Akaike causality in state space

We present a new approach of explaining instantaneous causality in multivariate fMRI time series by a state space model. A given single time series can be divided into two noise-driven processes, a common process shared among multivariate time series and a specific process refining the common process. By assuming that noises are independent, a causality map is drawn using Akaike noise contribution ratio theory. The method is illustrated by an application to fMRI data recorded under visual stimulation.     

Analyzing multiple spike trains with nonparametric granger causality

Simultaneous recordings of spike trains from multiple single neurons are becoming commonplace. Understanding the interaction patterns among these spike trains remains a key research area. A question of interest is the evaluation of information flow between neurons through the analysis of whether one spike train exerts causal influence on another. For continuous-valued time series data, Granger causality has proven an effective method for this purpose. However, the basis for Granger causality estimation is autoregressive data modeling, which is not directly applicable to spike trains. Various filtering options distort the properties of spike trains as point processes. Here we propose a new nonparametric approach to estimate Granger causality directly from the Fourier transforms of spike train data. We validate the method on synthetic spike trains generated by model networks of neurons with known connectivity patterns and then apply it to neurons simultaneously recorded from the thalamus and the primary somatosensory cortex of a squirrel monkey undergoing tactile stimulation.     

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.     

Perceptual learning with perceptions

In this work we present an approach to understand neuronal mechanisms underlying perceptual learning. Experimental results achieved with stimulus patterns of coherently moving dots are considered to build a simple neuronal model. The design of the model is made transparent and underlying behavioral assumptions made explicit. The key aspect of the suggested neuronal model is the learning algorithm used: We evaluated an implementation of Hebbian learning and are thus able to provide a straight-forward model capable to explain the neuronal dynamics underlying perceptual learning. Moreover, the simulation results suggest a very simple explanation for the aspect of “sub-threshold” learning (Watanabe et al. in Nature 413:844–884, 2001) as well as the relearning of motion discrimination after damage to primary visual cortex as recently reported (Huxlin et al. in J Neurosci 29:3981–3991, 2009) and at least indicate that perceptual learning might only occur when accompanied by conscious percepts.     

收敛交叉映射方法及其在生态学中的应用

收敛交叉映射(CCM)是一种分析非线性系统中时间序列变量间因果关系的方法。其不同于传统的线性系统分析方法,是通过对变量进行状态空间重构来获取变量的历史信息,随着时间序列不断增长,当其估计性能呈现收敛的性质时,可以判断因果关系的存在。本文介绍了CCM的发展史及其较传统的格兰杰因果检验的优点,详细阐明了CCM的原理、算法过程和实现途径。CCM作为一种针对变量间具有弱到中等强度耦合关系的系统分析方法,可以用来有效地解决非线性生态系统多变量间复杂的因果关系问题。将该方法应用于具有空间信息的多点位时间序列变量间因果分析时,应充分考虑点位间的空间自相关性,与可以去除变量及序列间空间相关性的方法相结合,从而确保CCM对变量因果关系的分析更加准确,结果也更具有信服力。     

Measuring frequency domain granger causality for multiple blocks of interacting time series

In the past years, several frequency-domain causality measures based on vector autoregressive time series modeling have been suggested to assess directional connectivity in neural systems. The most followed approaches are based on representing the considered set of multiple time series as a realization of two or three vector-valued processes, yielding the so-called Geweke linear feedback measures, or as a realization of multiple scalar-valued processes, yielding popular measures like the directed coherence (DC) and the partial DC (PDC). In the present study, these two approaches are unified and generalized by proposing novel frequency-domain causality measures which extend the existing measures to the analysis of multiple blocks of time series. Specifically, the block DC (bDC) and block PDC (bPDC) extend DC and PDC to vector-valued processes, while their logarithmic counterparts, denoted as multivariate total feedback $f^\mathrm{m}$ and direct feedback $g^\mathrm{m}$ , represent into a full multivariate framework the Geweke’s measures. Theoretical analysis of the proposed measures shows that they: (i) possess desirable properties of causality measures; (ii) are able to reflect either direct causality (bPDC, $g^\mathrm{m})$ or total (direct + indirect) causality (bDC, $f^\mathrm{m})$ between time series blocks; (iii) reduce to the DC and PDC measures for scalar-valued processes, and to the Geweke’s measures for pairs of processes; (iv) are able to capture internal dependencies between the scalar constituents of the analyzed vector processes. Numerical analysis showed that the proposed measures can be efficiently estimated from short time series, allow to represent in an objective, compact way the information derived from the causal analysis of several pairs of time series, and may detect frequency domain causality more accurately than existing measures. The proposed measures find their natural application in the evaluation of directional interactions in neurophysiological settings where several brain activity signals are simultaneously recorded from multiple regions of interest.     

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.     

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.     

Leader neurons in leaky integrate and fire neural network simulations

In this paper, we highlight the topological properties of leader neurons whose existence is an experimental fact. Several experimental studies show the existence of leader neurons in population bursts of activity in 2D living neural networks (Eytan and Marom, J Neurosci 26(33):8465–8476, 2006; Eckmann et al., New J Phys 10(015011), 2008). A leader neuron is defined as a neuron which fires at the beginning of a burst (respectively network spike) more often than we expect by chance considering its mean firing rate. This means that leader neurons have some burst triggering power beyond a chance-level statistical effect. In this study, we characterize these leader neuron properties. This naturally leads us to simulate neural 2D networks. To build our simulations, we choose the leaky integrate and fire (lIF) neuron model (Gerstner and Kistler 2002; Cessac, J Math Biol 56(3):311–345, 2008), which allows fast simulations (Izhikevich, IEEE Trans Neural Netw 15(5):1063–1070, 2004; Gerstner and Naud, Science 326:379–380, 2009). The dynamics of our lIF model has got stable leader neurons in the burst population that we simulate. These leader neurons are excitatory neurons and have a low membrane potential firing threshold. Except for these two first properties, the conditions required for a neuron to be a leader neuron are difficult to identify and seem to depend on several parameters involved in the simulations themselves. However, a detailed linear analysis shows a trend of the properties required for a neuron to be a leader neuron. Our main finding is: A leader neuron sends signals to many excitatory neurons as well as to few inhibitory neurons and a leader neuron receives only signals from few other excitatory neurons. Our linear analysis exhibits five essential properties of leader neurons each with different relative importance. This means that considering a given neural network with a fixed mean number of connections per neuron, our analysis gives us a way of predicting which neuron is a good leader neuron and which is not. Our prediction formula correctly assesses leadership for at least ninety percent of neurons.     

A novel symbolization scheme for multichannel recordings with emphasis on phase information and its application to differentiate EEG activity from different mental tasks

Symbolic dynamics is a powerful tool for studying complex dynamical systems. So far many techniques of this kind have been proposed as a means to analyze brain dynamics, but most of them are restricted to single-sensor measurements. Analyzing the dynamics in a channel-wise fashion is an invalid approach for multisite encephalographic recordings, since it ignores any pattern of coordinated activity that might emerge from the coherent activation of distinct brain areas. We suggest, here, the use of neural-gas algorithm (Martinez et al. in IEEE Trans Neural Netw 4:558–569, 1993) for encoding brain activity spatiotemporal dynamics in the form of a symbolic timeseries. A codebook of k prototypes, best representing the instantaneous multichannel data, is first designed. Each pattern of activity is then assigned to the most similar code vector. The symbolic timeseries derived in this way is mapped to a network, the topology of which encapsulates the most important phase transitions of the underlying dynamical system. Finally, global efficiency is used to characterize the obtained topology. We demonstrate the approach by applying it to EEG-data recorded from subjects while performing mental calculations. By working in a contrastive-fashion, and focusing in the phase aspects of the signals, we show that the underlying dynamics differ significantly in their symbolic representations.     

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.     

A PDE Model for Imatinib-Treated Chronic Myelogenous Leukemia

We derive a model for describing the dynamics of imatinib-treated chronic myelogenous leukemia (CML). This model is a continuous extension of the agent-based CML model of Roeder et al. (Nat. Med. 12(10), 1181–1184, 2006) and of its recent formulation as a system of difference equations (Kim et al. in Bull. Math. Biol. 70(3), 728–744, 2008). The new model is formulated as a system of partial differential equations that describe various stages of differentiation and maturation of normal hematopoietic cells and of leukemic cells. An imatinib treatment is also incorporated into the model. The simulations of the new PDE model are shown to qualitatively agree with the results that were obtained with the discrete-time (difference equation and agent-based) models. At the same time, for a quantitative agreement, it is necessary to adjust the values of certain parameters, such as the rates of imatinib-induced inhibition and degradation.     

Forensic Ecotoxicology: Establishing Causality between Contaminants and Biological Effects in Field Studies

Twelve papers in this series were derived from two conference sessions focusing on causality in field studies. Eight of these papers involve case studies examining biological effects of chemical contaminants in field situations. Using a weight-of-evidence approach, these case studies were evaluated against seven proposed criteria for establishing causality. The seven criteria were: strength of association; consistency of association; specificity of association; time order; biological gradient; experimental evidence; and biological plausibility. One of these seven criteria, ‘specificity of association’ was found to be of little utility for establishing causality in these field studies. The case studies are presented in approximate order of increasing levels of biological organization (i.e., going from endpoints at the suborganismal level to endpoints at the population or community level). In case studies examining higher levels of biological organization, it appears that the ‘biological gradient’ criterion was also not useful in establishing causality. These results, together with suggestions from other papers in the series, are used to recommend a set of modified criteria for establishing causality in field studies of the biological effects of chemical contaminants.     

A Computational Analysis of Localized Ca<Superscript>2+</Superscript>-Dynamics Generated by Heterogeneous Release Sites

We investigate the role of heterogeneous expression of IP₃R and RyR in generating diverse elementary Ca²⁺ signals. It has been shown empirically (Wojcikiewicz and Luo in Mol. Pharmacol. 53(4):656–662, 1998; Newton et al. in J. Biol. Chem. 269(46):28613–28619, 1994; Smedt et al. in Biochem. J. 322(Pt. 2):575–583, 1997) that tissues express various proportions of IP₃ and RyR isoforms and this expression is dynamically regulated (Parrington et al. in Dev. Biol. 203(2):451–461, 1998; Fissore et al. in Biol. Reprod. 60(1):49–57, 1999; Tovey et al. in J. Cell Sci. 114(Pt. 22):3979–3989, 2001). Although many previous theoretical studies have investigated the dynamics of localized calcium release sites (Swillens et al. in Proc. Natl. Acad. Sci. U.S.A. 96(24):13750–13755, 1999; Shuai and Jung in Proc. Natl. Acad. Sci. U.S.A. 100(2):506–510, 2003a; Shuai and Jung in Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 67(3 Pt. 1):031905, 2003b; Thul and Falcke in Biophys. J. 86(5):2660–2673, 2004; DeRemigio and Smith in Cell Calcium 38(2):73–86, 2005; Nguyen et al. in Bull. Math. Biol. 67(3):393–432, 2005), so far all such studies focused on release sites consisting of identical channel types. We have extended an existing mathematical model (Nguyen et al. in Bull. Math. Biol. 67(3):393–432, 2005) to release sites with two (or more) receptor types, each with its distinct channel kinetics. Mathematically, the release site is represented by a transition probability matrix for a collection of nonidentical stochastically gating channels coupled through a shared Ca²⁺ domain. We demonstrate that under certain conditions a previously defined mean-field approximation of the coupling strength does not accurately reproduce the release site dynamics. We develop a novel approximation and establish that its performance in these instances is superior. We use this mathematical framework to study the effect of heterogeneity in the Ca²⁺-regulation of two colocalized channel types on the release site dynamics. We consider release sites consisting of channels with both Ca²⁺-activation and inactivation (“four-state channels”) and channels with Ca²⁺-activation only (“two-state channels”) and show that for the appropriate parameter values, synchronous channel openings within a release site with any proportion of two-state to four-state channels are possible, however, the larger the proportion of two-state channels, the more sensitive the dynamics are to the exact spatial positioning of the channels and the distance between channels. Specifically, the clustering of even a small number of two-state channels interferes with puff/spark termination and increases puff durations or leads to a tonic response.     

Using genetic variation for establishing causality of cardiovascular risk factors: overcoming confounding and reverse causality

Cardiovascular disease (CVD) remains the leading cause of death in developed countries, despite the decline of CVD mortality over the last two decades. From observational, predictive research, efforts have been made to find causal risk factors for CVD. However, in recent years, some of these findings have been shown to be mistaken. Possible explanations for the discrepant findings are confounding and reverse causation. Genetic epidemiology has tried to address these problems through the use of Mendelian randomisation. In this paper, we discuss the promise and limitations of using genetic variation for establishing causality of cardiovascular risk factors.     

Detecting causality from short time-series data based on prediction of topologically equivalent attractors

Background

Detecting causality for short time-series data such as gene regulation data is quite important but it is usually very difficult. This can be used in many fields especially in biological systems. Recently, several powerful methods have been set up to solve this problem. However, it usually needs very long time-series data or much more samples for the existing methods to detect causality among the given or observed data. In our real applications, such as for biological systems, the obtained data or samples are short or small. Since the data or samples are highly depended on experiment or limited resource.

Results

In order to overcome these limitations, here we propose a new method called topologically equivalent position method which can detect causality for very short time-series data or small samples. This method is mainly based on attractor embedding theory in nonlinear dynamical systems. By comparing with inner composition alignment, we use theoretical models and real gene expression data to show the effectiveness of our method.

Conclusions

As a result, it shows our method can be effectively used in biological systems. We hope our method can be useful in many other fields in near future such as complex networks, ecological systems and so on.

     

Cointegration and nonlinear causality among ethanol‐related prices: evidence from Brazil

The objective of this study was to investigate the causal relationships among crude oil, ethanol and sugar prices in the context of Brazil. In doing so, we consider the application of ARDL bound tests to examine whether these variables comove in the long run. Besides, we employ a recently developed nonlinear symmetric and asymmetric test for noncausality which assists us to explore the short‐run ‘lead–lag’ associations among the price indexes under review. The results of the ARDL bound test indicate that cointegration exists only when the ethanol price is used as the dependent variable. This finding suggests that oil and sugar prices lead the Brazilian ethanol prices in the long run. Moreover, the results of nonlinear causality test also confirm the existence of a short‐term unidirectional causality running from sugar to ethanol market. We also document that the impact of sugar prices on ethanol prices appears to be positive indicating that rising sugar prices will cause a growth in the ethanol prices. Our findings further demonstrate that sugar prices are not affected by the fluctuations in ethanol price. The results carry important implications for policymakers.     

Extended causal modeling to assess Partial Directed Coherence in multiple time series with significant instantaneous interactions

The Partial Directed Coherence (PDC) and its generalized formulation (gPDC) are popular tools for investigating, in the frequency domain, the concept of Granger causality among multivariate (MV) time series. PDC and gPDC are formalized in terms of the coefficients of an MV autoregressive (MVAR) model which describes only the lagged effects among the time series and forsakes instantaneous effects. However, instantaneous effects are known to affect linear parametric modeling, and are likely to occur in experimental time series. In this study, we investigate the impact on the assessment of frequency domain causality of excluding instantaneous effects from the model underlying PDC evaluation. Moreover, we propose the utilization of an extended MVAR model including both instantaneous and lagged effects. This model is used to assess PDC either in accordance with the definition of Granger causality when considering only lagged effects (iPDC), or with an extended form of causality, when we consider both instantaneous and lagged effects (ePDC). The approach is first evaluated on three theoretical examples of MVAR processes, which show that the presence of instantaneous correlations may produce misleading profiles of PDC and gPDC, while ePDC and iPDC derived from the extended model provide here a correct interpretation of extended and lagged causality. It is then applied to representative examples of cardiorespiratory and EEG MV time series. They suggest that ePDC and iPDC are better interpretable than PDC and gPDC in terms of the known cardiovascular and neural physiologies.     

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.     

Estimation of Cell Proliferation Dynamics Using CFSE Data

Advances in fluorescent labeling of cells as measured by flow cytometry have allowed for quantitative studies of proliferating populations of cells. The investigations (Luzyanina et al. in J. Math. Biol. 54:57–89, 2007; J. Math. Biol., 2009; Theor. Biol. Med. Model. 4:1–26, 2007) contain a mathematical model with fluorescence intensity as a structure variable to describe the evolution in time of proliferating cells labeled by carboxyfluorescein succinimidyl ester (CFSE). Here, this model and several extensions/modifications are discussed. Suggestions for improvements are presented and analyzed with respect to statistical significance for better agreement between model solutions and experimental data. These investigations suggest that the new decay/label loss and time dependent effective proliferation and death rates do indeed provide improved fits of the model to data. Statistical models for the observed variability/noise in the data are discussed with implications for uncertainty quantification. The resulting new cell dynamics model should prove useful in proliferation assay tracking and modeling, with numerous applications in the biomedical sciences.