Information theoretic interpretation of frequency domain connectivity measures

In order to provide adequate multivariate measures of information flow between neural structures, modified expressions of partial directed coherence (PDC) and directed transfer function (DTF), two popular multivariate connectivity measures employed in neuroscience, are introduced and their formal relationship to mutual information rates are proved.     

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     

Diastolic time fraction as a determinant of subendocardial perfusion

Diastolic time fraction (DTF) has been recognized as an important determinant for subendocardial perfusion, but microsphere studies in which DTF was the independent variable are practically absent. In 21 anesthetized goats, the left coronary main stem was artificially perfused at controlled pressure. DTF was varied by pacing the heart, vagus stimulation, or administration of dobutamine. Regional coronary flow was measured with fluorescent microspheres under full adenosine dilation. Perfusion pressure (P(c)) was defined as mean coronary arterial pressure minus minimal left ventricular pressure. Regional flow conductances (flow/P(c)) were as follows: for the subendocardium, C(endo) = -0.103 + 0.197 DTF + 0.00074 P(c) (P < 0.001); for the midmyocardium, conductance = -0.048 + 0.126 DTF + 0.00049 P(c) (P < 0.001); and for the subepicardium, C(epi) was not significant. C(endo)-DTF relations demonstrated a finite value for DTF at which flow is zero, implying that, at physiological pressures, systolic subendocardial flow limitation extends into diastole. The DTF corresponding to an equal conductance in subendocardium and subepicardium (DTF1) was inversely related to P(c): DTF1 = 0.78 - 0.003 P(c) (P < 0.01). When heart rate and P(c) were held constant and dobutamine was administered (5 goats), contractility doubled and DTF increased by 39%, resulting in an increase of C(endo) of 40%. It is concluded that 1) DTF is a determinant of subendocardial perfusion, 2) systolic compression exerts a flow-limiting effect into diastole, and 3) corresponding to clinical findings on inducible ischemia we predict that, under hyperemic conditions, C(endo) < C(epi) if P(c) is lower than approximately 75% of a normal aortic pressure and heart rate >80 beats/min.     

Information flow in interaction networks II: channels, path lengths, and potentials

In our previous publication, a framework for information flow in interaction networks based on random walks with damping was formulated with two fundamental modes: emitting and absorbing. While many other network analysis methods based on random walks or equivalent notions have been developed before and after our earlier work, one can show that they can all be mapped to one of the two modes. In addition to these two fundamental modes, a major strength of our earlier formalism was its accommodation of context-specific directed information flow that yielded plausible and meaningful biological interpretation of protein functions and pathways. However, the directed flow from origins to destinations was induced via a potential function that was heuristic. Here, with a theoretically sound approach called the channel mode, we extend our earlier work for directed information flow. This is achieved by constructing a potential function facilitating a purely probabilistic interpretation of the channel mode. For each network node, the channel mode combines the solutions of emitting and absorbing modes in the same context, producing what we call a channel tensor. The entries of the channel tensor at each node can be interpreted as the amount of flow passing through that node from an origin to a destination. Similarly to our earlier model, the channel mode encompasses damping as a free parameter that controls the locality of information flow. Through examples involving the yeast pheromone response pathway, we illustrate the versatility and stability of our new framework.     

Increased diastolic time fraction as beneficial adjunct of alpha1-adrenergic receptor blockade after percutaneous coronary intervention

The effect of alpha1-receptor blockade with urapidil on coronary blood flow and left ventricular function has been attributed to relief of diffuse coronary vasoconstriction following percutaneous coronary intervention (PCI). We hypothesized that an increase in diastolic time fraction (DTF) contributes to the beneficial action of urapidil. In eleven patients with a 63% (SD 13) diameter stenosis, ECG, aortic pressure (Pa) and distal intracoronary pressure (Pd), and blood flow velocity were recorded at baseline and throughout adenosine-induced hyperemia. Measurements were obtained before and after PCI and after subsequent alpha1-receptor blockade with urapidil (10 mg ic). DTF was determined from the ECG and the Pa waveform. Functional parameters such as coronary flow velocity reserve, fractional flow reserve, and an index of hyperemic microvascular resistance (HMR) were assessed. Urapidil administration after PCI induced an upward shift in the DTF-heart rate relationship, resulting in a 3.1% (SD 2.7) increase in hyperemic DTF at a constant heart rate (P < 0.005) due to a shorter duration of systole. Hyperemic Pa and Pd decreased, respectively, by 6.1% (SD 6.6; P < 0.05) and 5.7% (SD 5.8; P < 0.01) after alpha1-blockade. Although epicardially measured functional parameters were on average not altered by alpha1-blockade due to concurrent changes in pressure and heart rate, HMR decreased by urapidil in those patients where coronary pressure remained constant. In conclusion, alpha1-receptor blockade after PCI produced a modest but significant prolongation of DTF at a given heart rate, thereby providing an adjunctive beneficial mechanism for improving subendocardial perfusion, which critically depends on DTF.     

Face to phase: pitfalls in time delay estimation from coherency phase

Coherency phase is often interpreted as a time delay reflecting a transmission delay between spatially separated neural populations. However, time delays estimated from corticomuscular coherency are conflicting and often shorter than expected physiologically. Recent work suggests that corticomuscular coherence is influenced by afferent sensory feedback and bidirectional interactions. We investigated how bidirectional interaction affects time delay estimated from coherency, using a feedback model of the corticomuscular system. We also evaluated the effect of bidirectional interaction on two popular directed connectivity measures: directed transfer function (DTF) and partial directed coherence (PDC). The model is able to reproduce the range of time delays found experimentally from coherency phase by varying the strengths of the efferent and afferent pathways and the recording of sensory feedback in the cortical signal. Both coherency phase and DTF phase were affected by sensory feedback, resulting in an underestimation of the transmission delay. Coherency phase was altered by the recording of sensory feedback in the cortical signals and both measures were affected by the presence of a closed loop feedback system. Only PDC phase led to the correct estimation of efferent transmission delay in all simulated model configurations. Coherency and DTF phase should not be used to estimate transmission delays in neural networks as the estimated time delays are meaningless in the presence of sensory feedback and closed feedback loops.     

Quantitative measures of cortical functional connectivity: A state-of-the-art brief survey

The modern concepts of quantitative measurement of the strength of functional and effective cortical connectivity in neurocognitive networks (large-scale distributed brain systems of interacting neuronal populations that are believed to underlie cognitive processing) are reviewed. The two main classes of the methods of connectivity assessment (linear and nonlinear) are discussed. In the class of linear methods, in addition to coherence routinely used to measure the strength of functional links, the vector autoregressive modeling of multichannel EEG is discussed in detail. The latter technique allows the estimation of both functional and effective connectivity, i.e., the measures such as directed transfer function (DTF) and direct partial coherence (PDC) extensively used in cognitive neuroscience. The impact of volume conduction on different estimates of connectivity is considered and the imaginary part of complex-valued coherence as a way to reduce the artificial influence of volume conduction is discussed. Independent component analysis (ICA) and transfer entropy as a method of estimating direct influence are reviewed in the class of nonlinear methods.     

A Bayesian Alternative to Mutual Information for the Hierarchical Clustering of Dependent Random Variables

The use of mutual information as a similarity measure in agglomerative hierarchical clustering (AHC) raises an important issue: some correction needs to be applied for the dimensionality of variables. In this work, we formulate the decision of merging dependent multivariate normal variables in an AHC procedure as a Bayesian model comparison. We found that the Bayesian formulation naturally shrinks the empirical covariance matrix towards a matrix set a priori (e.g., the identity), provides an automated stopping rule, and corrects for dimensionality using a term that scales up the measure as a function of the dimensionality of the variables. Also, the resulting log Bayes factor is asymptotically proportional to the plug-in estimate of mutual information, with an additive correction for dimensionality in agreement with the Bayesian information criterion. We investigated the behavior of these Bayesian alternatives (in exact and asymptotic forms) to mutual information on simulated and real data. An encouraging result was first derived on simulations: the hierarchical clustering based on the log Bayes factor outperformed off-the-shelf clustering techniques as well as raw and normalized mutual information in terms of classification accuracy. On a toy example, we found that the Bayesian approaches led to results that were similar to those of mutual information clustering techniques, with the advantage of an automated thresholding. On real functional magnetic resonance imaging (fMRI) datasets measuring brain activity, it identified clusters consistent with the established outcome of standard procedures. On this application, normalized mutual information had a highly atypical behavior, in the sense that it systematically favored very large clusters. These initial experiments suggest that the proposed Bayesian alternatives to mutual information are a useful new tool for hierarchical clustering.     

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.     

Estimating the cortex and autonomic nervous activity during a mental arithmetic task

The cerebral cortex has massive connections with autonomic nervous system and then arouses cardiovascular events, but the coupling mechanism between brain and heart is not clear. In this study the heart rate variability (HRV) and directed transfer function (DTF) methods are used to investigate the cortico-cortical functional coupling and direction of information flow between brain and heart during a mental arithmetic (MA) task. Electroencephalogram (EEG) and ECG were used for measuring neural/cardiac activity. Forty-three healthy male subjects were voluntarily participated in the study. Our results showed compared with control, LF/HF and LFn significantly increased while HF, HFn and total power significantly decreased (P < 0.05) during MA task. HR (79 ± 1.7 beats/min) was also significantly higher compared with the control (71 ± 1.4 beats/min). Moreover, MA task trigger the neurons of pre-central and central areas and then information transmit from front to back, and finished information integration at parietal and occipital locations. Our findings suggested that MA task caused an increase of the coupling of brain regions and quickened heart rate by virtue of increasing sympathetic activity and decreasing parasympathetic activity. The regulation from post-central areas to heart as well as feedback regulation from heart to central areas exists in the MA task.     

A Multivariate Granger Causality Concept towards Full Brain Functional Connectivity

Detecting changes of spatially high-resolution functional connectivity patterns in the brain is crucial for improving the fundamental understanding of brain function in both health and disease, yet still poses one of the biggest challenges in computational neuroscience. Currently, classical multivariate Granger Causality analyses of directed interactions between single process components in coupled systems are commonly restricted to spatially low- dimensional data, which requires a pre-selection or aggregation of time series as a preprocessing step. In this paper we propose a new fully multivariate Granger Causality approach with embedded dimension reduction that makes it possible to obtain a representation of functional connectivity for spatially high-dimensional data. The resulting functional connectivity networks may consist of several thousand vertices and thus contain more detailed information compared to connectivity networks obtained from approaches based on particular regions of interest. Our large scale Granger Causality approach is applied to synthetic and resting state fMRI data with a focus on how well network community structure, which represents a functional segmentation of the network, is preserved. It is demonstrated that a number of different community detection algorithms, which utilize a variety of algorithmic strategies and exploit topological features differently, reveal meaningful information on the underlying network module structure.     

Growth period QTL-allele constitution of global soybeans and its differential evolution changes in geographic adaptation versus maturity group extension

Soybean (Glycine max (L.) Merr.) has been disseminated globally as a photoperiod/temperature-sensitive crop with extremely diverse days to flowering (DTF) and days to maturity (DTM) values. A population with 371 global varieties covering 13 geographic regions and 13 maturity groups (MGs) was analyzed for its DTF and DTM QTL-allele constitution using restricted two-stage multi-locus genome-wide association study (RTM-GWAS). Genotypes with 20 701 genome-wide SNPLDBs (single-nucleotide polymorphism linkage disequilibrium blocks) containing 55 404 haplotypes were observed, and 52 DTF QTLs and 59 DTM QTLs (including 29 and 21 new ones) with 241 and 246 alleles (two to 13 per locus) were detected, explaining 84.8% and 74.4% of the phenotypic variance, respectively. The QTL-allele matrix characterized with all QTL-allele information of each variety in the global population was established and subsequently separated into geographic and MG set submatrices. Direct comparisons among them revealed that the genetic adaptation from the origin to geographic subpopulations was characterized by new allele/new locus emergence (mutation) but little allele exclusion (selection), while that from the primary MG set to emerged early and late MG sets was characterized by allele exclusion without allele emergence. The evolutionary changes involved mainly 72 DTF and 71 DTM alleles on 28 respective loci, 10–12 loci each with three to six alleles being most active. Further recombination potential for faster maturation (12–21 days) or slower maturation (14–56 days) supported allele convergence (recombination) as a constant genetic factor in addition to migration (inheritance). From the QTLs, 44 DTF and 36 DTM candidate genes were annotated and grouped respectively into nine biological processes, indicating multi-functional DTF/DTM genes are involved in a complex gene network. In summary, we identified QTL-alleles relatively thoroughly using RTM-GWAS for direct matrix comparisons and subsequent analysis.     

Total ancestry measure: quantifying the similarity in tree-like classification, with genomic applications

MOTIVATION: Many classifications of protein function such as Gene Ontology (GO) are organized in directed acyclic graph (DAG) structures. In these classifications, the proteins are terminal leaf nodes; the categories 'above' them are functional annotations at various levels of specialization and the computation of a numerical measure of relatedness between two arbitrary proteins is an important proteomics problem. Moreover, analogous problems are important in other contexts in large-scale information organization--e.g. the Wikipedia online encyclopedia and the Yahoo and DMOZ web page classification schemes. RESULTS: Here we develop a simple probabilistic approach for computing this relatedness quantity, which we call the total ancestry method. Our measure is based on counting the number of leaf nodes that share exactly the same set of 'higher up' category nodes in comparison to the total number of classified pairs (i.e. the chance for the same total ancestry). We show such a measure is associated with a power-law distribution, allowing for the quick assessment of the statistical significance of shared functional annotations. We formally compare it with other quantitative functional similarity measures (such as, shortest path within a DAG, lowest common ancestor shared and Azuaje's information-theoretic similarity) and provide concrete metrics to assess differences. Finally, we provide a practical implementation for our total ancestry measure for GO and the MIPS functional catalog and give two applications of it in specific functional genomics contexts. AVAILABILITY: The implementations and results are available through our supplementary website at: http://gersteinlab.org/proj/funcsim. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.     

Synergy,redundancy, and multivariate information measures: an experimentalist’s perspective

Information theory has long been used to quantify interactions between two variables. With the rise of complex systems research, multivariate information measures have been increasingly used to investigate interactions between groups of three or more variables, often with an emphasis on so called synergistic and redundant interactions. While bivariate information measures are commonly agreed upon, the multivariate information measures in use today have been developed by many different groups, and differ in subtle, yet significant ways. Here, we will review these multivariate information measures with special emphasis paid to their relationship to synergy and redundancy, as well as examine the differences between these measures by applying them to several simple model systems. In addition to these systems, we will illustrate the usefulness of the information measures by analyzing neural spiking data from a dissociated culture through early stages of its development. Our aim is that this work will aid other researchers as they seek the best multivariate information measure for their specific research goals and system. Finally, we have made software available online which allows the user to calculate all of the information measures discussedwithin this paper.     

Detection of directed information flow in biosignals.

Several analysis techniques have been developed for time series to detect interactions in multidimensional dynamic systems. When analyzing biosignals generated by unknown dynamic systems, awareness of the different concepts upon which these analysis techniques are based, as well as the particular aspects the methods focus on, is a basic requirement for drawing reliable conclusions. For this purpose, we compare four different techniques for linear time series analysis. In general, these techniques detect the presence of interactions, as well as the directions of information flow, in a multidimensional system. We review the different conceptual properties of partial coherence, a Granger causality index, directed transfer function, and partial directed coherence. The performance of these tools is demonstrated by application to linear dynamic systems.     

Optimal Information Transfer in the Cortex through Synchronization

In recent experimental work it has been shown that neuronal interactions are modulated by neuronal synchronization and that this modulation depends on phase shifts in neuronal oscillations. This result suggests that connections in a network can be shaped through synchronization. Here, we test and expand this hypothesis using a model network. We use transfer entropy, an information theoretical measure, to quantify the exchanged information. We show that transferred information depends on the phase relation of the signal, that the amount of exchanged information increases as a function of oscillations in the signal and that the speed of the information transfer increases as a function of synchronization. This implies that synchronization makes information transport more efficient. In summary, our results reinforce the hypothesis that synchronization modulates neuronal interactions and provide further evidence that gamma band synchronization has behavioral relevance.     

Multivariate information-theoretic measures reveal directed information structure and task relevant changes in fMRI connectivity

The human brain undertakes highly sophisticated information processing facilitated by the interaction between its sub-regions. We present a novel method for interregional connectivity analysis, using multivariate extensions to the mutual information and transfer entropy. The method allows us to identify the underlying directed information structure between brain regions, and how that structure changes according to behavioral conditions. This method is distinguished in using asymmetric, multivariate, information-theoretical analysis, which captures not only directional and non-linear relationships, but also collective interactions. Importantly, the method is able to estimate multivariate information measures with only relatively little data. We demonstrate the method to analyze functional magnetic resonance imaging time series to establish the directed information structure between brain regions involved in a visuo-motor tracking task. Importantly, this results in a tiered structure, with known movement planning regions driving visual and motor control regions. Also, we examine the changes in this structure as the difficulty of the tracking task is increased. We find that task difficulty modulates the coupling strength between regions of a cortical network involved in movement planning and between motor cortex and the cerebellum which is involved in the fine-tuning of motor control. It is likely these methods will find utility in identifying interregional structure (and experimentally induced changes in this structure) in other cognitive tasks and data modalities.     

Development of EST-SSR markers through de novo RNA sequencing and application for biomass productivity in kenaf (<Emphasis Type="Italic">Hibiscus cannabinus</Emphasis> L.)

Kenaf is a multipurpose crop, but a lack of genetic information hinders genetic and molecular research. In this study, we aimed to develop EST-SSR markers from mutant and wild-type cultivars, and to assess the genetic diversity of the kenaf resources. A total of 33 Gb of sequence data comprising 130,480 unigenes was assembled by de novo RNA-sequencing of six kenaf cultivars, and 5619 SSRs were identified. Tri-nucleotide motifs occurred most frequently (82.67%) followed by di-, tetra-, and penta-motifs. In total, 515 polymorphic EST-SSRs were derived by pairwise comparisons of the cultivars based on in silico analyses. Of these, 70 markers were successfully validated among six cultivars. We used the six cultivars, together with 39 kenaf accessions from worldwide to assess genetic diversity and to characterize the EST-SSRs. The number of alleles per locus ranged from 2 to 8. PIC and genetic diversity values ranged from 0.08 to 0.79 and 0.08–0.82, respectively. The phylogenetic and population structure showed that the 45 accessions could be clearly divided into three groups based on different days to flowering (DTF). Genetic differentiation among the DTF groups showed a proportionally high level of variance. Association analysis between the DTF and the markers revealed three significant associations. Furthermore, using a multiplex PCR with three markers, DTF could be perfectly discriminated. These markers will be useful in marker-assisted selection after further validation with segregating populations of kenaf.     

Causal pattern recovery from neural spike train data using the Snap Shot Score

We present a new approach to learning directed information flow networks from multi-channel spike train data. A novel scoring function, the Snap Shot Score, is used to assess potential networks with respect to their quality of causal explanation for the data. Additionally, we suggest a generic concept of plausibility in order to assess network learning techniques under partial observability conditions. Examples demonstrate the assessment of networks with the Snap Shot Score, and neural network simulations show its performance in complex situations with partial observability. We discuss the application of the new score to real data and indicate how it can be modified to suit other neural data types.