Causal inference in nonlinear systems: Granger causality versus time-delayed mutual information

Mapping effective connectivity by virtually perturbing a surrogate brain

Effective connectivity (EC), which reflects the causal interactions between brain regions, is fundamental to understanding information processing in the brain; however, traditional methods for obtaining EC, which rely on neural responses to stimulation, are often invasive or limited in spatial coverage, making them unsuitable for whole-brain EC mapping in humans. Here, to address this gap, we introduce Neural Perturbational Inference (NPI), a data-driven framework for mapping whole-brain EC. NPI employs an artificial neural network trained to model large-scale neural dynamics, serving as a computational surrogate of the brain. By systematically perturbing all regions in the surrogate brain and analyzing the resulting responses in other regions, NPI maps the directionality, strength and excitatory/inhibitory properties of brain-wide EC. Validation of NPI on generative models with known ground-truth EC demonstrates its superiority over existing methods such as Granger causality and dynamic causal modeling. When applied to resting-state functional magnetic resonance imaging data across diverse datasets, NPI reveals consistent, structurally supported EC patterns. Furthermore, comparisons with cortico-cortical evoked potential data show a strong resemblance between NPI-inferred EC and real stimulation propagation patterns. By transitioning from correlational to causal understandings of brain functionality, NPI marks a stride in decoding the brain's functional architecture and facilitating both neuroscience studies and clinical applications.

Probing Asymmetric Interactions with Time-Separated Mutual Information: A Case Study Using Golden Shiners

Leader-follower modalities and other asymmetric interactions that drive the collective motion of organisms are often quantified using information theory metrics like transfer or causation entropy. These metrics are difficult to accurately evaluate without a much larger number of data than is typically available from a time series of animal trajectories collected in the field or from experiments. In this paper, we use a generalized leader-follower model to argue that the time-separated mutual information between two organism positions can serve as an alternative metric for capturing asymmetric correlations that is much less data intensive and more accurately estimated by popular k-nearest neighbor algorithms than transfer entropy. Our model predicts a local maximum of this mutual information at a time separation value corresponding to the fundamental reaction timescale of the follower organism. We confirm this prediction by analyzing time series trajectories recorded for a pair of golden shiner fish circling an annular tank.

Causal connectivity measures for pulse-output network reconstruction: Analysis and applications

The causal connectivity of a network is often inferred to understand network function. It is arguably acknowledged that the inferred causal connectivity relies on the causality measure one applies, and it may differ from the network's underlying structural connectivity. However, the interpretation of causal connectivity remains to be fully clarified, in particular, how causal connectivity depends on causality measures and how causal connectivity relates to structural connectivity. Here, we focus on nonlinear networks with pulse signals as measured output, e.g., neural networks with spike output, and address the above issues based on four commonly utilized causality measures, i.e., time-delayed correlation coefficient, time-delayed mutual information, Granger causality, and transfer entropy. We theoretically show how these causality measures are related to one another when applied to pulse signals. Taking a simulated Hodgkin-Huxley network and a real mouse brain network as two illustrative examples, we further verify the quantitative relations among the four causality measures and demonstrate that the causal connectivity inferred by any of the four well coincides with the underlying network structural connectivity, therefore illustrating a direct link between the causal and structural connectivity. We stress that the structural connectivity of pulse-output networks can be reconstructed pairwise without conditioning on the global information of all other nodes in a network, thus circumventing the curse of dimensionality. Our framework provides a practical and effective approach for pulse-output network reconstruction.

Estimating hidden relationships in dynamical systems: Discovering drivers of infection rates of COVID-19

Discovering causal influences among internal variables is a fundamental goal of complex systems research. This paper presents a framework for uncovering hidden relationships from limited time-series data by combining methods from nonlinear estimation and information theory. The approach is based on two sequential steps: first, we reconstruct a more complete state of the underlying dynamical system, and second, we calculate mutual information between pairs of internal state variables to detail causal dependencies. Equipped with time-series data related to the spread of COVID-19 from the past three years, we apply this approach to identify the drivers of falling and rising infections during the three main waves of infection in the Chicago metropolitan region. The unscented Kalman filter nonlinear estimation algorithm is implemented on an established epidemiological model of COVID-19, which we refine to include isolation, masking, loss of immunity, and stochastic transition rates. Through the systematic study of mutual information between infection rate and various stochastic parameters, we find that increased mobility, decreased mask use, and loss of immunity post sickness played a key role in rising infections, while falling infections were controlled by masking and isolation.

Modeling transcriptional regulation of the cell cycle using a novel cybernetic-inspired approach

Quantitative understanding of cellular processes, such as cell cycle and differentiation, is impeded by various forms of complexity ranging from myriad molecular players and their multilevel regulatory interactions, cellular evolution with multiple intermediate stages, lack of elucidation of cause-effect relationships among the many system players, and the computational complexity associated with the profusion of variables and parameters. In this paper, we present a modeling framework based on the cybernetic concept that biological regulation is inspired by objectives embedding rational strategies for dimension reduction, process stage specification through the system dynamics, and innovative causal association of regulatory events with the ability to predict the evolution of the dynamical system. The elementary step of the modeling strategy involves stage-specific objective functions that are computationally determined from experiments, augmented with dynamical network computations involving endpoint objective functions, mutual information, change-point detection, and maximal clique centrality. We demonstrate the power of the method through application to the mammalian cell cycle, which involves thousands of biomolecules engaged in signaling, transcription, and regulation. Starting with a fine-grained transcriptional description obtained from RNA sequencing measurements, we develop an initial model, which is then dynamically modeled using the cybernetic-inspired method, based on the strategies described above. The cybernetic-inspired method is able to distill the most significant interactions from a multitude of possibilities. In addition to capturing the complexity of regulatory processes in a mechanistically causal and stage-specific manner, we identify the functional network modules, including novel cell cycle stages. Our model is able to predict future cell cycles consistent with experimental measurements. We posit that this innovative framework has the promise to extend to the dynamics of other biological processes, with a potential to provide novel mechanistic insights.

Upper Limb Cortical-Muscular Coupling Analysis Based on Time-Delayed Back Maximum Information Coefficient Model

In musculoskeletal systems, describing accurately the coupling direction and intensity between physiological electrical signals is crucial. The maximum information coefficient (MIC) can effectively quantify the coupling strength, especially for short time series. However, it cannot identify the direction of information transmission. This paper proposes an effective time-delayed back maximum information coefficient (TDBackMIC) analysis method by introducing a time delay parameter to measure the causal coupling. Firstly, the effectiveness of TDBackMIC is verified on simulations, and then it is applied to the analysis of functional cortical-muscular coupling and intermuscular coupling networks to explore the difference of coupling characteristics under different grip force intensities. Experimental results show that functional cortical-muscular coupling and intermuscular coupling are bidirectional. The average coupling strength of EEG → EMG and EMG → EEG in beta band is 0.86 ± 0.04 and 0.81 ± 0.05 at 10% maximum voluntary contraction (MVC) condition, 0.83 ± 0.05 and 0.76 ± 0.04 at 20% MVC, and 0.76 ± 0.03 and 0.73 ± 0.04 at 30% MVC. With the increase of grip strength, the strength of functional cortical-muscular coupling in beta frequency band decreases, the intermuscular coupling network exhibits enhanced connectivity, and the information exchange is closer. The results demonstrate that TDBackMIC can accurately judge the causal coupling relationship, and functional cortical-muscular coupling and intermuscular coupling network under different grip forces are different, which provides a certain theoretical basis for sports rehabilitation.

Description length guided nonlinear unified Granger causality analysis

Most Granger causality analysis (GCA) methods still remain a two-stage scheme guided by different mathematical theories; both can actually be viewed as the same generalized model selection issues. Adhering to Occam's razor, we present a unified GCA (uGCA) based on the minimum description length principle. In this research, considering the common existence of nonlinearity in functional brain networks, we incorporated the nonlinear modeling procedure into the proposed uGCA method, in which an approximate representation of Taylor's expansion was adopted. Through synthetic data experiments, we revealed that nonlinear uGCA was obviously superior to its linear representation and the conventional GCA. Meanwhile, the nonlinear characteristics of high-order terms and cross-terms would be successively drowned out as noise levels increased. Then, in real fMRI data involving mental arithmetic tasks, we further illustrated that these nonlinear characteristics in fMRI data may indeed be drowned out at a high noise level, and hence a linear causal analysis procedure may be sufficient. Next, involving autism spectrum disorder patients data, compared with conventional GCA, the network property of causal connections obtained by uGCA methods appeared to be more consistent with clinical symptoms.

Modeling transcriptional regulation of the cell cycle using a novel cybernetic-inspired approach

Quantitative understanding of cellular processes, such as cell cycle and differentiation, is impeded by various forms of complexity ranging from myriad molecular players and their multilevel regulatory interactions, cellular evolution with multiple intermediate stages, lack of elucidation of cause-effect relationships among the many system players, and the computational complexity associated with the profusion of variables and parameters. In this paper, we present an elegant modeling framework based on the cybernetic concept that biological regulation is inspired by objectives embedding entirely novel strategies for dimension reduction, process stage specification through the system dynamics, and innovative causal association of regulatory events with the ability to predict the evolution of the dynamical system. The elementary step of the modeling strategy involves stage-specific objective functions that are computationally-determined from experiments, augmented with dynamical network computations involving end point objective functions, mutual information, change point detection, and maximal clique centrality. We demonstrate the power of the method through application to the mammalian cell cycle, which involves thousands of biomolecules engaged in signaling, transcription, and regulation. Starting with a fine-grained transcriptional description obtained from RNA sequencing measurements, we develop an initial model, which is then dynamically modeled using the cybernetic-inspired method (CIM), utilizing the strategies described above. The CIM is able to distill the most significant interactions from a multitude of possibilities. In addition to capturing the complexity of regulatory processes in a mechanistically causal and stage-specific manner, we identify the functional network modules, including novel cell cycle stages. Our model is able to predict future cell cycles consistent with experimental measurements. We posit that this state-of-the-art framework has the promise to extend to the dynamics of other biological processes, with a potential to provide novel mechanistic insights.

STATEMENT OF SIGNIFICANCE

Cellular processes like cell cycle are overly complex, involving multiple players interacting at multiple levels, and explicit modeling of such systems is challenging. The availability of longitudinal RNA measurements provides an opportunity to "reverse-engineer" for novel regulatory models. We develop a novel framework, inspired using goal-oriented cybernetic model, to implicitly model transcriptional regulation by constraining the system using inferred temporal goals. A preliminary causal network based on information-theory is used as a starting point, and our framework is used to distill the network to temporally-based networks containing essential molecular players. The strength of this approach is its ability to dynamically model the RNA temporal measurements. The approach developed paves the way for inferring regulatory processes in many complex cellular processes.

Data-driven causal analysis of observational biological time series

Complex systems are challenging to understand, especially when they defy manipulative experiments for practical or ethical reasons. Several fields have developed parallel approaches to infer causal relations from observational time series. Yet, these methods are easy to misunderstand and often controversial. Here, we provide an accessible and critical review of three statistical causal discovery approaches (pairwise correlation, Granger causality, and state space reconstruction), using examples inspired by ecological processes. For each approach, we ask what it tests for, what causal statement it might imply, and when it could lead us astray. We devise new ways of visualizing key concepts, describe some novel pathologies of existing methods, and point out how so-called 'model-free' causality tests are not assumption-free. We hope that our synthesis will facilitate thoughtful application of methods, promote communication across different fields, and encourage explicit statements of assumptions. A video walkthrough is available (Video 1 or https://youtu.be/AIV0ttQrjK8).

Partial event coincidence analysis for distinguishing direct and indirect coupling in functional network construction

Correctly identifying interaction patterns from multivariate time series presents an important step in functional network construction. In this context, the widespread use of bivariate statistical association measures often results in a false identification of links because strong similarity between two time series can also emerge without the presence of a direct interaction due to intermediate mediators or common drivers. In order to properly distinguish such direct and indirect links for the special case of event-like data, we present here a new generalization of event coincidence analysis to a partial version thereof, which is aimed at excluding possible transitive effects of indirect couplings. Using coupled chaotic systems and stochastic processes on two generic coupling topologies (star and chain configuration), we demonstrate that the proposed methodology allows for the correct identification of indirect interactions. Subsequently, we apply our partial event coincidence analysis to multi-channel EEG recordings to investigate possible differences in coordinated alpha band activity among macroscopic brain regions in resting states with eyes open (EO) and closed (EC) conditions. Specifically, we find that direct connections typically correspond to close spatial neighbors while indirect ones often reflect longer-distance connections mediated via other brain regions. In the EC state, connections in the frontal parts of the brain are enhanced as compared to the EO state, while the opposite applies to the posterior regions. In general, our approach leads to a significant reduction in the number of indirect connections and thereby contributes to a better understanding of the alpha band desynchronization phenomenon in the EO state.

Transfer Entropy of West Pacific Earthquakes to Inner Van Allen Belt Electron Bursts

Lithosphere-ionosphere non-linear interactions create a complex system where links between different phenomena can remain hidden. The statistical correlation between West Pacific strong earthquakes and high-energy electron bursts escaping trapped conditions was demonstrated in past works. Here, it is investigated from the point of view of information. Starting from the conditional probability statistical model, which was deduced from the correlation, the Shannon entropy, the joint entropy, and the conditional entropy are calculated. Time-delayed mutual information and transfer entropy have also been calculated analytically here for binary events: by including correlations between consecutive earthquake events, and between consecutive earthquakes and electron bursts. These quantities have been evaluated for the complex dynamical system of lithosphere-ionosphere; although the expressions calculated by probabilities resulted in being valid for each pair of binary events. Peaks occurred for the same time delay as in the correlations, Δt = 1.5-3.5 h, and as well as for a new time delay, Δt = -58.5--56.5 h, for the transfer entropy; this last is linked to EQ self-correlations from the analysis. Even if the low number of self-correlated EQs makes this second peak insignificant in this case, it is of interest to separate the non-linear contribution of the transfer entropy of binary events in the study of a complex system.

A Review on Data-Driven Process Monitoring Methods: Characterization and Mining of Industrial Data

Safe and stable operation plays an important role in the chemical industry. Fault detection and diagnosis (FDD) make it possible to identify abnormal process deviations early and assist operators in taking proper action against fault propagation. After decades of development, data-driven process monitoring technologies have gradually attracted attention from process industries. Although many promising FDD methods have been proposed from both academia and industry, challenges remain due to the complex characteristics of industrial data. In this work, classical and recent research on data-driven process monitoring methods is reviewed from the perspective of characterizing and mining industrial data. The implementation framework of data-driven process monitoring methods is first introduced. State of art of process monitoring methods corresponding to common industrial data characteristics are then reviewed. Finally, the challenges and possible solutions for actual industrial applications are discussed.

Dissecting cascade computational components in spiking neural networks

Finding out the physical structure of neuronal circuits that governs neuronal responses is an important goal for brain research. With fast advances for large-scale recording techniques, identification of a neuronal circuit with multiple neurons and stages or layers becomes possible and highly demanding. Although methods for mapping the connection structure of circuits have been greatly developed in recent years, they are mostly limited to simple scenarios of a few neurons in a pairwise fashion; and dissecting dynamical circuits, particularly mapping out a complete functional circuit that converges to a single neuron, is still a challenging question. Here, we show that a recent method, termed spike-triggered non-negative matrix factorization (STNMF), can address these issues. By simulating different scenarios of spiking neural networks with various connections between neurons and stages, we demonstrate that STNMF is a persuasive method to dissect functional connections within a circuit. Using spiking activities recorded at neurons of the output layer, STNMF can obtain a complete circuit consisting of all cascade computational components of presynaptic neurons, as well as their spiking activities. For simulated simple and complex cells of the primary visual cortex, STNMF allows us to dissect the pathway of visual computation. Taken together, these results suggest that STNMF could provide a useful approach for investigating neuronal systems leveraging recorded functional neuronal activity.

It is well known that the computation of neuronal circuits is carried out through the staged and cascade structure of different types of neurons. Nevertheless, the information, particularly sensory information, is processed in a network primarily with feedforward connections through different pathways. A peculiar example is the early visual system, where light is transcoded by the retinal cells, routed by the lateral geniculate nucleus, and reached the primary visual cortex. One meticulous interest in recent years is to map out these physical structures of neuronal pathways. However, most methods so far are limited to taking snapshots of a static view of connections between neurons. It remains unclear how to obtain a functional and dynamical neuronal circuit beyond the simple scenarios of a few randomly sampled neurons. Using simulated spiking neural networks of visual pathways with different scenarios of multiple stages, mixed cell types, and natural image stimuli, we demonstrate that a recent computational tool, named spike-triggered non-negative matrix factorization, can resolve these issues. It enables us to recover the entire structural components of neural networks underlying the computation, together with the functional components of each individual neuron. Utilizing it for complex cells of the primary visual cortex allows us to reveal every underpinning of the nonlinear computation. Our results, together with other recent experimental and computational efforts, show that it is possible to systematically dissect neural circuitry with detailed structural and functional components.

Directed Information Flow Analysis Reveals Muscle Fatigue-Related Changes in Muscle Networks and Corticomuscular Coupling

As a common neurophysiological phenomenon, voluntary muscle fatigue is accompanied by changes in both the central nervous system and peripheral muscles. Considering the effectiveness of the muscle network and the functional corticomuscular coupling (FCMC) in analyzing motor function, muscle fatigue can be analyzed by quantitating the intermuscular coupling and corticomuscular coupling. However, existing coherence-based research on muscle fatigue are limited by the inability of the coherence algorithm to identify the coupling direction, which cannot further reveal the underlying neural mechanism of muscle fatigue. To address this problem, we applied the time-delayed maximal information coefficient (TDMIC) method to quantitate the directional informational interaction in the muscle network and FCMC during a right-hand stabilized grip task. Eight healthy subjects were recruited to the present study. For the muscle networks, the beta-band information flow increased significantly due to muscle fatigue, and the information flow between the synergist muscles were stronger than that between the synergist and antagonist muscles. The information flow in the muscle network mainly flows to flexor digitorum superficialis (FDS), flexor carpi ulnar (FCU), and brachioradialis (BR). For the FCMC, muscle fatigue caused a significant decrease in the beta- and gamma-band bidirectional information flow. Further analysis revealed that the beta-band information flow was significantly stronger in the descending direction [electroencephalogram (EEG) to surface electromyography (sEMG)] than that in the ascending direction (sEMG to EEG) during pre-fatigue tasks. After muscle fatigue, the beta-band information flow in the ascending direction was significantly stronger than that in the descending direction. The present study demonstrates the influence of muscle fatigue on information flow in muscle networks and FCMC. We proposes that beta-band intermuscular and corticomuscular informational interaction plays an adjusting role in autonomous movement completion under muscle fatigue. Directed information flow analysis can be used as an effective method to explore the neural mechanism of muscle fatigue on the macroscopic scale.

Real-Time Industrial Process Fault Diagnosis Based on Time Delayed Mutual Information Analysis

Causal relations among variables may change significantly due to different control strategies and fault types. Off line-based knowledge is not adequate for fault diagnosis, and existing causal models obtained from data driven methods are mostly based on historical data only. However, variable correlation would not remain identical, and could be very different under certain industrial operation conditions. To deal with this problem, a fault diagnosis framework is proposed based on information solely extracted from process data. By this method, mutual information (MI) between each pair of variables is first calculated to obtain thresholds using historical data, as variable correlation under normal conditions is mostly contributed by random noises, which is often neglected in existing causal analysis models. Once a process deviation is detected, each pair of variables with mutual information beyond these thresholds are further investigated by time delayed mutual information (TDMI) analysis using current data, so as to determine the causal logic between them, which is represented as fault propagation paths, can be tracked all the way back to the root cause. The proposed method is first applied to a simulated process and the Tennessee Eastman process. The results show that the difference in variable correlation under diverse operation or control response conditions can be captured in real time, and fault propagation path can be objectively identified, together with the root cause. Then, the method has been successfully applied to a whole year data in an industrial process, which proves the feasibility of industrial application.

Transfer entropy dependent on distance among agents in quantifying leader-follower relationships

Synchronized movement of (both unicellular and multicellular) systems can be observed almost everywhere. Understanding of how organisms are regulated to synchronized behavior is one of the challenging issues in the field of collective motion. It is hypothesized that one or a few agents in a group regulate(s) the dynamics of the whole collective, known as leader(s). The identification of the leader (influential) agent(s) is very crucial. This article reviews different mathematical models that represent different types of leadership. We focus on the improvement of the leader-follower classification problem. It was found using a simulation model that the use of interaction domain information significantly improves the leader-follower classification ability using both linear schemes and information-theoretic schemes for quantifying influence. This article also reviews different schemes that can be used to identify the interaction domain using the motion data of agents.

Identifying bidirectional total and non-linear information flow in functional corticomuscular coupling during a dorsiflexion task: a pilot study

Background

The key challenge to constructing functional corticomuscular coupling (FCMC) is to accurately identify the direction and strength of the information flow between scalp electroencephalography (EEG) and surface electromyography (SEMG). Traditional TE and TDMI methods have difficulty in identifying the information interaction for short time series as they tend to rely on long and stable data, so we propose a time-delayed maximal information coefficient (TDMIC) method. With this method, we aim to investigate the directional specificity of bidirectional total and nonlinear information flow on FCMC, and to explore the neural mechanisms underlying motor dysfunction in stroke patients.

Methods

We introduced a time-delayed parameter in the maximal information coefficient to capture the direction of information interaction between two time series. We employed the linear and non-linear system model based on short data to verify the validity of our algorithm. We then used the TDMIC method to study the characteristics of total and nonlinear information flow in FCMC during a dorsiflexion task for healthy controls and stroke patients.

Results

The simulation results showed that the TDMIC method can better detect the direction of information interaction compared with TE and TDMI methods. For healthy controls, the beta band (14–30 Hz) had higher information flow in FCMC than the gamma band (31–45 Hz). Furthermore, the beta-band total and nonlinear information flow in the descending direction (EEG to EMG) was significantly higher than that in the ascending direction (EMG to EEG), whereas in the gamma band the ascending direction had significantly higher information flow than the descending direction. Additionally, we found that the strong bidirectional information flow mainly acted on Cz, C3, CP3, P3 and CPz. Compared to controls, both the beta-and gamma-band bidirectional total and nonlinear information flows of the stroke group were significantly weaker. There is no significant difference in the direction of beta- and gamma-band information flow in stroke group.

Conclusions

The proposed method could effectively identify the information interaction between short time series. According to our experiment, the beta band mainly passes downward motor control information while the gamma band features upward sensory feedback information delivery. Our observation demonstrate that the center and contralateral sensorimotor cortex play a major role in lower limb motor control. The study further demonstrates that brain damage caused by stroke disrupts the bidirectional information interaction between cortex and effector muscles in the sensorimotor system, leading to motor dysfunction.

The Human Organism as an Integrated Interaction Network: Recent Conceptual and Methodological Challenges

The field of Network Physiology aims to advance our understanding of how physiological systems and sub-systems interact to generate a variety of behaviors and distinct physiological states, to optimize the organism's functioning, and to maintain health. Within this framework, which considers the human organism as an integrated network, vertices are associated with organs while edges represent time-varying interactions between vertices. Likewise, vertices may represent networks on smaller spatial scales leading to a complex mixture of interacting homogeneous and inhomogeneous networks of networks. Lacking adequate analytic tools and a theoretical framework to probe interactions within and among diverse physiological systems, current approaches focus on inferring properties of time-varying interactions-namely strength, direction, and functional form-from time-locked recordings of physiological observables. To this end, a variety of bivariate or, in general, multivariate time-series-analysis techniques, which are derived from diverse mathematical and physical concepts, are employed and the resulting time-dependent networks can then be further characterized with methods from network theory. Despite the many promising new developments, there are still problems that evade from a satisfactory solution. Here we address several important challenges that could aid in finding new perspectives and inspire the development of theoretic and analytical concepts to deal with these challenges and in studying the complex interactions between physiological systems.

The extended Granger causality analysis for Hodgkin-Huxley neuronal models

How to extract directions of information flow in dynamical systems based on empirical data remains a key challenge. The Granger causality (GC) analysis has been identified as a powerful method to achieve this capability. However, the framework of the GC theory requires that the dynamics of the investigated system can be statistically linearized; i.e., the dynamics can be effectively modeled by linear regressive processes. Under such conditions, the causal connectivity can be directly mapped to the structural connectivity that mediates physical interactions within the system. However, for nonlinear dynamical systems such as the Hodgkin-Huxley (HH) neuronal circuit, the validity of the GC analysis has yet been addressed; namely, whether the constructed causal connectivity is still identical to the synaptic connectivity between neurons remains unknown. In this work, we apply the nonlinear extension of the GC analysis, i.e., the extended GC analysis, to the voltage time series obtained by evolving the HH neuronal network. In addition, we add a certain amount of measurement or observational noise to the time series to take into account the realistic situation in data acquisition in the experiment. Our numerical results indicate that the causal connectivity obtained through the extended GC analysis is consistent with the underlying synaptic connectivity of the system. This consistency is also insensitive to dynamical regimes, e.g., a chaotic or non-chaotic regime. Since the extended GC analysis could in principle be applied to any nonlinear dynamical system as long as its attractor is low dimensional, our results may potentially be extended to the GC analysis in other settings.

Inferring domain of interactions among particles from ensemble of trajectories

An information-theoretic scheme is proposed to estimate the underlying domain of interactions and the timescale of the interactions for many-particle systems. The crux is the application of transfer entropy which measures the amount of information transferred from one variable to another, and the introduction of a "cutoff distance variable" which specifies the distance within which pairs of particles are taken into account in the estimation of transfer entropy. The Vicsek model often studied as a metaphor of collectively moving animals is employed with introducing asymmetric interactions and an interaction timescale. Based on ensemble data of trajectories of the model system, it is shown that using the interaction domain significantly improves the performance of classification of leaders and followers compared to the approach without utilizing knowledge of the domain. Given an interaction timescale estimated from an ensemble of trajectories, the first derivative of transfer entropy averaged over the ensemble with respect to the cutoff distance is presented to serve as an indicator to infer the interaction domain. It is shown that transfer entropy is superior for inferring the interaction radius compared to cross correlation, hence resulting in a higher performance for inferring the leader-follower relationship. The effects of noise size exerted from environment and the ratio of the numbers of leader and follower on the classification performance are also discussed.

Ordinal partition transition network based complexity measures for inferring coupling direction and delay from time series

It has been demonstrated that the construction of ordinal partition transition networks (OPTNs) from time series provides a prospective approach to improve our understanding of the underlying dynamical system. In this work, we introduce a suite of OPTN based complexity measures to infer the coupling direction between two dynamical systems from pairs of time series. For several examples of coupled stochastic processes, we demonstrate that our approach is able to successfully identify interaction delays of both unidirectional and bidirectional coupling configurations. Moreover, we show that the causal interaction between two coupled chaotic Hénon maps can be captured by the OPTN based complexity measures for a broad range of coupling strengths before the onset of synchronization. Finally, we apply our method to two real-world observational climate time series, disclosing the interaction delays underlying the temperature records from two distinct stations in Oxford and Vienna. Our results suggest that ordinal partition transition networks can be used as complementary tools for causal inference tasks and provide insights into the potentials and theoretical foundations of time series networks.