Causal measures of structure and plasticity in simulated and living neural networks

A major goal of neuroscience is to understand the relationship between neural structures and their function. Recording of neural activity with arrays of electrodes is a primary tool employed toward this goal. However, the relationships among the neural activity recorded by these arrays are often highly complex making it problematic to accurately quantify a network's structural information and then relate that structure to its function. Current statistical methods including cross correlation and coherence have achieved only modest success in characterizing the structural connectivity. Over the last decade an alternative technique known as Granger causality is emerging within neuroscience. This technique, borrowed from the field of economics, provides a strong mathematical foundation based on linear auto-regression to detect and quantify "causal" relationships among different time series. This paper presents a combination of three Granger based analytical methods that can quickly provide a relatively complete representation of the causal structure within a neural network. These are a simple pairwise Granger causality metric, a conditional metric, and a little known computationally inexpensive subtractive conditional method. Each causal metric is first described and evaluated in a series of biologically plausible neural simulations. We then demonstrate how Granger causality can detect and quantify changes in the strength of those relationships during plasticity using 60 channel spike train data from an in vitro cortical network measured on a microelectrode array. We show that these metrics can not only detect the presence of causal relationships, they also provide crucial information about the strength and direction of that relationship, particularly when that relationship maybe changing during plasticity. Although we focus on the analysis of multichannel spike train data the metrics we describe are applicable to any stationary time series in which causal relationships among multiple measures is desired. These techniques can be especially useful when the interactions among those measures are highly complex, difficult to untangle, and maybe changing over time.     

Economic growth,electricity consumption,urbanization and environmental degradation relationship in United Arab Emirates

The present study explores the relationship between economic growth, electricity consumption, urbanization and environmental degradation in case of United Arab Emirates (UAE). The study covers the quarter frequency data over the period of 1975–2011. We have applied the ARDL bounds testing approach to examine the long run relationship between the variables in the presence of structural breaks. The VECM Granger causality is applied to investigate the direction of causal relationship between the variables. Our empirical exercise reported the existence of cointegration among the series. Further, we found an inverted U-shaped relationship between economic growth and CO₂ emissions i.e. economic growth raises energy emissions initially and declines it after a threshold point of income per capita (EKC exists). Electricity consumption declines CO₂ emissions. The relationship between urbanization and CO₂ emissions is positive. Exports seem to improve the environmental quality by lowering CO₂ emissions. The causality analysis validates the feedback effect between CO₂ emissions and electricity consumption. Economic growth and urbanization Granger cause CO₂ emissions.     

The linkages between deforestation,energy and growth for environmental degradation in Pakistan

This study explores the validation of the Environmental Kuznets Curve (EKC) hypothesis for Pakistan using time series data from 1980–2013 with deforestation as an indicator (dependent variable) for environmental degradation, and four independent variables (economic growth, energy consumption, trade openness, and population) were also examined. The Autoregressive Distributed Lag (ARDL) bounds testing approach to cointegration and the VECM–Granger causality test were applied. The results confirmed the existence of cointegration among the variables both in long- and short-run paths. However, the diminishing negative impact of economic growth on deforestation in the long-run confirms the EKC hypothesis for deforestation in Pakistan. Moreover, economic growth and energy consumption Granger cause deforestation. A bidirectional causal effect is detected between economic growth and energy consumption, however, in the long-run, economic growth and trade openness Granger cause energy consumption. This study was designed with several significant tests to ensure the reliability of results for policy use and to contribute to future studies on the environment-growth-energy nexus.     

Testing environmental Kuznets curve hypothesis: The role of renewable and non-renewable energy consumption and trade in OECD countries

This paper investigates the causal relationships between per capita CO₂ emissions, gross domestic product (GDP), renewable and non-renewable energy consumption, and international trade for a panel of 25 OECD countries over the period 1980–2010. Short-run Granger causality tests show the existence of bidirectional causality between: renewable energy consumption and imports, renewable and non-renewable energy consumption, non-renewable energy and trade (exports or imports); and unidirectional causality running from: exports to renewable energy, trade to CO₂ emissions, output to renewable energy. There are also long-run bidirectional causalities between all our considered variables. Our long-run fully modified ordinary least squares (FMOLS) and dynamic ordinary least squares (DOLS) estimates show that the inverted U-shaped environmental Kuznets curve (EKC) hypothesis is verified for this sample of OECD countries. They also show that increasing non-renewable energy increases CO₂ emissions. Interestingly, increasing trade or renewable energy reduces CO₂ emissions. According to these results, more trade and more use of renewable energy are efficient strategies to combat global warming in these countries.     

The impact of trade openness on global carbon dioxide emissions: Evidence from the top ten emitters among developing countries

This study aims to analyze the relationship between carbon dioxide (CO₂) emissions, trade openness, real income and energy consumption in the top ten CO₂ emitters among the developing countries; namely China, India, South Korea, Brazil, Mexico, Indonesia, South Africa, Turkey, Thailand and Malaysia over the period of 1971–2011. In addition, the possible presence of the EKC hypothesis is investigated for the analyzed countries. The Zivot–Andrews unit root test with structural break, the bounds testing for cointegration in the presence of structural break and the VECM Granger causality method are employed. The empirical results indicate that (i) the analyzed variables are co-integrated for Thailand, Turkey, India, Brazil, China, Indonesia and Korea, (ii) real income, energy consumption and trade openness are the main determinants of carbon emissions in the long run, (iii) there exists a number of causal relations between the analyzed variables, (iv) the EKC hypothesis is validated for Turkey, India, China and Korea. Robust policy implications can be derived from this study since the estimated models pass several diagnostic and stability tests.     

基于格兰杰因果实证的城市绿地与经济发展互动机制研究

城市绿地与经济发展的互动机制是社会-经济-自然复合生态系统和生态文明理论研究的重要内容。利用面板格兰杰因果关系检验和面板数据回归分析方法,对中国城市绿地与经济发展之间存在的因果互动关系开展实证研究。研究结果表明:(1)全国尺度上,城市建成区绿地率与人均GDP互为格兰杰原因,存在"正向循环反馈"的因果互动关系,即城市绿地水平的提高能够推动城市经济发展,而经济发展能够拉动城市绿地建设。城市绿地率与城市人均GDP的因果关系可以概括为"绿磁效应"(GME)和"需求效应"(GDE)。(2)区域尺度上,东部地区人均GDP与城市建成区绿地率存在单向格兰杰因果关系,人均GDP是城市绿地率增长的格兰杰原因,而城市绿地率不是人均GDP的格兰杰原因;(3)西部地区城市绿地率与人均GDP的交互强度最大,即西部"绿地-经济弹性"和"经济-绿地弹性"高于其他地区。东部地区通过扩大城市绿地率来提升经济发展的空间比较小,应着眼于提高城市绿地质量,中、西部地区城市绿地率较小,应在兼顾城市绿地质量的基础上提升城市建成区绿地率。本研究能够为城市绿地与经济发展的关系研究和城市生态规划与经济发展规划提供参考。     

Evaluating causal relations in neural systems: granger causality, directed transfer function and statistical assessment of significance

 We consider the question of evaluating causal relations among neurobiological signals. In particular, we study the relation between the directed transfer function (DTF) and the well-accepted Granger causality, and show that DTF can be interpreted within the framework of Granger causality. In addition, we propose a method to assess the significance of causality measures. Finally, we demonstrate the applications of these measures to simulated data and actual neurobiological recordings. Received: 6 June 2000 / Accepted in revised form: 4 December 2000     

Is Granger Causality a Viable Technique for Analyzing fMRI Data?

Multivariate neural data provide the basis for assessing interactions in brain networks. Among myriad connectivity measures, Granger causality (GC) has proven to be statistically intuitive, easy to implement, and generate meaningful results. Although its application to functional MRI (fMRI) data is increasing, several factors have been identified that appear to hinder its neural interpretability: (a) latency differences in hemodynamic response function (HRF) across different brain regions, (b) low-sampling rates, and (c) noise. Recognizing that in basic and clinical neuroscience, it is often the change of a dependent variable (e.g., GC) between experimental conditions and between normal and pathology that is of interest, we address the question of whether there exist systematic relationships between GC at the fMRI level and that at the neural level. Simulated neural signals were convolved with a canonical HRF, down-sampled, and noise-added to generate simulated fMRI data. As the coupling parameters in the model were varied, fMRI GC and neural GC were calculated, and their relationship examined. Three main results were found: (1) GC following HRF convolution is a monotonically increasing function of neural GC; (2) this monotonicity can be reliably detected as a positive correlation when realistic fMRI temporal resolution and noise level were used; and (3) although the detectability of monotonicity declined due to the presence of HRF latency differences, substantial recovery of detectability occurred after correcting for latency differences. These results suggest that Granger causality is a viable technique for analyzing fMRI data when the questions are appropriately formulated.     

Granger causality between multiple interdependent neurobiological time series: blockwise versus pairwise methods

Granger causality is becoming an important tool for determining causal relations between neurobiological time series. For multivariate data, there is often the need to examine causal relations between two blocks of time series, where each block could represent a brain region of interest. Two alternative methods are available. In the pairwise method, bivariate autoregressive models are fit to all pairwise combinations involving one time series from the first block and one from the second. The total Granger causality between the two blocks is then derived by summing pairwise causality values from each of these models. This approach is intuitive but computationally cumbersome. Theoretically, a more concise method can be derived, which we term the blockwise Granger causality method. In this method, a single multivariate model is fit to all the time series, and the causality between the two blocks is then computed from this model. We compare these two methods by applying them to cortical local field potential recordings from monkeys performing a sensorimotor task. The obtained results demonstrate consistency between the two methods and point to the significance potential of utilizing Granger causality analysis in understanding coupled neural systems.     

CO2, economic growth,and energy consumption in China's provinces: Investigating the spatiotemporal and econometric characteristics of China's CO2 emissions

This study addresses the spatiotemporal variations at play in China's CO₂ emissions, based on an estimation of emission levels in the period 1995–2012 and an provincial analysis of the relationship of CO₂ emissions to economic growth and energy consumption. Using a series of econometric models and data on the combustion of fossil fuels and cement manufacturing, the study first estimated CO₂ emission levels during the study period, exploring their spatiotemporal pattern. The results indicate that both China's total and its per capita CO₂ emissions have increased significantly over the study period, with both measures evidencing a similar evolution (albeit one that is characterized by noticeable regional discrepancies at the provincial level and which displays properties of convergence). From a geographical perspective, we found both total and per capita CO₂ emissionsto be higher in China's eastern region than in the country's central and western regions. Panel data analysis was subsequently undertaken in order to quantify the dynamic casual relationship between economic growth, energy consumption, and CO₂ emissions. The empirical results indicated that the variables were in fact cointegrated and exhibited a long-run positive relationship. The results of further Granger causality tests indicated the existence of a bidirectional positive causality between economic growth and energy consumption, as well as between energy consumption and CO₂ emissions, and a unidirectional positive causality running from economic growth to CO₂ emissions. The findings of this study suggest that China is, in the long run, dependent on carbon energy consumption for its rapid economic growth, a dependency which is the cause of considerable increases in CO₂ emissions. China should therefore make greater efforts to develop low-carbon technologies and renewable energy, and improve energy efficiency in order to reduce emissions and achieve green economic growth.     

A Granger causality measure for point process models of ensemble neural spiking activity

The ability to identify directional interactions that occur among multiple neurons in the brain is crucial to an understanding of how groups of neurons cooperate in order to generate specific brain functions. However, an optimal method of assessing these interactions has not been established. Granger causality has proven to be an effective method for the analysis of the directional interactions between multiple sets of continuous-valued data, but cannot be applied to neural spike train recordings due to their discrete nature. This paper proposes a point process framework that enables Granger causality to be applied to point process data such as neural spike trains. The proposed framework uses the point process likelihood function to relate a neuron's spiking probability to possible covariates, such as its own spiking history and the concurrent activity of simultaneously recorded neurons. Granger causality is assessed based on the relative reduction of the point process likelihood of one neuron obtained excluding one of its covariates compared to the likelihood obtained using all of its covariates. The method was tested on simulated data, and then applied to neural activity recorded from the primary motor cortex (MI) of a Felis catus subject. The interactions present in the simulated data were predicted with a high degree of accuracy, and when applied to the real neural data, the proposed method identified causal relationships between many of the recorded neurons. This paper proposes a novel method that successfully applies Granger causality to point process data, and has the potential to provide unique physiological insights when applied to neural spike trains.     

Estimating the directed information to infer causal relationships in ensemble neural spike train recordings

Advances in recording technologies have given neuroscience researchers access to large amounts of data, in particular, simultaneous, individual recordings of large groups of neurons in different parts of the brain. A variety of quantitative techniques have been utilized to analyze the spiking activities of the neurons to elucidate the functional connectivity of the recorded neurons. In the past, researchers have used correlative measures. More recently, to better capture the dynamic, complex relationships present in the data, neuroscientists have employed causal measures—most of which are variants of Granger causality—with limited success. This paper motivates the directed information, an information and control theoretic concept, as a modality-independent embodiment of Granger’s original notion of causality. Key properties include: (a) it is nonzero if and only if one process causally influences another, and (b) its specific value can be interpreted as the strength of a causal relationship. We next describe how the causally conditioned directed information between two processes given knowledge of others provides a network version of causality: it is nonzero if and only if, in the presence of the present and past of other processes, one process causally influences another. This notion is shown to be able to differentiate between true direct causal influences, common inputs, and cascade effects in more two processes. We next describe a procedure to estimate the directed information on neural spike trains using point process generalized linear models, maximum likelihood estimation and information-theoretic model order selection. We demonstrate that on a simulated network of neurons, it (a) correctly identifies all pairwise causal relationships and (b) correctly identifies network causal relationships. This procedure is then used to analyze ensemble spike train recordings in primary motor cortex of an awake monkey while performing target reaching tasks, uncovering causal relationships whose directionality are consistent with predictions made from the wave propagation of simultaneously recorded local field potentials.     

On the spectral formulation of Granger causality

Spectral measures of causality are used to explore the role of different rhythms in the causal connectivity between brain regions. We study several spectral measures related to Granger causality, comprising the bivariate and conditional Geweke measures, the directed transfer function, and the partial directed coherence. We derive the formulation of dependence and causality in the spectral domain from the more general formulation in the information-theory framework. We argue that the transfer entropy, the most general measure derived from the concept of Granger causality, lacks a spectral representation in terms of only the processes associated with the recorded signals. For all the spectral measures we show how they are related to mutual information rates when explicitly considering the parametric autoregressive representation of the processes. In this way we express the conditional Geweke spectral measure in terms of a multiple coherence involving innovation variables inherent to the autoregressive representation. We also link partial directed coherence with Sims' criterion of causality. Given our results, we discuss the causal interpretation of the spectral measures related to Granger causality and stress the necessity to explicitly consider their specific formulation based on modeling the signals as linear Gaussian stationary autoregressive processes.     

Causality and persistence in ecological systems: a nonparametric spectral granger causality approach

Abstract Directionality in coupling, defined as the linkage relating causes to their effects at a later time, can be used to explain the core dynamics of ecological systems by untangling direct and feedback relationships between the different components of the systems. Inferring causality from measured ecological variables sampled through time remains a formidable challenge further made difficult by the action of periodic drivers overlapping the natural dynamics of the system. Periodicity in the drivers can often mask the self-sustained oscillations originating from the autonomous dynamics. While linear and direct causal relationships are commonly addressed in the time domain, using the well-established machinery of Granger causality (G-causality), the presence of periodic forcing requires frequency-based statistics (e.g., the Fourier transform), able to distinguish coupling induced by oscillations in external drivers from genuine endogenous interactions. Recent nonparametric spectral extensions of G-causality to the frequency domain pave the way for the scale-by-scale decomposition of causality, which can improve our ability to link oscillatory behaviors of ecological networks to causal mechanisms. The performance of both spectral G-causality and its conditional extension for multivariate systems is explored in quantifying causal interactions within ecological networks. Through two case studies involving synthetic and actual time series, it is demonstrated that conditional G-causality outperforms standard G-causality in identifying causal links and their concomitant timescales.     

Adaptive Thresholding for Reconstructing Regulatory Networks from Time-Course Gene Expression Data

Discovering regulatory interactions from time-course gene expression data constitutes a canonical problem in functional genomics and systems biology. The framework of graphical Granger causality allows one to estimate such causal relationships from these data. In this study, we propose an adaptively thresholding estimates of Granger causal effects obtained from the lasso penalization method. We establish the asymptotic properties of the proposed technique, and discuss the advantages it offers over competing methods, such as the truncating lasso. Its performance and that of its competitors is assessed on a number of simulated settings and it is applied on a data set that captures the activation of T-cells.     

Granger causality analysis of steady-state electroencephalographic signals during propofol-induced anaesthesia

Changes in conscious level have been associated with changes in dynamical integration and segregation among distributed brain regions. Recent theoretical developments emphasize changes in directed functional (i.e., causal) connectivity as reflected in quantities such as 'integrated information' and 'causal density'. Here we develop and illustrate a rigorous methodology for assessing causal connectivity from electroencephalographic (EEG) signals using Granger causality (GC). Our method addresses the challenges of non-stationarity and bias by dividing data into short segments and applying permutation analysis. We apply the method to EEG data obtained from subjects undergoing propofol-induced anaesthesia, with signals source-localized to the anterior and posterior cingulate cortices. We found significant increases in bidirectional GC in most subjects during loss-of-consciousness, especially in the beta and gamma frequency ranges. Corroborating a previous analysis we also found increases in synchrony in these ranges; importantly, the Granger causality analysis showed higher inter-subject consistency than the synchrony analysis. Finally, we validate our method using simulated data generated from a model for which GC values can be analytically derived. In summary, our findings advance the methodology of Granger causality analysis of EEG data and carry implications for integrated information and causal density theories of consciousness.     

Transfer entropy—a model-free measure of effective connectivity for the neurosciences

Understanding causal relationships, or effective connectivity, between parts of the brain is of utmost importance because a large part of the brain’s activity is thought to be internally generated and, hence, quantifying stimulus response relationships alone does not fully describe brain dynamics. Past efforts to determine effective connectivity mostly relied on model based approaches such as Granger causality or dynamic causal modeling. Transfer entropy (TE) is an alternative measure of effective connectivity based on information theory. TE does not require a model of the interaction and is inherently non-linear. We investigated the applicability of TE as a metric in a test for effective connectivity to electrophysiological data based on simulations and magnetoencephalography (MEG) recordings in a simple motor task. In particular, we demonstrate that TE improved the detectability of effective connectivity for non-linear interactions, and for sensor level MEG signals where linear methods are hampered by signal-cross-talk due to volume conduction.     

Change in cereal production caused by climate change in Malaysia

This study attempts to estimate the effects of climate change variables, such as average temperature, CO₂ emissions and average rainfall, on cereal production in Malaysia from 1969 to 2018. After preliminary tests on time series data, we employed a novel autoregressive distributed lag (ARDL) method known as the dynamic ARDL simulations technique. The results showed that a long-run co-integration relationship exists between cereal production and climatic and non-climatic factors. All climate variables have a negative impact on cereal yield, while energy consumption and cultivated land have a positive effect on cereal yield in the short- and long-run. Granger causality analyses also showed that a unidirectional causality link exists between rainfall and temperature with cereal production and between CO₂ emissions and cereal production. Energy consumption, as a proxy for technology, has a one-way Granger cause with cereal production. The results of the dynamic ARDL simulations suggest that cereal yield was most sensitive to CO₂ emissions, rainfall and temperature. In the long run, a 1% increase in temperature is associated with a 2.87% and 3.52% decrease in general and predicted estimates of cereal production, respectively. The dynamic ARDL simulations methodology provides a better understanding of the variability of cereal production in Malaysia as a result of climate change.     

A graphical approach for evaluating effective connectivity in neural systems

The identification of effective connectivity from time-series data such as electroencephalogram (EEG) or time-resolved function magnetic resonance imaging (fMRI) recordings is an important problem in brain imaging. One commonly used approach to inference effective connectivity is based on vector autoregressive models and the concept of Granger causality. However, this probabilistic concept of causality can lead to spurious causalities in the presence of latent variables. Recently, graphical models have been used to discuss problems of causal inference for multivariate data. In this paper, we extend these concepts to the case of time-series and present a graphical approach for discussing Granger-causal relationships among multiple time-series. In particular, we propose a new graphical representation that allows the characterization of spurious causality and, thus, can be used to investigate spurious causality. The method is demonstrated with concurrent EEG and fMRI recordings which are used to investigate the interrelations between the alpha rhythm in the EEG and blood oxygenation level dependent (BOLD) responses in the fMRI. The results confirm previous findings on the location of the source of the EEG alpha rhythm.     

Causality between pillars of sustainable development: Global stylized facts or regional phenomena?

Since the Brundtland report, most governments have committed themselves to sustainable development. For this purpose, different global and national organizations and institutions have tried to find the relationships, and especially causalities, between sustainability pillars, which are interesting for them from a policy point of view. With respect to their findings, some questions need to be answered before appropriate policies can be formulated. Are causalities between sustainability pillars global stylized facts or regional phenomena? Can countries with different characteristics follow the same rules, or are causalities between the pillars sensitive to the regional and intrinsic features of countries? Using principal component analysis for the construction of sustainability indicators and the Granger causality model (GMM approach) for testing the causalities between sustainability pillars in different samples, this study finds that causal patterns among the pillars of sustainability are completely sensitive to the characteristics of the countries that are grouped. Therefore, it is recommended that researchers concentrate more on homogeneous case studies and avoid generalizations of causal relationships between sets of heterogeneous countries.