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This technical paper offers a critical re-evaluation of (spectral) Granger causality measures in the analysis of biological timeseries. Using realistic (neural mass) models of coupled neuronal dynamics, we evaluate the robustness of parametric and nonparametric Granger causality. Starting from a broad class of generative (state-space) models of neuronal dynamics, we show how their Volterra kernels prescribe the second-order statistics of their response to random fluctuations; characterised in terms of cross-spectral density, cross-covariance, autoregressive coefficients and directed transfer functions. These quantities in turn specify Granger causality — providing a direct (analytic) link between the parameters of a generative model and the expected Granger causality. We use this link to show that Granger causality measures based upon autoregressive models can become unreliable when the underlying dynamics is dominated by slow (unstable) modes — as quantified by the principal Lyapunov exponent. However, nonparametric measures based on causal spectral factors are robust to dynamical instability. We then demonstrate how both parametric and nonparametric spectral causality measures can become unreliable in the presence of measurement noise. Finally, we show that this problem can be finessed by deriving spectral causality measures from Volterra kernels, estimated using dynamic causal modelling. [ABSTRACT FROM AUTHOR]

This paper examines intrinsic brain networks in light of recent developments in the characterisation of resting state fMRI timeseries -- and simulations of neuronal fluctuations based upon the connectome. Its particular focus is on patterns or modes of distributed activity that underlie functional connectivity. We first demonstrate that the eigenmodes of functional connectivity - or covariance among regions or nodes - are the same as the eigenmodes of the underlying effective connectivity, provided we limit ourselves to symmetrical connections. This symmetry constraint is motivated by appealing to proximity graphs based upon multidimensional scaling. Crucially, the principal modes of functional connectivity correspond to the dynamically unstable modes of effective connectivity that decay slowly and show long term memory. Technically, these modes have small negative Lyapunov exponents that approach zero from below. Interestingly, the superposition of modes - whose exponents are sampled from a power law distribution - produces classical 1/f (scale free) spectra. We conjecture that the emergence of dynamical instability - that underlies intrinsic brain networks - is inevitable in any system that is separated from external states by a Markov blanket. This conjecture appeals to a free energy formulation of nonequilibrium steady-state dynamics. The common theme that emerges from these theoretical considerations is that endogenous fluctuations are dominated by a small number of dynamically unstable modes. We use this as the basis of a dynamic causal model (DCM) of resting state fluctuations -- as measured in terms of their complex cross spectra. In this model, effective connectivity is parameterised in terms of eigenmodes and their Lyapunov exponents -- that can also be interpreted as locations in a multidimensional scaling space. Model inversion provides not only estimates of edges or connectivity but also the topography and dimensionality of the underlying scaling space. Here, we focus on conceptual issues with simulated fMRI data and provide an illustrative application using an empirical multi-region timeseries. [ABSTRACT FROM AUTHOR]

Abstract: This technical note introduces a dynamic causal model (DCM) for resting state fMRI time series based upon observed functional connectivity—as measured by the cross spectra among different brain regions. This DCM is based upon a deterministic model that generates predicted crossed spectra from a biophysically plausible model of coupled neuronal fluctuations in a distributed neuronal network or graph. Effectively, the resulting scheme finds the best effective connectivity among hidden neuronal states that explains the observed functional connectivity among haemodynamic responses. This is because the cross spectra contain all the information about (second order) statistical dependencies among regional dynamics. In this note, we focus on describing the model, its relationship to existing measures of directed and undirected functional connectivity and establishing its face validity using simulations. In subsequent papers, we will evaluate its construct validity in relation to stochastic DCM and its predictive validity in Parkinson's and Huntington's disease. [Copyright &y& Elsevier]

Abstract: In this work, we address the problem of using dynamic causal modelling (DCM) to estimate the coupling parameters (effective connectivity) of large models with many regions. This is a potentially important problem because meaningful graph theoretic analyses of effective connectivity rest upon the statistics of the connections (edges). This calls for characterisations of networks with an appreciable number of regions (nodes). The problem here is that the number of coupling parameters grows quadratically with the number of nodes—leading to severe conditional dependencies among their estimates and a computational load that quickly becomes unsustainable. Here, we describe a simple solution, in which we use functional connectivity to provide prior constraints that bound the effective number of free parameters. In brief, we assume that priors over connections between individual nodes can be replaced by priors over connections between modes (patterns over nodes). By using a small number of modes, we can reduce the dimensionality of the problem in an informed way. The modes we use are the principal components or eigenvectors of the functional connectivity matrix. However, this approach begs the question of how many modes to use. This question can be addressed using Bayesian model comparison to optimise the number of modes. We imagine that this form of prior – over the extrinsic (endogenous) connections in large DCMs – may be useful for people interested in applying graph theory to distributed networks in the brain or to characterise connectivity beyond the subgraphs normally examined in DCM. [Copyright &y& Elsevier]

Abstract: Estimating the precision or uncertainty associated with sensory signals is an important part of perception. Based on a previous computational model, we tested the hypothesis that increasing visual contrast increased the precision encoded in early visual areas by the gain or excitability of superficial pyramidal cells. This hypothesis was investigated using electroencephalography and dynamic causal modelling (DCM); a biologically constrained modelling of the cortical processes underlying EEG activity. Source localisation identified the electromagnetic sources of visually evoked responses and DCM was used to characterise the coupling among these sources. Bayesian model selection was used to select the most likely connectivity pattern and contrast-dependent changes in connectivity. As predicted, the model with the highest evidence entailed increased superficial pyramidal cell gain in higher-contrast trials. As predicted theoretically, contrast-dependent increases were reduced at higher levels of the hierarchy. These results demonstrate that increased signal-to-noise ratio in sensory signals produce (or are represented by) increased superficial pyramidal cell gain, and that synaptic parameters encoding statistical properties like sensory precision can be quantified using EEG and dynamic causal modelling. [Copyright &y& Elsevier]

Metastability is a key source of itinerant dynamics in the brain; namely, spontaneous spatiotemporal reorganization of neuronal activity. This itinerancy has been the focus of numerous dynamic functional connectivity (DFC) analyses – developed to characterize the formation and dissolution of distributed functional patterns over time, using resting state fMRI. However, aside from technical and practical controversies, these approaches cannot recover the neuronal mechanisms that underwrite itinerant (e.g., metastable) dynamics—due to their descriptive, model-free nature. We argue that effective connectivity (EC) analyses are more apt for investigating the neuronal basis of metastability. To this end, we appeal to biologically-grounded models (i.e., dynamic causal modelling, DCM) and dynamical systems theory (i.e., heteroclinic sequential dynamics) to create a probabilistic, generative model of haemodynamic fluctuations. This model generates trajectories in the parametric space of EC modes (i.e., states of connectivity) that characterize functional brain architectures. In brief, it extends an established spectral DCM, to generate functional connectivity data features that change over time. This foundational paper tries to establish the model's face validity by simulating non-stationary fMRI time series and recovering key model parameters (i.e., transition probabilities among connectivity states and the parametric nature of these states) using variational Bayes. These data are further characterized using Bayesian model comparison (within and between subjects). Finally, we consider practical issues that attend applications and extensions of this scheme. Importantly, the scheme operates within a generic Bayesian framework – that can be adapted to study metastability and itinerant dynamics in any non-stationary time series. [ABSTRACT FROM AUTHOR]

Abstract Parkinson's disease (PD) is a neurodegenerative condition in which aberrant oscillatory synchronization of neuronal activity at beta frequencies (15–35 Hz) across the cortico-basal ganglia-thalamocortical circuit is associated with debilitating motor symptoms, such as bradykinesia and rigidity. Mounting evidence suggests that the magnitude of beta synchrony in the parkinsonian state fluctuates over time, but the mechanisms by which thalamocortical circuitry regulates the dynamic properties of cortical beta in PD are poorly understood. Using the recently developed generic Dynamic Causal Modelling (DCM) framework, we recursively optimized a set of plausible models of the thalamocortical circuit (n = 144) to infer the neural mechanisms that best explain the transitions between low and high beta power states observed in recordings of field potentials made in the motor cortex of anesthetized Parkinsonian rats. Bayesian model comparison suggests that upregulation of cortical rhythmic activity in the beta-frequency band results from changes in the coupling strength both between and within the thalamus and motor cortex. Specifically, our model indicates that high levels of cortical beta synchrony are mainly achieved by a delayed (extrinsic) input from thalamic relay cells to deep pyramidal cells and a fast (intrinsic) input from middle pyramidal cells to superficial pyramidal cells. From a clinical perspective, our study provides insights into potential therapeutic strategies that could be utilized to modulate the network mechanisms responsible for the enhancement of cortical beta in PD. Specifically, we speculate that cortical stimulation aimed to reduce the enhanced excitatory inputs to either the superficial or deep pyramidal cells could be a potential non-invasive therapeutic strategy for PD. Highlights • Coupling changes within and between circuit nodes lead to cortical beta enhancement. • Input propagation delays play a crucial role in the up-regulation of cortical beta. • Beta power could be modulated by altering lamina specific inputs. [ABSTRACT FROM AUTHOR]

The development of large-scale network models that infer the effective (directed) connectivity among neuronal populations from neuroimaging data represents a key challenge for computational neuroscience. Dynamic causal models (DCMs) of neuroimaging and electrophysiological data are frequently used for inferring effective connectivity but are presently restricted to small graphs (typically up to 10 regions) in order to keep model inversion computationally feasible. Here, we present a novel variant of DCM for functional magnetic resonance imaging (fMRI) data that is suited to assess effective connectivity in large (whole-brain) networks. The approach rests on translating a linear DCM into the frequency domain and reformulating it as a special case of Bayesian linear regression. This paper derives regression DCM (rDCM) in detail and presents a variational Bayesian inversion method that enables extremely fast inference and accelerates model inversion by several orders of magnitude compared to classical DCM. Using both simulated and empirical data, we demonstrate the face validity of rDCM under different settings of signal-to-noise ratio (SNR) and repetition time (TR) of fMRI data. In particular, we assess the potential utility of rDCM as a tool for whole-brain connectomics by challenging it to infer effective connection strengths in a simulated whole-brain network comprising 66 regions and 300 free parameters. Our results indicate that rDCM represents a computationally highly efficient approach with promising potential for inferring whole-brain connectivity from individual fMRI data. [ABSTRACT FROM AUTHOR]

Effective connectivity is commonly assessed using blood oxygenation level-dependent (BOLD) signals. In ( Havlicek et al., 2015 ), we presented a novel, physiologically informed dynamic causal model (P-DCM) that extends current generative models. We demonstrated the improvements afforded by P-DCM in terms of the ability to model commonly observed neuronal and vascular transients in single regions. Here, we assess the ability of the novel and previous DCM variants to estimate effective connectivity among a network of five ROIs driven by a visuo-motor task. We demonstrate that connectivity estimates depend sensitively on the DCM used, due to differences in the modeling of hemodynamic response transients; such as the post-stimulus undershoot or adaptation during stimulation. In addition, using a novel DCM for arterial spin labeling (ASL) fMRI that measures BOLD and CBF signals simultaneously, we confirmed our findings (by using the BOLD data alone and in conjunction with CBF). We show that P-DCM provides better estimates of effective connectivity, regardless of whether it is applied to BOLD data alone or to ASL time-series, and that all new aspects of P-DCM (i.e. neuronal, neurovascular, hemodynamic components) constitute an improvement compared to those in the previous DCM variants. In summary, (i) accurate modeling of fMRI response transients is crucial to obtain valid effective connectivity estimates and (ii) any additional hemodynamic data, such as provided by ASL, increases the ability to disambiguate neuronal and vascular effects present in the BOLD signal. [ABSTRACT FROM AUTHOR]

Recently, there has been a lot of interest in characterising the connectivity of resting state brain networks. Most of the literature uses functional connectivity to examine these intrinsic brain networks. Functional connectivity has well documented limitations because of its inherent inability to identify causal interactions. Dynamic causal modelling (DCM) is a framework that allows for the identification of the causal (directed) connections among neuronal systems — known as effective connectivity. This technical note addresses the validity of a recently proposed DCM for resting state fMRI – as measured in terms of their complex cross spectral density – referred to as spectral DCM. Spectral DCM differs from (the alternative) stochastic DCM by parameterising neuronal fluctuations using scale free (i.e., power law) forms, rendering the stochastic model of neuronal activity deterministic. Spectral DCM not only furnishes an efficient estimation of model parameters but also enables the detection of group differences in effective connectivity, the form and amplitude of the neuronal fluctuations or both. We compare and contrast spectral and stochastic DCM models with endogenous fluctuations or state noise on hidden states. We used simulated data to first establish the face validity of both schemes and show that they can recover the model (and its parameters) that generated the data. We then used Monte Carlo simulations to assess the accuracy of both schemes in terms of their root mean square error. We also simulated group differences and compared the ability of spectral and stochastic DCMs to identify these differences. We show that spectral DCM was not only more accurate but also more sensitive to group differences. Finally, we performed a comparative evaluation using real resting state fMRI data (from an open access resource) to study the functional integration within default mode network using spectral and stochastic DCMs. [ABSTRACT FROM AUTHOR]

Abstract: Parkinson''s disease is a common and debilitating condition, caused by aberrant activity in a complex basal ganglia–thalamocortical circuit. Therapeutic advances rely on characterising interactions in this circuit. However, recording electrophysiological responses over the entire circuit is impractical. Dynamic causal modelling offers large-scale models of predictive value based on a limited or partial sampling of complex networks. Using dynamic causal modelling, we determined the network changes underlying the pathological excess of beta oscillations that characterise the Parkinsonian state. We modelled data from five patients undergoing surgery for deep brain stimulation of more than one target. We found that connections to and from the subthalamic nucleus were strengthened and promoted beta synchrony, in the untreated compared to the treated Parkinsonian state. Dynamic causal modelling was able to replicate the effects of lesioning this nucleus and may provide a new means of directing the search for therapeutic targets. [Copyright &y& Elsevier]

Abstract: Functional integration in the brain rests on anatomical connectivity (the presence of axonal connections) and effective connectivity (the causal influences mediated by these connections). The deployment of anatomical connections provides important constraints on effective connectivity, but does not fully determine it, because synaptic connections can be expressed functionally in a dynamic and context-dependent fashion. Although it is generally assumed that anatomical connectivity data is important to guide the construction of neurobiologically realistic models of effective connectivity; the degree to which these models actually profit from anatomical constraints has not yet been formally investigated. Here, we use diffusion weighted imaging and probabilistic tractography to specify anatomically informed priors for dynamic causal models (DCMs) of fMRI data. We constructed 64 alternative DCMs, which embodied different mappings between the probability of an anatomical connection and the prior variance of the corresponding of effective connectivity, and fitted them to empirical fMRI data from 12 healthy subjects. Using Bayesian model selection, we show that the best model is one in which anatomical probability increases the prior variance of effective connectivity parameters in a nonlinear and monotonic (sigmoidal) fashion. This means that the higher the likelihood that a given connection exists anatomically, the larger one should set the prior variance of the corresponding coupling parameter; hence making it easier for the parameter to deviate from zero and represent a strong effective connection. To our knowledge, this study provides the first formal evidence that probabilistic knowledge of anatomical connectivity can improve models of functional integration. [Copyright &y& Elsevier]

Abstract: Models of effective connectivity characterize the influence that neuronal populations exert over each other. Additionally, some approaches, for example Dynamic Causal Modelling (DCM) and variants of Structural Equation Modelling, describe how effective connectivity is modulated by experimental manipulations. Mathematically, both are based on bilinear equations, where the bilinear term models the effect of experimental manipulations on neuronal interactions. The bilinear framework, however, precludes an important aspect of neuronal interactions that has been established with invasive electrophysiological recording studies; i.e., how the connection between two neuronal units is enabled or gated by activity in other units. These gating processes are critical for controlling the gain of neuronal populations and are mediated through interactions between synaptic inputs (e.g. by means of voltage-sensitive ion channels). They represent a key mechanism for various neurobiological processes, including top-down (e.g. attentional) modulation, learning and neuromodulation. This paper presents a nonlinear extension of DCM that models such processes (to second order) at the neuronal population level. In this way, the modulation of network interactions can be assigned to an explicit neuronal population. We present simulations and empirical results that demonstrate the validity and usefulness of this model. Analyses of synthetic data showed that nonlinear and bilinear mechanisms can be distinguished by our extended DCM. When applying the model to empirical fMRI data from a blocked attention to motion paradigm, we found that attention-induced increases in V5 responses could be best explained as a gating of the V1→V5 connection by activity in posterior parietal cortex. Furthermore, we analysed fMRI data from an event-related binocular rivalry paradigm and found that interactions amongst percept-selective visual areas were modulated by activity in the middle frontal gyrus. In both practical examples, Bayesian model selection favoured the nonlinear models over corresponding bilinear ones. [Copyright &y& Elsevier]

Abstract: The classical model of blood oxygen level-dependent (BOLD) responses by Buxton et al. [Buxton, R.B., Wong, E.C., Frank, L.R., 1998. Dynamics of blood flow and oxygenation changes during brain activation: the Balloon model. Magn. Reson. Med. 39, 855–864] has been very important in providing a biophysically plausible framework for explaining different aspects of hemodynamic responses. It also plays an important role in the hemodynamic forward model for dynamic causal modeling (DCM) of fMRI data. A recent study by Obata et al. [Obata, T., Liu, T.T., Miller, K.L., Luh, W.M., Wong, E.C., Frank, L.R., Buxton, R.B., 2004. Discrepancies between BOLD and flow dynamics in primary and supplementary motor areas: application of the Balloon model to the interpretation of BOLD transients. NeuroImage 21, 144–153] linearized the BOLD signal equation and suggested a revised form for the model coefficients. In this paper, we show that the classical and revised models are special cases of a generalized model. The BOLD signal equation of this generalized model can be reduced to that of the classical Buxton model by simplifying the coefficients or can be linearized to give the Obata model. Given the importance of hemodynamic models for investigating BOLD responses and analyses of effective connectivity with DCM, the question arises which formulation is the best model for empirically measured BOLD responses. In this article, we address this question by embedding different variants of the BOLD signal equation in a well-established DCM of functional interactions among visual areas. This allows us to compare the ensuing models using Bayesian model selection. Our model comparison approach had a factorial structure, comparing eight different hemodynamic models based on (i) classical vs. revised forms for the coefficients, (ii) linear vs. non-linear output equations, and (iii) fixed vs. free parameters, ε, for region-specific ratios of intra- and extravascular signals. Using fMRI data from a group of twelve subjects, we demonstrate that the best model is a non-linear model with a revised form for the coefficients, in which ε is treated as a free parameter. [Copyright &y& Elsevier]