Homological scaffolds of brain functional networks

Dynamic cycles between brain states during creative storytelling

Many theories suggest that creative thinking involves a dynamic transition between different mental states, yet empirical evidence supporting this notion remains scarce. The dual process model proposes that spontaneous thinking and deliberate thinking drive the dwell in and the transitions between different mental states during creative thinking, but there is a debate over whether the two types of thinking operate in parallel or in sequence. To address these gaps, we conducted a functional magnetic resonance imaging (fMRI) study in 41 college students during a creative storytelling task. We then compared the dynamic brain states in creative versus uncreative storytelling to identify key brain states associated with creative thinking. And we further performed correlation analysis between these key brain states with performance of various creative tasks, trying to link the key brain states with different cognitive processes. The results showed that two key brain states are associated with creative thinking, with one involving whole-brain synchronization and the other involving the synchronization of four networks, including the default mode network and the control network. The transition patterns between the key brain states provide tentative evidence for dynamic circulation between different mental states during creative storytelling. Using a deep learning approach, we demonstrate an alternating interaction between spontaneous and deliberate thinking, driving dwelling in and the transitions between different brain states. These findings deepen our understanding of the cognitive and neural mechanisms underlying creative thinking.

Heterogeneous and higher-order cortical connectivity undergirds efficient, robust, and reliable neural codes

We hypothesized that the heterogeneous architecture of biological neural networks provides a substrate to regulate the well-known tradeoff between robustness and efficiency, thereby allowing different subpopulations of the same network to optimize for different objectives. To distinguish between subpopulations, we developed a metric based on the mathematical theory of simplicial complexes that captures the complexity of their connectivity by contrasting its higher-order structure to a random control and confirmed its relevance in several openly available connectomes. Using a biologically detailed cortical model and an electron microscopic dataset, we showed that subpopulations with low simplicial complexity exhibit efficient activity. Conversely, subpopulations of high simplicial complexity play a supporting role in boosting the reliability of the network as a whole, softening the robustness-efficiency tradeoff. Crucially, we found that both types of subpopulations can and do coexist within a single connectome in biological neural networks, due to the heterogeneity of their connectivity.

Synchronization stability of epileptic brain network with higher-order interactions

Generally, epilepsy is considered as abnormally enhanced neuronal excitability and synchronization. So far, previous studies on the synchronization of epileptic brain networks mainly focused on the synchronization strength, but the synchronization stability has not yet been explored as deserved. In this paper, we propose a novel idea to construct a hypergraph brain network (HGBN) based on phase synchronization. Furthermore, we apply the synchronization stability framework of the nonlinear coupled oscillation dynamic model (generalized Kuramoto model) to investigate the HGBNs of epilepsy patients. Specifically, the synchronization stability of the epileptic brain is quantified by calculating the eigenvalue spectrum of the higher-order Laplacian matrix in HGBN. Results show that synchronization stability decreased slightly in the early stages of seizure but increased significantly prior to seizure termination. This indicates that an emergency self-regulation mechanism of the brain may facilitate the termination of seizures. Moreover, the variation in synchronization stability during epileptic seizures may be induced by the topological changes of epileptogenic zones (EZs) in HGBN. Finally, we verify that the higher-order interactions improve the synchronization stability of HGBN. This study proves the validity of the synchronization stability framework with the nonlinear coupled oscillation dynamical model in HGBN, emphasizing the importance of higher-order interactions and the influence of EZs on the termination of epileptic seizures.

Classic psychedelics and the treatment for alcoholism

Alcohol is a harmful drug, and reducing its consumption is a significant challenge for users. Furthermore, alcohol dependence is often treatment-resistant, and no completely effective treatment model is available for chemical dependence. Classic psychedelics, such as LSD, psilocybin, and ayahuasca have been used in different clinical and pre-clinical trials, demonstrating promising pharmacotherapeutic effects in the treatment of treatment-resistant psychopathological conditions, such as addiction, especially related to alcohol dependence. In this work, we conducted a narrative review of the emerging research regarding the potential of psychedelics for alcohol use disorder treatment. Psychedelic substances have demonstrated potential for treating drug addiction, especially AUD, mostly by modulating neuroplasticity in the brain. Given that serotonergic psychedelics do not produce physical dependence or withdrawal symptoms with repeated use, they may be considered promising treatment options for managing drug use disorders. However, certain limitations could be found. Although many participants achieve positive results with only one treatment dose in clinical studies, great inter-individual variability exists in the duration of these effects. Therefore, further studies using different doses and experimental protocols should be conducted to enhance evidence about psychedelic substances.

Bifurcations in the Kuramoto model with external forcing and higher-order interactions

Synchronization is an important phenomenon in a wide variety of systems comprising interacting oscillatory units, whether natural (like neurons, biochemical reactions, and cardiac cells) or artificial (like metronomes, power grids, and Josephson junctions). The Kuramoto model provides a simple description of these systems and has been useful in their mathematical exploration. Here, we investigate this model by combining two common features that have been observed in many systems: External periodic forcing and higher-order interactions among the elements. We show that the combination of these ingredients leads to a very rich bifurcation scenario that produces 11 different asymptotic states of the system, with competition between forced and spontaneous synchronization. We found, in particular, that saddle-node, Hopf, and homoclinic manifolds are duplicated in regions of parameter space where the unforced system displays bi-stability.

Psychedelic 5-HT2A receptor agonism: neuronal signatures and altered neurovascular coupling

Psychedelics hold therapeutic promise for mood disorders due to rapid, sustained results. Human neuroimaging studies have reported dramatic serotonin-2A receptor-(5-HT2AR)-dependent changes in functional brain reorganization that presumably reflect neuromodulation. However, the potent vasoactive effects of serotonin have been overlooked. We found psilocybin-mediated alterations to fMRI-HRFs in humans, suggesting potentially altered NVC. To assess the neuronal, hemodynamic, and neurovascular coupling (NVC) effects of the psychedelic 5-HT2AR agonist, 2,5-Dimethoxy-4-iodoamphetamine (DOI), wide-field optical imaging (WFOI) was used in awake Thy1-jRGECO1a mice during stimulus-evoked and resting-state conditions. While DOI partially altered tasked-based NVC, more pronounced NVC alterations occurred under resting-state conditions and were strongest in association regions. Further, calcium and hemodynamic activity reported different accounts of RSFC changes under DOI. Co-administration of DOI and the 5-HT2AR antagonist, MDL100907, reversed many of these effects. Dissociation between neuronal and hemodynamic signals emphasizes a need to consider neurovascular effects of psychedelics when interpreting blood-oxygenation-dependent neuroimaging measures.

Functional Hypergraphs of Stock Markets

In stock markets, nonlinear interdependencies between various companies result in nontrivial time-varying patterns in stock prices. A network representation of these interdependencies has been successful in identifying and understanding hidden signals before major events like stock market crashes. However, these studies have revolved around the assumption that correlations are mediated in a pairwise manner, whereas, in a system as intricate as this, the interactions need not be limited to pairwise only. Here, we introduce a general methodology using information-theoretic tools to construct a higher-order representation of the stock market data, which we call functional hypergraphs. This framework enables us to examine stock market events by analyzing the following functional hypergraph quantities: Forman-Ricci curvature, von Neumann entropy, and eigenvector centrality. We compare the corresponding quantities of networks and hypergraphs to analyze the evolution of both structures and observe features like robustness towards events like crashes during the course of a time period.

Neuroimaging and the Investigation of Drug-Drug Interactions Involving Psychedelics

Psychedelic therapies are an emerging class of treatments in psychiatry with great potential, however relatively little is known about their interactions with other commonly used psychiatric medications. As psychedelic therapies become more widespread and move closer to the clinic, they likely will need to be integrated into existing treatment models which may include one or more traditional pharmacological therapies, meaning an awareness of potential drug-drug interactions will become vital. This commentary outlines some of the issues surrounding the study of drug-drug interactions of this type, provides a summary of some of the relevant key results to date, and charts a way forward which relies crucially on multimodal neuroimaging investigations. Studies in humans which combine Positron Emission Tomography (PET) and functional Magnetic Resonance Imaging (fMRI), plus ancillary measures, are likely to provide the most comprehensive assessment of drug-drug interactions involving psychedelics and the relevant effects at multiple levels of the drug response (molecular, functional, and clinical).

Deeper but smaller: Higher-order interactions increase linear stability but shrink basins

A key challenge of nonlinear dynamics and network science is to understand how higher-order interactions influence collective dynamics. Although many studies have approached this question through linear stability analysis, less is known about how higher-order interactions shape the global organization of different states. Here, we shed light on this issue by analyzing the rich patterns supported by identical Kuramoto oscillators on hypergraphs. We show that higher-order interactions can have opposite effects on linear stability and basin stability: They stabilize twisted states (including full synchrony) by improving their linear stability, but also make them hard to find by markedly reducing their basin size. Our results highlight the importance of understanding higher-order interactions from both local and global perspectives.

Chimera-inspired dynamics: When higher-order interactions are expressed differently

The exploration of chimera-inspired dynamics in nonlocally coupled networks of Kuramoto oscillators with higher-order interactions is still in its nascent stages. Concurrently, the investigation of collective phenomena in higher-order interaction networks is gaining attraction. Here, we observe that hypergraph networks tend to synchronize through lower-order interactions, whereas simplicial complex networks exhibit a preference for higher-order interactions. This observation suggests that higher-order representations manifest substantial differences in chimera-inspired synchronization regions. Moreover, we introduce an explicit expression for identifying the chimera state. With a comprehensive basin stability analysis and the interplay of pairwise and higher-order interaction strengths, the emergence of the chimera state is inherent in high-order interaction networks. Our findings contribute to the understanding of chimera-inspired dynamics in higher-order interaction networks.

Persistent homology reveals robustness loss in inhaled substance abuse rs-fMRI networks

Analyzing functional brain activity through functional magnetic resonance imaging (fMRI) is commonly done using tools from graph theory for the analysis of the correlation matrices. A drawback of these methods is that the networks must be restricted to values of the weights of the edges within certain thresholds and there is no consensus about the best choice of such thresholds. Topological data analysis (TDA) is a recently-developed tool in algebraic topology which allows us to analyze networks through combinatorial spaces obtained from them, with the advantage that all the possible thresholds can be considered at once. In this paper we applied TDA, in particular persistent homology, to study correlation matrices from rs-fMRI, and through statistical analysis, we detected significant differences between the topological structures of adolescents with inhaled substance abuse disorder (ISAD) and healthy controls. We interpreted the topological differences as indicative of a loss of robustness in the functional brain networks of the ISAD population.

Dynamical Fluctuations of Random Walks in Higher-Order Networks

Although higher-order interactions are known to affect the typical state of dynamical processes giving rise to new collective behavior, how they drive the emergence of rare events and fluctuations is still an open problem. We investigate how fluctuations of a dynamical quantity of a random walk exploring a higher-order network arise over time. In the quenched case, where the hypergraph structure is fixed, through large deviation theory we show that the appearance of rare events is hampered in nodes with many higher-order interactions, and promoted elsewhere. Dynamical fluctuations are further boosted in an annealed scenario, where both the diffusion process and higher-order interactions evolve in time. Here, extreme fluctuations generated by optimal higher-order configurations can be predicted in the limit of a saddle-point approximation. Our study lays the groundwork for a wide and general theory of fluctuations and rare events in higher-order networks.

Higher-order interaction induced chimeralike state in a bipartite network

We report higher-order coupling induced stable chimeralike state in a bipartite network of coupled phase oscillators without any time-delay in the coupling. We show that the higher-order interaction breaks the symmetry of the homogeneous synchronized state to facilitate the manifestation of symmetry breaking chimeralike state. In particular, such symmetry breaking manifests only when the pairwise interaction is attractive and higher-order interaction is repulsive, and vice versa. Further, we also demonstrate the increased degree of heterogeneity promotes homogeneous symmetric states in the phase diagram by suppressing the asymmetric chimeralike state. We deduce the low-dimensional evolution equations for the macroscopic order parameters using Ott-Antonsen ansatz and obtain the bifurcation curves from them using the software xppaut, which agrees very well with the simulation results. We also deduce the analytical stability conditions for the incoherent state, in-phase and out-of-phase synchronized states, which match with the bifurcation curves.

Acute effects of different types of cannabis on young adult and adolescent resting-state brain networks

Adolescence is a time of rapid neurodevelopment and the endocannabinoid system is particularly prone to change during this time. Cannabis is a commonly used drug with a particularly high prevalence of use among adolescents. The two predominant phytocannabinoids are Delta-9-tetrahydrocannabinol (THC) and cannabidiol (CBD), which affect the endocannabinoid system. It is unknown whether this period of rapid development makes adolescents more or less vulnerable to the effects of cannabis on brain-network connectivity, and whether CBD may attenuate the effects of THC. Using fMRI, we explored the impact of vaporized cannabis (placebo, THC: 8 mg/75 kg, THC + CBD: 8 mg/75 kg THC & 24 mg/75 kg CBD) on resting-state networks in groups of semi-regular cannabis users (usage frequency between 0.5 and 3 days/week), consisting of 22 adolescents (16-17 years) and 24 young adults (26-29 years) matched for cannabis use frequency. Cannabis caused reductions in within-network connectivity in the default mode (F[2,88] = 3.97, P = 0.022, η² = 0.018), executive control (F[2,88] = 18.62, P < 0.001, η² = 0.123), salience (F[2,88] = 12.12, P < 0.001, η² = 0.076), hippocampal (F[2,88] = 14.65, P < 0.001, η² = 0.087), and limbic striatal (F[2,88] = 16.19, P < 0.001, η² = 0.102) networks compared to placebo. Whole-brain analysis showed cannabis significantly disrupted functional connectivity with cortical regions and the executive control, salience, hippocampal, and limbic striatal networks compared to placebo. CBD did not counteract THC's effects and further reduced connectivity both within networks and the whole brain. While age-related differences were observed, there were no interactions between age group and cannabis treatment in any brain network. Overall, these results challenge the assumption that CBD can make cannabis safer, as CBD did not attenuate THC effects (and in some cases potentiated them); furthermore, they show that cannabis causes similar disruption to resting-state connectivity in the adolescent and adult brain.

Stochastic block hypergraph model

The stochastic block model is widely used to generate graphs with a community structure, but no simple alternative currently exists for hypergraphs, in which more than two nodes can be connected together through a hyperedge. We discuss here such a hypergraph generalization, based on the clustering connection probability P_{ij} between nodes of communities i and j, and that uses an explicit and modulable hyperedge formation process. We focus on the standard case where P_{ij}=pδ_{ij}+q(1-δ_{ij}) when 0≤q≤p (δ_{ij} is the Kronecker symbol). We propose a simple model that satisfies three criteria: it should be as simple as possible, when p=q the model should be equivalent to the standard hypergraph random model, and it should use an explicit and modulable hyperedge formation process so that the model is intuitive and can easily express different real-world formation processes. We first show that for such a model the degree distribution and hyperedge size distribution can be approximated by binomial distributions with effective parameters that depend on the number of communities and q/p. Also, the composition of hyperedges goes for q=0 from 'pure' hyperedges (comprising nodes belonging to the same community) to 'mixed' hyperedges that comprise nodes from different communities for q=p. We test various formation processes and our results suggest that when they depend on the composition of the hyperedge, they tend to favor the dominant community and lead to hyperedges with a smaller diversity. In contrast, for formation processes that are independent from the hyperedge structure, we obtain hyperedges comprising a larger diversity of communities. The advantages of the model proposed here are its simplicity and flexibility that make it a good candidate for testing community-related problems, such as their detection, impact on various dynamics, and visualization.

Robustness of higher-order interdependent networks with reinforced nodes

In reality, pairwise interactions are no longer sufficient to describe the higher-order interactions between nodes, such as brain networks, social networks, etc., which often contain groups of three or more nodes. Since the failure of one node in a high-order network can lead to the failure of all simplices in which it is located and quickly propagates to the whole system through the interdependencies between networks, multilayered high-order interdependent networks are challenged with high vulnerability risks. To increase the robustness of higher-order networks, in this paper, we proposed a theoretical model of a two-layer partial high-order interdependent network, where a proportion of reinforced nodes are introduced that can function and support their simplices and components, even losing connection with the giant component. We study the order parameter of the proposed model, including the giant component and functional components containing at least one reinforced node, via theoretical analysis and simulations. Rich phase transition phenomena can be observed by varying the density of 2-simplices and the proportion of the network's reinforced nodes. Increasing the density of 2-simplices makes a double transition appear in the network. The proportion of reinforced nodes can alter the type of second transition of the network from discontinuous to continuous or transition-free, which is verified on the double random simplicial complex, double scale-free simplicial complex, and real-world datasets, indicating that reinforced nodes can significantly enhance the robustness of the network and can prevent networks from abrupt collapse. Therefore, the proposed model provides insights for designing robust interdependent infrastructure networks.

Effects of high-order interactions on synchronization of a fractional-order neural system

In this study, effects of high-order interactions on synchronization of the fractional-order Hindmarsh-Rose neuron models have been examined deeply. Three different network situations in which first-order coupling, high-order couplings and first-plus second-order couplings included in the neuron models, have been considered, respectively. In order to find the optimal values of the first- and high-order coupling parameters by minimizing the cost function resulted from pairwise and triple interactions, the particle swarm optimization algorithm is employed. It has been deduced from the numerical simulation results that the first-plus second-order couplings induce the synchronization with both reduced first-order coupling strength and total cost compared to the first-order coupled case solely. When the only first-order coupled case is compared with the only second-order coupled case, it is determined that the neural network with only second-order couplings involved could achieve synchronization with lower coupling strength and, as a natural result, lower cost. On the other hand, solely second- and first-plus second-order coupled networks give very similar results each other. Therefore, high-order interactions have a positive effect on the synchronization. Additionally, increasing the network size decreases the values of the both first- and high-order coupling strengths to reach synchronization. However, in this case, total cost should be kept in the mind. Decreasing the fractional order parameter causes slower synchronization due to the decreased frequency of the neural response. On the other hand, more synchronous network is possible with increasing the fractional order parameter. Thus, the neural network with higher fractional order as well as high-order coupled is a good candidate in terms of the neural synchronization.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-023-10055-z.

Breathwork-induced psychedelic experiences modulate neural dynamics

Breathwork is an understudied school of practices involving intentional respiratory modulation to induce an altered state of consciousness (ASC). We simultaneously investigate the phenomenological and neural dynamics of breathwork by combining Temporal Experience Tracing, a quantitative methodology that preserves the temporal dynamics of subjective experience, with low-density portable EEG devices. Fourteen novice participants completed a course of up to 28 breathwork sessions-of 20, 40, or 60 min-in 28 days, yielding a neurophenomenological dataset of 301 breathwork sessions. Using hypothesis-driven and data-driven approaches, we found that "psychedelic-like" subjective experiences were associated with increased neural Lempel-Ziv complexity during breathwork. Exploratory analyses showed that the aperiodic exponent of the power spectral density-but not oscillatory alpha power-yielded similar neurophenomenological associations. Non-linear neural features, like complexity and the aperiodic exponent, neurally map both a multidimensional data-driven composite of positive experiences, and hypothesis-driven aspects of psychedelic-like experience states such as high bliss.

A simplex path integral and a simplex renormalization group for high-order interactions. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2024;87:087601. [PMID: 39077989 DOI: 10.1088/1361-6633/ad5c99] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

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.

Volume-optimal persistence homological scaffolds of hemodynamic networks covary with MEG theta-alpha aperiodic dynamics

Higher-order properties of functional magnetic resonance imaging (fMRI) induced connectivity have been shown to unravel many exclusive topological and dynamical insights beyond pairwise interactions. Nonetheless, whether these fMRI-induced higher-order properties play a role in disentangling other neuroimaging modalities' insights remains largely unexplored and poorly understood. In this work, by analyzing fMRI data from the Human Connectome Project Young Adult dataset using persistent homology, we discovered that the volume-optimal persistence homological scaffolds of fMRI-based functional connectomes exhibited conservative topological reconfigurations from the resting state to attentional task-positive state. Specifically, while reflecting the extent to which each cortical region contributed to functional cycles following different cognitive demands, these reconfigurations were constrained such that the spatial distribution of cavities in the connectome is relatively conserved. Most importantly, such level of contributions covaried with powers of aperiodic activities mostly within the theta-alpha (4-12 Hz) band measured by magnetoencephalography (MEG). This comprehensive result suggests that fMRI-induced hemodynamics and MEG theta-alpha aperiodic activities are governed by the same functional constraints specific to each cortical morpho-structure. Methodologically, our work paves the way toward an innovative computing paradigm in multimodal neuroimaging topological learning. The code for our analyses is provided in https://github.com/ngcaonghi/scaffold_noise.

Dynamic Functional Hyperconnectivity After Psilocybin Intake Is Primarily Associated With Oceanic Boundlessness

BACKGROUND

Psilocybin is a widely studied psychedelic substance that leads to the psychedelic state, a specific altered state of consciousness. To date, the relationship between the psychedelic state's neurobiological and experiential patterns remains undercharacterized because they are often analyzed separately. We investigated the relationship between neurobiological and experiential patterns after psilocybin by focusing on the link between dynamic cerebral connectivity and retrospective questionnaire assessment.

METHODS

Healthy participants were randomized to receive either psilocybin (n = 22) or placebo (n = 27) and scanned for 6 minutes in an eyes-open resting state during the peak subjective drug effect (102 minutes posttreatment) in ultrahigh field 7T magnetic resonance imaging. The 5-Dimensional Altered States of Consciousness Rating Scale was administered 360 minutes after drug intake.

RESULTS

Under psilocybin, there were alterations across all dimensions of the 5-Dimensional Altered States of Consciousness Rating Scale and widespread increases in averaged brain functional connectivity. Time-varying functional connectivity analysis unveiled a recurrent hyperconnected pattern characterized by low blood oxygen level-dependent signal amplitude, suggesting heightened cortical arousal. In terms of neuroexperiential links, canonical correlation analysis showed higher transition probabilities to the hyperconnected pattern with feelings of oceanic boundlessness and secondly with visionary restructuralization.

CONCLUSIONS

Psilocybin generates profound alterations at both the brain and the experiential levels. We suggest that the brain's tendency to enter a hyperconnected-hyperarousal pattern under psilocybin represents the potential to entertain variant mental associations. These findings illuminate the intricate interplay between brain dynamics and subjective experience under psilocybin, thereby providing insights into the neurophysiology and neuroexperiential qualities of the psychedelic state.

Global topological synchronization of weighted simplicial complexes

Higher-order networks are able to capture the many-body interactions present in complex systems and to unveil fundamental phenomena revealing the rich interplay between topology, geometry, and dynamics. Simplicial complexes are higher-order networks that encode higher-order topology and dynamics of complex systems. Specifically, simplicial complexes can sustain topological signals, i.e., dynamical variables not only defined on nodes of the network but also on their edges, triangles, and so on. Topological signals can undergo collective phenomena such as synchronization, however, only some higher-order network topologies can sustain global synchronization of topological signals. Here we consider global topological synchronization of topological signals on weighted simplicial complexes. We demonstrate that topological signals can globally synchronize on weighted simplicial complexes, even if they are odd-dimensional, e.g., edge signals, thus overcoming a limitation of the unweighted case. These results thus demonstrate that weighted simplicial complexes are more advantageous for observing these collective phenomena than their unweighted counterpart. In particular, we present two weighted simplicial complexes: the weighted triangulated torus and the weighted waffle. We completely characterize their higher-order spectral properties and demonstrate that, under suitable conditions on their weights, they can sustain global synchronization of edge signals. Our results are interpreted geometrically by showing, among the other results, that in some cases edge weights can be associated with the lengths of the sides of curved simplices.

Higher-order correlations reveal complex memory in temporal hypergraphs

Many real-world complex systems are characterized by interactions in groups that change in time. Current temporal network approaches, however, are unable to describe group dynamics, as they are based on pairwise interactions only. Here, we use time-varying hypergraphs to describe such systems, and we introduce a framework based on higher-order correlations to characterize their temporal organization. The analysis of human interaction data reveals the existence of coherent and interdependent mesoscopic structures, thus capturing aggregation, fragmentation and nucleation processes in social systems. We introduce a model of temporal hypergraphs with non-Markovian group interactions, which reveals complex memory as a fundamental mechanism underlying the emerging pattern in the data.

[Psilocybin-Assisted Treatment of Depression, Anxiety and Substance use Disorders: Neurobiological Basis and Clinical Application]

Successful therapy of mental disorders is very important in view of the high level of suffering of those affected. Since established pharmaceutical and psychotherapeutic approaches do not lead to the desired improvement in all cases, complementary or alternative treatment methods are intensively researched. Psilocybin-assisted psychotherapy seems particularly promising, and has been approved in the USA for larger clinical trials. Psilocybin belongs to the group of psychedelics and influences psychological experiences. In assisted therapy, psilocybin is administered in controlled doses under medical supervision to patients with different mental disorders. In the studies conducted so far, longer-term positive effects could be shown after just one or a few doses. In order to provide a better understanding of the potential therapeutic mechanisms, this article will first describe neurobiological and psychological effects of psilocybin. To better assess the potential of psilocybin-assisted psychotherapy for various disorders, clinical studies conducted so far with patients administered psilocybin are reviewed.

Topological Embedding of Human Brain Networks with Applications to Dynamics of Temporal Lobe Epilepsy

We introduce a novel, data-driven topological data analysis (TDA) approach for embedding brain networks into a lower-dimensional space in quantifying the dynamics of temporal lobe epilepsy (TLE) obtained from resting-state functional magnetic resonance imaging (rs-fMRI). This embedding facilitates the orthogonal projection of 0D and 1D topological features, allowing for the visualization and modeling of the dynamics of functional human brain networks in a resting state. We then quantify the topological disparities between networks to determine the coordinates for embedding. This framework enables us to conduct a coherent statistical inference within the embedded space. Our results indicate that brain network topology in TLE patients exhibits increased rigidity in 0D topology but more rapid flections compared to that of normal controls in 1D topology.

Giftedness and atypical sexual differentiation: enhanced perceptual functioning through estrogen deficiency instead of androgen excess

Syndromic autism spectrum conditions (ASC), such as Klinefelter syndrome, also manifest hypogonadism. Compared to the popular Extreme Male Brain theory, the Enhanced Perceptual Functioning model explains the connection between ASC, savant traits, and giftedness more seamlessly, and their co-emergence with atypical sexual differentiation. Overexcitability of primary sensory inputs generates a relative enhancement of local to global processing of stimuli, hindering the abstraction of communication signals, in contrast to the extraordinary local information processing skills in some individuals. Weaker inhibitory function through gamma-aminobutyric acid type A (GABAA) receptors and the atypicality of synapse formation lead to this difference, and the formation of unique neural circuits that process external information. Additionally, deficiency in monitoring inner sensory information leads to alexithymia (inability to distinguish one's own emotions), which can be caused by hypoactivity of estrogen and oxytocin in the interoceptive neural circuits, comprising the anterior insular and cingulate gyri. These areas are also part of the Salience Network, which switches between the Central Executive Network for external tasks and the Default Mode Network for self-referential mind wandering. Exploring the possibility that estrogen deficiency since early development interrupts GABA shift, causing sensory processing atypicality, it helps to evaluate the co-occurrence of ASC with attention deficit hyperactivity disorder, dyslexia, and schizophrenia based on phenotypic and physiological bases. It also provides clues for understanding the common underpinnings of these neurodevelopmental disorders and gifted populations.

Topological state-space estimation of functional human brain networks

We introduce an innovative, data-driven topological data analysis (TDA) technique for estimating the state spaces of dynamically changing functional human brain networks at rest. Our method utilizes the Wasserstein distance to measure topological differences, enabling the clustering of brain networks into distinct topological states. This technique outperforms the commonly used k-means clustering in identifying brain network state spaces by effectively incorporating the temporal dynamics of the data without the need for explicit model specification. We further investigate the genetic underpinnings of these topological features using a twin study design, examining the heritability of such state changes. Our findings suggest that the topology of brain networks, particularly in their dynamic state changes, may hold significant hidden genetic information.

Catastrophic cascade of failures in interdependent hypergraphs

The failures of individual agents can significantly impact the functionality of associated groups in interconnected systems. To reveal these impacts, we develop a threshold model to investigate cascading failures in double-layer hypergraphs with interlayer interdependence. We hypothesize that a hyperedge disintegrates when the proportion of failed nodes within it exceeds a threshold. Due to the interdependence between a node and its replica in the other layer, the disintegrations of these hyperedges could trigger a cascade of events, leading to an iterative collapse across these two layers. We find that double-layer hypergraphs undergo abrupt, discontinuous first-order phase transitions during systemic collapse regardless of the specific threshold value. Additionally, the connectivity measured by average cardinality and hyperdegree plays a crucial role in shaping system robustness. A higher average hyperdegree always strengthens system robustness. However, the relationship between system robustness and average cardinality exhibits non-monotonic behaviors. Specifically, both excessively small and large average cardinalities undermine system robustness. Furthermore, a higher threshold value can boost the system's robustness. In summary, our study provides valuable insights into cascading failure dynamics in double-layer hypergraphs and has practical implications for enhancing the robustness of complex interdependent systems across domains.

The unique neural signature of your trip: Functional connectome fingerprints of subjective psilocybin experience

The emerging neuroscientific frontier of brain fingerprinting has recently established that human functional connectomes (FCs) exhibit fingerprint-like idiosyncratic features, which map onto heterogeneously distributed behavioral traits. Here, we harness brain-fingerprinting tools to extract FC features that predict subjective drug experience induced by the psychedelic psilocybin. Specifically, in neuroimaging data of healthy volunteers under the acute influence of psilocybin or a placebo, we show that, post psilocybin administration, FCs become more idiosyncratic owing to greater intersubject dissimilarity. Moreover, whereas in placebo subjects idiosyncratic features are primarily found in the frontoparietal network, in psilocybin subjects they concentrate in the default mode network (DMN). Crucially, isolating the latter revealed an FC pattern that predicts subjective psilocybin experience and is characterized by reduced within-DMN and DMN-limbic connectivity, as well as increased connectivity between the DMN and attentional systems. Overall, these results contribute to bridging the gap between psilocybin-mediated effects on brain and behavior, while demonstrating the value of a brain-fingerprinting approach to pharmacological neuroimaging.

Altered topological structure of the brain white matter in maltreated children through topological data analysis

Childhood maltreatment may adversely affect brain development and consequently influence behavioral, emotional, and psychological patterns during adulthood. In this study, we propose an analytical pipeline for modeling the altered topological structure of brain white matter in maltreated and typically developing children. We perform topological data analysis (TDA) to assess the alteration in the global topology of the brain white matter structural covariance network among children. We use persistent homology, an algebraic technique in TDA, to analyze topological features in the brain covariance networks constructed from structural magnetic resonance imaging and diffusion tensor imaging. We develop a novel framework for statistical inference based on the Wasserstein distance to assess the significance of the observed topological differences. Using these methods in comparing maltreated children with a typically developing control group, we find that maltreatment may increase homogeneity in white matter structures and thus induce higher correlations in the structural covariance; this is reflected in the topological profile. Our findings strongly suggest that TDA can be a valuable framework to model altered topological structures of the brain. The MATLAB codes and processed data used in this study can be found at https://github.com/laplcebeltrami/maltreated.

Psilocybin enhances insightfulness in meditation: a perspective on the global topology of brain imaging during meditation

In this study, for the first time, we explored a dataset of functional magnetic resonance images collected during focused attention and open monitoring meditation before and after a five-day psilocybin-assisted meditation retreat using a recently established approach, based on the Mapper algorithm from topological data analysis. After generating subject-specific maps for two groups (psilocybin vs. placebo, 18 subjects/group) of experienced meditators, organizational principles were uncovered using graph topological tools, including the optimal transport (OT) distance, a geometrically rich measure of similarity between brain activity patterns. This revealed characteristics of the topology (i.e. shape) in space (i.e. abstract space of voxels) and time dimension of whole-brain activity patterns during different styles of meditation and psilocybin-induced alterations. Most interestingly, we found that (psilocybin-induced) positive derealization, which fosters insightfulness specifically when accompanied by enhanced open-monitoring meditation, was linked to the OT distance between open-monitoring and resting state. Our findings suggest that enhanced meta-awareness through meditation practice in experienced meditators combined with potential psilocybin-induced positive alterations in perception mediate insightfulness. Together, these findings provide a novel perspective on meditation and psychedelics that may reveal potential novel brain markers for positive synergistic effects between mindfulness practices and psilocybin.

Framework to generate hypergraphs with community structure

In recent years hypergraphs have emerged as a powerful tool to study systems with multibody interactions which cannot be trivially reduced to pairs. While highly structured methods to generate synthetic data have proved fundamental for the standardized evaluation of algorithms and the statistical study of real-world networked data, these are scarcely available in the context of hypergraphs. Here we propose a flexible and efficient framework for the generation of hypergraphs with many nodes and large hyperedges, which allows specifying general community structures and tune different local statistics. We illustrate how to use our model to sample synthetic data with desired features (assortative or disassortative communities, mixed or hard community assignments, etc.), analyze community detection algorithms, and generate hypergraphs structurally similar to real-world data. Overcoming previous limitations on the generation of synthetic hypergraphs, our work constitutes a substantial advancement in the statistical modeling of higher-order systems.

Topological features of spike trains in recurrent spiking neural networks that are trained to generate spatiotemporal patterns

In this study, we focus on training recurrent spiking neural networks to generate spatiotemporal patterns in the form of closed two-dimensional trajectories. Spike trains in the trained networks are examined in terms of their dissimilarity using the Victor-Purpura distance. We apply algebraic topology methods to the matrices obtained by rank-ordering the entries of the distance matrices, specifically calculating the persistence barcodes and Betti curves. By comparing the features of different types of output patterns, we uncover the complex relations between low-dimensional target signals and the underlying multidimensional spike trains.

Adaptation rules inducing synchronization of heterogeneous Kuramoto oscillator network with triadic couplings

A class of adaptation functions is found for which a synchronous mode with different number of phase clusters exists in a network of phase oscillators with triadic couplings. This mode is implemented in a fairly wide range of initial conditions and the maximum number of phase clusters is four. The joint influence of coupling strength and adaptation parameters on synchronization in the network has been studied. The desynchronization transition under variation of the adaptation parameter occurs abruptly and begins with the highest-frequency oscillator, spreading hierarchically to all other elements.

Phase chimera states on nonlocal hyperrings

Chimera states are dynamical states where regions of synchronous trajectories coexist with incoherent ones. A significant amount of research has been devoted to studying chimera states in systems of identical oscillators, nonlocally coupled through pairwise interactions. Nevertheless, there is increasing evidence, also supported by available data, that complex systems are composed of multiple units experiencing many-body interactions that can be modeled by using higher-order structures beyond the paradigm of classic pairwise networks. In this work we investigate whether phase chimera states appear in this framework, by focusing on a topology solely involving many-body, nonlocal, and nonregular interactions, hereby named nonlocal d-hyperring, (d+1) being the order of the interactions. We present the theory by using the paradigmatic Stuart-Landau oscillators as node dynamics, and we show that phase chimera states emerge in a variety of structures and with different coupling functions. For comparison, we show that, when higher-order interactions are "flattened" to pairwise ones, the chimera behavior is weaker and more elusive.

Determining bifurcations to explosive synchronization for networks of coupled oscillators with higher-order interactions

We determine bifurcations from gradual to explosive synchronization in coupled oscillator networks with higher-order coupling using self-consistency analysis. We obtain analytic bifurcation values for generic symmetric natural frequency distributions. We show that nonsynchronized, drifting, oscillators are non-negligible and play a crucial role in bifurcation. As such, the entire natural frequency distribution must be accounted for, rather than just the shape at the center. We verify our results for Lorentzian- and Gaussian-distributed natural frequencies.

Neural Geometrodynamics, Complexity, and Plasticity: A Psychedelics Perspective

We explore the intersection of neural dynamics and the effects of psychedelics in light of distinct timescales in a framework integrating concepts from dynamics, complexity, and plasticity. We call this framework neural geometrodynamics for its parallels with general relativity's description of the interplay of spacetime and matter. The geometry of trajectories within the dynamical landscape of "fast time" dynamics are shaped by the structure of a differential equation and its connectivity parameters, which themselves evolve over "slow time" driven by state-dependent and state-independent plasticity mechanisms. Finally, the adjustment of plasticity processes (metaplasticity) takes place in an "ultraslow" time scale. Psychedelics flatten the neural landscape, leading to heightened entropy and complexity of neural dynamics, as observed in neuroimaging and modeling studies linking increases in complexity with a disruption of functional integration. We highlight the relationship between criticality, the complexity of fast neural dynamics, and synaptic plasticity. Pathological, rigid, or "canalized" neural dynamics result in an ultrastable confined repertoire, allowing slower plastic changes to consolidate them further. However, under the influence of psychedelics, the destabilizing emergence of complex dynamics leads to a more fluid and adaptable neural state in a process that is amplified by the plasticity-enhancing effects of psychedelics. This shift manifests as an acute systemic increase of disorder and a possibly longer-lasting increase in complexity affecting both short-term dynamics and long-term plastic processes. Our framework offers a holistic perspective on the acute effects of these substances and their potential long-term impacts on neural structure and function.

Percolation and Topological Properties of Temporal Higher-Order Networks

Many complex systems that exhibit temporal nonpairwise interactions can be represented by means of generative higher-order network models. Here, we propose a hidden variable formalism to analytically characterize a general class of higher-order network models. We apply our framework to a temporal higher-order activity-driven model, providing analytical expressions for the main topological properties of the time-integrated hypergraphs, depending on the integration time and the activity distributions characterizing the model. Furthermore, we provide analytical estimates for the percolation times of general classes of uncorrelated and correlated hypergraphs. Finally, we quantify the extent to which the percolation time of empirical social interactions is underestimated when their higher-order nature is neglected.

Psilocybin and Other Classic Psychedelics in Depression

Psychedelic drugs such as psilocybin and ketamine are returning to clinical research and intervention across several disorders including the treatment of depression. This chapter focusses on psychedelics that specifically target the 5-HT2A receptor such as psilocybin and DMT. These produce plasma-concentration related psychological effects such as hallucinations and out of body experiences, insightful and emotional breakthroughs as well as mystical-type experiences. When coupled with psychological support, effects can produce a rapid improvement in mood among people with depression that can last for months. In this chapter, we summarise the scientific studies to date that explore the use of psychedelics in depressed individuals, highlighting key clinical, psychological and neuroimaging features of psychedelics that may account for their therapeutic effects. These include alterations in brain entropy that disrupt fixed negative ruminations, a period of post-treatment increased cognitive flexibility, and changes in self-referential psychological processes. Finally, we propose that the brain mechanisms underlying the therapeutic effect of serotonergic psychedelics might be distinct from those underlying classical serotonin reuptake-blocking antidepressants.

Higher-order interactions induce anomalous transitions to synchrony

We analyze the simplest model of identical coupled phase oscillators subject to two-body and three-body interactions with permutation symmetry and phase lags. This model is derived from an ensemble of weakly coupled nonlinear oscillators by phase reduction, where the first and second harmonic interactions with phase lags naturally appear. Our study indicates that the higher-order interactions induce anomalous transitions to synchrony. Unlike the conventional Kuramoto model, higher-order interactions lead to anomalous phenomena such as multistability of full synchronization, incoherent, and two-cluster states, and transitions to synchrony through slow switching and clustering. Phase diagrams of the dynamical regimes are constructed theoretically and verified by direct numerical simulations. We also show that similar transition scenarios are observed even if a small heterogeneity in the oscillators' frequency is included.

Self-organized bistability on globally coupled higher-order networks

Self-organized bistability (SOB) stands as a critical behavior for the systems delicately adjusting themselves to the brink of bistability, characterized by a first-order transition. Its essence lies in the inherent ability of the system to undergo enduring shifts between the coexisting states, achieved through the self-regulation of a controlling parameter. Recently, SOB has been established in a scale-free network as a recurrent transition to a short-living state of global synchronization. Here, we embark on a theoretical exploration that extends the boundaries of the SOB concept on a higher-order network (implicitly embedded microscopically within a simplicial complex) while considering the limitations imposed by coupling constraints. By applying Ott-Antonsen dimensionality reduction in the thermodynamic limit to the higher-order network, we derive SOB requirements under coupling limits that are in good agreement with numerical simulations on systems of finite size. We use continuous synchronization diagrams and statistical data from spontaneous synchronized events to demonstrate the crucial role SOB plays in initiating and terminating temporary synchronized events. We show that under weak-coupling consumption, these spontaneous occurrences closely resemble the statistical traits of the epileptic brain functioning.

Homological landscape of human brain functional sub-circuits

Human whole-brain functional connectivity networks have been shown to exhibit both local/quasilocal (e.g., set of functional sub-circuits induced by node or edge attributes) and non-local (e.g., higher-order functional coordination patterns) properties. Nonetheless, the non-local properties of topological strata induced by local/quasilocal functional sub-circuits have yet to be addressed. To that end, we proposed a homological formalism that enables the quantification of higher-order characteristics of human brain functional sub-circuits. Our results indicated that each homological order uniquely unravels diverse, complementary properties of human brain functional sub-circuits. Noticeably, the H 1 homological distance between rest and motor task were observed at both whole-brain and sub-circuit consolidated level which suggested the self-similarity property of human brain functional connectivity unraveled by homological kernel. Furthermore, at the whole-brain level, the rest-task differentiation was found to be most prominent between rest and different tasks at different homological orders: i) Emotion task H 0 , ii) Motor task H 1 , and iii) Working memory task H 2 . At the functional sub-circuit level, the rest-task functional dichotomy of default mode network is found to be mostly prominent at the first and second homological scaffolds. Also at such scale, we found that the limbic network plays a significant role in homological reconfiguration across both task- and subject- domain which sheds light to subsequent Investigations on the complex neuro-physiological role of such network. From a wider perspective, our formalism can be applied, beyond brain connectomics, to study non-localized coordination patterns of localized structures stretching across complex network fibers.

Unified topological inference for brain networks in temporal lobe epilepsy using the Wasserstein distance

Persistent homology offers a powerful tool for extracting hidden topological signals from brain networks. It captures the evolution of topological structures across multiple scales, known as filtrations, thereby revealing topological features that persist over these scales. These features are summarized in persistence diagrams, and their dissimilarity is quantified using the Wasserstein distance. However, the Wasserstein distance does not follow a known distribution, posing challenges for the application of existing parametric statistical models. To tackle this issue, we introduce a unified topological inference framework centered on the Wasserstein distance. Our approach has no explicit model and distributional assumptions. The inference is performed in a completely data driven fashion. We apply this method to resting-state functional magnetic resonance images (rs-fMRI) of temporal lobe epilepsy patients collected from two different sites: the University of Wisconsin-Madison and the Medical College of Wisconsin. Importantly, our topological method is robust to variations due to sex and image acquisition, obviating the need to account for these variables as nuisance covariates. We successfully localize the brain regions that contribute the most to topological differences. A MATLAB package used for all analyses in this study is available at https://github.com/laplcebeltrami/PH-STAT.

Altered dynamical integration/segregation balance during anesthesia-induced loss of consciousness

In recent years, brain imaging studies have begun to shed light on the neural correlates of physiologically-reversible altered states of consciousness such as deep sleep, anesthesia, and psychedelic experiences. The emerging consensus is that normal waking consciousness requires the exploration of a dynamical repertoire enabling both global integration i.e., long-distance interactions between brain regions, and segregation, i.e., local processing in functionally specialized clusters. Altered states of consciousness have notably been characterized by a tipping of the integration/segregation balance away from this equilibrium. Historically, functional MRI (fMRI) has been the modality of choice for such investigations. However, fMRI does not enable characterization of the integration/segregation balance at sub-second temporal resolution. Here, we investigated global brain spatiotemporal patterns in electrocorticography (ECoG) data of a monkey (Macaca fuscata) under either ketamine or propofol general anesthesia. We first studied the effects of these anesthetics from the perspective of band-specific synchronization across the entire ECoG array, treating individual channels as oscillators. We further aimed to determine whether synchrony within spatially localized clusters of oscillators was differently affected by the drugs in comparison to synchronization over spatially distributed subsets of ECoG channels, thereby quantifying changes in integration/segregation balance on physiologically-relevant time scales. The findings reflect global brain dynamics characterized by a loss of long-range integration in multiple frequency bands under both ketamine and propofol anesthesia, most pronounced in the beta (13-30 Hz) and low-gamma bands (30-80 Hz), and with strongly preserved local synchrony in all bands.

Emergence and Control of Synchronization in Networks with Directed Many-Body Interactions

The emergence of collective behaviors in networks of dynamical units in pairwise interaction has been explained as the effect of diffusive coupling. How does the presence of higher-order interaction impact the onset of spontaneous or induced synchronous behavior? Inspired by actuation and measurement constraints typical of physical and engineered systems, we propose a diffusion mechanism over hypergraphs that explains the onset of synchronization through a clarifying analogy with signed graphs. Our findings are mathematically backed by general conditions for convergence to the synchronous state.

Altered Topological Structure of the Brain White Matter in Maltreated Children through Topological Data Analysis

Childhood maltreatment may adversely affect brain development and consequently influence behavioral, emotional, and psychological patterns during adulthood. In this study, we propose an analytical pipeline for modeling the altered topological structure of brain white matter in maltreated and typically developing children. We perform topological data analysis (TDA) to assess the alteration in the global topology of the brain white-matter structural covariance network among children. We use persistent homology, an algebraic technique in TDA, to analyze topological features in the brain covariance networks constructed from structural magnetic resonance imaging (MRI) and diffusion tensor imaging (DTI). We develop a novel framework for statistical inference based on the Wasserstein distance to assess the significance of the observed topological differences. Using these methods in comparing maltreated children to a typically developing control group, we find that maltreatment may increase homogeneity in white matter structures and thus induce higher correlations in the structural covariance; this is reflected in the topological profile. Our findings strongly suggest that TDA can be a valuable framework to model altered topological structures of the brain. The MATLAB codes and processed data used in this study can be found at https://github.com/laplcebeltrami/maltreated.

The complementary contribution of each order topology into the synchronization of multi-order networks

Higher-order interactions improve our capability to model real-world complex systems ranging from physics and neuroscience to economics and social sciences. There is great interest nowadays in understanding the contribution of higher-order terms to the collective behavior of the network. In this work, we investigate the stability of complete synchronization of complex networks with higher-order structures. We demonstrate that the synchronization level of a network composed of nodes interacting simultaneously via multiple orders is maintained regardless of the intensity of coupling strength across different orders. We articulate that lower-order and higher-order topologies work together complementarily to provide the optimal stable configuration, challenging previous conclusions that higher-order interactions promote the stability of synchronization. Furthermore, we find that simply adding higher-order interactions based on existing connections, as in simple complexes, does not have a significant impact on synchronization. The universal applicability of our work lies in the comprehensive analysis of different network topologies, including hypergraphs and simplicial complexes, and the utilization of appropriate rescaling to assess the impact of higher-order interactions on synchronization stability.

A Comprehensive Review of the Current Status of the Cellular Neurobiology of Psychedelics

Psychedelic substances have gained significant attention in recent years for their potential therapeutic effects on various psychiatric disorders. This review delves into the intricate cellular neurobiology of psychedelics, emphasizing their potential therapeutic applications in addressing the global burden of mental illness. It focuses on contemporary research into the pharmacological and molecular mechanisms underlying these substances, particularly the role of 5-HT2A receptor signaling and the promotion of plasticity through the TrkB-BDNF pathway. The review also discusses how psychedelics affect various receptors and pathways and explores their potential as anti-inflammatory agents. Overall, this research represents a significant development in biomedical sciences with the potential to transform mental health treatments.

First-order transition to oscillation death in coupled oscillators with higher-order interactions

We investigate the dynamical evolution of Stuart-Landau oscillators globally coupled through conjugate or dissimilar variables on simplicial complexes. We report a first-order explosive phase transition from an oscillatory state to oscillation death, with higher-order (2-simplex triadic) interactions, as opposed to the second-order transition with only pairwise (1-simplex) interactions. Moreover, the system displays four distinct homogeneous steady states in the presence of triadic interactions, in contrast to the two homogeneous steady states observed with dyadic interactions. We calculate the backward transition point analytically, confirming the numerical results and providing the origin of the dynamical states in the transition region. The results are robust against the application of noise. The study will be useful in understanding complex systems, such as ecological and epidemiological, having higher-order interactions and coupling through conjugate variables.

Unified Topological Inference for Brain Networks in Temporal Lobe Epilepsy Using the Wasserstein Distance

Persistent homology offers a powerful tool for extracting hidden topological signals from brain networks. It captures the evolution of topological structures across multiple scales, known as filtrations, thereby revealing topological features that persist over these scales. These features are summarized in persistence diagrams, and their dissimilarity is quantified using the Wasserstein distance. However, the Wasserstein distance does not follow a known distribution, posing challenges for the application of existing parametric statistical models. To tackle this issue, we introduce a unified topological inference framework centered on the Wasserstein distance. Our approach has no explicit model and distributional assumptions. The inference is performed in a completely data driven fashion. We apply this method to resting-state functional magnetic resonance images (rs-fMRI) of temporal lobe epilepsy patients collected from two different sites: the University of Wisconsin-Madison and the Medical College of Wisconsin. Importantly, our topological method is robust to variations due to sex and image acquisition, obviating the need to account for these variables as nuisance covariates. We successfully localize the brain regions that contribute the most to topological differences. A MATLAB package used for all analyses in this study is available at https://github.com/laplcebeltrami/PH-STAT.