N = 16) and cognitively "supernormal" controls (SNC, N = 10) subjects. Using complementary state-based and state-free approaches, we find that resting-state fluctuations of different functional links are not independent but are constrained by high-order correlations between triplets or quadruplets of functionally connected regions. When contrasting patients with healthy subjects, we find that dFC between cingulate and other limbic regions is increasingly bursty and intermittent when ranking the four groups from SNC to NC, aMCI and AD. Furthermore, regions affected at early stages of AD pathology are less involved in higher order interactions in patient than in control groups, while pairwise interactions are not significantly reduced. Our analyses thus suggest that the spatiotemporal complexity of dFC organization is precociously degraded in AD and provides a richer window into the underlying neurobiology than time-averaged FC connections.

High-Order Interdependencies in the Aging Brain.

Background: Brain interdependencies can be studied from either a structural/anatomical perspective ("structural connectivity") or by considering statistical interdependencies ("functional connectivity" [FC]). Interestingly, while structural connectivity is by definition pairwise (white-matter fibers project from one region to another), FC is not. However, most FC analyses only focus on pairwise statistics and they neglect higher order interactions. A promising tool to study high-order interdependencies is the recently proposed O-Information, which can quantify the intrinsic statistical synergy and the redundancy in groups of three or more interacting variables. Methods: We analyzed functional magnetic resonance imaging (fMRI) data obtained at rest from 164 healthy subjects with ages ranging in 10 to 80 years and used O-Information to investigate how high-order statistical interdependencies are affected by age. Results: Older participants (from 60 to 80 years old) exhibited a higher predominance of redundant dependencies compared with younger participants, an effect that seems to be pervasive as it is evident for all orders of interaction. In addition, while there is strong heterogeneity across brain regions, we found a "redundancy core" constituted by the prefrontal and motor cortices in which redundancy was evident at all the interaction orders studied. Discussion: High-order interdependencies in fMRI data reveal a dominant redundancy in functions such as working memory, executive, and motor functions. Our methodology can be used for a broad range of applications, and the corresponding code is freely available. Impact statement Past research has showcased multiple changes to the brain's structural and functional properties caused by aging. Here we expand prior work through recent advancements in multivariate information theory, which provide richer and more theoretically principled analyses than existing alternatives. We show that the brains of older participants contain more redundant information at multiple spatial scales-that is, activation in different brain regions is less diverse, compared with younger participants-and identify a "redundancy core" constituted by prefrontal and motor cortices, which might explained impaired performance in the old population in functions such as working memory and executive control.

BowSaw: Inferring Higher-Order Trait Interactions Associated With Complex Biological Phenotypes.

Machine learning is helping the interpretation of biological complexity by enabling the inference and classification of cellular, organismal and ecological phenotypes based on large datasets, e.g., from genomic, transcriptomic and metagenomic analyses. A number of available algorithms can help search these datasets to uncover patterns associated with specific traits, including disease-related attributes. While, in many instances, treating an algorithm as a black box is sufficient, it is interesting to pursue an enhanced understanding of how system variables end up contributing to a specific output, as an avenue toward new mechanistic insight. Here we address this challenge through a suite of algorithms, named BowSaw, which takes advantage of the structure of a trained random forest algorithm to identify combinations of variables ("rules") frequently used for classification. We first apply BowSaw to a simulated dataset and show that the algorithm can accurately recover the sets of variables used to generate the phenotypes through complex Boolean rules, even under challenging noise levels. We next apply our method to data from the integrative Human Microbiome Project and find previously unreported high-order combinations of microbial taxa putatively associated with Crohn's disease. By leveraging the structure of trees within a random forest, BowSaw provides a new way of using decision trees to generate testable biological hypotheses.

Neural Correlates of Social Behavior in Mushroom Body Extrinsic Neurons of the Honeybee Apis mellifera.

The social behavior of honeybees (Apis mellifera) has been extensively investigated, but little is known about its neuronal correlates. We developed a method that allowed us to record extracellularly from mushroom body extrinsic neurons (MB ENs) in a freely moving bee within a small but functioning mini colony of approximately 1,000 bees. This study aimed to correlate the neuronal activity of multimodal high-order MB ENs with social behavior in a close to natural setting. The behavior of all bees in the colony was video recorded. The behavior of the recorded animal was compared with other hive mates and no significant differences were found. Changes in the spike rate appeared before, during or after social interactions. The time window of the strongest effect on spike rate changes ranged from 1 s to 2 s before and after the interaction, depending on the individual animal and recorded neuron. The highest spike rates occurred when the experimental animal was situated close to a hive mate. The variance of the spike rates was analyzed as a proxy for high order multi-unit processing. Comparing randomly selected time windows with those in which the recorded animal performed social interactions showed a significantly increased spike rate variance during social interactions. The experimental set-up employed for this study offers a powerful opportunity to correlate neuronal activity with intrinsically motivated behavior of socially interacting animals. We conclude that the recorded MB ENs are potentially involved in initiating and controlling social interactions in honeybees.