Partial cross mapping eliminates indirect causal influences

Decomposing causality into its synergistic, unique, and redundant components

Causality lies at the heart of scientific inquiry, serving as the fundamental basis for understanding interactions among variables in physical systems. Despite its central role, current methods for causal inference face significant challenges due to nonlinear dependencies, stochastic interactions, self-causation, collider effects, and influences from exogenous factors, among others. While existing methods can effectively address some of these challenges, no single approach has successfully integrated all these aspects. Here, we address these challenges with SURD: Synergistic-Unique-Redundant Decomposition of causality. SURD quantifies causality as the increments of redundant, unique, and synergistic information gained about future events from past observations. The formulation is non-intrusive and applicable to both computational and experimental investigations, even when samples are scarce. We benchmark SURD in scenarios that pose significant challenges for causal inference and demonstrate that it offers a more reliable quantification of causality compared to previous methods.

On formal limitations of causal ecological networks

Causal multivariate time-series analysis, combined with network theory, provide a powerful tool for studying complex ecological interactions. However, these methods have limitations often underestimated when used in graphical modelling of ecological systems. In this opinion article, I examine the relationship between formal logic methods used to describe causal networks and their inherent statistical and epistemological limitations. I argue that while these methods offer valuable insights, they are restricted by axiomatic assumptions, statistical constraints and the incompleteness of our knowledge. To prove that, I first consider causal networks as formal systems, define causality and formalize their axioms in terms of modal logic and use ecological counterexamples to question the axioms. I also highlight the statistical limitations when using multivariate time-series analysis and Granger causality to develop ecological networks, including the potential for spurious correlations among other data characteristics. Finally, I draw upon Gödel's incompleteness theorems to highlight the inherent limits of fully understanding complex networks as formal systems and conclude that causal ecological networks are subject to initial rules and data characteristics and, as any formal system, will never fully capture the intricate complexities of the systems they represent. This article is part of the theme issue 'Connected interactions: enriching food web research by spatial and social interactions'.

Associating frailty and dynamic dysregulation between motor and cardiac autonomic systems

Frailty is a geriatric syndrome associated with the lack of physiological reserve and consequent adverse outcomes (therapy complications and death) in older adults. Recent research has shown associations between heart rate (HR) dynamics (HR changes during physical activity) with frailty. The goal of the present study was to determine the effect of frailty on the interconnection between motor and cardiac systems during a localized upper-extremity function (UEF) test. Fifty-six individuals aged 65 or above were recruited and performed the previously developed UEF test consisting of 20-s rapid elbow flexion with the right arm. Frailty was assessed using the Fried phenotype. Wearable gyroscopes and electrocardiography were used to measure motor function and HR dynamics. In this study, the interconnection between motor (angular displacement) and cardiac (HR) performance was assessed, using convergent cross-mapping (CCM). A significantly weaker interconnection was observed among pre-frail and frail participants compared to non-frail individuals (p < 0.01, effect size = 0.81 ± 0.08). Using logistic models, pre-frailty and frailty were identified with sensitivity and specificity of 82%-89%, using motor, HR dynamics, and interconnection parameters. Findings suggested a strong association between cardiac-motor interconnection and frailty. Adding CCM parameters in a multimodal model may provide a promising measure of frailty.

Robust inference of causality in high-dimensional dynamical processes from the Information Imbalance of distance ranks

We introduce an approach which allows detecting causal relationships between variables for which the time evolution is available. Causality is assessed by a variational scheme based on the Information Imbalance of distance ranks, a statistical test capable of inferring the relative information content of different distance measures. We test whether the predictability of a putative driven system Y can be improved by incorporating information from a potential driver system X, without explicitly modeling the underlying dynamics and without the need to compute probability densities of the dynamic variables. This framework makes causality detection possible even between high-dimensional systems where only few of the variables are known or measured. Benchmark tests on coupled chaotic dynamical systems demonstrate that our approach outperforms other model-free causality detection methods, successfully handling both unidirectional and bidirectional couplings. We also show that the method can be used to robustly detect causality in human electroencephalography data.

Partial Cross Mapping Based on Sparse Variable Selection for Direct Fault Root Cause Diagnosis for Industrial Processes

Root cause diagnosis of process industry is of significance to ensure safe production and improve production efficiency. Conventional contribution plot methods have challenges in root cause diagnosis due to the smearing effect. Other traditional root cause diagnosis methods, such as Granger causality (GC) and transfer entropy, have unsatisfactory performance in root cause diagnosis for complex industrial processes due to the existence of indirect causality. In this work, a regularization and partial cross mapping (PCM)-based root cause diagnosis framework is proposed for efficient direct causality inference and fault propagation path tracing. First, generalized Lasso-based variable selection is performed. The Hotelling T2 statistic is formulated and the Lasso-based fault reconstruction is applied to select candidate root cause variables. Second, the root cause is diagnosed through the PCM and the propagation path is drawn out according to the diagnosis result. The proposed framework is studied in four cases to verify its rationality and effectiveness, including a numerical example, the Tennessee Eastman benchmark process, the wastewater treatment process (WWTP), and the decarburization process of high-speed wire rod spring steel.

Imitating and exploring the human brain's resting and task-performing states via brain computing: scaling and architecture

A computational human brain model with the voxel-wise assimilation method was established based on individual structural and functional imaging data. We found that the more similar the brain model is to the biological counterpart in both scale and architecture, the more similarity was found between the assimilated model and the biological brain both in resting states and during tasks by quantitative metrics. The hypothesis that resting state activity reflects internal body states was validated by the interoceptive circuit's capability to enhance the similarity between the simulation model and the biological brain. We identified that the removal of connections from the primary visual cortex (V1) to downstream visual pathways significantly decreased the similarity at the hippocampus between the model and its biological counterpart, despite a slight influence on the whole brain. In conclusion, the model and methodology present a solid quantitative framework for a digital twin brain for discovering the relationship between brain architecture and functions, and for digitally trying and testing diverse cognitive, medical and lesioning approaches that would otherwise be unfeasible in real subjects.

Higher-order Granger reservoir computing: simultaneously achieving scalable complex structures inference and accurate dynamics prediction

Recently, machine learning methods, including reservoir computing (RC), have been tremendously successful in predicting complex dynamics in many fields. However, a present challenge lies in pushing for the limit of prediction accuracy while maintaining the low complexity of the model. Here, we design a data-driven, model-free framework named higher-order Granger reservoir computing (HoGRC), which owns two major missions: The first is to infer the higher-order structures incorporating the idea of Granger causality with the RC, and, simultaneously, the second is to realize multi-step prediction by feeding the time series and the inferred higher-order information into HoGRC. We demonstrate the efficacy and robustness of the HoGRC using several representative systems, including the classical chaotic systems, the network dynamical systems, and the UK power grid system. In the era of machine learning and complex systems, we anticipate a broad application of the HoGRC framework in structure inference and dynamics prediction.

Causal effect of air pollution and meteorology on the COVID-19 pandemic: A convergent cross mapping approach

Environmental factors have been suspected to influence the propagation and lethality of COVID-19 in the global population. However, most of the studies have been limited to correlation analyses and did not use specific methods to address the dynamic of the causal relationship between the virus and its external drivers. This work focuses on inferring and understanding the causal effect of critical air pollutants and meteorological parameters on COVID-19 by using an Empirical Dynamic Modeling approach called Convergent Cross Mapping. This technique allowed us to identify the time-delayed causation and the sign of interactions. Considering its remarkable urban environment and mortality rate during the pandemic, Quito, Ecuador, was chosen as a case study. Our results show that both urban air pollution and meteorology have a causal impact on COVID-19. Even if the strength and the sign of the causality vary over time, a general trend can be drawn. NO2, SO2, CO and PM2.5 have a positive causation for COVID-19 infections (ρ > 0.35 and ∂ > 9.1). Contrary to current knowledge, this study shows a rapid effect of pollution on COVID-19 cases (1 < lag days <24) and a negative impact of O3 on COVID-19-related deaths (ρ = 0.53 and ∂ = -0.3). Regarding the meteorology, temperature (ρ = 0.24 and ∂ = -0.4) and wind speed (ρ = 0.34 and ∂ = -3.9) tend to mitigate the epidemiological consequences of SARS-CoV-2, whereas relative humidity seems to increase the excess deaths (ρ = 0.4 and ∂ = 0.05). A causal network is proposed to synthesize the interactions between the studied variables and to provide a simple model to support the management of coronavirus outbreaks.

The geometry of synchronization: quantifying the coupling direction of physiological signals of stress between individuals using inter-system recurrence networks

In the study of synchronization dynamics between interacting systems, several techniques are available to estimate coupling strength and coupling direction. Currently, there is no general 'best' method that will perform well in most contexts. Inter-system recurrence networks (IRN) combine auto-recurrence and cross-recurrence matrices to create a graph that represents interacting networks. The method is appealing because it is based on cross-recurrence quantification analysis, a well-developed method for studying synchronization between 2 systems, which can be expanded in the IRN framework to include N > 2 interacting networks. In this study we examine whether IRN can be used to analyze coupling dynamics between physiological variables (acceleration, blood volume pressure, electrodermal activity, heart rate and skin temperature) observed in a client in residential care with severe to profound intellectual disabilities (SPID) and their professional caregiver. Based on the cross-clustering coefficients of the IRN conclusions about the coupling direction (client or caregiver drives the interaction) can be drawn, however, deciding between bi-directional coupling or no coupling remains a challenge. Constructing the full IRN, based on the multivariate time series of five coupled processes, reveals the existence of potential feedback loops. Further study is needed to be able to determine dynamics of coupling between the different layers.

A general model-based causal inference method overcomes the curse of synchrony and indirect effect

To identify causation, model-free inference methods, such as Granger Causality, have been widely used due to their flexibility. However, they have difficulty distinguishing synchrony and indirect effects from direct causation, leading to false predictions. To overcome this, model-based inference methods that test the reproducibility of data with a specific mechanistic model to infer causality were developed. However, they can only be applied to systems described by a specific model, greatly limiting their applicability. Here, we address this limitation by deriving an easily testable condition for a general monotonic ODE model to reproduce time-series data. We built a user-friendly computational package, General ODE-Based Inference (GOBI), which is applicable to nearly any monotonic system with positive and negative regulations described by ODE. GOBI successfully inferred positive and negative regulations in various networks at both the molecular and population levels, unlike existing model-free methods. Thus, this accurate and broadly applicable inference method is a powerful tool for understanding complex dynamical systems.

Tipping Point Detection Using Reservoir Computing

Detection in high fidelity of tipping points, the emergence of which is often induced by invisible changes in internal structures or/and external interferences, is paramountly beneficial to understanding and predicting complex dynamical systems (CDSs). Detection approaches, which have been fruitfully developed from several perspectives (e.g., statistics, dynamics, and machine learning), have their own advantages but still encounter difficulties in the face of high-dimensional, fluctuating datasets. Here, using the reservoir computing (RC), a recently notable, resource-conserving machine learning method for reconstructing and predicting CDSs, we articulate a model-free framework to accomplish the detection only using the time series observationally recorded from the underlying unknown CDSs. Specifically, we encode the information of the CDS in consecutive time durations of finite length into the weights of the readout layer in an RC, and then we use the learned weights as the dynamical features and establish a mapping from these features to the system's changes. Our designed framework can not only efficiently detect the changing positions of the system but also accurately predict the intensity change as the intensity information is available in the training data. We demonstrate the efficacy of our supervised framework using the dataset produced by representative physical, biological, and real-world systems, showing that our framework outperforms those traditional methods on the short-term data produced by the time-varying or/and noise-perturbed systems. We believe that our framework, on one hand, complements the major functions of the notable RC intelligent machine and, on the other hand, becomes one of the indispensable methods for deciphering complex systems.

Conditional cross-map-based technique: From pairwise dynamical causality to causal network reconstruction

Causality detection methods based on mutual cross mapping have been fruitfully developed and applied to data originating from nonlinear dynamical systems, where the causes and effects are non-separable. However, these pairwise methods still have shortcomings in discriminating typical network structures, including common drivers, indirect dependencies, and facing the curse of dimensionality, when they are stepping to causal network reconstruction. A few endeavors have been devoted to conquer these shortcomings. Here, we propose a novel method that could be regarded as one of these endeavors. Our method, named conditional cross-map-based technique, can eliminate third-party information and successfully detect direct dynamical causality, where the detection results can exactly be categorized into four standard normal forms by the designed criterion. To demonstrate the practical usefulness of our model-free, data-driven method, data generated from different representative models covering all kinds of network motifs and measured from real-world systems are investigated. Because correct identification of the direct causal links is essential to successful modeling, predicting, and controlling the underlying complex systems, our method does shed light on uncovering the inner working mechanisms of real-world systems only using the data experimentally obtained in a variety of disciplines.

Detecting the causal influence of thermal environments among climate regions in the United States

The quantification of cross-regional interactions for the atmospheric transport processes is of crucial importance to improve the predictive capacity of climatic and environmental system modeling. The dynamic interactions in these complex systems are often nonlinear and non-separable, making conventional approaches of causal inference, such as statistical correlation or Granger causality, infeasible or ineffective. In this study, we applied an advanced approach, based on the convergent cross mapping algorithm, to detect and quantify the causal influence among different climate regions in the contiguous U.S. in response to temperature perturbations using the long-term (1901-2018) climatology of near surface air temperature record. Our results show that the directed causal network constructed by convergent cross mapping algorithm, enables us to distinguish the causal links from spurious ones rendered by statistical correlation. We also find that the Ohio Valley region, as an atmospheric convergent zone, acts as the regional gateway and mediator to the long-term thermal environments in the U.S. In addition, the temporal evolution of dynamic causality of temperature exhibits superposition of periodicities at various time scales, highlighting the impact of prominent low frequency climate variabilities such as El Niño-Southern Oscillation. The proposed method in this work will help to promote novel system-based and data-driven framework in studying the integrated environmental system dynamics.

Decomposing predictability to identify dominant causal drivers in complex ecosystems

Ecosystems are complex systems of various physical, biological, and chemical processes. Since ecosystem dynamics are composed of a mixture of different levels of stochasticity and nonlinearity, handling these data is a challenge for existing methods of time series-based causal inferences. Here, we show that, by harnessing contemporary machine learning approaches, the concept of Granger causality can be effectively extended to the analysis of complex ecosystem time series and bridge the gap between dynamical and statistical approaches. The central idea is to use an ensemble of fast and highly predictive artificial neural networks to select a minimal set of variables that maximizes the prediction of a given variable. It enables decomposition of the relationship among variables through quantifying the contribution of an individual variable to the overall predictive performance. We show how our approach, EcohNet, can improve interaction network inference for a mesocosm experiment and simulated ecosystems. The application of the method to a long-term lake monitoring dataset yielded interpretable results on the drivers causing cyanobacteria blooms, which is a serious threat to ecological integrity and ecosystem services. Since performance of EcohNet is enhanced by its predictive capabilities, it also provides an optimized forecasting of overall components in ecosystems. EcohNet could be used to analyze complex and hybrid multivariate time series in many scientific areas not limited to ecosystems.

Inferring structural and dynamical properties of gene networks from data with deep learning

The reconstruction of gene regulatory networks (GRNs) from data is vital in systems biology. Although different approaches have been proposed to infer causality from data, some challenges remain, such as how to accurately infer the direction and type of interactions, how to deal with complex network involving multiple feedbacks, as well as how to infer causality between variables from real-world data, especially single cell data. Here, we tackle these problems by deep neural networks (DNNs). The underlying regulatory network for different systems (gene regulations, ecology, diseases, development) can be successfully reconstructed from trained DNN models. We show that DNN is superior to existing approaches including Boolean network, Random Forest and partial cross mapping for network inference. Further, by interrogating the ensemble DNN model trained from single cell data from dynamical system perspective, we are able to unravel complex cell fate dynamics during preimplantation development. We also propose a data-driven approach to quantify the energy landscape for gene regulatory systems, by combining DNN with the partial self-consistent mean field approximation (PSCA) approach. We anticipate the proposed method can be applied to other fields to decipher the underlying dynamical mechanisms of systems from data.

Gene Differential Co-Expression Networks Based on RNA-Seq: Construction and Its Applications

Gene co-expression network (GCN) becomes an increasingly important tool in omics data analysis. A great challenge for GCN construction is that the sample size is far lower than the number of genes. Traditional methods rely on considerable samples. Moreover, association signals are likely weak, nonlinear and stochastic, which are difficult to be identified among thousands of candidates. In this paper, the gray correlation coefficient (GCC) is introduced, and a novel method to construct gene differential co-expression networks (GDCNs) is proposed. Based on the GDCNs, three measures are proposed to explore informative genes. The proposed method can make full use of the information provided by a handful of samples and overcome the shortages of GCNs, which can evaluate the changes of co-expression relationships that are possibly triggered by treatments. Based on RNA-seq data of Brassica napus, GDCNs under multiple experimental conditions are constructed and investigated. It is found that the GCC-based method is very robust to data processing. The GDCNs facilitate the inference of gene functions and the identification of informative genes that are responsible for stress responsiveness. The GDCN-based approaches integrate the 'guilt by association' and the 'guilt by rewiring' rules, which provide alternative tools for omics data analysis.

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).

Brain-inspired chaotic backpropagation for MLP

Backpropagation (BP) algorithm is one of the most basic learning algorithms in deep learning. Although BP has been widely used, it still suffers from the problem of easily falling into the local minima due to its gradient dynamics. Inspired by the fact that the learning of real brains may exploit chaotic dynamics, we propose the chaotic backpropagation (CBP) algorithm by integrating the intrinsic chaos of real neurons into BP. By validating on multiple datasets (e.g. cifar10), we show that, for multilayer perception (MLP), CBP has significantly better abilities than those of BP and its variants in terms of optimization and generalization from both computational and theoretical viewpoints. Actually, CBP can be regarded as a general form of BP with global searching ability inspired by the chaotic learning process in the brain. Therefore, CBP not only has the potential of complementing or replacing BP in deep learning practice, but also provides a new way for understanding the learning process of the real brain.

Causation inference in complicated atmospheric environment

Reliable attribution is crucial for understanding various climate change issues. However, complicated inner-interactions between various factors make causation inference in atmospheric environment highly challenging. Taking PM_2.5-Meteorology causation, which involves a large number of non-Linear and uncertain interactions between many meteorological factors and PM_2.5, as a case, we examined the performance of a series of mainstream statistical models, including Correlation Analysis (CA), Partial Correlation Analysis (PCA), Structural Equation Model (SEM), Convergent Cross Mapping (CCM), Partial Cross Mapping (PCM) and Geographical Detector (GD). From a coarse perspective, the Top 3 major meteorological factors for PM_2.5 in 190 cities across China extracted using different models were generally consistent. From a strict perspective, the extracted dominant meteorological factor for PM_2.5 demonstrated large model variations and shared a limited consistence. Such models as SEM and PCM, which are capable of further separating direct and indirect causation in simple systems, performed poorly to identify the direct and indirect PM_2.5-Meteorology causation. The notable inconsistence denied the feasibility of employing multiple models for better causation inference in atmospheric environment. Instead, the sole use of CCM, which is advantageous in dealing with non-linear causation and removing disturbing factors, is a preferable strategy for causation inference in complicated ecosystems. Meanwhile, given the multi-direction, uncertain interactions between many variables, we should be more cautious and less ambitious on the separation of direct and indirect causation. For better causation inference in the complicated atmospheric environment, the combination of statistical models and atmospheric models, and the further exploration of Deep Neural Network can be promising strategies.

Continuity Scaling: A Rigorous Framework for Detecting and Quantifying Causality Accurately

Data-based detection and quantification of causation in complex, nonlinear dynamical systems is of paramount importance to science, engineering, and beyond. Inspired by the widely used methodology in recent years, the cross-map-based techniques, we develop a general framework to advance towards a comprehensive understanding of dynamical causal mechanisms, which is consistent with the natural interpretation of causality. In particular, instead of measuring the smoothness of the cross-map as conventionally implemented, we define causation through measuring the scaling law for the continuity of the investigated dynamical system directly. The uncovered scaling law enables accurate, reliable, and efficient detection of causation and assessment of its strength in general complex dynamical systems, outperforming those existing representative methods. The continuity scaling-based framework is rigorously established and demonstrated using datasets from model complex systems and the real world.

Embedding entropy: a nonlinear measure of dynamical causality

Research on concepts and computational methods of causality has a long history, and there are various concepts of causality as well as corresponding computing algorithms based on measured data. Here, by considering causes and effects from a dynamical perspective, we present a unified mathematical framework for the so-called dynamical causality (DC), which can detect causal interactions over time; notably, this framework covers Granger causality, transfer entropy, embedding causality and their conditional versions. Based on this framework, we further propose a causality criterion called embedding entropy (EE) to measure the DC between two variables. Moreover, its conditional version, conditional embedding entropy (cEE), is also derived for detecting conditional/direct causality. The significant advantages of EE and cEE are that they can be employed for solving not only nonlinear causal inference but also the non-separability problem, and they can reduce the scale bias in numerical calculation. The performance and robustness of EE and cEE were demonstrated through numerical simulations, and causal inference on various real-world datasets validated their effectiveness.

Interpreting Functional Impact of Genetic Variations by Network QTL for Genotype–Phenotype Association Study

An enormous challenge in the post-genome era is to annotate and resolve the consequences of genetic variation on diverse phenotypes. The genome-wide association study (GWAS) is a well-known method to identify potential genetic loci for complex traits from huge genetic variations, following which it is crucial to identify expression quantitative trait loci (eQTL). However, the conventional eQTL methods usually disregard the systematical role of single-nucleotide polymorphisms (SNPs) or genes, thereby overlooking many network-associated phenotypic determinates. Such a problem motivates us to recognize the network-based quantitative trait loci (QTL), i.e., network QTL (nQTL), which is to detect the cascade association as genotype → network → phenotype rather than conventional genotype → expression → phenotype in eQTL. Specifically, we develop the nQTL framework on the theory and approach of single-sample networks, which can identify not only network traits (e.g., the gene subnetwork associated with genotype) for analyzing complex biological processes but also network signatures (e.g., the interactive gene biomarker candidates screened from network traits) for characterizing targeted phenotype and corresponding subtypes. Our results show that the nQTL framework can efficiently capture associations between SNPs and network traits (i.e., edge traits) in various simulated data scenarios, compared with traditional eQTL methods. Furthermore, we have carried out nQTL analysis on diverse biological and biomedical datasets. Our analysis is effective in detecting network traits for various biological problems and can discover many network signatures for discriminating phenotypes, which can help interpret the influence of nQTL on disease subtyping, disease prognosis, drug response, and pathogen factor association. Particularly, in contrast to the conventional approaches, the nQTL framework could also identify many network traits from human bulk expression data, validated by matched single-cell RNA-seq data in an independent or unsupervised manner. All these results strongly support that nQTL and its detection framework can simultaneously explore the global genotype–network–phenotype associations and the underlying network traits or network signatures with functional impact and importance.

A framework for causal discovery in non-intervenable systems

Many frameworks exist to infer cause and effect relations in complex nonlinear systems, but a complete theory is lacking. A new framework is presented that is fully nonlinear, provides a complete information theoretic disentanglement of causal processes, allows for nonlinear interactions between causes, identifies the causal strength of missing or unknown processes, and can analyze systems that cannot be represented on directed acyclic graphs. The basic building blocks are information theoretic measures such as (conditional) mutual information and a new concept called certainty that monotonically increases with the information available about the target process. The framework is presented in detail and compared with other existing frameworks, and the treatment of confounders is discussed. While there are systems with structures that the framework cannot disentangle, it is argued that any causal framework that is based on integrated quantities will miss out potentially important information of the underlying probability density functions. The framework is tested on several highly simplified stochastic processes to demonstrate how blocking and gateways are handled and on the chaotic Lorentz 1963 system. We show that the framework provides information on the local dynamics but also reveals information on the larger scale structure of the underlying attractor. Furthermore, by applying it to real observations related to the El-Nino-Southern-Oscillation system, we demonstrate its power and advantage over other methodologies.

Inferring the connectivity of coupled chaotic oscillators using Kalman filtering

Inferring the interactions between coupled oscillators is a significant open problem in complexity science, with multiple interdisciplinary applications. While the Kalman filter (KF) technique is a well-known tool, widely used for data assimilation and parameter estimation, to the best of our knowledge, it has not yet been used for inferring the connectivity of coupled chaotic oscillators. Here we demonstrate that KF allows reconstructing the interaction topology and the coupling strength of a network of mutually coupled Rössler-like chaotic oscillators. We show that the connectivity can be inferred by considering only the observed dynamics of a single variable of the three that define the phase space of each oscillator. We also show that both the coupling strength and the network architecture can be inferred even when the oscillators are close to synchronization. Simulation results are provided to show the effectiveness and applicability of the proposed method.

Inferring causality in biological oscillators

MOTIVATION

Fundamental to biological study is identifying regulatory interactions. The recent surge in time-series data collection in biology provides a unique opportunity to infer regulations computationally. However, when components oscillate, model-free inference methods, while easily implemented, struggle to distinguish periodic synchrony and causality. Alternatively, model-based methods test the reproducibility of time series given a specific model but require inefficient simulations and have limited applicability.

RESULTS

We develop an inference method based on a general model of molecular, neuronal and ecological oscillatory systems that merges the advantages of both model-based and model-free methods, namely accuracy, broad applicability and usability. Our method successfully infers the positive and negative regulations within various oscillatory networks, e.g. the repressilator and a network of cofactors at the pS2 promoter, outperforming popular inference methods.

AVAILABILITY AND IMPLEMENTATION

We provide a computational package, ION (Inferring Oscillatory Networks), that users can easily apply to noisy, oscillatory time series to uncover the mechanisms by which diverse systems generate oscillations. Accompanying MATLAB code under a BSD-style license and examples are available at https://github.com/Mathbiomed/ION. Additionally, the code is available under a CC-BY 4.0 License at https://doi.org/10.6084/m9.figshare.16431408.v1.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Fast and effective pseudo transfer entropy for bivariate data-driven causal inference

Identifying, from time series analysis, reliable indicators of causal relationships is essential for many disciplines. Main challenges are distinguishing correlation from causality and discriminating between direct and indirect interactions. Over the years many methods for data-driven causal inference have been proposed; however, their success largely depends on the characteristics of the system under investigation. Often, their data requirements, computational cost or number of parameters limit their applicability. Here we propose a computationally efficient measure for causality testing, which we refer to as pseudo transfer entropy (pTE), that we derive from the standard definition of transfer entropy (TE) by using a Gaussian approximation. We demonstrate the power of the pTE measure on simulated and on real-world data. In all cases we find that pTE returns results that are very similar to those returned by Granger causality (GC). Importantly, for short time series, pTE combined with time-shifted (T-S) surrogates for significance testing strongly reduces the computational cost with respect to the widely used iterative amplitude adjusted Fourier transform (IAAFT) surrogate testing. For example, for time series of 100 data points, pTE and T-S reduce the computational time by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$82\%$$\end{document}82% with respect to GC and IAAFT. We also show that pTE is robust against observational noise. Therefore, we argue that the causal inference approach proposed here will be extremely valuable when causality networks need to be inferred from the analysis of a large number of short time series.

Seasonal dynamics of the activities of dissolved 137Cs and the 137Cs of fish in a shallow, hypereutrophic lake: Links to bottom-water oxygen concentrations

Remobilization of radiocesium from anoxic sediments can be an important mechanism responsible for long-term contaminations of lakes. However, it is unclear whether such remobilization occurs in shallow lakes, where concentrations of dissolved oxygen in the hypolimnion (bottom DO) change temporally in response to meteorological conditions, and whether remobilized radiocesium influences the activity in fish. We examined the seasonal dynamics of the activities of dissolved ¹³⁷Cs and ¹³⁷Cs in fish (pond smelt and crucian carp) from Lake Kasumigaura, a shallow, hypereutrophic lake, five years after the Fukushima Daiichi Nuclear Power Plant accident. The activities of both dissolved ¹³⁷Cs and ¹³⁷Cs in fish declined during that time, but the declines showed a clear seasonal pattern that included a summer peak of ¹³⁷Cs activity. The activity of dissolved ¹³⁷Cs increased when the bottom DO concentration decreased, and a nonlinear causality test revealed significant causal forcing of dissolved ¹³⁷Cs activity by bottom DO. The fact that NH₄-N concentrations in bottom waters were higher in the summer suggested that remobilization of ¹³⁷Cs from sediments could result from highly selective ion-exchange with NH₄-N. Despite the shallow depth of Lake Kasumigaura, winds had little influence bottom DO concentrations or dissolved ¹³⁷Cs activities. The fact that seasonal means of ¹³⁷Cs activities in pond smelt and crucian carp were positively correlated with the seasonal means of dissolved ¹³⁷Cs activities suggested that remobilized ¹³⁷Cs may have influenced the seasonal dynamics of radiocesium in fish through food-chain transfer, but higher feeding rates in warm water could may have also contributed to the seasonal dynamics of ¹³⁷Cs activity in fish. Our findings suggest that in shallow lakes, intermittent but repeated hypoxic events may enhance remobilization of radiocesium from sediments, and remobilized radiocesium may contributed to long-term retention of radiocesium in aquatic organisms.

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.