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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: 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: This paper is about inferring or discovering the functional architecture of distributed systems using Dynamic Causal Modelling (DCM). We describe a scheme that recovers the (dynamic) Bayesian dependency graph (connections in a network) using observed network activity. This network discovery uses Bayesian model selection to identify the sparsity structure (absence of edges or connections) in a graph that best explains observed time-series. The implicit adjacency matrix specifies the form of the network (e.g., cyclic or acyclic) and its graph-theoretical attributes (e.g., degree distribution). The scheme is illustrated using functional magnetic resonance imaging (fMRI) time series to discover functional brain networks. Crucially, it can be applied to experimentally evoked responses (activation studies) or endogenous activity in task-free (resting state) fMRI studies. Unlike conventional approaches to network discovery, DCM permits the analysis of directed and cyclic graphs. Furthermore, it eschews (implausible) Markovian assumptions about the serial independence of random fluctuations. The scheme furnishes a network description of distributed activity in the brain that is optimal in the sense of having the greatest conditional probability, relative to other networks. The networks are characterised in terms of their connectivity or adjacency matrices and conditional distributions over the directed (and reciprocal) effective connectivity between connected nodes or regions. We envisage that this approach will provide a useful complement to current analyses of functional connectivity for both activation and resting-state studies. [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]